**1. Introduction**

The mechanism of nanostructural carbides synthesis, and chemical activity of nanoparticles in oxidizing environments, even occurring in trace amounts in high-purity gases, require application of precise methods and performing investigations for a wide range of parameters. The methodology of investigation and the way of determination of selected synthesis conditions have been described in this work. Appropriate selection of investigation methodology enables understanding of process mechanisms, performing quantitative analysis and then correct determination of their conditions.

Good basis for analysis of the processes proceeding with participation of solids are kinetic studies. Kinetic studies can be carried out under isothermal or non-isothermal conditions. The transitions with participation of solid reagents usually proceed in many stages. Each step should be treated as an independent process. The measurements indispensable for identification of process stages and reagents are usually carried out by methods of thermal analysis. There are elaborated different methods of kinetic studies. They are the subject of ongoing discussion [1].

Methods of process kinetics are of great significance for materials engineering. In work [2], for example, kinetics of carbothermal synthesis of β-SiC was investigated; in work [3,4] kinetics and mechanism of carbothermal reduction of MoO3 to Mo2C; in work [5] kinetics of thermal decomposition of NH4VO3; in work [6] kinetics of nanometric ceramic materials synthesis in argon and their oxidation in dry air were investigated.

Kinetic studies of manufacturing process of carbides of the metals, e.g. titanium, vanadium, niobium, tantalum or silicon, are of particular significance. These metal carbides belong to the group of ceramic materials known as conventionally hard materials, high wear and oxidation resistance. This results from the character of chemical bound and crystallographic structure. The fabrication of ultrafine-grain ceramics by powder- metallurgical processes

#### 136 Heat Treatment – Conventional and Novel Applications

involves a number of serious difficulties. In particular, it is necessary to use ultrafine powders and to optimize sintering conditions so as to prevent grain growth. Due to their high reactivity of such powders, the process must be run in an inert atmosphere. The composites containing nanostructural carbides show higher strengthening in comparison to their microstructural equivalents [7].

Methodology of Thermal Research in Materials Engineering 137

(1)

(2)

(3)

= − (4)

= − (5)

artificial neuron networks. Kinetics and by the same, the mechanism of the process of oxidation of nanocrystalline carbides in form of powder were tested, and they were

The possibility to remove the carbon matrix in reaction with oxygen was considered during analysis of the kinetics of the process of TiCx/C nanocomposite oxidation in dry air. It was proved that it was not possible to purify the obtained nc-TiC by burning in the air the

It is assumed that the rate of non-catalytic, heterophase reactions depends on temperature,

( ) , , *<sup>d</sup> r TP*

() ( )( ) *<sup>d</sup> kT f hP dt*

()() *<sup>d</sup> kT f dt*

Separation of variables means that k(T) function should not depend on conversion degree, and function f(α) should not depend on temperature. Function k(T) is described by

k(T) function maintains its exponential character in relatively narrow range of temperature. Assuming T=0 as integration limit this range is exceeded since k (T) tends asymptotically to

exp ( ) *d E A f dt RT*

α

( ) exp *<sup>E</sup> kT A*

α

α

*RT*

α

*dt* α = = φ α

α=

> α=

Under thermogravimetric measurements conditions *h P*( ) ≈ 1 [18]. Then

subjected to evaluation based on the comparison of the oxidation rate values.

carbon matrix, contained in the system.

**2.1. Methods kinetic analysis** 

conversion degree and pressure

Arrhenius equation

After separation of variables we obtain

the limit values (at T → *o* and T → ∞ ).

Thus, for isothermal conditions we have

By combining (3) and (4) we obtain

**2. Thermal decomposition of NH4VO3** 

Synthesis of these materials is most often carried out by carbothermal reduction of oxides or precursors containing metal-oxygen-carbon conjugation [1,2,8,9,10]. Other attractive synthesis routes are processes of decomposition of organometallic precursors or synthesis carried out with participation of hydrocarbons, eg. CH4 or C6H6, and salts of transition metals [11,12]. Carbothermal synthesis of the carbides proceeds through stages of oxides formation and than oxycarbides formation being the intermediate products of the syntheses. In the latter case the intermediate products are low-stoichiometry carbides of high vacancies concentration resulting from stability range of these phases in equilibrium metal-carbide system.

Presence of oxygen atoms in MCxO1-x lattice or high concentration of carbon vacancies in MC1-x lattice cause reduction in hardness and wear and oxidation resistance [12,13].

The synthesis stages leading to elimination of above mentioned lattice defects and obtainment of high-stoichiometry, oxygen stripped carbides, characterized by best properties, proceeds in temperature range above 1000ºC.

At the example of decomposition processes investigations process kinetics of the decomposition of NH4VO3 has been analysed. The issues concerning kinetics of those processes have been presented at the example of thermal decomposition of NH4VO3 to V2O5 in dry air. Vanadium carbides, carbonitrides and borides are known ceramic materials [14- 16]. NH4VO3 could be used in these processes as a precursor. The results of investigations on thermal decomposition of NH4VO3 in dry air have been presented. The base of kinetic description of these processes were termoanalytical TG-DSC measurements. They allowed identifying of intermediate and final products, distinguishing stages of the process, determination of their temperature ranges and acquisition the quantitative description. Process kinetics of the stages were described by Kissinger's method, isoconversional method and applying Coats- Redfern equation.

Obtaining the carbides of high carbonization degree, remaining the right grains size and properties, is essential. Selection of parameters meeting these requirements is difficult. Kinetic studies have significance during investigations of conversions proceeding with use of nanomaterials [17]. In the process of TiC synthesis by sol-gel method, described in work [6], the intermediate product is low stoichiometric MC1-x (x≤ 0.3). Carbides nucleate and grow in the carbon matrix. The process is carried out in argon. Describing kinetics, Coats-Redfern's equation was applied. Kinetic models of stages were identified basing on statistical evaluation and compliance to a large extent of conversion degrees of stages calculated and determined from termoanalytical measurements. While building the kinetic models of the processes, the results of measurements were treated as statistic values. A system of a complex analysis of measurements results was developed with the use of artificial neuron networks. Kinetics and by the same, the mechanism of the process of oxidation of nanocrystalline carbides in form of powder were tested, and they were subjected to evaluation based on the comparison of the oxidation rate values.

The possibility to remove the carbon matrix in reaction with oxygen was considered during analysis of the kinetics of the process of TiCx/C nanocomposite oxidation in dry air. It was proved that it was not possible to purify the obtained nc-TiC by burning in the air the carbon matrix, contained in the system.

## **2. Thermal decomposition of NH4VO3**

#### **2.1. Methods kinetic analysis**

136 Heat Treatment – Conventional and Novel Applications

their microstructural equivalents [7].

system.

involves a number of serious difficulties. In particular, it is necessary to use ultrafine powders and to optimize sintering conditions so as to prevent grain growth. Due to their high reactivity of such powders, the process must be run in an inert atmosphere. The composites containing nanostructural carbides show higher strengthening in comparison to

Synthesis of these materials is most often carried out by carbothermal reduction of oxides or precursors containing metal-oxygen-carbon conjugation [1,2,8,9,10]. Other attractive synthesis routes are processes of decomposition of organometallic precursors or synthesis carried out with participation of hydrocarbons, eg. CH4 or C6H6, and salts of transition metals [11,12]. Carbothermal synthesis of the carbides proceeds through stages of oxides formation and than oxycarbides formation being the intermediate products of the syntheses. In the latter case the intermediate products are low-stoichiometry carbides of high vacancies concentration resulting from stability range of these phases in equilibrium metal-carbide

Presence of oxygen atoms in MCxO1-x lattice or high concentration of carbon vacancies in

The synthesis stages leading to elimination of above mentioned lattice defects and obtainment of high-stoichiometry, oxygen stripped carbides, characterized by best

At the example of decomposition processes investigations process kinetics of the decomposition of NH4VO3 has been analysed. The issues concerning kinetics of those processes have been presented at the example of thermal decomposition of NH4VO3 to V2O5 in dry air. Vanadium carbides, carbonitrides and borides are known ceramic materials [14- 16]. NH4VO3 could be used in these processes as a precursor. The results of investigations on thermal decomposition of NH4VO3 in dry air have been presented. The base of kinetic description of these processes were termoanalytical TG-DSC measurements. They allowed identifying of intermediate and final products, distinguishing stages of the process, determination of their temperature ranges and acquisition the quantitative description. Process kinetics of the stages were described by Kissinger's method, isoconversional method

Obtaining the carbides of high carbonization degree, remaining the right grains size and properties, is essential. Selection of parameters meeting these requirements is difficult. Kinetic studies have significance during investigations of conversions proceeding with use of nanomaterials [17]. In the process of TiC synthesis by sol-gel method, described in work [6], the intermediate product is low stoichiometric MC1-x (x≤ 0.3). Carbides nucleate and grow in the carbon matrix. The process is carried out in argon. Describing kinetics, Coats-Redfern's equation was applied. Kinetic models of stages were identified basing on statistical evaluation and compliance to a large extent of conversion degrees of stages calculated and determined from termoanalytical measurements. While building the kinetic models of the processes, the results of measurements were treated as statistic values. A system of a complex analysis of measurements results was developed with the use of

MC1-x lattice cause reduction in hardness and wear and oxidation resistance [12,13].

properties, proceeds in temperature range above 1000ºC.

and applying Coats- Redfern equation.

It is assumed that the rate of non-catalytic, heterophase reactions depends on temperature, conversion degree and pressure

$$r = \frac{d\alpha}{dt} = \phi\left(T, \alpha, P\right) \tag{1}$$

After separation of variables we obtain

$$\frac{d\alpha}{dt} = k(T)f\left(\alpha\right)h(P) \tag{2}$$

Under thermogravimetric measurements conditions *h P*( ) ≈ 1 [18]. Then

$$\frac{d\alpha}{dt} = k(T)f(\alpha)\tag{3}$$

Separation of variables means that k(T) function should not depend on conversion degree, and function f(α) should not depend on temperature. Function k(T) is described by Arrhenius equation

$$k\left(T\right) = A\exp\left(-\frac{E}{RT}\right) \tag{4}$$

k(T) function maintains its exponential character in relatively narrow range of temperature. Assuming T=0 as integration limit this range is exceeded since k (T) tends asymptotically to the limit values (at T → *o* and T → ∞ ).

By combining (3) and (4) we obtain

$$\frac{d\alpha}{dt} = A \exp\left(-\frac{E}{RT}\right) f(\alpha) \tag{5}$$

Thus, for isothermal conditions we have

#### 138 Heat Treatment – Conventional and Novel Applications

$$\log\left(\alpha\right) = \prod\_{o}^{\alpha} \frac{d\alpha}{f\left(\alpha\right)} = k\left(T\right)\mathbf{t} \tag{6}$$

Methodology of Thermal Research in Materials Engineering 139

charts are prepared. Then the activation

− = are calculated by linear

(12)

(11)

By transforming (10) and adapting the result to the measurements performed at different

( ) ' <sup>2</sup> , ,

α

*m i m i AR <sup>E</sup> <sup>f</sup> <sup>T</sup> <sup>E</sup> RT*

> ( ) ( ) ' exp *<sup>E</sup> Af <sup>B</sup> <sup>R</sup>* − = α

> > 1

=

α

α

depending on the conversion degree, for different βi, should be the same. The ranges

and HF charts. After determining the form of the function f(α) the derivative ( ) '

In the the isoconversional method equation (5), obtained after separation of variables, is also

= = ÷ for given βi must be determined experimentally. At higher heating rates of the

α

 α

Arrhenius coefficient A is calculated.

'( ) ( ) exp[ ] *T m i T Af <sup>E</sup> Y T dT*

α

β

0

=

*RT*

≈ *const* The integral was calculated numerically. There should be

= − (13)

1

*T*

α

*T*

α β

1

=

exp

*i RT*

*<sup>E</sup> dT*

*<sup>m</sup> f* α

is

 <sup>−</sup> functions

= = ÷ . This is consistent with DTG

0

=

To calculate A one needs to know the kinetic model for a given stage (form of the function f(α)). The kinetic model most consistent with the experimental data, of the tested models, was established by analyzing the trajectories of Y(T) function depending on α(T). The conversion degrees α(T) for the stages were estimated on the basis of experimental data,

=− −

, , <sup>1000</sup> ln *<sup>i</sup> mi mi T T*

*m*

*AR f B <sup>E</sup>* α

heating rates of samples the Kissinger equation (11) is obtained

energy E and the value of expression ( ) ' ln *<sup>m</sup>*

whereas the Y(T) function was calculated from the formula

added that at constant activation energy, trajectories of

samples there should be wider temperature range 0 1 *T T*

 *<sup>m</sup>* and ( ) ' *<sup>m</sup> f* α

α

used. Differentiating (5) with respect to <sup>1</sup> *T*<sup>−</sup> ∂ we obtain

*<sup>m</sup>* − is calculated from equation (12)

While performing calculations, 2

α

For the stage A ( ) ' . *<sup>m</sup> f*

calculated. Knowing ( ) ' *Af*

*2.1.2. Isoconversional method*

0 1 *T T* α

 α α

regression method.

Knowing B, ( ) ' *Af*

ln ln *<sup>i</sup>*

β<sup>÷</sup>

β

For non-isothermal conditions, at a linear heating rate of sample *dT dt* <sup>=</sup> βwe obtain

$$\log\left(\alpha\right) = \frac{A}{\beta} \int\_{-T\alpha = 0}^{T\alpha = 1} \exp\left(-\frac{E}{RT}\right) dT\tag{7}$$

The integral has no analytical solution. An important approximate solution is the Coats-Redfern equation [12]

$$\ln\left[\frac{g(\alpha)}{T^2}\right] = F\left(T\right) = \ln\left[\frac{AR}{\beta E}\left(1 - \frac{2RT\_m}{E}\right)\right] - \frac{E}{RT} \tag{8}$$

Based on experimental data for each stage the form of the f(α) or g(α) function most consistent with the experimental data and the parameters of Arrhenius equation A and E should be determined.

During standard thermogravimetric measurements the temperature of the sample, the TG, DTG and HF functions and the mass spectra of evolved gaseous products are recorded. Solid products are identified by XRD method. On this basis the division of the process into stages is made and α(T) dependencies are determined for the stages. The methods of measurement results elaboration and kinetic models recommended by ICTAC Kinetics Comittee are given in work [1]. These include in particular Kissinger's method and isoconversional method.

#### *2.1.1. Kissinger's method*

The basis of Kissinger's method are the parameters describing the process rate maxima (Tm, αm), determined at different heating rates of samples [14]. For maximum 2 <sup>2</sup> 0. *<sup>d</sup> dt* α = By differentiating equation (5) we obtain

$$\frac{d^2\alpha}{dt^2} = \left\{\frac{E\beta}{RT\_m^2} + A f\left(\alpha\_m\right) \exp\left[-\frac{E}{RT\_m}\right]\right\} \left(\frac{d\alpha}{dt}\right)\_m = 0\tag{9}$$

Since 0 *m d dt* α ≠ , therefore

$$\frac{E\mathcal{B}}{RT\_m^2} = -A\dot{f}\left(\alpha\_m\right)\exp\left[-\frac{E}{RT\_m}\right] \tag{10}$$

There should be added that the formula (10) requires that ( ) ' *<sup>m</sup> f* α <0. In this method therefore only kinetic models fulfilling this condition can be used.

By transforming (10) and adapting the result to the measurements performed at different heating rates of samples the Kissinger equation (11) is obtained

$$\ln\left(\frac{\mathcal{B}\_i}{T\_{m,i}^2}\right) = \ln\left[-\frac{AR}{E}f^\cdot\left(\alpha\_m\right)\right] - \frac{E}{RT\_{m,i}}\tag{11}$$

While performing calculations, 2 , , <sup>1000</sup> ln *<sup>i</sup> mi mi T T* β <sup>÷</sup> charts are prepared. Then the activation

energy E and the value of expression ( ) ' ln *<sup>m</sup> AR f B <sup>E</sup>* α − = are calculated by linear regression method.

Knowing B, ( ) ' *Af* α*<sup>m</sup>* − is calculated from equation (12)

138 Heat Treatment – Conventional and Novel Applications

Redfern equation [12]

should be determined.

isoconversional method.

*2.1.1. Kissinger's method* 

Since 0 *m*

≠

*d dt* α

differentiating equation (5) we obtain

, therefore

( ) ( ) ( )<sup>t</sup> *o*

α

= = (6)

*dt* <sup>=</sup> β

(7)

we obtain

(8)

2 <sup>2</sup> 0. *<sup>d</sup> dt* α= By

(10)

<0. In this method

(9)

*d g k T f* α α

1

=

*A E g dT*

*RT*

*T*

α

*T*

β

α

0 exp

The integral has no analytical solution. An important approximate solution is the Coats-

<sup>2</sup> ln ln 1 *<sup>m</sup> <sup>g</sup> AR E RT*

== − −

*T E E RT*

β

Based on experimental data for each stage the form of the f(α) or g(α) function most consistent with the experimental data and the parameters of Arrhenius equation A and E

During standard thermogravimetric measurements the temperature of the sample, the TG, DTG and HF functions and the mass spectra of evolved gaseous products are recorded. Solid products are identified by XRD method. On this basis the division of the process into stages is made and α(T) dependencies are determined for the stages. The methods of measurement results elaboration and kinetic models recommended by ICTAC Kinetics Comittee are given in work [1]. These include in particular Kissinger's method and

The basis of Kissinger's method are the parameters describing the process rate maxima (Tm,

2 2 exp 0 *<sup>m</sup> m m m*

> ( ) '

= − <sup>−</sup>

α

<sup>2</sup> exp *<sup>m</sup> m m E E Af RT RT*

α

'

*<sup>m</sup> f* α

αm), determined at different heating rates of samples [14]. For maximum

2 '

 β

> β

There should be added that the formula (10) requires that ( )

therefore only kinetic models fulfilling this condition can be used.

α

( )

*d E E d Af dt RT RT dt*

α =+ − =

= = −

α

For non-isothermal conditions, at a linear heating rate of sample *dT*

( )

( ) ( ) <sup>2</sup>

α

*F T*

α

$$-A\vec{f}\left(\alpha\right) = \frac{E}{R} \exp\left(\mathcal{B}\right) \tag{12}$$

To calculate A one needs to know the kinetic model for a given stage (form of the function f(α)). The kinetic model most consistent with the experimental data, of the tested models, was established by analyzing the trajectories of Y(T) function depending on α(T). The conversion degrees α(T) for the stages were estimated on the basis of experimental data, whereas the Y(T) function was calculated from the formula

$$Y(T) = \frac{Af'(\alpha\_m)}{\beta\_i} \int\_{T\_{a=0}}^{T\_{a=1}} \exp[-\frac{E}{RT}] dT \tag{13}$$

For the stage A ( ) ' . *<sup>m</sup> f* α ≈ *const* The integral was calculated numerically. There should be added that at constant activation energy, trajectories of 1 0 1 exp *T T <sup>E</sup> dT i RT* α α β = = <sup>−</sup> functions depending on the conversion degree, for different βi, should be the same. The ranges 0 1 *T T* α α = = ÷ for given βi must be determined experimentally. At higher heating rates of the samples there should be wider temperature range 0 1 *T T* α α = = ÷ . This is consistent with DTG and HF charts. After determining the form of the function f(α) the derivative ( ) ' *<sup>m</sup> f* α is calculated. Knowing ( ) ' *Af* α *<sup>m</sup>* and ( ) ' *<sup>m</sup> f* αArrhenius coefficient A is calculated.

#### *2.1.2. Isoconversional method*

In the the isoconversional method equation (5), obtained after separation of variables, is also used. Differentiating (5) with respect to <sup>1</sup> *T*<sup>−</sup> ∂ we obtain

#### 140 Heat Treatment – Conventional and Novel Applications

$$\left[\frac{\partial \ln\left(\frac{d\alpha}{dt}\right)}{\partial T^{-1}}\right]\_{\alpha} = \left[\frac{\partial \ln k\left(T\right)}{\partial T^{-1}}\right]\_{\alpha} + \left[\frac{\partial f\left(\alpha\right)}{\partial T^{-1}}\right]\_{\alpha} \tag{14}$$

For α = const we have

$$\left[\frac{\partial \ln\left(\frac{d\alpha}{dt}\right)}{\partial T^{-1}}\right]\_{\alpha} = -\frac{E\_{\alpha}}{R} \tag{15}$$

Methodology of Thermal Research in Materials Engineering 141

(20)

(21)

are constructed. The parameters of this equation

÷ are constructed. The parameters of equation

The time of obtaining the assumed conversion degree at different temperatures was

(19) are calculated by linear regression method. For non-isothermal conditions there is no analytical solution. During the calculations of activation energy there are used approximate

> *<sup>E</sup> const C T RT*

*const T RT*

= −

NH4VO3 from Fluka was used as a substrate. Decomposition process was carried out in dry air (Messer, Germany) containing 20,5% vol. O2 rest N2. Impurities occurred in amounts: H2O < 10 vpm, CO2 < 0,5 vpm, NOx < 0,1 vpm, hydrocarbons < 0,1 vpm. Thermogravimetric measurements were carried out on TG–DSC Q600 (TA Instruments) apparatus. Gaseous products of proceeding transitions were identified by mass spectrometry method. Pfeifer Vacuum ThermoStar GDS 301 apparatus was used. Solid products were identified by XRD method. X'Pert Pro apparatus from PANalytical with a copper X-ray tube with current voltage 35 kV and intensity 40 mA was used. Spectra processing and analysis

During the TG-DSC measurements weighed amounts of the sample in the order of 20 mg were used. The temperature of the sample, TG, DTG and HF functions, and mass spectra of gaseous products were registered in time. In all series the temperature of samples changed linearly in time. It was found that thermal decomposition of NH4VO3 in dry air proceeds in

α

*E*

α

α

Pert HighScore 1.0 software with incorporated ICDD spectra

() () 4 3 4 6 16 4 6 16 2 5 3 2 6 NH VO NH V O NH V O V O →→→ (22)

α

*i*

= −

*T*

<sup>1000</sup> ln *<sup>i</sup>*

, ln *<sup>i</sup> B i*

> 2 , ln *<sup>i</sup> i*

β

α

α

α

In this work the Kissinger's-Akoshira-Sunose equation was used

, <sup>1000</sup> ln *<sup>i</sup> <sup>i</sup> T T*

α

β<sup>÷</sup> β

*t*

α

designated by tα,i. The charts of ,

equations of general form

In this case the charts of 2

**2.2. Experimental** 

was performed using X'

library.

**2.3. Results** 

are calculated by linear regression method.

the three endothermic stages according to

Integrating (15), taking into account that *dT dt* <sup>=</sup> β , and generalizing the result for different β<sup>i</sup> we obtain

$$\ln\left[\beta\_i \left(\frac{d\alpha}{dt}\right)\_{\alpha,i}\right] = \text{const} - \frac{E\_\alpha}{RT\_{\alpha,i}}\tag{16}$$

Integration constant, according to equation (5), is equal ln *f A* ( ) α α . Equation (16) therefore takes the form

$$\ln\left[\beta\_i \left(\frac{d\alpha}{dt}\right)\_{\alpha,i}\right] = \ln\left[f\left(\alpha\right)A\_{\alpha}\right] - \frac{E\_{\alpha}}{RT\_{\alpha,i}}\tag{17}$$

It should be added that the introduction of the coefficients A<sup>α</sup> and Eα depending on conversion degree does not comply with the principle of separation of variables. Equation (17) is used in the differential isoconversional method. The basis of the reaction rate calculations are reaction rates ,*i d dt* α α determined for selected conversion degrees and

different βi and the corresponding to them temperature values Tα,i.

In the case of the integral isoconversional method the basis for calculations, for isothermal conditions, is equation

$$\log\left(\alpha\right) = A \exp\left(-\frac{E}{RT}\right)t\tag{18}$$

Activation energy, in this case, is calculated from the formula

$$\ln t\_{\alpha,i} = \ln \left[ \frac{g(\alpha)}{A\_{\alpha}} \right] + \frac{E\_{\alpha}}{RT\_{\alpha}} \tag{19}$$

The time of obtaining the assumed conversion degree at different temperatures was designated by tα,i. The charts of , <sup>1000</sup> ln *<sup>i</sup> i t T* α÷ are constructed. The parameters of equation

(19) are calculated by linear regression method. For non-isothermal conditions there is no analytical solution. During the calculations of activation energy there are used approximate equations of general form

$$\ln\left(\frac{\beta\_i}{T\_{\alpha,i}^{\mathcal{B}}}\right) = const - C\frac{E\_{\alpha}}{RT\_{\alpha}}\tag{20}$$

In this work the Kissinger's-Akoshira-Sunose equation was used

$$\ln\left(\frac{\mathcal{B}\_i}{T\_{\alpha,i}^2}\right) = const - \frac{E\_{\alpha}}{RT\_{\alpha}}\tag{21}$$

In this case the charts of 2 , <sup>1000</sup> ln *<sup>i</sup> <sup>i</sup> T T*α α β <sup>÷</sup> are constructed. The parameters of this equation

are calculated by linear regression method.

#### **2.2. Experimental**

140 Heat Treatment – Conventional and Novel Applications

Integrating (15), taking into account that *dT*

For α = const we have

therefore takes the form

calculations are reaction rates

conditions, is equation

we obtain

( ) ( )

α

 α

, and generalizing the result for different β<sup>i</sup>

α

(18)

(19)

α

(16)

. Equation (16)

(17)

(14)

(15)

α

α

1 11

*dt E T R*

α

*d E const dt RT*

, .

( ) , ,

 α

*i i <sup>d</sup> <sup>E</sup> f A dt RT*

*E*

*RT*

α

*i i*

α

α

α

α

determined for selected conversion degrees and

− −− <sup>∂</sup> ∂ ∂ = + ∂ ∂∂

*dt kT f T TT*

1

= − <sup>∂</sup>

*dt* <sup>=</sup> β

α

= −

= −

It should be added that the introduction of the coefficients A<sup>α</sup> and Eα depending on conversion degree does not comply with the principle of separation of variables. Equation (17) is used in the differential isoconversional method. The basis of the reaction rate

In the case of the integral isoconversional method the basis for calculations, for isothermal

*g A t*

= −

( )

 α

= + 

α

*g E*

*A RT*

α

 α

( ) exp

α

, ln ln *<sup>i</sup>*

*t*

α

α β

−

α

ln *<sup>d</sup>*

<sup>∂</sup>

ln ln

α

*d*

ln *<sup>i</sup>*

Integration constant, according to equation (5), is equal ln *f A* ( )

ln ln *<sup>i</sup>*

α β

> *d dt* α

different βi and the corresponding to them temperature values Tα,i.

Activation energy, in this case, is calculated from the formula

 α

α

,*i*

α

NH4VO3 from Fluka was used as a substrate. Decomposition process was carried out in dry air (Messer, Germany) containing 20,5% vol. O2 rest N2. Impurities occurred in amounts: H2O < 10 vpm, CO2 < 0,5 vpm, NOx < 0,1 vpm, hydrocarbons < 0,1 vpm. Thermogravimetric measurements were carried out on TG–DSC Q600 (TA Instruments) apparatus. Gaseous products of proceeding transitions were identified by mass spectrometry method. Pfeifer Vacuum ThermoStar GDS 301 apparatus was used. Solid products were identified by XRD method. X'Pert Pro apparatus from PANalytical with a copper X-ray tube with current voltage 35 kV and intensity 40 mA was used. Spectra processing and analysis was performed using X' Pert HighScore 1.0 software with incorporated ICDD spectra library.

#### **2.3. Results**

During the TG-DSC measurements weighed amounts of the sample in the order of 20 mg were used. The temperature of the sample, TG, DTG and HF functions, and mass spectra of gaseous products were registered in time. In all series the temperature of samples changed linearly in time. It was found that thermal decomposition of NH4VO3 in dry air proceeds in the three endothermic stages according to

$$\text{(6 NH}\_4\text{VO}\_3 \rightarrow \text{(NH}\_4\text{)}\_3\text{ V}\_6\text{O}\_{16} \rightarrow \text{(NH}\_4\text{)}\_2\text{ V}\_6\text{O}\_{16} \rightarrow \text{V}\_2\text{O}\_5\tag{22}$$

#### 142 Heat Treatment – Conventional and Novel Applications

The theoretical, total mass loss of the sample equals 22,29 wt%. The mass losses in the stages, referred to the initial mass of the sample, equal: 11,48 wt% in stage I, 4,404 wt% in stage II, and 6,66 wt% in stage III. The intermediate products (for control also the final product) were obtained under isothermal conditions, in the temperature ranges of their occurrence. As the final product V2O5 (ICDD card 85-0601), and as the second intermediate product (NH4)2V6O16 (ICDD card 79-205) were obtained. The intermediate product formed in step I was identified on the basis of the mass balance (there is no pattern of this compound in ICDD directory). In all the stages and at different heating rates of the samples evolved: NH3, H2O, NO and N2O resulting from oxidation of NH3. In the gas phase NO2 did not occur. The results of this step of research are given in [19].

Methodology of Thermal Research in Materials Engineering 143

The linear segments correspond to the isothermal conditions. The samples were heated

In case of the investigated process, under isothermal conditions, the results for the stage could by obtained only at a few temperatures, while at higher temperatures the results were obtained at high conversion degrees. For these reasons, the course of investigated process

**Figure 2.** Plot of TGu functions in temperature. Decomposition of NH4VO3 in dry air [19].

Figure 3 shows an example of DTG plots for selected heating rates of the samples.

It follows that at higher βi the temperature range 0 1. *T T*

temperature range of the process in stage is an important parameter.

stages.

In Figure 2 the plots of TGu function in temperature obtained under non-isothermal conditions are presented. The single curve is formed of about 20 thousand points. The determined TGu values should depend only on temperature and heating rate of the samples. This was confirmed by neural networks method in reference [24]. All the series were considered simultaneously. The computer software Statistica Neural Network was used.

Along with the increase of sample heating rates the DTG function plots are shifted into the higher temperature range. It is also visible that at the higher sample heating rates stages I and II are overlapped to a larger extent, and the final segments of plots, corresponding to the stage III, are not monotonic. As mentioned before, this was attributed to the oxidation of formed earlier V2O5-x to V2O5. The Tm temperatures corresponding to peaks of DTG function plots (necessary for Kissinger's method) have been determined. The results are listed in Table 1.

The values of αm, for Tm temperatures given in Table1 were retrieved from data sets of α(T). There are also given the temperature ranges, determined basing on DTG plots, for the

α

 α

= = ÷ increases (Fig.3). The

isothermally until the stable mass has been reached (about 30 min).

was not further examined under isothermal conditions.

## *2.3.1. Analysis of influence of sample heating rate on the course of the process*

The influence of sample heating rate on the course of the process was examined on the basis of the TGu , DTG and HF functions. The measurements were carried out at sample heating rates of: 1; 1,5; 2; 2,5; 3; 3,5; 4; 4,5; 5; 6; 7 ; 8 and 10 K min-1. The isothermal measurements were carried out at the sample heating rate equal to 2 K min-1. Basing on the TG-DSC measurements the division of the process into stages was made and the conversion degrees for the stages were determined. There should be added that the data sets concerning conversion degrees, determined for the stages, are the basis for the description of process kinetics as in [1,17,20-23].

#### *2.3.2. Analysis of TGu, DTG and HF charts*

The trajectories of TGu, DTG and HF plots are presented in separate figures. In Figure 1 TGu functions in temperature registered during the measurements are presented.

**Figure 1.** TGu functions in temperature. Decomposition of NH4VO3 in dry air [19].

The linear segments correspond to the isothermal conditions. The samples were heated isothermally until the stable mass has been reached (about 30 min).

142 Heat Treatment – Conventional and Novel Applications

kinetics as in [1,17,20-23].

*2.3.2. Analysis of TGu, DTG and HF charts* 

0,75

0,80

0,85

0,90

**TGu /mg mg-1**

0,95

1,00

not occur. The results of this step of research are given in [19].

*2.3.1. Analysis of influence of sample heating rate on the course of the process* 

The theoretical, total mass loss of the sample equals 22,29 wt%. The mass losses in the stages, referred to the initial mass of the sample, equal: 11,48 wt% in stage I, 4,404 wt% in stage II, and 6,66 wt% in stage III. The intermediate products (for control also the final product) were obtained under isothermal conditions, in the temperature ranges of their occurrence. As the final product V2O5 (ICDD card 85-0601), and as the second intermediate product (NH4)2V6O16 (ICDD card 79-205) were obtained. The intermediate product formed in step I was identified on the basis of the mass balance (there is no pattern of this compound in ICDD directory). In all the stages and at different heating rates of the samples evolved: NH3, H2O, NO and N2O resulting from oxidation of NH3. In the gas phase NO2 did

The influence of sample heating rate on the course of the process was examined on the basis of the TGu , DTG and HF functions. The measurements were carried out at sample heating rates of: 1; 1,5; 2; 2,5; 3; 3,5; 4; 4,5; 5; 6; 7 ; 8 and 10 K min-1. The isothermal measurements were carried out at the sample heating rate equal to 2 K min-1. Basing on the TG-DSC measurements the division of the process into stages was made and the conversion degrees for the stages were determined. There should be added that the data sets concerning conversion degrees, determined for the stages, are the basis for the description of process

The trajectories of TGu, DTG and HF plots are presented in separate figures. In Figure 1 TGu

 441 K 448 K 453 K 470 K 539 K 545 K 673 K

300 400 500 600 700

**T/K**

**Figure 1.** TGu functions in temperature. Decomposition of NH4VO3 in dry air [19].

functions in temperature registered during the measurements are presented.

In case of the investigated process, under isothermal conditions, the results for the stage could by obtained only at a few temperatures, while at higher temperatures the results were obtained at high conversion degrees. For these reasons, the course of investigated process was not further examined under isothermal conditions.

**Figure 2.** Plot of TGu functions in temperature. Decomposition of NH4VO3 in dry air [19].

In Figure 2 the plots of TGu function in temperature obtained under non-isothermal conditions are presented. The single curve is formed of about 20 thousand points. The determined TGu values should depend only on temperature and heating rate of the samples. This was confirmed by neural networks method in reference [24]. All the series were considered simultaneously. The computer software Statistica Neural Network was used. Figure 3 shows an example of DTG plots for selected heating rates of the samples.

It follows that at higher βi the temperature range 0 1. *T T* α α = = ÷ increases (Fig.3). The temperature range of the process in stage is an important parameter.

Along with the increase of sample heating rates the DTG function plots are shifted into the higher temperature range. It is also visible that at the higher sample heating rates stages I and II are overlapped to a larger extent, and the final segments of plots, corresponding to the stage III, are not monotonic. As mentioned before, this was attributed to the oxidation of formed earlier V2O5-x to V2O5. The Tm temperatures corresponding to peaks of DTG function plots (necessary for Kissinger's method) have been determined. The results are listed in Table 1.

The values of αm, for Tm temperatures given in Table1 were retrieved from data sets of α(T). There are also given the temperature ranges, determined basing on DTG plots, for the stages.

Methodology of Thermal Research in Materials Engineering 145

In Figure 4 the HF function in temperature has been presented for selected example values

**Figure 4.** Plots of HF functions in temperature. Decomposition of NH4VO3 in dry air.

The conversion degrees for stages were calculated from the following formula

α

the process into stages also DTG and HF plots were taken into account [22-24].

( )

*2.3.3. Analysis of α(T) plots for determined stages* 

It is visible from the course of HF plots that in the case there are three endothermic stages referred to the NH4VO3 decomposition. The HF plots end with the exothermic transition attributed to the oxidation of small amount of V2O5-x to V2O5. Along with the increase of sample heating rate stages I and II overlap. Peaks of HF function are shifted, with respect to the DTG peaks, by a few degrees into the range of higher temperature. There should be added that NH3, H2O, NO and N2O evolved in all the stages, also at higher simple heating

> 0 0

*m m <sup>T</sup> m m*

The data concerning the TGu function were the basis for calculations. During the division of

In Figure 5 the results obtained for the stages are presented. One curve was formed of a few

*k*

<sup>−</sup> <sup>=</sup> − (23)

of β.

rates.

thousand points.

**Figure 3.** Plots of DTG function in temperature. Decomposition of NH4VO3 in dry air [19].


**Table 1.** List of data for Kissinger's method. Thermal decomposition of NH4VO3 in dry air.

In Figure 4 the HF function in temperature has been presented for selected example values of β.

**Figure 4.** Plots of HF functions in temperature. Decomposition of NH4VO3 in dry air.

It is visible from the course of HF plots that in the case there are three endothermic stages referred to the NH4VO3 decomposition. The HF plots end with the exothermic transition attributed to the oxidation of small amount of V2O5-x to V2O5. Along with the increase of sample heating rate stages I and II overlap. Peaks of HF function are shifted, with respect to the DTG peaks, by a few degrees into the range of higher temperature. There should be added that NH3, H2O, NO and N2O evolved in all the stages, also at higher simple heating rates.

#### *2.3.3. Analysis of α(T) plots for determined stages*

144 Heat Treatment – Conventional and Novel Applications

1 447,83 0,721 410,75-

1,5 452,62 0,742 405,55-

2 457,48 0,761 413,15-

2,5 458,06 0,698 412,85-

3 460,18 0,683 415,65-

3,5 462,25 0,703 415,65-

4 465,12 0,689 413,25-

4,5 465,32 0,689 417,25-

5 467,29 0,714 418,45-

6 471,44 0,698 416,45-

7 475,68 0,741 417,15-

8 476,22 0,71 421,65-

10 480,09 0,716 423,15-

β Kmin-1

**Figure 3.** Plots of DTG function in temperature. Decomposition of NH4VO3 in dry air [19].

Stage I Stage II Stage III Tm /K α<sup>m</sup> ∆T /K Tm /K α<sup>m</sup> ∆T /K Tm /K α<sup>m</sup> ∆T /K

490,65 543,93 0,617 509,25-

489,45 550,38 0,653 513,25-

491,35 558,2 0,705 516,25-

494,65 560,65 0,701 518,15-

496,15 565,44 0,729 522,95-

497,15 567,21 0,737 523,75-

503,15 572,81 0,744 526,55-

500,35 572,17 0,743 526,15-

479,35 574,3 0,736 530,45-

481,85 580,7 0,719 533,45-

485,25 588,2 0,83 537,85-

485,25 587,06 0,723 536,25-

490,15 591,89 0,73 538,85-

566,15

572,15

580,45

583,45

590,75

590,75

596,75

595,15

597,95

609,25

614,15

620,75

625,35

457,95 465,65 0,384 457,95-

462,85 469,87 0,415 462,85-

467,55 473,58 0,476 467,55-

469,65 475,6 0,443 469,65-

472,15 477,16 0,495 472,15-

474,15 479,51 0,474 474,15-

476,15 481,41 0,517 476,15-

476,95 481,9 0,38 476,95-

**Table 1.** List of data for Kissinger's method. Thermal decomposition of NH4VO3 in dry air.

The conversion degrees for stages were calculated from the following formula

$$\alpha(T) = \frac{m\_0 - m}{m\_0 - m\_k} \tag{23}$$

The data concerning the TGu function were the basis for calculations. During the division of the process into stages also DTG and HF plots were taken into account [22-24].

In Figure 5 the results obtained for the stages are presented. One curve was formed of a few thousand points.

Methodology of Thermal Research in Materials Engineering 147

for the stages. Thermal decomposition of NH4VO3 in dry air.

exp .

 <sup>−</sup> 

*RT*

*<sup>m</sup> f* α

≈ *const* . In figure 7 plots of

The kinetic models for the stages were determined analyzing the courses of the function

*T m i T*

α

=

α

=

There should be added, that the values of αm determined for the stage, changed marginally

Y(T) function for the stages depending on α(T), for β = 3K min-1, are presented as an

0 ,01

*Af <sup>E</sup> dT*

Y(T) = ( ) 0.99

for different β. Therefore, there could be assumed that ( ) '

β

**Figure 7.** Plots of Y(T) function for the stages depending on α(T) for β = 3K min-1. Thermal

'

α

**Figure 6.** Plots of 2

example.

, , <sup>1000</sup> ln *<sup>i</sup> m i m i T T*

β ÷ 

decomposition of NH4VO3 in dry air.

**Figure 5.** Plots of α(T) depending on temperature for the stages. Thermal decomposition of NH4VO3 in dry air.

### **2.4. Result analysis**

#### *2.4.1. Kissinger's method*

In Figure 6 the plots of 2 , , <sup>1000</sup> ln *<sup>i</sup> m i m i T T* β <sup>÷</sup> for the stages, obtained on the basis of the data from Table 1, are presented.

The parameters for Kissinger's equation (11) were calculated by linear regression method. The computer software Statistica 6.0 was used. The results are listed in Table 2.


**Table 2.** List of the results calculated by Kissinger's method. Thermal decomposition of NH4VO3 in dry air. \* constant in Kissinger's equation (11)

146 Heat Treatment – Conventional and Novel Applications

β=1 β=1,5 β=2 β=2,5 β=3

Stage I

β=3,5

β=4

β=4,5

β=6

β=2 β=2,5

β=1 β=1,5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Stage III

α **(T)**

400 410 420 430 440 450 460 470 480 490 500

**T /K**

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

α **(T)**

dry air.

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

α**(T)**

**2.4. Result analysis** 

Table 1, are presented.

*2.4.1. Kissinger's method* 

In Figure 6 the plots of 2 , ,

air. \* constant in Kissinger's equation (11)

<sup>1000</sup> ln *<sup>i</sup> m i m i T T*

β <sup>÷</sup> 

**Figure 5.** Plots of α(T) depending on temperature for the stages. Thermal decomposition of NH4VO3 in

**T /K**

β=3 β=4,5

500 520 540 560 580 600

The parameters for Kissinger's equation (11) were calculated by linear regression method.

Stage E kJ mol-1 B\* Af'(αm) 1 min-1 rp model -f '(αm) A 1 min-1 I 117,66 19,496 4,133 E12 0,979 A2 2,476 1,669 E12 II 164,48 30,178 2,37 E17 0,989 A4 6,762 3,504 E16 III 113,75 12,623 4,137 E9 0,985 A2 2,596 1,594 E9 **Table 2.** List of the results calculated by Kissinger's method. Thermal decomposition of NH4VO3 in dry

The computer software Statistica 6.0 was used. The results are listed in Table 2.

for the stages, obtained on the basis of the data from

β=8

β=6

β=3,5

β=4

450 455 460 465 470 475 480 485 490 495 500 505

Stage II

β=1 β=1,5

β=2,5 β=3

<sup>β</sup>=2 <sup>β</sup>=4

β=6

β=4,5

β=3,5

**T /K**

**Figure 6.** Plots of 2 , , <sup>1000</sup> ln *<sup>i</sup> m i m i T T* β ÷ for the stages. Thermal decomposition of NH4VO3 in dry air.

The kinetic models for the stages were determined analyzing the courses of the function

$$\text{Y(T)} = \frac{A f^{\stackrel{\cdot}{\cdot}} (\alpha\_m)}{\beta\_i} \prod\_{T\_{\alpha=0, 01}}^{T\_{\alpha=0, 09}} \exp \left( - \frac{E}{RT} \right) dT.$$

There should be added, that the values of αm determined for the stage, changed marginally for different β. Therefore, there could be assumed that ( ) ' *<sup>m</sup> f* α ≈ *const* . In figure 7 plots of Y(T) function for the stages depending on α(T), for β = 3K min-1, are presented as an example.

**Figure 7.** Plots of Y(T) function for the stages depending on α(T) for β = 3K min-1. Thermal decomposition of NH4VO3 in dry air.

#### 148 Heat Treatment – Conventional and Novel Applications

For the stages I, III the most consistent with experimental data, of the tested models, was the A2 model.

$$\mathcal{S}\left(\alpha\right) = \left[ -\ln\left(1 - \alpha\right) \right]^{\frac{1}{2}} \tag{24}$$

Methodology of Thermal Research in Materials Engineering 149

for the stages. Isoconversional method. Conversion degrees equal

α

kJ mol-1 B\* rp

E

could not be

The calculations for other values of conversion degree were also performed. Should be

determined with sufficient accuracy. Therefore, the calculation was performed 0,1<α < 0,90.

for the stages are shown in Figure 9.

Stage I Stage II Stage III

kJ mol-1 B\* rp

0,1 140,45 27,00 0,986 163,02 30,083 0,996 147,58 21,299 0,991 0,2 131,69 24,303 0,988 161,22 29,493 0,998 137,71 18,709 0,990 0,3 128,45 23,06 0,991 158,71 28,788 0,992 126,45 16,012 0,991 0,4 125,65 22,099 0,995 155,72 27,987 0,994 130,05 16,577 0,989 0,5 123,13 21,256 0,999 156,50 28,187 0,994 127,46 15,849 0,986 0,6 120,29 20,351 0,989 154,54 27,58 0,989 125,48 15,276 0,987 0,7 117,56 19,489 0,988 156,70 28,065 0,985 122,94 14,597 0,985 0,8 114,75 18,61 0,988 158,2 28,365 0,982 121,83 14,227 0,987 0,9 112,38 17,826 0,989 119,92 13,679 0,983

**Table 4.** The activation energies calculated for the stages by the isoconversional method. Thermal

The values of activation energies calculated by the isokinetic method are given in Table 4.

E

noted that α(T) tend asymptotically to the limit values. In these ranges ,*<sup>i</sup> T*

$$f\left(\alpha\right) = 2\left(1 - \alpha\right) \left[ -\ln\left(1 - \alpha\right) \right]^{\frac{1}{2}}\tag{25}$$

$$f'\left(\alpha\_m\right) = -2\left[-\ln\left(1-\alpha\_m\right)\right]^{\frac{1}{2}} - \frac{1-\alpha\_m}{\alpha\_m\left[-\ln\left(1-\alpha\_m\right)\right]^{\frac{1}{2}}}\tag{26}$$

Whereas for stage II this was the A4 model

$$\log\left(\alpha\right) = \left[ -\ln\left(1 - \alpha\right) \right]^{\frac{1}{4}} \tag{27}$$

**Figure 8.** Plots of

The plots of 2 , ,

E

to αm,i.

αi

''

α

, ,

÷ 

*m m*

α

decomposition of NH4VO3 in dry air.\* constant in equation (21)

2

<sup>1000</sup> ln *<sup>i</sup> <sup>i</sup> <sup>i</sup> T T* α

kJ mol-1 B\* rp

β<sup>÷</sup>

*i <sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>T</sup>* α

β

<sup>1000</sup> ln

$$f\left(\alpha\right) = 4\left(1 - \alpha\right) \left[ -\ln\left(1 - \alpha\right) \right]^{\frac{3}{4}} \tag{28}$$

$$f'\left(\alpha\_m\right) = -4\left[-\ln\left(1-\alpha\_m\right)\right]^{\frac{3}{4}} - \frac{3\left(1-\alpha\_m\right)}{\alpha\_m\left[-\ln\left(1-\alpha\_m\right)\right]^{\frac{1}{4}}}\tag{29}$$

The selection of the model was confirmed making calculations by Coats – Redfern method. The selected models and calculated values of ( ) ' *<sup>m</sup> f* αare given in Table 2.

#### *2.4.2. Isoconversional method*

The basis for calculations by this method were dependencies α(T) determined for the stages at different heating rates of the samples (Fig. 5). First, the activation energies were calculated for αm,i corresponding to the inflection points of α(T) curves (average values of αm given in Table 1). In this case, formula (21) should take the form of equation (11). In Figure 8 the

$$\text{obtained plots of } \ln \left( \frac{\beta\_i}{T\_{\dot{\alpha}\_m, i}^2} \right) + \frac{1000}{T\_{\dot{\alpha}\_m, i}} \text{ are presented.}$$

The results of calculations are given in Table 3. This method is less accurate than the Kissinger's method


**Table 3.** List of the results of activation energies calculations by isoconversional method for the inflection points of α(T) curves. \*) constant in equation (21).

**Figure 8.** Plots of ' ' 2 , , <sup>1000</sup> ln *m m i <sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>T</sup>* α α β ÷ for the stages. Isoconversional method. Conversion degrees equal

to αm,i.

148 Heat Treatment – Conventional and Novel Applications

'

'

α

The selected models and calculated values of ( )

'

*m m*

<sup>1000</sup> ln

inflection points of α(T) curves. \*) constant in equation (21).

, ,

α

2

α

*i <sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>T</sup>*

β <sup>÷</sup> 

*f*

*2.4.2. Isoconversional method* 

obtained plots of '

Kissinger's method

α

*f*

Whereas for stage II this was the A4 model

A2 model.

For the stages I, III the most consistent with experimental data, of the tested models, was the

() ( )

() ( ) ( )

<sup>1</sup> 2 ln 1

() ( )

() ( ) ( )

() ( ) ( )

<sup>−</sup> =− − − − 

The selection of the model was confirmed making calculations by Coats – Redfern method.

The basis for calculations by this method were dependencies α(T) determined for the stages at different heating rates of the samples (Fig. 5). First, the activation energies were calculated for αm,i corresponding to the inflection points of α(T) curves (average values of αm given in Table 1). In this case, formula (21) should take the form of equation (11). In Figure 8 the

The results of calculations are given in Table 3. This method is less accurate than the

Stage α<sup>m</sup> E /kJmol-1 B\* rp I 0,713 117,3 19,401 0,989 II 0,460 148,2 26,05 0,987 III 0,721 122,3 14,426 0,988

**Table 3.** List of the results of activation energies calculations by isoconversional method for the

are presented.

'

*<sup>m</sup> f* α

 α

4 1 ln 1 <sup>4</sup> *f*

=−− −

3

<sup>4</sup> *g*

<sup>−</sup> =− − − − 

 α

2 1 ln 1 <sup>2</sup> *f*

=−− −

1

α

αα

() ( )

2

*m m*

α

αα

4

*m m*

4 ln 1

<sup>2</sup> *g*

1

 α

1

 α

3 1

ln 1

*m m*

1

ln 1

*m m*

3

( )

 α

are given in Table 2.

*m*

α

− −

( )

 α

*m*

α

− −

=− − ln 1 (24)

(25)

(26)

(29)

1 2

=− − ln 1 (27)

(28)

1 4

 α

α

 α

α

The calculations for other values of conversion degree were also performed. Should be noted that α(T) tend asymptotically to the limit values. In these ranges ,*<sup>i</sup> T*α could not be determined with sufficient accuracy. Therefore, the calculation was performed 0,1<α < 0,90.

$$\text{The plots of } \ln\left(\frac{\beta\_i}{T\_{\alpha,i}^2}\right) \div \frac{1000}{T\_{\alpha,i}} \text{ for the stages are shown in Figure 9.}$$

The values of activation energies calculated by the isokinetic method are given in Table 4.


**Table 4.** The activation energies calculated for the stages by the isoconversional method. Thermal decomposition of NH4VO3 in dry air.\* constant in equation (21)

#### 150 Heat Treatment – Conventional and Novel Applications

In the case of investigated process, the E for the stages I and III (asymmetric plots of DTG and HF), changed continuously along with the change of conversion degree. In the case of stage II (symmetric plots of DTG and HF) E was practically constant, similar to that determined by Kissinger's method. That is the isokinetic method compensates the impact of βi on the course of the process by changing the activation energy. Theoretically [25] much more interesting is the possibility to compensate the impact of βi, at a constant activation energy, with use of temperature ranges 0 1 *T T* α α = = ÷ , the values determined experimentally.

Methodology of Thermal Research in Materials Engineering 151

<sup>2</sup> ln *<sup>g</sup> <sup>E</sup> T RT*

(30)

 α <sup>÷</sup>

(31)

. The values of Tm and

During the calculations by this method the sets of α(T) presented in Figure 5 were also used. The activation energies E, determined for the stages by Kissinger's method (for αm values

were constructed. Among the known kinetic models the ones the most consistent with the experimental data were selected; for stages I and III model A2, and for stage II model A4. For the selected model and different βi the values of the coefficient A were calculated for the

belonging to the sets of α(T)), were taken as the base values. The plots of ( )

( ) ( )

*<sup>E</sup> A ZT*

= + <sup>−</sup>

the temperature ranges for the stages are given in Table 1. However the calculated values of A are given in Table 5. For all the stages the obtained values of coefficient A were higher than the values determined by Kissinger's method. There should be added that the

Using average values of A the activation energies were tested. The criterion was the best conformity of trajectories of α(T) plots, calculated and determined experimentally, in the whole range. The conversion degrees were calculated as follows. First g(α) was calculated

() () <sup>2</sup> *g*

Formula (32) is suitable for different models. Whereas the method of calculating conversion

() ( )

<sup>2</sup> *g*

1

 α

{ ( ) } 2

 α

α

α

degrees depends on the form of kinetic model. For example, for model A2

α

<sup>2</sup> ln 1 ln *<sup>m</sup> AR E RT <sup>g</sup> Z T*

 = −= + 

*E E T RT*

<sup>2</sup> <sup>1</sup> *<sup>m</sup>*

*RT <sup>R</sup> E*

β

2

α

<sup>=</sup> *T FT* exp (32)

=− − ln 1 (33)

=− − 1 exp *<sup>g</sup>* (34)

α

stage in the following manner. First Z(T) was calculated from the formula

β

ln ( ) ln

The calculations of Z(T) were performed for the range 0,01 0,99 ≤ ≤

*2.4.3. Coats – Redfern method* 

Then lnA was calculated according to

proportion between the values was remained.

from the formula

thus

**Figure 9.** Plots of 2 , , 1000 ln *<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>T</sup>* α α β ÷ for the stages. Thermal decomposition of NH4VO3 in dry air.

#### *2.4.3. Coats – Redfern method*

150 Heat Treatment – Conventional and Novel Applications

experimentally.

**Figure 9.** Plots of 2




**ln (**β**i/T2**

α**,i)**


0,9 0,8 0,7 0,6 0,5


, ,

**1000/T**<sup>α</sup>**, i**

*<sup>i</sup> <sup>i</sup> <sup>T</sup> <sup>T</sup>* α

ln *<sup>i</sup>*

β ÷ 

0,243 0,248 0,253 0,258

1000


Stage II

0,1

0,2

0,3

0,4




**ln (**β**i/T2**

α**,i)**


0,9 0,8 0,7 0,6 0,5



α

In the case of investigated process, the E for the stages I and III (asymmetric plots of DTG and HF), changed continuously along with the change of conversion degree. In the case of stage II (symmetric plots of DTG and HF) E was practically constant, similar to that determined by Kissinger's method. That is the isokinetic method compensates the impact of βi on the course of the process by changing the activation energy. Theoretically [25] much more interesting is the possibility to compensate the impact of βi, at a constant

Stage I

0,245 0,25 0,255 0,26 0,265 0,27 0,275 0,28

**1000/T**<sup>α</sup>**, i**

α

0,4

0,1 0,2 0,3

for the stages. Thermal decomposition of NH4VO3 in dry air.

0,2 0,205 0,21 0,215 0,22 0,225 0,23

Stage III

0,1

0,2

0,3

0,4

**1000/T**α, **<sup>i</sup>**




**ln (**β**i/T2**

α**,i)**


0,9 0,8 0,7 0,6 0,5


 α

= = ÷ , the values determined

activation energy, with use of temperature ranges 0 1 *T T*

During the calculations by this method the sets of α(T) presented in Figure 5 were also used.

The activation energies E, determined for the stages by Kissinger's method (for αm values

belonging to the sets of α(T)), were taken as the base values. The plots of ( ) <sup>2</sup> ln *<sup>g</sup> <sup>E</sup> T RT* α <sup>÷</sup>

were constructed. Among the known kinetic models the ones the most consistent with the experimental data were selected; for stages I and III model A2, and for stage II model A4. For the selected model and different βi the values of the coefficient A were calculated for the stage in the following manner. First Z(T) was calculated from the formula

$$Z(T) = \ln\left[\frac{AR}{\beta E}\left(1 - \frac{2RT\_m}{E}\right)\right] = \ln\left[\frac{g\left(\alpha\right)}{T^2}\right] + \frac{E}{RT} \tag{30}$$

Then lnA was calculated according to

$$\ln A = Z\left(T\right) + \ln \left| \frac{\beta E}{R\left(1 - \frac{2RT\_m}{E}\right)} \right| \tag{31}$$

The calculations of Z(T) were performed for the range 0,01 0,99 ≤ ≤ α . The values of Tm and the temperature ranges for the stages are given in Table 1. However the calculated values of A are given in Table 5. For all the stages the obtained values of coefficient A were higher than the values determined by Kissinger's method. There should be added that the proportion between the values was remained.

Using average values of A the activation energies were tested. The criterion was the best conformity of trajectories of α(T) plots, calculated and determined experimentally, in the whole range. The conversion degrees were calculated as follows. First g(α) was calculated from the formula

$$\lg(\alpha) = T^2 \exp\left[F(T)\right] \tag{32}$$

Formula (32) is suitable for different models. Whereas the method of calculating conversion degrees depends on the form of kinetic model. For example, for model A2

$$\log\left(\alpha\right) = \left[ -\ln\left(1 - \alpha\right) \right]^{\frac{1}{2}} \tag{33}$$

thus

$$\alpha = 1 - \exp\left\{-\left[\mathcal{g}\left(\alpha\right)\right]^2\right\} \tag{34}$$

#### 152 Heat Treatment – Conventional and Novel Applications


Methodology of Thermal Research in Materials Engineering 153

Stage III alfa exp alfa calc r(a,T)

<sup>β</sup><sup>6</sup> β2

β4

500 520 540 560 580 600 620

**T /K**

0

0,2

0,4

0,6

**r(**α**,T) /min-1**

0,8

1

1,2

**Figure 10.** Plots of α(T) and r(T, α) for the stage I and III. Thermal decomposition of NH4VO3 in dry air.

0

0,2

0,4

0,6

α**(T)**

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

**r(a,T) /min-1**

The results are fairly well consistent with the experimental data; at the experimentally determined temperature ranges the process rates change from zero for α(T) = 0, reaching the maximum values at the points of inflection of α(T) curves, and then decrease to zero at

 ( ) *T* → 0 . The obtained results show that in our study Coats-Redfern equation was of great importance. Due to the analytical form it is also easy to use in the calculations of the kinetics

Manufacturing, storage and use of nc-TiC in form of powder for sintering or co-deposition processes involves the possibility of occurrence of free carbon in the system. Carbon can be a by-product of the synthesis process (pyrolysis of hydrocarbons) and remain in equilibrium in the TiC-C system or be the product of oxidation of TiC in the air according to the mechanism proposed by Schimada [26], whereby the oxygen dissolves in the carbide and

In second stage, in the layer amorphous TiO2 is formed and elemental carbon is produced

The produced carbon changes the state of the surface of TiC particles. A low oxygen content is the most important prerequisite for high sintering activity of the nanocrystalline TiC powders, especially in the sintering processes, synthesis of the nanocomposites for example

( ) 2 1x x 2 TiC 3 / 2 x O TiC O CO <sup>−</sup> + →+ (36)

( ) () 1x x 2 2 TiC O 1 x / 2 O TiO 1 x C <sup>−</sup> +− → +− (37)

**3. Heat treatment of TiCx/C. Carbonisation of nc-TiCx** 

α

0

0,2

0,4

0,6

α**(T)**

0,8

1

1,2

[27-31]

nc-TiC in metallic matrices.

of heterophase non-catalytic processes.

410 420 430 440 450 460 470 480 490

Stage I a(T) exp a(T) calc r(a,T)

β2

β4 β6

**T /K**

the layer containing oxycarbides is formed.

The values of activation energy calculated in this way are given in Table 5.

**Table 5.** The kinetic parameters obtained by Coats–Redfern method. Thermal decomposition of NH4VO3 in dry air.

The parameters A and E determined by this method for different βi remain nearly constant.

The corrected activation energies are slightly lower than the ones determined by Kissinger's method; in the case of stage II they are practically equal to the values calculated by isokinetic method. The result is interesting because it shows that the activation energy determined by Kissinger's method can be regarded as representative for the whole set of α(T) related to the stage. It should also be emphasized that determining the triad of g(α), A and E directly from the Coats-Redfern equation, usually the good results are not obtained [1,21,24]. That is it is a matter of calculation methods, and not of the Coats-Redfern equation.

Using the kinetic parameters given in table 5 the verifying calculations were performed. For the selected βi the trajectories of α(T) and r(α,T) functions were determined. The process rate was calculated from the formula

$$r\left(\alpha, T\right) = A \exp\left(-\frac{E}{RT}\right) f\left(\alpha\right) \tag{35}$$

While calculating f(α) the values of α(T) obtained from the Coats-Redfern equation (formulas (32)–(34)) were used. In Figure 10 the results obtained for stage I and II ( β= 2, 4 and 6 K min-1) are presented as an example.

**Figure 10.** Plots of α(T) and r(T, α) for the stage I and III. Thermal decomposition of NH4VO3 in dry air.

The results are fairly well consistent with the experimental data; at the experimentally determined temperature ranges the process rates change from zero for α(T) = 0, reaching the maximum values at the points of inflection of α(T) curves, and then decrease to zero at α ( ) *T* → 0 . The obtained results show that in our study Coats-Redfern equation was of great importance. Due to the analytical form it is also easy to use in the calculations of the kinetics of heterophase non-catalytic processes.

## **3. Heat treatment of TiCx/C. Carbonisation of nc-TiCx**

152 Heat Treatment – Conventional and Novel Applications

<sup>∆</sup>Z(T) A E12

min-1

β K min-1

NH4VO3 in dry air.

the Coats-Redfern equation.

was calculated from the formula

and 6 K min-1) are presented as an example.

constant.

The values of activation energy calculated in this way are given in Table 5.

kJ mol-1 <sup>∆</sup>Z(T) A E17

**Table 5.** The kinetic parameters obtained by Coats–Redfern method. Thermal decomposition of

The parameters A and E determined by this method for different βi remain nearly

The corrected activation energies are slightly lower than the ones determined by Kissinger's method; in the case of stage II they are practically equal to the values calculated by isokinetic method. The result is interesting because it shows that the activation energy determined by Kissinger's method can be regarded as representative for the whole set of α(T) related to the stage. It should also be emphasized that determining the triad of g(α), A and E directly from the Coats-Redfern equation, usually the good results are not obtained [1,21,24]. That is it is a matter of calculation methods, and not of

Using the kinetic parameters given in table 5 the verifying calculations were performed. For the selected βi the trajectories of α(T) and r(α,T) functions were determined. The process rate

( ) , exp ( ) *<sup>E</sup> rTA f RT*

While calculating f(α) the values of α(T) obtained from the Coats-Redfern equation (formulas (32)–(34)) were used. In Figure 10 the results obtained for stage I and II ( β= 2, 4

= −

 α

(35)

α

1 19,81 5,91 114,10 29,98 1,468 157,48 12,72 4,97 110,2 1,5 19,58 7,23 114,36 29,34 1,721 157,85 12,46 5,73 110,5 2 19,22 6,71 114,56 29,31 2,227 - - - - 2,5 19,12 7,61 114,60 28,85 1,436 157,00 12,06 6,52 110,6 3 19,11 9,02 115,06 28,87 2,150 158,20 12,05 7,54 111,1 3,5 18,82 7,85 114,56 28,57 1,860 157,90 11,76 6,70 110,5 4 18,83 9,15 115,26 28,51 1,908 158,50 11,76 7,59 111,5 4,5 18,61 8,27 114,56 28,31 1,845 157,88 11,57 7,10 110,5 5 18,57 8,84 114,76 28,34 2,113 158,80 11,50 7,31 110,6 6 18,41 8,99 115,16 27,98 1,770 158,10 11,33 7,48 111,1 7 18,24 8,85 115,36 27,81 1,742 158,90 11,19 7,56 111,3

E\*

Stage I; A2; E=117,66 Stage II; A4; E=164,48 Stage III; A2; E=113,75

min-1

E\*

kJ mol-1 <sup>∆</sup>Z(T) A E9

min-1

E\* kJ mol-1

> Manufacturing, storage and use of nc-TiC in form of powder for sintering or co-deposition processes involves the possibility of occurrence of free carbon in the system. Carbon can be a by-product of the synthesis process (pyrolysis of hydrocarbons) and remain in equilibrium in the TiC-C system or be the product of oxidation of TiC in the air according to the mechanism proposed by Schimada [26], whereby the oxygen dissolves in the carbide and the layer containing oxycarbides is formed.

$$\text{TiC} + \text{(3/2)} \times \text{O}\_2 \rightarrow \text{TiC}\_{1-x}\text{O}\_x + \text{CO}\_2 \tag{36}$$

In second stage, in the layer amorphous TiO2 is formed and elemental carbon is produced [27-31]

$$\text{TiC}\_{1-\text{x}}\text{O}\_{\text{x}} + \left(1-\text{x}/2\right)\text{O}\_{2} \rightarrow \text{TiO}\_{2} + \left(1-\text{x}\right)\text{C}\tag{37}$$

The produced carbon changes the state of the surface of TiC particles. A low oxygen content is the most important prerequisite for high sintering activity of the nanocrystalline TiC powders, especially in the sintering processes, synthesis of the nanocomposites for example nc-TiC in metallic matrices.

#### 154 Heat Treatment – Conventional and Novel Applications

In the manufacturing of nanomaterials by sol-gel method the second, high temperature, i.e. at temperatures above 1400 K, stage is essential. The first stage of this method is the sol-gel technique. This stage is carried out at lower temperature. The intermediate product of pyrolytic decomposition of PAN-DMF-TiCl3 is formed - the powder containing nanocrystalline TiCx in carbon matrix [6]. Nanocrystallites of titanium carbide are characterized by high fraction of lattice defects, i.e., vacant carbon sites, presence of oxygen and/or nitrogen. In the second stage carbonisation and purification of TiCx/C composite in argon takes place. It is essential to obtain the materials with high values of the C/Ti ratio, while maintaining the proper, nanometric grain size, in order to obtain the most favourable properties such as hardness and oxidation resistance. The selection of parameters meeting these requirements is difficult.

There was assumed that the good basis are kinetic studies. They allow to determine the intermediate and final products, distinguish the stages of the process, determine the temperature ranges of their courses and obtain a quantitative description.

The kinetic measurements were carried out using TG-DSC-MS technique. Nanoparticles size, lattice parameter, chemical and phase composition before and after heat-treatment were determined with the following techniques: XRD (Philips PW3040/x0 X'Pert Pro), HRTEM (JEM 3010), SEM (JEOL JSM 6100), EDX (Oxford Instruments, ISIS 300), XPS (SIA 100 Cameca), total carbon measurement (MULTI EA2000, AnalyticJena), and the presence of free carbon was estimated by Raman Spectroscopy. The measurements were carried out under non-isothermal and isothermal conditions. The advantage of this method is the possibility of continuous registration of measured values and use of small weighed amounts of samples during measurements. The procedure is illustrated at the example of heat treatment in argon of the nanocomposite powder containing nc-TiCx (x≤0.7) in carbon matrix obtained by sol-gel method. The method ensures maintaining the small size of crystallites, by physically limiting the volume available for their growth, and the matrix prevents agglomeration of particles and oxidation during storage and transport.

At a certain temperature range, and under certain conditions, the reaction of carbonisation is possible

$$\text{TiC}\_{\text{x}} + \text{yC} \rightarrow \text{TiC}\_{\text{x+y}} \tag{38}$$

Methodology of Thermal Research in Materials Engineering 155

conditions for 6 h. High-purity argon 'Alphagaz 2 Ar' from Air Liquide (H2O\ppm/mol, O2\0.1 ppm/mol, CnHm\0.1 ppm/ mol, CO\0.1 ppm/mol, CO2\0.1 ppm/mol, H2\0.1 ppm/mol) was used during experiments. With regard to reactivity of O2 the special attention during measurements was given to possibility of oxidation of components of the system during the process run. Argon purge flow rate during measurements was set at 100 cm3/min. The part of the results in works [17,32] were presented. Below the complete

The kinetic description of the process was based of thermogravimetric measurements. The results of the measurements are presented on the plots of sample temperature, TG, DTG and

Initially the measurements were carried out under non-isothermal conditions at a linear change in sample temperature, then at the transitional regime, and finally under isothermal conditions (Fig.11a). It should be added that these results were the basis of the description of the process. The theory of kinetics under non-isothermal conditions require that this function depends only on the sample heating rate and temperature. The results were evaluated by neural networks method. Neural networks, used to analyze the non-isothermal measurements, were previously described in [20-23]. The TGu function was the described (dependent) variable, and the sample heating rate and temperature were the describing (indepedent) variables. All the measurement series were examined simultaneously. The received network was GRNN 2/11310. Statistical analysis of this network is given in Table 6.

> Parameter Tr Ve Te S.D. Ratio 0.01716 0.01926 0.01734 Correlation 0.999874 0.999845 0.99987

In columns 2, 3, 4 the statistical evaluation of training (Tr), verification (Ve) and testing (Te)

A high accuracy was obtained. The TG dependencies determined experimentally could have

The essential operation is the division of the process into stages. Basing on the presented measurement results four stages of the process were identified (Fig. 12). In the endothermic stage I, proceeding with a weight loss, the release and desorption of volatile products, contained in the samples after the first stage, occurred. In the exothermic, second stage, marked with the symbol II, proceeding with a samples weight gain, the oxidation of noncarbonised nc-TiCx/C by oxygen present in trace amounts in argon occurred. In the endothermic third stage, labeled by III, simultaneously proceeded the pyrolysis of organic

analysis description were introduced.

HF function dependencies on time (Fig. 11).

**Table 6.** Statistical estimation of GRNN 2/11310 network

**3.1. Kinetic analysis** 

subset is listed.

been used in kinetic calculations.

Heating of the samples in an argon atmosphere can lead to the growth of crystallites. The aim of this study was to develop conditions for annealing of the composite, under which, as a result of carbonisation, TiCx reaches a high stoichiometric composition, i.e. x> 0.8, at the minimal growth of crystallites. Obtaining such optimal material properties can be controlled by selecting the appropriate temperature and rate of the process.

The thermogravimetric measurements were carried out at sample heating rates of 10, 20, and 50 Kmin-1. The mass of the samples were ca. 40 mg. During the measurements the samples were heated up to 1473, 1573, 1673 and 1773 K. These temperature values correspond to the isothermal conditions. The samples were heated under isothermal conditions for 6 h. High-purity argon 'Alphagaz 2 Ar' from Air Liquide (H2O\ppm/mol, O2\0.1 ppm/mol, CnHm\0.1 ppm/ mol, CO\0.1 ppm/mol, CO2\0.1 ppm/mol, H2\0.1 ppm/mol) was used during experiments. With regard to reactivity of O2 the special attention during measurements was given to possibility of oxidation of components of the system during the process run. Argon purge flow rate during measurements was set at 100 cm3/min. The part of the results in works [17,32] were presented. Below the complete analysis description were introduced.
