thermophysical properties of zrb2 and zrb2–sic ceramics

7
Thermophysical Properties of ZrB 2 and ZrB 2 –SiC Ceramics James W. Zimmermann,* ,w,z Gregory E. Hilmas,* and William G. Fahrenholtz** Department of Materials Science and Engineering, University of Missouri-Rolla, Rolla, Missouri 65409 Ralph B. Dinwiddie, Wallace D. Porter, and Hsin Wang* Oak Ridge National Laboratory, Oak Ridge, Tenneessee 37831 Thermophysical properties were investigated for zirconium dibo- ride (ZrB 2 ) and ZrB 2 –30 vol% silicon carbide (SiC) ceramics. Thermal conductivities were calculated from measured thermal diffusivities, heat capacities, and densities. The thermal conduc- tivity of ZrB 2 increased from 56 W (m K) 1 at room temperature to 67 W (m K) 1 at 1675 K, whereas the thermal conductivity of ZrB 2 –SiC decreased from 62 to 56 W (m K) 1 over the same temperature range. Electron and phonon contributions to ther- mal conductivity were determined using electrical resistivity mea- surements and were used, along with grain size models, to explain the observed trends. The results are compared with previously reported thermal conductivities for ZrB 2 and ZrB 2 –SiC. I. Introduction Z IRCONIUM diboride (ZrB 2 )-based ceramics are being investi- gated for ultra-high-temperature structural applications (42000 K) that would take advantage of the high melting tem- perature of ZrB 2 , which is approximately 3500 K. The addition of silicon carbide (SiC) has been reported to improve the specific strength, fracture toughness, and oxidation resistance of ZrB 2 . 1–5 Proposed applications for ZrB 2 -based materials include thermal protection components in hypersonic aerospace vehicles, current and future propulsion systems, thermocouple sheaths, and refractory crucibles. 6,7 ZrB 2 also has a high electrical conduc- tivity (B1 10 8 Sm 1 ), only slightly less than most metals, which allows complex shapes to be machined using electrical discharge machining (EDM). 8 Other applications that take advantage of the electrical conductivity include furnace heating elements, high-temperature electrodes, and metal evaporator boats. 7 Despite the development of ZrB 2 -based materials for use in high-temperature applications, the thermophysical properties of ZrB 2 have not been investigated extensively. In particular, ther- mal expansion and thermal conduction behavior of ZrB 2 and ZrB 2 –SiC ceramics are critical for determining the magnitude of the thermal stresses developed in ultrahigh-temperature ceramic (UHTC) components. 9 The coefficient of thermal expansion (CTE) for ZrB 2 has been reported previously to be B6.8 10 6 K 1 . 10 Additions of a-SiC (6H) particles are ex- pected to reduce the average CTE of ZrB 2 –SiC composites due to the lower CTE of SiC, which is 4.3 10 6 K 1 . 11 The differ- ence in thermal expansion between ZrB 2 and SiC may result in thermal stresses at the ZrB 2 –SiC interfaces during cooling from the processing temperature or in applications that involve ther- mal cycling. Specific heat capacity (c), J (g K) 1 , also affects thermal conductivity (Eq. (1)). The phonon contribution to the specific heat capacity value (c ph ) is the heat capacity value that takes into account all phonon modes (i.e., Debye theory). Electrically con- ductive materials have an electron contribution that results in a linear increase in specific heat capacity (c e ) with temperature. Room-temperature specific heat capacities (sum of c ph and c e ) of ZrB 2 and SiC are 0.422 and 0.668 J (g K) 1 , respectively. 12 A specific heat capacity value of 0.466 J (g K) 1 is predicted for ZrB 2 –30 vol% SiC (18 wt% SiC) using mass fraction averaging, which is 22% greater than ZrB 2 . The room-temperature specific heat capacity of ZrB 2 –20 vol% SiC has been reported previously to be 0.437 J (g K) 1 , B3% less than the mass fraction averaging prediction of 0.449 J (g K) 1 and slightly greater than ZrB 2 . 13 k ¼ X i¼e;ph 1 3 c i v i l i r (1) In addition to its dependence on heat capacity, thermal con- ductivity is also dependent on the bulk density (r), velocity (v), and the mean free path (l) of phonons and electrons (Eq. (1)). Phonon velocity is essentially independent of temperature and is assumed to be constant in the present study. The phonon mean free path decreases with temperature and reaches a minimum value, which is related to the distance between neighboring at- oms. 14 Electron velocity and free mean path are dependent on temperature and the electron contribution to the thermal con- ductivity can be determined using the Wiedmann–Franz law. At high temperatures, the phonon heat capacity (c v, ph ) reaches a maximum value as described by the Dulong–Petit law when the sample volume is constant. The thermal conductivity reaches a constant value at high temperatures because the heat capacity, phonon velocity, and phonon mean free path are constant at high temperatures. The plateau is often referred to as the satu- ration limit of phonon-controlled thermal conductivity. 14 Thermal conductivity value of single–crystal ZrB 2 is 140 W (m K) 1 in the basal direction and 100 W (m K) 1 along the c-axis. 15 Thermal conductivity of ZrB 2 –SiC is expected to be similar to or greater than polycrystalline ZrB 2 (85 W (m K) 1 ). 16 Thermal conductivity values for SiC range from 125 W (m K) 1 for sintered SiC to 490 W (m K) 1 for 6H single-crystal SiC. 8,17 Using the Maxwell–Eucken model, 18 the thermal conductivity of ZrB 2 with 30 vol% SiC is predicted to be between 96 and 143 W (m K) 1 depending on the ther- mal conductivity value of SiC used. Previously reported room– temperature thermal conductivity values for ZrB 2 and ZrB 2 –SiC ceramics are between 84 and 120 W (m K) 1 (Table I), similar to the predicted values. High thermal conductivity, which im- proves the thermal shock resistance of ceramic components due to the minimization of thermal gradients, is important for the applications discussed above. Y. Blum—contributing editor This work was financially supported by the U.S. Air Force Office of Scientific Research, under contract numbers F49620-03-0072 and FA9550-06-1-0125. Presented at the AFOSR Workshop on Ultra-High-Temperature Ceramic Materials hosted by SRI International, July 23–25, 2007. *Member, The American Ceramic Society. **Fellow, The American Ceramic Society. w Author to whom correspondence should be addressed. e-mail: zimmermajw@ corning.com z Present address: Corning Inc., Corning, NY 14831. Manuscript No. 23614. Received August 16, 2007; approved October 20, 2007. J ournal J. Am. Ceram. Soc., 91 [5] 1405–1411 (2008) DOI: 10.1111/j.1551-2916.2008.02268.x r 2008 The American Ceramic Society 1405

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Thermophysical Properties of ZrB2 and ZrB2–SiC Ceramics

James W. Zimmermann,*,w,z Gregory E. Hilmas,* and William G. Fahrenholtz**

Department of Materials Science and Engineering, University of Missouri-Rolla, Rolla, Missouri 65409

Ralph B. Dinwiddie, Wallace D. Porter, and Hsin Wang*

Oak Ridge National Laboratory, Oak Ridge, Tenneessee 37831

Thermophysical properties were investigated for zirconium dibo-ride (ZrB2) and ZrB2–30 vol% silicon carbide (SiC) ceramics.Thermal conductivities were calculated from measured thermaldiffusivities, heat capacities, and densities. The thermal conduc-tivity of ZrB2 increased from 56W (m K)�1 at room temperatureto 67 W (m K)

�1at 1675 K, whereas the thermal conductivity of

ZrB2–SiC decreased from 62 to 56 W (m K)�1

over the sametemperature range. Electron and phonon contributions to ther-mal conductivity were determined using electrical resistivity mea-surements and were used, along with grain size models, to explainthe observed trends. The results are compared with previouslyreported thermal conductivities for ZrB2 and ZrB2–SiC.

I. Introduction

ZIRCONIUM diboride (ZrB2)-based ceramics are being investi-gated for ultra-high-temperature structural applications

(42000 K) that would take advantage of the high melting tem-perature of ZrB2, which is approximately 3500 K. The additionof silicon carbide (SiC) has been reported to improve the specificstrength, fracture toughness, and oxidation resistance of ZrB2.

1–5

Proposed applications for ZrB2-based materials include thermalprotection components in hypersonic aerospace vehicles, currentand future propulsion systems, thermocouple sheaths, andrefractory crucibles.6,7 ZrB2 also has a high electrical conduc-tivity (B1� 108 S m�1), only slightly less than most metals,which allows complex shapes to be machined using electricaldischarge machining (EDM).8 Other applications that takeadvantage of the electrical conductivity include furnace heatingelements, high-temperature electrodes, and metal evaporatorboats.7

Despite the development of ZrB2-based materials for use inhigh-temperature applications, the thermophysical properties ofZrB2 have not been investigated extensively. In particular, ther-mal expansion and thermal conduction behavior of ZrB2 andZrB2–SiC ceramics are critical for determining the magnitude ofthe thermal stresses developed in ultrahigh-temperature ceramic(UHTC) components.9 The coefficient of thermal expansion(CTE) for ZrB2 has been reported previously to beB6.8� 10�6 K�1.10 Additions of a-SiC (6H) particles are ex-pected to reduce the average CTE of ZrB2–SiC composites dueto the lower CTE of SiC, which is 4.3� 10�6 K�1.11 The differ-ence in thermal expansion between ZrB2 and SiC may result in

thermal stresses at the ZrB2–SiC interfaces during cooling fromthe processing temperature or in applications that involve ther-mal cycling.

Specific heat capacity (c), J � (g �K)�1, also affects thermalconductivity (Eq. (1)). The phonon contribution to the specificheat capacity value (cph) is the heat capacity value that takes intoaccount all phonon modes (i.e., Debye theory). Electrically con-ductive materials have an electron contribution that results in alinear increase in specific heat capacity (ce) with temperature.Room-temperature specific heat capacities (sum of cph and ce) ofZrB2 and SiC are 0.422 and 0.668 J � (g �K)�1, respectively.12 Aspecific heat capacity value of 0.466 J � (g �K)�1 is predicted forZrB2–30 vol% SiC (18 wt% SiC) using mass fraction averaging,which is 22% greater than ZrB2. The room-temperature specificheat capacity of ZrB2–20 vol% SiC has been reported previouslyto be 0.437 J � (g �K)�1, B3% less than the mass fractionaveraging prediction of 0.449 J � (g �K)�1 and slightly greaterthan ZrB2.

13

k ¼Xi¼e;ph

1

3civi � lir (1)

In addition to its dependence on heat capacity, thermal con-ductivity is also dependent on the bulk density (r), velocity (v),and the mean free path (l) of phonons and electrons (Eq. (1)).Phonon velocity is essentially independent of temperature and isassumed to be constant in the present study. The phonon meanfree path decreases with temperature and reaches a minimumvalue, which is related to the distance between neighboring at-oms.14 Electron velocity and free mean path are dependent ontemperature and the electron contribution to the thermal con-ductivity can be determined using the Wiedmann–Franz law. Athigh temperatures, the phonon heat capacity (cv, ph) reaches amaximum value as described by the Dulong–Petit law when thesample volume is constant. The thermal conductivity reaches aconstant value at high temperatures because the heat capacity,phonon velocity, and phonon mean free path are constant athigh temperatures. The plateau is often referred to as the satu-ration limit of phonon-controlled thermal conductivity.14

Thermal conductivity value of single–crystal ZrB2 is140 W � (m �K)�1 in the basal direction and 100 W � (m �K)�1

along the c-axis.15 Thermal conductivity of ZrB2–SiC is expectedto be similar to or greater than polycrystalline ZrB2

(85 W � (m �K)�1).16 Thermal conductivity values for SiC rangefrom 125 W � (m �K)�1 for sintered SiC to 490 W � (m �K)�1 for6H single-crystal SiC.8,17 Using the Maxwell–Eucken model,18

the thermal conductivity of ZrB2 with 30 vol% SiC is predictedto be between 96 and 143 W � (m �K)�1 depending on the ther-mal conductivity value of SiC used. Previously reported room–temperature thermal conductivity values for ZrB2 and ZrB2–SiCceramics are between 84 and 120 W � (m �K)�1 (Table I), similarto the predicted values. High thermal conductivity, which im-proves the thermal shock resistance of ceramic components dueto the minimization of thermal gradients, is important for theapplications discussed above.

Y. Blum—contributing editor

This work was financially supported by the U.S. Air Force Office of Scientific Research,under contract numbers F49620-03-0072 and FA9550-06-1-0125.

Presented at the AFOSR Workshop on Ultra-High-Temperature Ceramic Materialshosted by SRI International, July 23–25, 2007.

*Member, The American Ceramic Society.**Fellow, The American Ceramic Society.wAuthor to whom correspondence should be addressed. e-mail: zimmermajw@

corning.comzPresent address: Corning Inc., Corning, NY 14831.

Manuscript No. 23614. Received August 16, 2007; approved October 20, 2007.

Journal

J. Am. Ceram. Soc., 91 [5] 1405–1411 (2008)

DOI: 10.1111/j.1551-2916.2008.02268.x

r 2008 The American Ceramic Society

1405

The purpose of this study was to characterize the thermo-physical properties of ZrB2 and ZrB2–30% SiC ceramics. Themicrostructure and mechanical properties of the ZrB2 andZrB2–30% SiC ceramics examined in the present study havebeen reported previously by Chamberlain et al.1 The constantpressure specific heat capacity (cp), density, thermal diffusivity,and electrical conductivity of ZrB2 and ZrB2–30% SiC were de-termined in this study over the range of 300–1675 K. The ther-mal conductivities were calculated using measured values forspecific heat capacity, CTE, and thermal diffusivity. The elec-tron and phonon contributions to the measured thermal con-ductivities were estimated as a function of temperature using theWiedemann–Franz law.

II. Experimental Procedure

(1) Sample Preparation

Billets were hot pressed by the method reported by Chamberlainet al.1 Commercially available ZrB2 (2 mm; Grade B, H.C.Starck, Newton, MA) and SiC (0.7 mm, Grade UF-10, H.C.Starck) powders were attrition milled (Model HD-01, UnionProcess, Akron, OH) using WCmilling media for 2 h at 600 rpmin hexane. The ceramic powder was recovered from the hexane–powder slurry using rotary evaporation (Roto-vapor R-124,Buchi, Flawil, Switzerland). The powder prepared was hotpressed (Thermal Technologies, HP-3060, Santa Rosa, CA) ingraphite dies lined with a boron nitride-coated graphite foil. Thefurnace was heated under vacuum (B130 mtorr) to 1923 K.The heating schedule included isothermal holds of 3 h at 1723 Kand 2 h at 1923 K. After the 1923 K hold, the hot-presschamber was filled with argon and heated to 2173 K at a rateof B15 K min�1. Billets were densified at 2173 K using apressure of 32 MPa and a holding time of 45 min.

Four specimens with different dimensions were required forthe ZrB2 and ZrB2–30% SiC materials. Two 22-mm-diameterdisks with a thickness of 6 mm were hot-pressed for each com-position to measure thermal diffusivity and heat capacity. Ther-mal diffusivity specimens, nominally 12.5 mm diameter� 3 mmin thickness, were cut from hot-pressed billets using diamondmachining. Heat capacity specimens were machined from thesecond 22 mm disk that was diamond surface ground to producea 1-mm-thick disk. A combination of diamond coring and cen-terless grinding was then used to prepare the 6-mm-diameterdisks. Two 4 mm� 3 mm� 25 mm test bars were machinedfrom the middle of a 45 mm� 32 mm� 6 mm rectangular billetusing diamond machining (400 grit final finish) for CTE andelectrical conductivity measurements.

(2) Characterization

Bulk density values were calculated from Archimedes measure-ments using water as the immersion medium at room temperature(293–299 K). The dry mass and suspended mass of the specimenwere measured to obtain the density of the bulk material. Thisdensity was used in the thermal conductivity calculation.

(3) Linear Thermal Expansion Coefficient

The linear thermal expansion behavior of ZrB2 and ZrB2–30vol% SiC was determined using a dual push-rod dilatometer(Dilatronic II, Theta Industries, Port Washington, NY). Mea-surements were made between 293 and 1673 K relative to a

sapphire standard. Differential length changes were measured toan accuracy of 71.5% and the temperature of the specimenswas measured with a Pt versus Pt–10% Rh (Type S) thermo-couple. Measurements were recorded every 0.5 s as the speci-mens were heated at 3 K min�1 and cooled at 3 K min�1.

(4) Specific Heat Capacity

Constant pressure specific heat capacity (cP) was measured ac-cording to ASTM E 1269 between 300 and 1125 K in a flowingargon atmosphere using differential scanning calorimetry (DSC404C, Netzsch, Selb, Germany) and a sapphire standard.19 Thetemperature was measured using a Pt versus Pt–10% Rh (TypeS) thermocouple. A heat capacity error of73% is estimated forthis measurement.

(5) Thermal Diffusivity

Thermal diffusivity (D) was measured using a laser flash tech-nique according to ASTM E 1461-01.20 The faces of the ZrB2

and ZrB2–30 vol% SiC disks were coated with a thin (B0.1 mm)layer of colloidal graphite to increase the absorption and radi-ation of thermal energy. Between 300 and 1273 K, a NetzschLFA 427 laser flash apparatus (Netzsch, Selb/Bavaria, Germa-ny) was used to measure the thermal diffusivity. Data was col-lected every 50 K during cooling. Between 1250 and 1675 K, onediffusivity measurment was obtained every 50 K while heatingusing an Anter Flashline 5000 (Anter Corp, Pittsburgh, PA).Argon atmospheres were used in all of the thermal diffusivitymeasurements.

(6) Thermal Conductivity

Thermal conductivities were not measured directly, but werecalculated as a function of temperature from the measured ther-mal diffusivities (D) and heat capacities (cp) using Eq. (2). Heatcapacity values were extrapolated from 1125 to 1675 K in orderto calculate the thermal conductivity up to 1675 K. The effect oftemperature on specimen density (r) was taken into account us-ing the measured thermal expansion behavior (Eq. (3)) whereDl(T) is the experimentally determined linear thermal extension.

k ¼ D � r � cp (2)

r Tð Þ ¼ r300K � 1þ DlðTÞl300

� ��3(3)

(7) Electrical Conductivity

Direct current potentiodynamic (�0.1–0.1 mV) four-point proberesistivity/conductivity measurements (1470 Cell Test, SolartronAnalytical, Farnborough, U.K.) were conducted in an argonatmosphere at elevated temperatures. Platinum electrodes wereused and wound into grooves machined into the test specimens.Measurements were taken every 50 K from 300 to 1350 K forZrB2. A smaller temperature range (300–1150 K) was used forZrB2–30 vol% SiC to minimize the reaction between the plat-inum electrode and the SiC. All measurements were recordedduring cooling after the specimens were allowed to equilibrate atthe test temperature for 15 min. Specimen temperature wasmeasured using a type S thermocouple placed B1 mm from thespecimen surface.

III. Results and Discussion

(1) Density and CTE

Measured thermal strains, Dl(T)l0�1, were plotted as a function

of temperature (Fig. 1) for ZrB2 and ZrB2–30 vol% SiC. Asobserved commonly in polycrystalline ceramics,21,22 a third-order polynomial curve best fit the thermal expansion curvesfor both materials from 300 to 1700 K. Equations (4) and (5)represent the thermal expansion of ZrB2 and ZrB2–SiC, respec-tively, and have R2 values 40.9998.

Table I. Reported Thermal Conductivity Values forZrB2-Based Ceramics at 298 and 1273 K

k298 k1273 Reference

ZrB2 83.8 81.8 16

ZrB2120% SiC 99.2 79.0 16

ZrB2120% SiC 103.8 76.2 (1227 K) 13

1406 Journal of the American Ceramic Society—Zimmermann et al. Vol. 91, No. 5

Dll0

� �ZrB2

¼ �1:64� 10�3 þ 4:92� 10�6T þ 1:72

� 10�9T2�2:31�10�13T3 ð300� 1675KÞ (4)

Dll0

� �ZrB2�30%SiC

¼� 1:40� 10�3 þ 3:99� 10�6T

þ 2:75� 10�9T2

� 6:80� 10�13T3 ð300� 1675KÞ(5)

Average thermal expansion coefficients (CTEs) were also de-termined for both ZrB2 and ZrB2–30% SiC. Owing to the non-linearity of the thermal expansion of the materials, twotemperature regions were used to represent the measured resultsmore accurately. The CTE values of ZrB2 and ZrB2–30% SiCwere 6.8� 10�6 K�1 between 300 and 1300 K. A difference wasmeasured between ZrB2 and ZrB2–30% SiC for the temperaturerange of 1300–1675 K with CTE values of 8.4� 10�6 and7.8� 10�6 K�1, respectively.

The similarity in the CTE values between ZrB2 and ZrB2–30% SiC at low temperatures (o1300 K) suggests that the SiCparticles are in compression over this temperature range and donot influence the expansion of ZrB2. SiC is expected to be incompression because the CTE of SiC (4.3� 10�6 K�1) is lessthan the CTE of ZrB2 (6.8� 10�6 K�1).23 At higher tempera-tures (41300 K), the thermal expansion of the materials di-verges, indicating a transition from compressive to tensile forceson the SiC particles. The measured CTE of ZrB2–30 vol% SiC(7.8� 10�6 K�1) was greater than the value predicted usingKerner’s24 and Turner’s models25 at these temperatures(7.2� 10�6 and 7.3� 10�6 K�1, respectively), indicating thatthe thermal expansion of ZrB2 is only partially being resisted bythe SiC particles. Young’s modulus values of 490 and 410 GPafor ZrB2 and SiC, respectively, were used in the thermal expan-sion calculations based on the literature values.1,10,26 Poisson’sratio values of 0.16 were used for both materials. The transitionfrom compressive forces to tensile forces in the SiC particle mayindicate that the thermal stresses relax during cooling from theprocessing temperature (2175 K) to B1300 K.

The room-temperature bulk density values for ZrB2 andZrB2–30% SiC were 6.27 and 5.33 g cm�3, similar to the val-ues reported previously.1 The densities of fabricated ZrB2 andZrB2–30 vol% SiC were 3% and 2% greater than the theoretical

values resulting from contamination from WC milling media.The contamination could be a possible cause of deviation of allthe thermophysical properties from pure ZrB2 and ZrB2–SiCmaterials. The densities of the fabricated specimens were calcu-lated as a function of temperature using the linear CTE values todetermine the volumetric thermal expansion (Eq. (3)). Thecalculated density changed by o0.16 g cm�3 (o3%) for ZrB2

and ZrB2–30% SiC between room temperature and 1675 K.

(2) Heat Capacity

The constant pressure-specific heat capacity for ZrB2 was similarto the values in the NIST-JANAF tables for ZrB2 (cr) (Fig. 2)(cp5�3.9161233.34T�0.545� 105T�2�196.082� 10�6T2).The Debye temperature (Y) and Gruneisen parameter of 965 Kand 1.0, respectively, were determined by fitting the experimen-tal curve to the cp Debye equation (Eqs. (6) and (7)), where a isthe linear CTE, kB is Boltzmann’s constant, N is Avogadro’snumber, n is the number of atoms in a unit cell, x is hvd(kB T)�1,and T is the absolute temperature. The measured Y is greaterthan the values reported previously that were calculated fromthe melting temperature, thermal expansion, and entropy, whichwere 730 K,27, 765 K27 and 834 K.14

cv ¼ 9kBnNT3Y�3Z YT�1

0

x4ex ex � 1ð Þ�2dx (6)

cp ¼ Cv 1þ 3agGT½ � (7)

The specific heat capacity data for ZrB2–30% SiC followedthe typical cp5 a1bT1cT�21dT2 relationship with a, b, c, andd values of 0.62246, 1.0� 10�4,�19834, and 0, respectively. TheR2 value for the fit was 0.9990 (Fig. 3). The curve isB5% lowerthan the values predicted using mass fraction averaging and datafrom the NIST-JANAF tables for ZrB2 (cr) and a SiC (cr).

(3) Thermal Diffusivity

Ambient temperature thermal diffusivity values of 22 and28 mm2 s�1 were measured for ZrB2 and ZrB2–30% SiC(Fig. 4). The higher value of diffusivity for ZrB2–30% SiC isdue to the higher room-temperature thermal diffusivity of SiC(160 mm2 s-1)28 compared with ZrB2. As expected, based on anincrease in phonon scattering with temperature, thermal diff-usivity decreased as a function of temperature for both materi-als. The large initial decrease in diffusivity for ZrB2–30 vol%SiC compared with ZrB2 is consistent with the diffusivity be-

Fig. 1. Thermal expansion curves for ZrB2 and ZrB2–30 vol% SiC.Values were measured up to 1675 K, and extrapolated up to the dens-ification temperature (2175 K).

Fig. 2. Heat capacity as a function of temperature for ZrB2. Measuredvalues were similar to the NIST-JANAF reference data. Fitting to aDebye heat capacity curve used a Debye temperature of 965 K.

May 2008 Thermophysical Properties of ZrB2 and ZrB2–SiC Ceramics 1407

havior of SiC.28 The thermal diffusivities at 1675 K were B16and 14 mm2 s�1 for ZrB2 and ZrB2–30% SiC, respectively.

A sharp decrease in the thermal diffusivity of ZrB2–30 vol%SiC was observed at approximately 1300 K, after the curve ap-peared to be reaching a constant saturation value (Fig. 4). Be-cause the decrease was not observed in ZrB2, the decrease wasattributed to impurities with the addition of SiC in ZrB2. Spe-cific heat capacity was extrapolated above 1125 K and was notmeasured in the temperature region in which the decrease inthermal diffusivity occurred. The specific heat capacity curvewould deviate from the extrapolated curve around 1300 K, cor-relating to the decrease in thermal diffusivity. The impurityphase may also contribute to the higher than expected CTEvalue for ZrB2–SiC above 1300 K. A softening of the impurityphase would reduce the traction between the SiC and ZrB2, re-ducing the effect of the SiC addition on the overall CTE of thematerial. The removal of the impurities could result in higherthermal diffusivity values and a lower CTE for ZrB2–SiC. Basedon extrapolation of the curve from below 1300 K, a diffusivity ofB14.1 mm2 s�1 is predicted at 1675 K for a material that doesnot undergo this presumed transition.

(4) Electrical Resistivity

Electrical resistivity increased linearly with increasing tempera-ture (Fig. 5) for both ZrB2 and ZrB2–30% SiC. The room-

temperature resistivity of ZrB2 was 22 mO � cm and the slopeof the resistivity versus temperature relationship was0.045 mO � cm �K�1 with an R2 fit of 0.9998. ZrB2–30% SiChad a room-temperature resistivity of 24 mO � cm and a resistivityversus temperature of 0.058 mO � cm �K�1. The correspondingtemperature coefficient of electrical resistance (TCR; Eq. (8))values were 2.06� 10�3 K�1 for ZrB2 and 2.52� 10�3 K�1 forZrB2–30% SiC. The measured resistivity values for ZrB2 arecomparable to values reported previously for room-temperatureresistivity, which range from 7.8 to 24 mO � cm, and, TCR, whichranged from 1.0� 10�3 to 6.4� 10�3 K�1.16,29–31 ZrB2–30 vol%SiC had higher room-temperature resistivity and lower TCRvalues than has been reported previously for ZrB2–20 vol% SiC(10.2 mO � cm and 4.8� 10�3 K�1).16 The electrical resistivity ofparticulate composites is dependent on several factors includingthe resistivity, concentration, shape, size, and distribution of thesecond phase.32–34 These parameters were not investigated fullyin this study and the effect of SiC on the electrical resistivity ofZrB2 is the subject of a separate investigation.

TCR ¼ se;300K �dð1=seÞdT

(8)

(5) Thermal Conductivity of ZrB2 and ZrB2–SiC

The thermal conductivities were calculated as a function of tem-perature for ZrB2 and ZrB2–30 vol% SiC using measured ther-mal diffusivity, bulk density, and constant pressure heat capacityvalues (Fig. 6). The calculated room-temperature thermalconductivity values, 53W (mK)�1 for ZrB2 and 62.0W (mK)�1

for ZrB2–30% SiC, were lower than the reported previouslyvalues for ZrB2 and ZrB2–SiC ceramics (Table I). The thermalconductivity of both materials increased sharply between roomtemperature and 350 K. The thermal conductivity of ZrB2 con-tinued to increase throughout the measured temperature range.In contrast, the thermal conductivity of ZrB2–30% SiC reacheda maximum value of 63.5 W (m K)�1 at B350 K and decreasedto an apparent saturation value of B56 W (m K)�1 before de-creasing again sharply at B1300 K. Again, the latter decreasewas attributed to a phase change of a secondary phase thoughtto be present at the grain boundaries.

The thermal conductivity calculated using the thermal diff-usivity, density, and heat capacity (Eq. (2)) has both electronand phonon contributions. Based on the desire to improvestrength and fracture toughness, some recent UHTC researchefforts have focused on reducing the grain size of both the ZrB2

and SiC,1,35–37 without considering the effect of grain size onthermal conductivity or other properties. The electron contri-

Fig. 4. Thermal diffusivity of ZrB2 (J, &) and ZrB2–30 vol% SiC(�, & ) as a function of temperature.

Fig. 5. Electrical resistivity of ZrB2 and ZrB2–vol% SiC as a functionof temperature.

Fig. 3. Heat capacity as a function of temperature for ZrB2–30 vol%SiC. Measured values were lower than the values predicted using massfraction averaging.

1408 Journal of the American Ceramic Society—Zimmermann et al. Vol. 91, No. 5

bution (ke) of the thermal conductivity can be predicted usingmeasured electrical conductivity (se), the absolute temperature(T), and the Lorentz constant (2.45� 10�8 W �O �K�2) using theWiedemann–Franz law (Eq. (9)). Grain boundary effects onelectrical conductivity are minimal due to the grain sizes beingsignificantly greater than the electron mean free path.38 Thephonon contribution to thermal conductivity (kp) at high tem-peratures is difficult to predict accurately due to its dependenceon phonon–phonon, phonon—lattice, and phonon–electronscattering. However, the phonon contribution can be estimatedby subtracting the calculated electron contribution from themeasured total thermal conductivity. Grain boundary effects onkp can be determined by the brick layer model using the k�1pversus T curve of a specimen with a known grain size.39

ke¼LseT (9)

The dependence of the electrical resistivity with temperatureis described by the TCR (Eq. (8)), which is a function of se,300 K

and T. Previously reported TCR values and the Wiedemann–Franz law were used to calculate the difference between the el-evated and room-temperature ke values (Fig. 7). A TCR value of3.33� 10�3 K�1 was required to have a Dke of zero. Larger TCRvalues resulted in a decrease in ke as shown by the curve plotted

using the TCR value of 6.4� 10�3. Smaller TCR values resultedin an increase in ke, resulting in increased thermal conductivity.

The TCR values for both ZrB2 and ZrB2–30% SiC wereo3.33� 10�3 (2.06� 10�3 and 2.52� 10�3 K�1, respectively)resulting in an increase in the electron contribution to the ther-mal conductivity. An increase in ke from 33W (mK)�1 at 300 Kto 48 W (mK)�1 at 1600 K was calculated for ZrB2 (Fig. 8). Theke difference was smaller for ZrB2–30% SiC as it increased from32 to 39 W (m K)�1 in the same temperature range (Fig. 9).

The phonon contribution to thermal conductivity decreasedfrom 22 to 18W (mK)�1 for ZrB2 and from 31 to 11W (mK)�1

for ZrB2–30% SiC between 300 and 1600 K (Figs. 8 and 9). Asnoted previously, the thermal diffusivity of ZrB2–30% SiC ap-peared to reach saturation at 1300 K before decreasing, possiblyindicating a phase change within. This decrease was carried overto the thermal conductivity and phonon contribution of thethermal conductivity. Because the specific heat capacity valueswere extrapolated at these temperatures, the actual thermal con-ductivity values may differ from the calculated values.

The room-temperature thermal conductivity of ZrB2 mea-sured in this study, which was 56.3 W � (m � K)�1, is lower thanthat reported previously (Table I). The grain size of ZrB2 fab-ricated from the same precursor powder and using the same hot

Fig. 6. Thermal conductivity as a function of temperature for ZrB2 andZrB2–30 vol% SiC.

Fig. 7. Calculated changes in the electrical contributions to thermalconductivity (ke) of ZrB2 vary significantly depending on the value of thethermal coefficient of electrical resistance (TCR) used. The change in keparameter remains constant with a TCR value of 3.33� 10�3 K�1.

Fig. 8. Total thermal conductivity of ZrB2 as a function of temperatureplotted with its electrical and phonon (ktotal–kelectron) contributions.

Fig. 9. Total thermal conductivity of ZrB2–30 vol% SiC plotted withits electrical and phonon (ktotal–kelectron) contributions. The Maxwell–Eucken model was used to predict the curve based on 3-mm-grain sizethermal conductivity values of ZrB2 obtained from this study and pre-viously reported SiC.

May 2008 Thermophysical Properties of ZrB2 and ZrB2–SiC Ceramics 1409

pressing schedule has previously been reported by Chamberlainet al.1 to be 6 mm, which is significantly smaller than the grainsizes reported for other ZrB2 materials, which range from 13 to27 mm.16,40 A grain boundary internal thermal resistance orKapitza resistance (R; Eq. (10)) of 0.20� 10�8 �m2 (K �W)�1

was calculated using the brick layer model by extrapolating thek�1p values between 400 and 900 K to absolute zero.39 The slopeof the curve (a; 2.18� 10�5 m W�1) is inherent to the phonon–phonon scattering within the ZrB2 crystal and is independent ofgrain size.39 Using the calculated R, Eq. (10) was used to com-pute the kp for different grain sizes (d) (Fig. 10).

1

kp¼ aT þ 1

dR (10)

The thermal conductivity values predicted in this study for15 mm grain size ZrB2 were similar to previously reported datafor ZrB2 with 13 mm grain size (Fig. 11).16 This similarity islikely to be coincidental, however, because the electron andphonon contributions to the thermal conductivity were signifi-cantly different from the contributions in this study (Fig. 12). Theelectron contributions to the thermal conductivity (and the elec-trical conductivity) of ZrB2 studied were approximately 50% low-er than the values reported previously by Tye and Clougherty.16

This may be due to a tungsten-rich phase from milling mediacontamination.41 The phonon contribution of the ZrB2 studiedwas greater than that of Tye and Clougherty, which was likelydue to a lower grain boundary thermal resistance.

A relationship between the measured thermal conductivitiesof ZrB2 and ZrB2–30% SiC was determined using the phononand electron contributions. The size of the ZrB2 grains in thefabricated ZrB2–30% SiC particulate composites has been re-ported previously to be 3 mm.1 The phonon contribution to thethermal conductivity of ZrB2–30% SiC was calculated using theMaxwell–Eucken model.18 Calculated thermal conductivity datafor 3-mm-grain size ZrB2 (Fig. 10) and previously reported SiCthermal conductivity data28 were used as the matrix and partic-ulate, respectively. The electron contribution (Fig. 9) was deter-mined using the experimental electrical resistivity (Fig. 5) andthe Wiedemann–Franz law. The calculated thermal conductivity(the sum of the phonon and electron contributions) was similarto the experimental data (Fig. 9).

IV. Conclusions

Thermophysical properties were measured and thermal conduc-tivities were calculated for ZrB2 and ZrB2 with 30 vol% SiCprepared by hot-pressed attrition-milled powders. The thermalconductivity of ZrB2 with a grain size of 6 mm increased from56W (mK)�1 at room temperature to 67 W (mK)�1 at 1600 K.The thermal conductivity of ZrB2–30vol% SiC with an averageZrB2 grain size of 3 mm decreased from 62 to 50W (mK)�1 overthe same temperature range. Electrical resistivities of 22 and24 mO � cm, and TCR values of 2.06� 10�3 and 2.52� 10�3 K�1,were measured for ZrB2 and ZrB2–30% SiC, respectively. Thethermal conductivity values for ZrB2 and ZrB2–30% SiC weredecoupled into electron and phonon contributions using theWeideman–Franz law. Electron contributions to the thermalconductivities measured increased from 33 W (m K)�1 at 300 Kto 48 W (m K)�1 at 1675 K for ZrB2 and from 32 W (m K)�1 at300 K to 38 W (m K)�1 at 1675 K for ZrB2–30% SiC.

The grain size effect on the phonon contribution to the thermalconductivity of ZrB2 was determined using the brick layer mod-el.39 The effect of grain size on thermal conductivity was calcu-lated by summing the calculated electron and phonon thermalconductivity contributions. Current thermal conductivity resultsare consistent with previous thermal conductivity measurements,but the electron and phonon contributions differ. The currentstudy provides additional insights into the mechanisms by whichparticle additions and grain size affect the temperature depen-dence of the thermophysical properties in ZrB2 ceramics.

Fig. 10. The effect of grain size on the phonon contribution of thethermal conductivity (kp) was predicted from the calculated internalthermal resistance (Rint).

Fig. 11. Thermal conductivity of ZrB2 as a function of temperature andits dependence on mean grain size. Previously reported thermal conduc-tivity values (J) are plotted for comparison.

Fig. 12. Comparison of ktotal, kp, and ke of data predicted in the currentstudy for 15 mm ZrB2 (& , �, m) and data reported previously for 13-mm-grain size ZrB2 (&, J, D).16

1410 Journal of the American Ceramic Society—Zimmermann et al. Vol. 91, No. 5

The results of this study imply that the thermal conductivityof ZrB2 materials varies with the grain size. Continual reductionof the grain size to improve the mechanical properties of ZrB2-based materials is predicted to decrease heat transport within thematerial. Thermal loading conditions should be considered todetermine the optimal grain size to balance the need for thermaltransport and strength.

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