On the Thermal Conductivity of Spark Plasma Sintered Alumina Hybrid Nanocomposites: Estimation Modeling and Experimental Validation

In the current work, an improved model to estimate the thermal conductivity of spark plasma sintered nanocomposites is presented. In the developed model, the thermal conductivity of the matrix was modeled as a function of the average matrix crystallite size rather than taking a constant matrix thermal conductivity. The model has been validated against experimentally measured thermal conductivity of Al2O3-SiC-CNT hybrid nanocomposites. Using the experimental and modeling results, it was shown that the addition of SiC and CNT inclusions to alumina resulted in a decrease in its thermal conductivity. The main reason for this decrease was found to be the reduction in the thermal conductivity of the alumina matrix itself because of the reduction in the crystallite size. Additional reduction in the composite thermal conductivity was due to the matrix-inclusion interface resistance and porosity. The predicted and measured thermal conductivities were found to be in good agreement.


Introduction
The brittleness of monolithic ceramics had limited their use in many applications [1].Fortunately, removal of this limitation was made possible through the development of composites [2], nanocomposites [3], and hybrid nanocomposites [4,5].Hybrid ceramic nanocomposites are advanced ceramic materials obtained by reinforcing a ceramic matrix with two nano-reinforcements that have different morphologies and/or attributes.The design goal for the development of these composites is to obtain materials with improved mechanical and/or functional properties [6,7].Alumina is a well-known advanced ceramic material with wide-ranging applications [8].Reinforcing alumina with two nanoscale phases has proven to be an efficient way for the improvement of its properties and performance.Analysis of literature shows that the majority of research work published on alumina hybrid nanocomposites has been devoted to the characterization of the microstructure and the evaluation of mechanical properties [5][6][7].Although, thermal properties in general and thermal conductivity in particular, are among the most important factors used in materials selection for engineering design, published work on the thermal conductivity of alumina hybrid nanocomposites produced by spark plasma sintering is very scarce [9].Moreover, estimation modeling of the thermal conductivity of these hybrid nanocomposites has also not been reported.
For conventional composites, several thermal conductivity estimation models have been reported in the literature.Early works on the problem were carried out by Maxwell [10] and Lord Rayleigh [11].Hasselman and Johnson [12] and Benveniste [13] independently extended these works to include the effect of thermal boundary resistance on the effective thermal conductivity.For composites with a high concentration of inclusions, a thermal conductivity model in the form of an implicit equation was formulated by Bruggeman [14] which was extended by Every et al. [15] to include the effect of thermal boundary resistance.Siddiqui and Arif [16] presented a generalized effective medium theory for the estimation of effective thermal conductivity particulate nanocomposites with multiple types of inclusions.Their model considers the size, shape, orientation and dispersion uniformity of the inclusions as inputs.
Several works reported in the literature show that the application of conventional thermal conductivity estimation techniques to spark plasma sintered composites leads to an overestimation of the composite's thermal conductivity [17,18].Wu et al. [17] and Ahmed et al. [18] studied the thermal conductivity of spark plasma sintered Al/MWCNT and Alumina/MWCNT nanocomposites respectively.In their respective works, they compared the estimated composite thermal conductivities with experimental measurements.Both concluded that the model over-predicted the composite thermal conductivity and that no significant improvement was achieved if the interfacial thermal resistance between the inclusion particles and matrix material was set to very high values.A possible reason for the overestimation of the composite thermal conductivity is the polycrystalline nature of the matrix material with small crystallite sizes.In a polycrystalline material, the grain boundary thermal resistance can lead to a significant reduction in the thermal conductivity of the material [19][20][21].The dependence of thermal conductivity of polycrystalline ceramic materials on the crystallite sizes has also been reported in the literature [22][23][24][25][26]. Several works have also been carried out to model the dependence of the thermal conductivity on crystallite size [20,21].
The aim of this work is to develop a prediction model for the estimation of the thermal conductivity of hybrid nanocomposites prepared by the spark plasma sintering process.The prediction model comprises an effective medium theory model for composites with multiple inclusions and a model of the thermal conductivity of the polycrystalline matrix.The accuracy of the model is analyzed by comparing its predictions to the experimentally measured thermal conductivity of Al 2 O 3 -SiC-CNT hybrid nanocomposite samples prepared using the spark plasma sintering process in the current work.Several parametric studies are also carried out to analyze the effects of various material parameters of the composite thermal conductivity.

Effective Medium Theory Model
The effective thermal conductivities of the hybrid nanocomposites are a function of the thermal conductivity of the matrix, the particle sizes, shapes, orientations and volume fractions of all inclusion types, the porosity fraction and the interfacial thermal resistance between the inclusion particles and the matrix.For the estimation of the effective thermal conductivity of the hybrid composites, the generalized effective medium theory model of Siddiqui and Arif [16] was used.The model is defined by equations (1).The parameters used in the model are defined by equations (2-9): ) ) where φ i is the volume fraction, and are the particle radii, p θ is a factor defining the orientation of inclusion of type i.K mat is the thermal conductivity of the matrix and is a function of its average crystallite size.Further details of the model are available in Siddiqui and Arif [16].

Effect of Polycrystalline Matrix
In the current work, the effective thermal conductivity of polycrystalline alumina matrix was modeled using equation (10) which is a modified form of Yang et al.'s model [21].The choice of Yang et al.'s model over Dong et al.'s model [20] which considers phonon scattering at grain boundaries, was based on the fact that the phonon means free path in alumina is around 4.8 nm [18].This length is much smaller than the average crystallite size of the alumina matrix considered in the current work.Therefore, the phonon scattering phenomenon does not need to be considered in the model: where K pc and K sc are the thermal conductivity of the polycrystalline material and the corresponding single crystal, R K is the thermal resistance at the grain boundary, d is the average crystallite size in the material and n is a sensitivity parameter.

Materials
α-Al 2 O 3 , 99.85 % pure with an average particle size of 150 nm, procured from ChemPUR, Germany, and β-SiC, 97.5 % pure with particle sizes between 45 and 55 nm, supplied by Nanostructured and Amorphous Material Inc., USA were used in this investigation.The carbon nanotubes (CNTs) were produced using chemical vapor deposition (CVD) and decorated with COOH -functional groups [4].

Sample Preparation
The nanocomposite powders were prepared as follows: (i) the required amount of alumina and SiC powders were magnetically stirred for 15 minutes in deionized water.(ii) the slurry was ultrasonicated for 2 hours using high-energy probe sonicator.(iii) the sonicated slurry was ball milled for 2 hours, using cylindrical alumina vials (250 ml in volume) and alumina balls (10 mm in diameter).A planetary ball mill (Fritsch Pulverisette Equipment P5, Germany) was used to mill the mixture at room temperature using a ball-to-powder weight ratio of 4:1 and a speed of 100 rpm.(iv) the required amount of functionalized CNTs, separately ultrasonicated for 10-15 mins, were added to the ball-milled slurry and further sonicated for 2 hours using a high energy probe sonicator.(v) finally, the mixture was dried in an oven at 120 o C for 15 hours.A fully automated spark plasma sintering equipment (FCT system, Model HDP 5, Germany) was used to consolidate the hybrid nanocomposite powders using a compaction pressure of 50 MPa, a heating rate of 100 o /min, a sintering temperature of 1500 o C, and a holding time of 10 minutes.In addition, monolithic alumina, as a reference sample, was sintered at the same conditions.More details on the consolidation process can be found elsewhere [5].To further understand the effect of average alumina crystallite size of its thermal conductivity, additional monolithic alumina samples were sintered under different sets of sintering conditions.For these monolithic Alumina samples, three different sintering temperatures i.e. 1000, 1300 and 1400 o C, and three different sintering times i.e. 1 min., 5 min.and 10 min.were used to make nine samples in total.The variation of sintering parameters was done to achieve different average crystallite sizes and therefore, different thermal conductivities of the samples.

Sample Characterization
The Al 2 O 3 nano-powder was characterized using a transmission electron microscope (Philips model CM200 200 kV).The microstructure of sintered samples was analyzed with FE-SEM.The XRD analysis of the synthesized powders and consolidated specimens was carried out using a model D8 x-ray diffractometer made by Bruker, USA with a characteristic wavelength of 0.15405 nm.A Metler Toledo balance density determination KIT model AG285 was used to measure the density of the sintered samples.The theoretical density of the hybrid composites was calculated following the rule of a mixture [27,28] and using density values of 3.97, 3.21, and 2 g/cm 3 for alumina [27], SiC [29], and CNTs [30], respectively.The thermal conductivity of sintered samples was measured using a Hotdisk Thermal Constants Analyser model TPS 2500S according to ISO standard (ISO/DIS 22007-2.2) [31].The crystallite size of the alumina phase in sintered composite samples as determined from XRD is presented in Tab.I.The average crystallite size of the monolithic alumina sample sintered at a temperature of 1500 o C and a pressure of 50 MPa for 10 min was 125 nm.The average crystallite size decreased to 98 and 93 nm for the Al 2 O 3 -5SiC-1CNT and Al 2 O 3 -5SiC-2CNT composites, respectively.The increase in SiC content to 10 wt.% led to a further decrease in the crystallite size which reached 88 and 85 nm for Al 2 O 3 -10SiC-1CNT and Al 2 O 3 -10SiC-2CNT composites, respectively.In monolithic alumina, densification, grain growth, an increase in crystallite size are due to grain boundary diffusion and grain boundary mobility [32,33].Reinforcements in composites are known to pin grain boundaries, which reduce grain boundary mobility during sintering.In the Al 2 O 3 -SiC-CNT hybrid nanocomposites, the presence of CNTs on the grain boundaries and SiC on the grain boundaries and within the grains [34] inhibited grain growth compared to monolithic alumina [35].

Microscopy, Density and Crystallite Size
Tab.I Crystallite Size Calculation using XRD peaks and Scherrer Equation for alumina matrix hybrid nanocomposites.SPS Conditions: 1500 o C, 100 o /min, 50 MPa, 10 min.

Sample Average Crystallite Size [nm]
The relative density of consolidated samples is presented in Tab.II.Monolithic alumina had a relative density of 99.85 %.In the case of pure alumina, dominant densification mechanisms are plastic flow during the initial stage and diffusion during the final stage of sintering [36,37].As for the composites, relative density values of 97.7 and 97.2 % were obtained for Al 2 O 3 -5SiC-1CNT and Al 2 O 3 -5SiC-2CNT composites, respectively.The increase in SiC content to 10 wt.% led to a further decrease in the relative density to reach 95.43 and 95.40 % for Al 2 O 3 -10SiC-1CNT and Al 2 O 3 -10SiC-2CNT, respectively.This reduction of the density due to the incorporation of SiC and CNTs in alumina can be attributed to the reduced ability of the composite to plastically deform and the reduced grain boundary diffusion due to the pinning effect of the inclusions at the grain boundaries [4,5].

Thermal Conductivity of Monolithic Alumina
The measured thermal conductivities of sintered alumina samples are listed in Tab.along with the respective porosity volume fractions.The Tab. also shows the estimated thermal conductivities of fully dense alumina samples.These were calculated by inputting the experimentally measured thermal conductivities and porosity volume fractions of the sintered samples into the generalized effective medium theory model and solving for the matrix thermal conductivity.Tab.IV shows a strong dependence of the thermal conductivity of sintered alumina on the sintering parameters which is still present when the effect of porosity has been removed by calculating the thermal conductivity of dense alumina.Using the thermal conductivity values of dense monolithic alumina from Tab. IV along with the corresponding average crystallite sizes from Tab. III, the unknown parameters in the equation (10) were estimated using the Nelder-Mead simplex search algorithm [38] implemented in the computational code MATLAB.The fitted model for the thermal conductivity of polycrystalline alumina is given by the equation (11).The single crystal thermal conductivity of alumina was determined to be 34.55W/m•K and the thermal resistance of the crystallite interfaces was estimated as •K/W [19] respectively.Fig. 2 shows the predicted alumina thermal conductivity as a function of the average crystallite size plotted against experimentally measured values.Good agreement is observed between the predicted and experimental thermal conductivity values which show that the model presented in equation ( 11) is able to accurately capture the dependence of thermal conductivity of polycrystalline alumina on its average crystallite size.

( ) (
) Fig. 2. Predicted and measured the thermal conductivity of alumina as a function of its average crystallite size.

Thermal Conductivity of Alumina-SiC-CNT Nanocomposite
Tab. V shows measured thermal conductivity values of sintered composite samples along with the thermal conductivity of monolithic alumina sintered at the same conditions.The thermal conductivity of monolithic alumina was found to be 34 The generalized effective medium theory model of Siddiqui and Arif, shown in equations (1)(2)(3)(4)(5)(6)(7)(8)(9), was used to estimate the thermal conductivity of the spark plasma sintered alumina matrix composites with SiC and CNT hybrid inclusions.The results are listed in Tab.VII.The thermal properties of inclusion phases used in the model are listed in Tab.VI.The thermal conductivity of alumina matrix was estimated using the equation (11)  shows that ignoring the crystallite size dependence of matrix thermal conductivity leads to very large errors and that the errors increase as the matrix crystallite size reduces.Considering the matrix thermal conductivity as a function of the matrix crystallite size improves the prediction accuracy for all cases.The improvement in prediction accuracy is especially high for the composites with 10 % SiC inclusion i.e.Al 2 O 3 -10SiC-1CNT and Al 2 O 3 -10SiC-2CNT for which the average alumina crystallite sizes were 88 and 85 nm respectively.The errors in the predicted thermal conductivity of these two samples reduced by 56 and 78 % respectively when the matrix thermal conductivity was considered a function of its average crystallite size.
Tab. Material properties of inclusions used in EMT model.Studies were also carried out to investigate the effects of inclusion volume fractions on the effective thermal conductivity of alumina matrix hybrid nanocomposite.For these studies, the matrix crystallite size was set at 100 nm.The results, presented in Fig. 5, show that although the thermal conductivities of CNT and SiC are higher than the thermal conductivity of the alumina matrix, their addition into the matrix results in a reduction in the effective thermal conductivity of the nanocomposite.This is due to the effect of interfacial thermal resistance between the inclusion particles and the matrix which causes the effective thermal conductivity of the inclusion particles to be reduced below the thermal conductivity of the matrix [40].

Conclusion
In the current work, a thermal conductivity model for estimating the thermal conductivity of spark plasma sintered composites has been presented.The model consists of two sub-models; one to estimate the thermal conductivity of the polycrystalline matrix a function of its average crystallite size and the second to estimate the effective thermal conductivity of a multi-inclusion composite.
Using the developed model, a study was conducted to compare the estimated thermal conductivity of spark plasma sintered monolithic alumina and ball milled and spark plasma sintered Al 2 O 3 -SiC-CNTs hybrid nanocomposites with experimental results.Following conclusions were drawn from the study: • The study on monolithic alumina showed that the inter-crystallite thermal interface resistance in alumina matrix was .The thermal conductivity of alumina single crystals was estimated as • Taking the thermal conductivity of the alumina matrix as a function of average matrix crystallite size provided a better prediction of composite thermal conductivity than the conventional approach of taking a constant matrix thermal conductivity.For the composite samples sintered in the current work, the prediction errors reduced by up to 78 % when the effect of crystallinity of matrix material was considered.
• A parametric study on the effect of matrix crystallite size on composite thermal conductivity showed that for spark-plasma sintered alumina matrix composites, the effect of matrix crystallite started to reduce above 100 nm and became negligible above 125 nm.
• It was found that the addition of SiC and CNT inclusions to alumina resulted in a decrease in its thermal conductivity.The main reason for this decrease was found to be the reduction in the thermal conductivity of the alumina matrix itself because of the reduction in its crystallite size.Additional reduction in the composite thermal conductivity was due to the matrix-inclusion interface resistance and porosity.

Fig. 1 (Fig. 1 .
Fig. 1 (a) and (b) show typical TEM and FE-SEM images of alumina powder and fracture surface of sintered alumina, respectively.The alumina powder has an average particle size of 200 nm.Sintering at 1500 o C for 10 min yielded full dense alumina.The fracture mode in alumina was mainly intergranular.Addition of SiC and CNTs to alumina did not only change the fracture mode from intergranular to intragranular but also restricted the growth of the alumina matrix.This can be clearly seen in Fig. 1 (c) and (d) showing typical FE-SEM images of Al 2 O 3 -5SiC-2CNT and Al 2 O 3 -10SiC-2CNT composites, respectively.

9 2 . 9 1
195 10 − × m 2 •K/W.As a reference, the crystallite interface thermal resistance of Silicon, Yttria stabilized Zirconia and Germanium-Silicon polycrystals have been reported to be m .44 W/m•K.The thermal conductivity decreased to 21.2 and 20.4 W/mK for the Al 2 O 3 -5SiC-1CNT and Al 2 O 3 -5SiC-2CNT composites, respectively.The increase in SiC content to 10 wt.% led to a further decrease in the thermal conductivity which reduced to 17.71 and 17.83 W/mK for Al 2 O 3 -10SiC-1CNT and Al 2 O 3 -10SiC-2CNT, respectively.Tab.V Thermal Conductivity of alumina matrix hybrid nanocomposites (at 25 o C).SPS Conditions: 1500 o C -10 min -50 MPa -100 o C/min.
in which the crystallite sizes of the sintered samples from Tab.I was used.The results are shown graphically in Fig which compares the estimated composite thermal conductivities with experimental values.The Fig.

Fig.
Fig. (a) Comparison of predicted thermal conductivities of sintered hybrid nanocomposite samples with experimentally measured values (b) Absolute errors in predicted thermal conductivities.

Fig. 4 .Fig. 5 .
Fig. 4. Effect of matrix crystallite size on the effective thermal conductivity of sintered hybrid nanocomposite samples.
Relative densities of alumina based hybrid nanocomposites (at 25 o C).SPS Conditions: 1500 o C -10 min -50 MPa -100 o C/min.Theoretical densities: Al 2 O 3 (3.97g/cm 3 ), SiC (3.21 g/cm 3 ), CNT (2 g/cm 3 ) For the additional monolithic alumina samples sintered at different sintering parameters, the relative densities and average crystallite sizes are tabulated in Tab.III.The samples sintered at 1000 o C showed very low relative densities (< 70 %) while the samples sintered at 1300 and 1400 o C showed relative densities greater than 99 %.Increase in sintering temperature from 1000 to 1400 o C also resulted in an increase in the average crystallite size from 66-73 nm to 119-127 nm.Tab.III Relative density and average crystallite size of sintered monolithic alumina samples.
Thermal conductivities of sintered monolithic alumina samples.