Non-isothermal oxidation of ceramic nanocomposites using the example of Ti–Si–C–N powder: Kinetic analysis method

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Acta Materialia 56 (2008) 3132–3145 www.elsevier.com/locate/actamat

Non-isothermal oxidation of ceramic nanocomposites using the example of Ti–Si–C–N powder: Kinetic analysis method A. Biedunkiewicz a, A. Strzelczak b,*, G. Mozdzen c, J. Lelatko d b

a Szczecin University of Technology, Institute of Materials Science and Engineering, Al. Piastow 19, 70-310 Szczecin, Poland Szczecin University of Technology, Institute of Chemistry and Environmental Protection, Al. Piastow 42, 71-065 Szczecin, Poland c Austrian Research Center – ARC GmbH, 2444 Seibersdorf, Austria d Silesian University of Technology, Institute of Materials Science, Bankowa Street 12, 40-007 Katowice, Poland

Received 20 July 2007; received in revised form 26 February 2008; accepted 27 February 2008 Available online 1 April 2008

Abstract A method of kinetic analysis applicable to non-isothermal oxidation processes of ceramic nanocomposites is presented using Ti–Si–C– N powder as the substrate. The nanoparticle size and phase composition were determined using high-resolution transmission electron microscopy and X-ray diffraction (XRD). Thermogravimetric measurements were carried out for powder samples in dry air in the temperature range 298–1770 K. The following heating rates were applied: 3, 5, 10, 20 K min1. Mass spectrometry was used to analyze gaseous oxidation products and solid products were identified by the XRD technique. The Coats–Redfern equation was applied for the kinetic analysis. For each stage of the oxidation kinetic models, the best accuracy was achieved using a series of criteria, and then the A and E parameters of the Arrhenius equations were estimated. Both linear regression and artificial neural networks were applied in testing kinetic models. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ceramics; Nanocomposites; Non-isothermal; Oxidation

1. Introduction Ceramic composites are widely used in various technologies such as tool production, aerospace technology and nuclear power engineering. Their key features are considerable hardness and temperature, and resistance to corrosion and wear. However, the application of these materials has limitations, such as brittleness and a catastrophic failure mode. Further innovation in the field of ceramic composites is associated with nanotechnology and the formation of composites containing nanoparticles, bringing a significant improvement in their mechanical properties due to the size effect [1–6] but also causing decreasing corrosion resistance [7,8]. Because many ceramic nanocomposites

*

Corresponding author. Tel.: +48 91449435. E-mail address: [email protected] (A. Strzelczak).

are designed to work in a high temperature, it is important to determine their high-temperature corrosion resistance. High-temperature corrosion is a form of corrosion that does not require the presence of a liquid electrolyte. Sometimes, this type of damage is called ‘‘dry corrosion” or ‘‘scaling”. The term ‘‘oxidation” is ambivalent, as it can refer either to the formation of oxides or to the mechanism of oxidation of a metal, i.e. its change to a higher valence than the metallic state. Strictly speaking, high-temperature oxidation is only one type of high-temperature corrosion. In fact, oxidation is the most important high-temperature corrosion reaction. Oxidation mechanisms differ depending on the type of environment, e.g. content of oxygen, steam and other gases. Studies on ceramic nanocomposites have a relatively short history and have focused mainly on fabrication technologies and structure research. Research techniques associated with those issues have already been well established,

1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.02.043

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Nomenclature pre-exponential Arrhenius factor (1 min1) apparent activation energy (J mol1, kJ mol1) conversion function dependent on mechanism of reaction F Snedecor’s statistics g(a) integral form of kinetic model h(T, a) temperature and conversion degree function k(T) reaction rate constant (1 min1) Dm current observed sample mass change (mg) Dm0 total observed sample mass change in a given stage (mg) Dm* actual current sample mass change (mg) Dm0 actual total sample mass change in a given stage (mg) r reaction rate (mol m3 min, mol m2 min, 1 min1) rP Pearson’s correlation coefficient R gas constant (J mol1 K) t time (min) A E f(a)

while the characterization of ceramic nanomaterial properties is still a challenge. Oxidation kinetics investigations have been based so far on the traditional kinetics of heterogeneous reactions. However, owing to the specific properties of nanoparticles, it should be confirmed in each case if this theory is applicable. Moreover, the majority of papers on the oxidation kinetics of ceramic nanocomposites have been based on thermogravimetric analysis under isothermal conditions. Non-isothermal measurements have usually been used to set appropriate temperatures for isothermal experiments [8–11]. Considering trends for forming multiphase nanocomposites which oxidize in a wide temperature range, isothermal studies are time consuming and prevent a quick and complex assessment of corrosion resistance. Non-isothermal measurements are more applicable in this case and correspond to the actual conditions in which ceramic nanocomposites are supposed to work. In this context, there is a need to create a system of kinetic analysis in these experiments based on the traditional kinetics of heterogeneous reactions applicable to ceramic nanocomposites. In kinetics, the reaction rate of a process proceeding with the participation of solid reacting substances can be determined by the following equation [12–16]: r¼

da dt

ð1Þ

The rate depends on temperature and conversion degree: r ¼ hðT ; aÞ

ð2Þ

After separation of variables, Eq. (2) has the form r ¼ kðT Þf ðaÞ

ð3Þ

T Tm T0 DT Te Tr Dw Dw0 Ve a Da b

temperature (K) maximum conversion rate temperature for a stage (K) initial temperature of a stage (K) temperature range (K) testing subset training subset current mass of educed gaseous products (mg) total mass of gaseous products educed in a stage (mg) verification subset conversion degree conversion degree range heating rate (K min1)

Subscripts el elementary matr matrix

In the theory of non-isothermal processes, the dependence of the reaction rate on the temperature is commonly described by the Arrhenius equation. Thus, Eq. (3) for isothermal conditions (in a developed form) can be written as follows:   da E ¼ A exp  f ðaÞ ð4Þ dt RT Assuming a linear increase in temperature and constant heating rate, one obtains [12–21]   da A E ¼ exp  f ðaÞ ð5Þ dT b RT After integration, Eq. (5) has the following form:   Z A T E gðaÞ ¼ exp  dT b T0 RT

ð6Þ

The low integration limit results from the assumption that, within a temperature range from 0 to T0, the reaction does not proceed. T0 denotes the initial temperature of a given stage. The integral on the right-hand side of Eq. (6) does not have an analytical solution. The Coats–Redfern approximation is most often applied [12]. As a result, the Coats–Redfern equation is obtained      gðaÞ AR 2RT E 1  ð7Þ ln  ¼ ln 2 bE E RT T The foregoing formula is a theoretical linear model and may not be sufficiently fulfilled for real processes, owing to different particle size and geometry, heat balance or gas phase mass transfer resistance. If the influence of these factors is not big, and they do not alter the kinetic model, they can be treated as internal randomness for series at the same

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as well as at different heating rates. In this case, linearization of non-linear equations, as is widely used to treat various problems [22,23], can be applied. That means that measurements should be treated as a stochastic process with a deterministic equation but random error associated with the coefficients. Linear equation parameters should be estimated so that they best approximate the non-linear model. Calculations are based on minimization of the error, which is treated as a stochastic process [22,23]. This procedure can be applied for not large deviations from linearity. During the kinetic analysis of multi-stages processes, for each stage the appropriate form of the g(a) function (kinetic model) is determined, and the parameters of the Arrhenius equation (A, E) are calculated. Usually identification of kinetic models is performed using statistical assessment. However, in many cases, models cannot be distinguished in this way, and additional criteria are needed [14–16,24]. There is also a danger that kinetic models elaborated for microcrystalline materials would not be appropriate for nanomaterials. Mesoscopic structures lie between the world of single atoms and molecules, which is governed by quantum mechanics, and the macroscopic world, governed by classical physics. Nanoparticles contain too many atoms to be described only by quantum mechanics, but still they are too small to pass over quantum effects. Therefore, they do not always behave in accordance with classical physics [7]. The aim of the authors’ work (part of which is presented in this paper) was to elaborate a methodology of kinetic analysis appropriate for non-isothermal oxidation processes of ceramic nanocomposites. The system will be presented for the case of a Ti–Si–C–N nanocomposite. This multiphase powder is designed for tribological coatings used at high temperatures and fulfills the requirements of the proper research goal. In order to avoid the influence of other corrosive agents, the measurements were carried out in dry air.

2.2. Characterization of the oxidation process of a Ti–Si–C– N nanocomposite

2. Experimental

Dm ¼ Dm þ Dw

2.1. Structural characterization of Ti–Si–C–N nanocomposite

and after finishing a stage as

The nanocomposite studied was a Ti–Si–C–N powder, produced by the sol–gel method. The analysed samples had the same qualitative (but different quantitative) phase composition. Dispersion and nanoparticle sizes were assessed by transmission electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM) using JEOL JEM 1200EX and JEM 3010 apparatus. In turn, the phase composition was determined by X-ray diffraction (XRD). Measurements were carried out on a Philips X’Pert apparatus with a copper X-ray tube with wavelength Cu Ka = 0.15418 nm, current voltage 35 kV and intensity 30 mA. Spectrum analysis was performed using X’Pert HighScore 1.0 software.

The conversion degree is then calculated as follows:

Thermogravimetric analysis (TG) in non-isothermal conditions was performed on a SETARAM 92–15 apparatus in dry air and under atmospheric pressure. The air used was of class 5.0 (Messer, Germany) with 20.5 vol.% O2 and the rest N2. The impurities contained were as follows: H2O < 10 vpm, CO2 < 0.5 vpm, NOx < 0.1 vpm, hydrocarbons < 0.1 vpm. The following linear heating rates were applied: 3 K min1 (sample mass 5.939 mg); 5 K min1 (5.130 mg); 10 K min1 (4.738 mg); 20 K min1 (5.7979 mg). TG, differential thermogravimetric (DTG) and heat flow (HF) curves were recorded. The apparatus used was of high-resolution: 0.4 lW for HF and 0.03 lg for TG. The identification of reacting substances and the division of the process into stages were supported by additional techniques, which were applied to samples oxidized with a heating rate of 5 K min1. Gaseous products were determined by mass spectroscopy (MS). Measurements were carried out on a Thermostar GSD 301 type Pfeiffer Vacuum apparatus. Solid products were identified by XRD in samples heated up to 573, 783, 823, 1103 and 1573 K. These temperatures were chosen so that they fell at the beginning and the end of a given stage. Apparatus, measurement conditions and software for XRD spectrum analysis were the same as those identified in Section 2.1. 2.3. Kinetic analysis A quantitative description of the process studied was based on normalized TG curves obtained from a division of the TG by sample mass. Because the oxidation of Ti + Si + C + N nanocomposite proceeds with both mass gain (solid products) and mass loss (gaseous products), the conversion degree was calculated in the following way [14–27,24]. The current gain of a sample mass can be expressed as

Dm0 ¼ Dm0 þ Dw0

aðT Þ ¼

Dm Dm0

ð8Þ

ð9Þ

ð10Þ

Inserting Eqs. (8) and (9) into Eq. (10), one obtains  Dw Dm 1 þ Dm

 aðT Þ ¼ ð11Þ Dm0 1 þ Dw0 Dm0 Assuming a constant ratio of gaseous products to the recorded change of the sample mass or when the amount of gaseous products is very low compared with the recorded sample mass, one obtains

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aðT Þ ¼

Dm Dm0

ð12Þ

Eq. (12) was applied for kinetic calculations in this study. In order to verify its correctness, the normalized TG curve for stage I was compared with the integral curve calculated on the basis of MS measurements (TG MS). Because there is a much lower ion current in the case of spectral line m/e 30, and its considerable fluctuations result in low confidence in the calculations, only spectral line m/e 44 was taken into account. That spectral line determined the main gaseous product concentration (CO2) in the air flowing through the apparatus. The amount of CO2 produced was calculated by numerical integration. The assessment criterion was similarity of the curves, which usually happens when the mass increase (attributed to metal oxidation) and mass loss (connected with gaseous oxidation products) start, achieve maximum rate and stop at the same time. The applied method of kinetic analysis consisted of several steps [14–27,24]. Calculated a(T) functions were assessed by artificial neural networks (ANN) [25–28] using StatSoft STATISTICA Neural Networks 4.0A software to check whether, in accordance with theory, the temperature was sufficient for a functional description of the conversion degree at different heating rates. All the series of measurements were considered jointly. The performance of a multilayer perceptron (MLP) model was assessed by Pearson’s correlation coefficient between experimental and predicted data, and also by the SD ratio (the ratio of prediction error standard deviation to standard deviation of experimental data). These parameters were calculated separately for training (Tr), verification (Ve) and testing (Te) subsets [29]. In a subsequent analysis, the following kinetic models were tested: D1, D2, D3, D4, F1, F2, F3, R1, R2, R3 (Table 1). Preliminary selection was based on the assumption

of  fulfilling linearity in the coordinate system gðaÞ ln T 2  1000 . The goodness of fit was assessed by linear T regression using Statgraf software (separately for a consec-

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utive series of measurements). Additionally, the fulfillment of the Coats–Redfern equation was evaluated with linear neural network models. In that case, all the series of measurements were considered jointly under the assumption that A and E parameters should be constant and independent of process conditions, and therefore not only particular but also overall deviations from linearity should be minimal. Parameters of the Arrhenius equation calculated from a given model by means of linear regression were adjusted by stochastic linearization, i.e. correction of the E value so that the mean percentage error in a series approached zero. The identification of the final kinetic model was supported by additional criteria. It was required that linearity in all the series of measurements was fulfilled for the same g(a) function. Plots of k(T) dependences had to be convergent in spite of differences in A and E values between series. Finally, consistency between a(T) values calculated from the Coats–Redfern equation and determined from measurements over a wide range were necessary.

3. Results and discussion 3.1. Structural characterization of the Ti–Si–C–N nanocomposite The XRD powder diagram of a Ti–Si–C–N nanocomposite is shown in Fig. 1. The following phases were identified: Ti(C,N) (ICDD card no. 42-1488 [30]); Si(C,N) (ICDD card no. 74-2308 [31]); Si3N4 (ICDD card no. 410360 [32]). Crystalline carbon was not observed. Thus, it was concluded that a slight amount of carbon present in the samples had an amorphous form. TEM analysis revealed the polycrystalline structure of the powder studied. In the TEM picture (Fig. 2a), whiskers of Si3N4 and cubic crystals of Ti(C,N) and Si(C,N) are visible. They give an electron diffraction pattern (Fig. 2b and c) as linearly arranged diffraction points and rings, respectively [33,34].

Table 1 List of tested kinetic models Mechanism

Symbol

f(a)

g(a)

One-dimensional diffusion Two-dimensional diffusion, cylindrical symmetry

D1 D2

1 2a

Three-dimensional diffusion, spherical symmetry, Jander equation

D3

ð1aÞ2=3 1ð1aÞ1=3

Three-dimensional diffusion, spherical symmetry, Ginstling–Brounshtein equation

D4

½ð1  aÞ1=3  11

a2 (1  a)ln(1  a) + a h i 2=3 3 2 2 1  3 a  ð1  aÞ h i 2=3 3 2 2 1  3 a  ð1  aÞ

First-order reaction Second-order reaction Third-order reaction Random nucleation, Avrami I equation Random nucleation, Avramiego II equation Phase-boundary reaction, zero-order reaction Phase-boundary reaction, cylindrical symmetry Phase-boundary reaction, spherical symmetry

F1 F2 F3 A2 A3 R1 R2 R3

(1  a) (1  a)2 3 1 2 ð1  aÞ (1  a)[ln(1  a)]1/2 (1  a)[ln(1  a)]2/3 1 (1  a)1/2 (1  a)2/3

[ln(1  a)] (1  a)1  1 (1  a)2  1 [ln(1  a)]1/2 [ln(1  a)]1/3 a 2[1  (1  a)1/2] 3[1  (1  a)1/3]

[ln (1  a)]1

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Fig. 1. XRD powder diagram of Ti–Si–C–N nanocomposite.

Fig. 2. (a) TEM picture and (b, c) electron diffraction images of Ti–Si–C–N nanocomposite.

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3.2. Characterization of oxidation process of Ti–Si–C–N nanocomposite In a Ti–Si–C–N powder sample heated to 573 K, no change in the phase composition was observed. At 783 K, crystalline substrates still existed. Additionally, crystalline carbon of unidentified form and anatase (ICDD (International Centre for Diffraction Data) card no. 84-1285 [35]) appeared (Fig. 3). This situation indicated oxidation of the Ti(C,N) phase. A sample heated to 823 K did not contain titanium carbonitride; rutile (ICDD card no. 77-0441 [36]) appeared as a result of anatase phase transition. At 1103 K Si(C,N), Si3N4, anatase and rutile were observed,

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while at 1573 K, only Si3N4, rutile and SiO2 (ICDD card no. 01-0424 [37]) were observed (Fig. 4), indicating oxidation of Si(C,N) to silicon dioxide [33,34] and the final transition of anatase to rutile. Under applied conditions, the oxidation of silicon nitride did not take place or did so only to a small extent. During oxidation the following gaseous products were detected with MS: CO2, CO, NO and NO2. Owing to similarities between the mass spectrum of CO and CO2 in Figs. 5 and 6 only spectral lines m/e 44 (CO2) and m/e 30 (NO, NO2) are juxtaposed with the normalized TG curves. As the apparatus used was of high-resolution, the error bars are not visible in the plots.

Fig. 3. XRD powder diagram of Ti–Si–C–N nanocomposite heated to 783 K with a heating rate of 5 K min1.

Fig. 4. XRD powder diagram of Ti–Si–C–N nanocomposite heated to 1573 K with a heating rate of 5 K min1.

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Fig. 5. Spectral line m/e 44 and normalized TG curve recorded during oxidation of Ti–Si–C–N powder up to 1573 K with a heating rate of 5 K min1.

Fig. 6. Spectral line m/e 30 and normalized TG curve recorded during oxidation of Ti–Si–C–N powder up to 1573 K with a heating rate of 5 K min1.

In the initial phase of the process, a slight decrease in sample mass was not accompanied by the formation of gaseous products of oxidation. This stage was identified as desorption of impurities and was not considered in the kinetic analysis. The first peak in the spectral line m/e 44 indicated oxidation of Ti(C,N) with the formation of carbon oxides. At the end of this stage, nitrogen oxides also appeared. The shift of spectral line m/e 30 at a higher temperature confirms that carbon in carbonitrides is substituted by oxygen faster than by nitrogen [38]. The maximum of the first peak for m/e 44 fell upon a mass decrease connected with the combustion of elementary carbon formed during the oxidation of Ti(C,N), while the second peak was assigned to the combustion of carbon impurities contained in the original samples. This assignment was chosen according to earlier research, which had shown that elementary carbon oxidizes at a lower temperature than carbon matrix or impurities. The proposed

mechanism corresponds also to the oxidation of transition metal carbides and carbon nanocomposites [39,40]. The third peak of m/e 44 and the second of m/e 30 spectral lines indicated the oxidation of Si(C,N). Normalized TG, DTG and HF curves recorded during oxidation of four samples at consecutive heating rates are shown in Figs. 7–9. The error bars are not visible owing to the high accuracy of the measurements and very long measurement series. At higher heating rates, the last stage was not completed; the total mass increase in this step was calculated as double the value of the mass increase for the maximum in DTG curves. Oxidation of Ti(C,N), combustion of elementary carbon and carbon impurities were exothermal processes. Theoretically, oxidation of Si(C,N) should also be exothermic, but an endothermic peak was observed, probably due to simultaneous transformation of TiO2 from anatase to rutile, which dominated the thermal effect.

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Fig. 7. Normalized TG curves from oxidation of Ti–Si–C–N nanocomposite.

Fig. 8. DTG curves from oxidation of Ti–Si–C–N nanocomposite.

Fig. 9. HF curves from oxidation of Ti–Si–C–N nanocomposite.

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From the foregoing results, one can conclude that the oxidation of a Ti–Si–C–N nanocomposite occurred in four stages: (I) oxidation of Ti(C,N); (II) combustion of elementary carbon formed in stage I; (III) combustion of carbon impurities contained in the original samples and (IV) oxidation of Si(C,N). 3.3. Kinetic analysis The assumption of constant ratio of gaseous oxidation products to solid weight change was verified by comparison of the normalized TG curve recorded for step I with the integral curve calculated on the basis of m/e 44 spectral line (Fig. 10, the error bars are smaller than the plotted dots). The observed shift of TG MS plot with relation to normalized TG, often observed for transition metal carbides and carbonitrides [8,38,39], caused deviations from the assumption for low and high conversion degrees. However, the assumption was still fulfilled in a wide range of conversion degree. Moreover, if the foregoing assumptions were not fulfilled, the identification of kinetic models would be difficult. Conversion degrees calculated for consecutive stages at different heating rates are shown in Fig. 11 (the error bars are smaller than the plotted dots). The a(T) dependences obtained, with all the series of measurements considered jointly, were assessed using a neural network technique. The architecture and performance of the neural models are given in Table 2. High accuracy was achieved for all the stages. Therefore, any errors occurring in the analyses would be caused by insufficient fulfillment of the Coats– Redfern equation. The applied method of kinetic analysis will be presented using the example of stage I (oxidation of Ti(C,N)). In this

case, on

the basis of the preliminary assessment of linearity 

in the ln gðaÞ  1000 coordinate system, the following modT T2 els fitted well to experimental data: D3, F1 and R3. The results of linear regression calculations with Snedecor’s F statistics are shown in Table 3. Kinetic parameters differed between each series of measurements because of deviations from  linearity rather than gðaÞ physical factors. Plots of ln T 2  1000 during linear T regression calculations had different slopes between series and therefore different A and E values. Statistical parameters were at the same level for all the models, and an unambiguous choice of one of them was not possible. Using linear neural network models, which considered all the series of measurements jointly, similar results were obtained (Table 4) and further criteria had to be used. Plots of k(T) dependence were highly convergent for models F1 and R3 (Fig. 12; the error bars are smaller than the plotted dots) and on that basis they were chosen for further calculations. Using the A and E parameters listed in Table 3, values of the ln(g(a)/T2) function were calculated. Considerable systematic error was corrected by a slight change in parameter E in order to improve fitting of the linear model. The objective function was a minimal error in a series (close to 0). The results are given in Table 5. Using the corrected values of the Arrhenius parameters, dependence a(T) was calculated for both models and compared with those determined from measurements (Fig. 13). Deviations from experimental conversion degrees were higher for model F1, while for R3 good conformity was obtained in the whole range of a(T). The results of all the kinetic calculations are listed in Table 6. In stage II (combustion of carbon formed in

Fig. 10. Comparison of the normalized TG curve recorded for step I with the integral curve calculated on the basis of m/e 44 spectral line.

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Fig. 11. Dependence of conversion degree on temperature in the distinguished stages of Ti–Si–C–N oxidation at the consecutive heating rates: (a) stage I; (b) stage II; (c) stage III and (d) stage IV.

Table 2 Statistical assessment of a = f(T) neural models at different heating rates for the consecutive stages of Ti–Si–C–N oxidation Parameter

Tr

Ve

Te

Tr

SD ratio Correlation

Stage I, MLP 2/4 0.0120 0.0127 0.999 0.999

SD ratio Correlation

Ve

Te

0.0129 0.999

Stage II, MLP 2/3 0.0172 0.0183 0.0181 0.999 0.999 0.999

Stage III, MLP 2/3 0.0240 0.0243 0.0240 0.999 0.999 0.999

Stage IV, MLP 2/3 0.0190 0.0194 0.0189 0.999 0.999 0.999

Neural network architecture: input neurons/hidden neurons.

stage I), only model F3 was fitted relatively well. In turn, the combustion of carbon impurities (stage III) contained in the original samples proceeded according to model F1. It should be noted that, in the case of steps II and III, a considerable discrepancy of k(T) functions appeared in spite of the good fit of the same kinetic model for all the series. This situation indicated possible crystallization of the carbon phases, which might have changed the conversion characteristics. High values of the E parameter in stage II were caused by a small amount of this reacting substance, resulting in a rapid course of combustion in a narrow temperature range. In the last stage (oxidation of

Si(C,N)), three kinetic models fitted well to experimental data: D3, F1 and R3. Only after application of the k(T) convergence criterion was model D3 chosen. The final verification of the kinetic parameters (given in Table 6) is shown by the comparison of experimental a(T) functions and the functions calculated from kinetic models (Fig. 14). For all the remaining stages, a good fit was also obtained. Assuming the process parameters to be stochastic, deviations from linearity should be of normal distribution. Therefore, mean kinetic parameters, given in Table 6, properly describe the oxidation process of Ti–Si–C–N nanocomposite. Because often several kinetic models fulfilled linearity with similar accuracy, it is difficult to indicate unambiguously a mechanism according to which a reaction proceeds. This problem may reflect the peculiarity of nanostructural materials and processes in the nanoworld. However, the results obtained confirm some of the earlier reports. The kinetics of Ti(C,N) oxidation was described by model R3 of phase-boundary reaction with spherical symmetry, which has not been reported so far. Sparse investigations revealed that the oxidation of microcrystalline Ti(C,N) proceeded according to parabolic model [41,43]. The best fit of model F3 and F1 (connected with the number of reaction nuclei) in the combustion of elementary carbon and

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Table 3 Kinetic parameters and model fitting in the consecutive series of measurements b (K min1)

Kinetic model

rP

F

Tm (K)

E (kJ mol1)

A (min1)

Da

3

D3 F1 R3

0.999 0.999 0.9995

1,104,873.00 836,943.20 981,947.41

720.22

313.61 163.05 151.06

7.19E+20 5.79E+10 6.03E+09

0.01–0.99

5

D3 F1 R3

0.998 0.996 0.994

378,981.10 128,909.22 362,069.83

733.68

307.54 158.76 147.98

1.67E+20 2.78E+10 3.69E+09

0.01–0.99

10

D3 F1 R3

0.998 0.992 0.987

299,371.72 808,295.10 277,739.00

755.49

420.58 221.48 204.19

6.44E+27 7.31E+14 3.57E+13

0.01–0.99

20

D3 F1 R3

0.999 0.995 0.999

418,1764.20 235,153.74 3,763,900.05

777.15

406.06 211.82 196.82

2.14E+26 1.17E+14 8.88E+12

0.01–0.99

Oxidation of Ti–Si–C–N nanocomposite in stage I.

Table 4 Statistical assessment of ln(g(a)/T2) = f(H) neural models at different heating rates Parameter

Model D3

Model F1

Model R3

Tr

We

Te

Tr

We

Te

Tr

We

Te

SD ratio Correlation

0.1516 0.988

0.1663 0.986

0.1734 0.985

0.1649 0.986

0.1760 0.984

0.1827 0.983

0.1590 0.987

0.1762 0.984

0.1696 0.986

Oxidation of Ti–Si–C–N nanocomposite in stage I.

Fig. 12. Dependencies k(T) for the consecutive series of measurements. Oxidation of Ti–Si–C–N nanocomposite in stage I: (a) model D3; (b) model F1 and (c) model R3.

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Table 5 Values of E parameter and errors for F1 and R3 models in the consecutive series of measurements b (K min1)

Model

Before correction E (kJ mol1)

Error (%)

After correction E (kJ mol1)

Error (%)

3

F1 R3

163.05 151.06

6.6 5.8

157.52 146.14

0.0003 0.0007

5

F1 R3

158.76 147.98

6.2 5.4

153.47 143.28

0.0002 0.0045

10

F1 R3

221.49 204.19

8.3 7.6

214.09 197.34

0.0031 0.0003

20

F1 R3

211.82 196.82

7.7 7.1

204.69 190.23

0.0029 0.0002

Oxidation of Ti–Si–C–N nanocomposite in stage I.

Fig. 13. Comparison of experimental conversion degrees and those calculated from kinetic models. Oxidation of Ti–Si–C–N nanocomposite in stage I.

Table 6 Kinetic parameters obtained for all stages of the Ti–Si–C–N oxidation process Stage

Model

b (K min1)

A (1 min1)

E (kJ mol1)

(I) Ti(C,N)

R3

3 5 10 20 Mean

6.03E+09 3.69E+09 3.57E+13 8.88E+12 1.11E+13

146.14 143.28 197.34 190.23 169.25

720.2 733.7 755.5 777.2

626–762 630–772 681–790 691–809

0.01–0.99 0.01–0.99 0.01–0.99 0.01–0.99

(II) Cel

F3

3 5 10 20 Mean

2.52E+88 5.48E+113 2.77E+125 1.43E+164 3.58E+163

1306.86 1699.05 1907.83 2550.91 1866.16

783.0 791.6 807.0 824.9

771–816 780–818 797–830 815–833

0.02–0.99 0.02–0.99 0.02–0.99 0.02–0.96

(III) Cmatr

F1

3 5 10 20 Mean

2.66E+21 5.55E+19 1.88E+16 1.41E+17 6.79E+20

358.64 334.88 280.26 300.67 318.61

863.2 880.4 903.0 935.5

823–898 833–914 845–943 871–975

0.02–0.99 0.02–0.99 0.02–0.99 0.02–0.99

(IV) Si(C,N)

D3

3 5 10 20 Mean

3468.11 7108.99 55033.5 86694.5 38076.3

175.87 183.28 205.12 211.08 183.84

1104.9 1481.3 1536.5 1581.4

956–1617 978–1671 1041–1720 1093–1608

0.01–0.99 0.01–0.99 0.01–0.99 0.01–0.57

Tm (K)

DT (K)

Da

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A. Biedunkiewicz et al. / Acta Materialia 56 (2008) 3132–3145

Fig. 14. Comparison of experimental a(T) functions and those calculated from kinetic models for the remaining stages of Ti–Si–C–N oxidation: (a) stage II; (b) stage III and (c) stage IV.

carbon impurities can be explained by the low amount of substrate [8,11,41,42]. And finally it was confirmed that the oxidation of Si(C,N) is diffusion controlled [8].

range of temperatures – it meets the requirements of an efficient engineering tool. Acknowledgements

4. Conclusions This study confirmed that methods of traditional kinetics of heterogeneous reactions in non-isothermal conditions based on the Coats–Redfern equation can be successfully applied to the oxidation processes of nanocomposites. The oxidation of Ti–Si–C–N nanocomposite occurred in four stages: the kinetics of Ti(C,N) oxidation (stage I) was described by model R3 of phase-boundary reaction with spherical symmetry; combustion of elementary carbon (stage II) and carbon impurities (stage III) proceeded according to models F3 and F1, connected with number of reaction nuclei; the oxidation of Si(C,N) (stage IV) was diffusion controlled (model D3). It was demonstrated that statistical tests commonly used in the identification of kinetic models and the accuracy assessment of the Arrhenius parameters were insufficient. The application of additional criteria was necessary. The method of kinetic analysis proposed for the oxidation of ceramic nanocomposites in non-isothermal conditions allows a high accuracy to be achieved and can be applied to multiphase materials which oxidize in a wide

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