CHEMICAL ENGINEERING TRANSACTIONS VOL. 52, 2016 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam Copyright © 2016, AIDIC Servizi S.r.l., ISBN 978-88-95608-42-6; ISSN 2283-9216 Spark Plasma Sinering Simulation of Alumina Composite Modified with Carbon Nanotubes Natalia A. Fedosova, Eleonora M. Koltsova*, Evgeniy V. Zharikov, Ivan I. Mitrichev, Anna S. Shaneva Dmitry Mendeleev University of Chemical Technology of Russia, Miusskaya sq. 9, 125047 Moscow, Russia kolts@muctr.ru Ceramic alumina composites with carbon nanotubes (20 - 50 %vol.) were prepared by spark plasma sintering method. Dependence of the composite strength from porosity and the amount of CNTs was found. The process of spark plasma sintering was investigated to develop mathematical model of this process. Resulting model of sintering process can predict porosity of composite Al2O3-CNT after thermal exposure. Optimization of sintering parameters was allowed to find heating mode for obtain composite samples with porosity less than 0.1 % 1. Introduction Ceramic materials are widely used as a basis for protective coatings, thermal insulation and construction materials. Often they are used for the production of machinery parts exposed high temperature and high mechanical loads. Creating ceramic composites is effective way to reduce the brittleness of ceramics. Type of toughening agent depends on the kind of ceramic matrix and desirable properties of the final composite material (Cho et al., 2009). Carbon nanotubes (CNT) and carbon nanofiber (CNF) are the most common and most effective reinforcing particles for ceramic matrixes (Ghobadi et al., 2014). CNTs are carbon structures which have extraordinary properties (Kasperski et al., 2013).The ratio of length to diameter and high strength CNTs (1 TPa) can significantly improve the composite strength parameters. Reinforcement the ceramic matrix may be due to several toughening mechanisms such as crack deflection, fiber pull-out and crack bridging mechanism (Xia et al., 2004). Composite ceramic materials modified carbon nanotubes exhibit excellent mechanical properties and have a lower density than the unreinforced ceramics. Spark plasma sintering (SPS) is the most modern and efficient technology consolidation of powder materials. It is based on the use of directional pulsed electric current at a low atmospheric pressure. High energy and low voltage pulsating current generate spark plasma at high local temperatures between particles (up to 10,000 °C) (Wang et al., 2004). This leads to instantaneous thermal and electrolyte diffusion. Use SPS technique can reduce the maximum temperature of heating the material at 200 - 500 °C while the sintering duration is reduced to several minutes. The high heating and cooling rates increase the density of the final composite structure (Riggs et al., 2000) and reduce the rate of grain growth of the ceramic matrix (Suarez Riggs et al., 2013). The aim of this work is a computer simulation of the spark plasma sintering ceramic composite material reinforced by CNTs, optimization of sintering parameters in order to obtain composites with zero porosity and high strength characteristics. 2. Experimental investigations Initial composite powder obtained from a commercial alumin (α-Al2O3 – 99 %, a grain size of 3-4 μm) and CNT obtained by pyrolysis of mixture hydrogen and methane (Fedosova et al., 2014). Ultrasonic dispersing CNTs was carried out in an aqueous solution of polyvinyl alcohol (1 %) (Fedosova et al., 2015). The homogeneous distribution of nanotubes in the bulk alumina matrix provides intensive mixing suspension of CNTs and DOI: 10.3303/CET1652164 Please cite this article as: Fedosova N. A., Koltsova E. M., Zharikov E. V., Mitrichev I. I., Shaneva A. S., 2016, Spark plasma sinering simulation of alumina composite modified with carbon nanotubes, Chemical Engineering Transactions, 52, 979-984 DOI:10.3303/CET1652164 979 alumina powder in a planetary mill. After drying composite suspension and pelleting of the composite powder was obtained composite powder Al2O3-CNT with 20-50 %vol. CNT. SPS of a composite powder Al2O3-CNT produced on installing HP D25 (FCT Systeme GmbH, Germany). Pressure load, atmosphere, heating rate, maximum temperature and time of thermal action were set using of the program software HP D25. During the SPS experiments heating rate was varied from 200 to 380 °C/min, the maximum sintering temperature ranged from 1,500 to 1,600 °C. Isothermal holding time at the maximum temperature and load were constant (3 minutes and 20 kN respectively). Table 1 shows the results of experimental work, SPS modes and properties of the ceramic matrix composites Al2O3-CNT (porosity, flexural strength, fracture toughness). Measurement of flexural strength was carried out on a tensile testing machine Istron 5581 (Istron, USA). The fracture toughness was evaluated by indentation of a diamond Vickers pyramid under load. A more detailed description of a series of experiments to obtain composite samples presented in (Yi et al., 2015) and (Inbaraj et al., 2012). Table 1: Spark plasma sintering conditions and properties of composite samples. CNT volume, %vol. Heating rate, °C/min Max temperature, °C Flexural strength, MPa Fracture toughness, MPa·m½ Porosity, % 50 350 1,500 580 5.9 3.05 50 367 1,550 640 6.9 0.26 50 383 1,600 630 7.2 0.20 30 200 1,500 250 4.5 10.33 30 200 1,550 470 5.1 3.21 30 200 1,600 550 6.9 < 0.1 20 300 1,500 390 4.6 5.78 20 300 1,550 430 4.8 2.12 20 300 1,600 520 6.2 < 0.1 0 300 1,500 380 2.4 6.98 0 367 1,550 410 2.9 0.11 0 200 1,600 430 3.5 < 0.1 The results of the studies of the properties of the ceramic composite Al2O3-CNT samples can be concluded that the strength characteristics are directly dependent on the amount of the CNT and porosity of composite material. By increasing the amount of CNT is observed a gradual increase of the flexural strength and fracture toughness. Conversely, increasing the porosity of the composite tends to reduce the strength characteristics. Consequently, to obtain a composite Al2O3-CNT is necessary to provide the absence of porosity after the spark plasma sintering composite powder. Below is a mathematical model describing the SPS process, which allows numerical experiments to identify the sintering parameters provide a final porosity of the composite material less than 0.1 % and a high value of flexural strength and fracture toughness, respectively. 3. Mathematical modeling and optimization As a result of experiments, it was found that the strength of the composite depends on the porosity value. It allows us to use this characteristic as the basis of a mathematical model. In the process of developing a mathematical description of changes in composite compact porosity, we take into account physical parameters of the processes: heating rate, maximum temperature, holding time at the maximum temperature, CNTs volume and the current state of composite powder compact (current temperature and current size of pores). To describe the process of porosity decrease during sintering we introduce function of pore size distribution f(t, l), where t – time, l – pore diameter. This function shows the state of powder compact in moment t. The equation describing the process of decreasing pores is:      maxl;0l;maxt;0t;0 l l,tf t f ∈∈ ∂ ∂ - ∂ ∂   (1) f(t, l) - function of the pore size distribution, t - process time, η - speed pore size change speed overgrowing of pores, l - pore diameter. For the first stage of sintering (heating stage) driving force reduction process depends on the heating rate ΔT/Δt, current pore diameter l, current temperature T and the volume of CNTs VCNT. For the second stage 980 driving force depends on the difference in maximum temperature and recrystallization temperature (Tmax - Trecr), current pore diameter l and the volume of CNTs VCNT.                2 1010101 1 3 111 11 ; 1 C N TC N T m VcVbaa Tclbak t T k (2)          2 2020202 3 222 max22 ;2 C N TC N T m r ecr VcVbaa lbak TTk (3) η1 and η2 – the rate of decrease pore size on the 1st and 2nd stage of sintering; k1 and k2 - phenomenological coefficients considering dependence of the speed of pore size decrease from the CNTs volume VCNT (parameters a1 and a2); the current size of the pores l and the current temperature in the sintering chamber T; m1 and m2 – constants characterizing the degree of deviation of system from balance state for the 1st and 2nd stage of sintering. Looking for solutions of differential equations are used schemes with different error value. Most of the schemes are numerically stable. This fact creates a restriction on the ratio of the time step and step in space used in the calculation. We have worked to developed absolutely numerical stable method for solving differential equations. The numerical errors of this new scheme are proportional to the square of the time step and the square of the space step. Figure 1 shows stencil of new differential scheme, we called “Z-scheme”. Figure 1: Stencil of new absolutely numerical stable “Z-scheme” We used this new differential scheme “Z-scheme” to solve the Eq(1): 0 1 - 1 - - 11 - 1 1 1 1 2 1 - - 1                     l n j n j f n j n j f l n j n j f n j n j f t n j f n j f  (4) Δt – time step, Δl – a space step (pore size), index n is responsible for the time step, index j - per space step. Also we used in the initial condition and the right boundary condition:       0,;,0 ma x 0  lltflfltf (5) Initial pore size distribution f0(l) is set according the law of normal distribution with pore diameter l from 0 to 5 μm and a total porosity of 59 - 62 %. Right boundary condition corresponds to the absence of pores with maximum diameter (lmax). Current porosity ε was calculated according to the formulas:   %1 0 0; 23 4 3 0 ∫ max          s olid l VV V dllf l V      (6) Vε – total pore volume, Vsolid – total solid volume, ε – porosity of the composite. 981 To determine the final form of the mathematical model, we carried out numerical search for values of the constants of the Eq(2)-(3). For this purpose was written a computer program which can be used to evaluate and specify the values of the kinetic parameters of the mathematical model. The search results are represented in Table 2. The search criteria are performed sum of the relative errors between the calculated and experimental porosity for each experiment: ∑ 1 exp - exp N i i calc iiS     (7) S –criteria of optimization, εexp – experimental porosity value, εcalc – calculated porosity value, N – number of experiments for constant search. Table 2: Values of the constant of mathematical SPS model. Constant of first sintering stage Numeric value Constant of second sintering stage Numeric value a10 1.3 × 10-3 a20 5.2 × 10-2 b10 -5.7 × 10-3 b20 -3.6 × 10-2 c10 5.8 × 10-3 c20 2.5 × 10-2 b1 5.7 × 10-3 b2 16.1 × 10-2 c1 12.2 × 10-3 m2 3.5 m1 1.7 The adequacy of the resulting mathematical model was estimated according to the equation: %100∑ 1 exp - exp 1 mod    K i i calc ii K E   (7) Emod – total error of mathematical model, εexp – experimental porosity value, εcalc – calculated porosity value, K – number of experiments for adequacy calculate. The relative error Emod of the resulting mathematical model accounted for 10.6 %. The mathematical model Eq(1)-(3) is used to numerical experiments on SPS ceramic composite Al2O3-CNT (0-50 %vol.). Modeling used for calculating the current values of the function of the pore size distribution and current porosity of the composite material. Figure 2 and Figure 3 show results of numerical experiment for composite Al2O3-CNT (20 %vol.) which was sintered with heating rate of 300 °C/min and maximum temperature 1,600 °C. Figure 2: Porosity decrease in numerical experiment for composite Al2O3-CNT (20 %vol.). 982 Figure 3: The results of numerical experiment for composite Al2O3-CNT (20 %vol.): function of pore sizes distribution: 1 – the initial distribution, 2 – distribution after the first sintering stage, 3 - distribution after the second sintering stage. As the result of analysis of numerical experiments using a mathematical model SPS Eq(1)-(3), was found that the optimum temperature is 1,600 °C. It allows to reach the non-porous composite Al2O3-CNT (0 - 50 %vol.). Figure 4 shows the results of optimization of the CNT volume in composite powder with a sintering temperature of 1,600 °C and different heating rates: 200 °C/min, 300 °C/min, 383 °C/min. Increase in the volume fraction of CNTs increases the porosity of the composite, but the increase in the heating rate on the contrary decreases it. For receiving a non-porous Al2O3-CNT composite (porosity is less 0.1 %) at a heating rate of 383 °C/min CNT content should be less than about 30 %vol., for a heating rate of 300 °C/min – 20 %vol., the heating rate. 200 °C/min – 15 % vol. Figure 4: Porosity of the composite Al2O3-CNT at the differing heating rate and maximum temperature 1,600 °C. Heating rate: 1 – 383 °C/min, 2 – 300 °C/min, 3 – 200 °C/min. 983 4. Conclusions The mathematical model of spark plasma sintering ceramic composite Al2O3-CNT (20 - 50 %vol.) was developed. It is based on the equation of reduce composite powder porosity and is included the equations describing the heating stage by pulsed current and the stage of holding at the maximum temperature. Thermodynamic fluxes and driving forces of spark plasma sintering were identified and the differential equation describing the process was made up. The driving force of the first stage of the process the heating rate is defined. For the second stage it is the difference between the maximum temperature and the temperature of matrix recrystallization. The resulting equations include sintering parameters: the temperature, the current size of the pores and the amount of CNTs in the composite. The authors developed the new differential scheme “Z-scheme”. It is the absolutely numerical stable method of solving differential equations. The numerical errors are proportional to the square of the time step and the square of the space step. The pore size distribution and porosity value in each moment of the spark plasma sintering was calculated by “Z-scheme”. Values of model constant parameters were calculated with an error of 10.6 %. Computational experiments allowed to establish the dependence of the pore size distribution function and porosity of composite Al2O3-CNT with different volume of CNTs. It was found that to obtain a dense non- porous composite Al2O3-CNT (20-50 %vol.) necessary to carry out the SPS at 1,600°C. CNT volume must not exceed 30 %vol., 20 %vol. and 15 %vol. if the heating rate is 383 °C/min, 300 °C/min and 200 °C/min, respectively. Acknowledgments The work was supported by the Russian Science Foundation (RSF), project 14-19-00522. Reference Cho J., Boccaccini A. R., Shaffer M. 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