A kinetic method in simultaneous thermal analysis

Abstract

Ordinarily, Thermogravimetry (TG) or Differential Thermal Analysis (DTA) or Differential Scanning Calorimetry (DSC) can describe only the change of α or △T or q with T(t), and Simultaneous Thermal Analysis (STA) provides two curves, (α∼ T(t) curve and △T or q∼ T(t) curve, in a single test (T is temperature; q is the heat flow; t is time). This work proposes a Simultaneous Kinetic Method, or simply SKM method aimed at STA tests. In the SKM method, the values of the activation energy (E) and reaction mechanism function can be calculated and determined by a trial and error method using the differential kinetic equation. When the values of α_m at the different heating rate are different, for example the heat degradation of polyamide 6 (PA6), the SKM method provides a more accurate activation energy than that calculated by the Kissinger equation.

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Introduction

In 1958, Paulik¹ introduced a multi-purpose method by which thermogravimetric TG and DTA curves could be obtained from a single analysis, a technique called simultaneous thermal analysis (STA). Historically the Mettler-Toledo Company developed the first commercial equipment to achieve STA in 1964. Until now, STA can be carried out in many types of machines, such as the TGA/SDTA851^e (Mettler-Toledo), DSC/TGAQ600 (TA Instruments), DTG-60/60H (Shimadzu) and STA449 series (Netzsch).

The Kissinger method² is a well known differential method in thermal analysis. The main supposition is that ln[−f′(α_p)] is zero or constant. But many researchers have proved that ln[−f′(α_p)] is not a constant.^3–9Ref. 10 suggests the use of differential methods when the experimental curve is fluctuating.

Since the values of α_p and T_p can be obtained directly from STA, a differential method similar to the Kissinger method, but even more exact than that, is proposed here.

Theoretical consideration

Generally, the kinetic equation of a chemical reaction undergoing linear heating may be described by the basic rate equation as follows: where f(α) is the reaction mechanism function, α is the degree of conversion in reaction, T is the temperature, A is the pre-exponential factor, β is the linear heating rate (β = dT/dt), E is the activation energy and R is the universal gas constant.

At a peak temperature (T_p) of a curve of either differential thermal analysis DTA or differential scanning calorimetry DSC, d²α/dT² = 0, the differential rate equation, eqn (1), may be expressed as:

where α_p is the degree of conversion corresponding to the peak temperature (T = T_p).

Usually, because f(α)is unknown and the value of α_p cannot be obtained directly from just a DTA or DSC curve, eqn (2) cannot be solved.

Kissinger¹ suggested that choosing f(α) = (1 − α)ⁿ and assuming n(1 − α_p)ⁿ⁻¹ ≈ 1, then ln[−f′(α)] = 0. Hence, eqn (2) can be re-expressed as:

which is usually called the Kissinger equation. However, many researchers have found that n(1 − α_p)^n − 1 ≠ 1 when n ≠ 1, and each mechanism has a specific constant value of α_p.^3–9 If we consider that the values of α_p at different heating rates are constant or quasi-constant, that is f′(α_p) ≈ C₁, eqn (2) can be rewritten as: where C = ln[−f′(α_p)]. Eqn (4) may be thought of as an “outstretched” Kissinger equation.

In fact, n(1 − α_p)ⁿ⁻¹ ≠ 1 or f′(α_p) ≠ C₁(constant) in many reactions. Hence, the activation energy can not calculated exactly by Kissinger eqn (3) and (4) for many reactions.

Now, with the development of thermal analysis techniques, the STA technique can give TG and DTA (or DSC) curves in one experimental set-up, which are α ∼ T(t) and either ΔT ∼ T(t) or q ∼ T(t) (q is the heat flow mw/mg). In this case, the values of T_p and corresponding α_p can easily be determined.

By varying the linear heating rate, a series of α_p and T_p values are obtained. From any of the f(α), the reaction active energy (E) can be calculated from eqn (2). The f(α) can assume any reaction mechanism function, such as the reaction order equation, Johnson–Mehl–Avrami equation, the one-dimensional diffusion equation, the two-dimensional diffusion equation, Jander equation, Ginstling–Brounshtein equation, random equation, etc.¹⁰ From the different f(α), we can get a series of E values. Comparing the deviation of E for different f(α) and analyzing the rationality of f(α), E and f(α) can be determined. This method is called the Simultaneous Kinetic Method (SKM).

To illustrate SKM's application, it is applied to analysis degradation of polyamide 6.

Experimental

The STA analysis was performed on a Mettler-Toledo TGA/SDTA851 with selected heating rates of 5, 10 and 20 K min⁻¹ in the range 30–600 °C. Samples of polyamide 6 (PA6), all of mass 8 mg, were put in alumina 30 ul pans in an atmosphere of dry nitrogen at a flow rate of 20 ml min⁻¹.

Results and discussion

TG-DTA curve of PA6 degradation

The TG-DTA curves of PA6 are shown in Fig.1. It can be found that PA6 has two peaks on DTA curves: the first peak is the melting of PA6; the second peak is the main degradation reaction. The values of α_p and T_p of the second peak are listed in Table 1. It is clear that the values of α_p at the different heating rates are not constant or even quasi-constant. The TG data are listed in Table 2.

Fig. 1 The STA curves of PA6 at 5, 10 and 20 K min⁻¹ (the heating rate increasing from left to right).

Table 1 DSC data of degradation of PA6 at STA

β/K min^−1	5	10	20
1 − αp	0.431	0.372	0.333
T p/K	712.24	729.65	745.48

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Table 2 TG data of degradation of PA6

β T/K	α
	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90
5	665.5	685.2	695.2	702.5	708.5	713.9	719.0	724.4	731.0
10	677.7	698.2	709.0	716.5	722.9	728.2	733.4	738.5	745.0
20	689.5	710.5	721.9	729.9	736.2	741.9	747.2	752.5	758.9

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The kinetic calculation of PA6 by SKM

If f(α) = (1 − α)ⁿ, eqn (2) can be re-expressed as follows:E_s is the activation energy calculated by SKM method. Alternatively if f(1 − α) = (α)^L−1[1 + (L − 1) × (1 − α)], where L is the least number of repeating units of polymer not volatilized, L ≥ 2,¹¹ then take this equation into eqn (2), eqn (2) at L = 2, 3 and 4 are listed below:

ln(β/T²_p) − ln(2 − 2α_p) = ln(AR/E_s2) − E_s2/RT_p (L = 2) (6-1)

ln(β/T²_p) − ln(6α_p − 6α²_p) = ln(AR/E_s2) − E_s2/RT_p (L = 3) (6-2)

By the SKM method, the activation energies for different reaction orders and reaction mechanism functions can be calculated separately.

ΔE_max is the maximum deviation between the activation energy at same reaction order. From Table 3, ΔE_max is seen to decrease with increasing n when f(α) = (1 − α)ⁿ. Usually, it is impossible for n to exceed 5, so it is unsuitable to assume f(α) = (1 − α)ⁿ in describing the thermal degradation of PA6. For the alternative f(α) function, the smallest value of ΔE_max was obtained when L = 2 and the value of E_s of PA6 should then be 206.79 kJ mol⁻¹.

Table 3 The reaction active energy E from eqn (5) and (6) at different n and L

n						L
	1	2	3	4	5	2	3	4
E 5/10(kJ mol^−1)	160	197	233	270	307	197	172	148
E 5/20(kJ mol^−1)	172	206	241	275	309	206	185	164
E 10/20(kJ mol^−1)	186	217	249	281	312	217	200	183
ΔEmax(kJ mol^−1)	26	21	16	11	6	21	28	35
E s/kJ mol^−1	173	207	241	275	309	207	186	165

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The kinetic calculation of PA6 by Ozawa and Kissinger method

To compare the kinetic results calculated by the SKM method with that of other methods, the kinetic parameters of PA6 are also calculated by Ozawa¹¹ and the Kissinger method.

The Ozawa equation is given below:

The activation energy of PA6 calculation by Ozawa, Kissinger and SKM method are listed in Table 4 and 5. Where E₀ is calculated by Ozawa method according to the data in Table 2; E_k is calculated by Kissinger method according to the data in Table 1; E_s is calculated by SKM method according to the data in Table 1.

Table 4 The activation energy of PA6 calculated by Ozawa method

E/kJ mol^−1	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90
E 5/10	203	202	196	197	195	199	201	208	213	201
E 5/20	210	212	206	205	206	207	209	213	218	209
E 10/20	217	221	217	214	219	216	218	218	223	218

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Table 5 The average activation energy of PA6 calculated by Ozawa, Kissinger and SKM method

E/kJ mol^−1		E k	E s1(n = 5)	E s2 (L = 2)
E 5/10	201	160	307	197
E 5/20	209	172	309	207
E 10/20	218	186	312	217
ΔEmax	17	26	6	21
E average	210	173	309	207

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Because the Ozawa equation is a model-free kinetic equation, the value of E_o is considered reasonable, implying that the value of E_s2 in Table 3 is also reasonable. Furthermore, it can be determined from Table 3 that the reaction mechanism function for PA6 degradation is f(1 − α) = (α)^L−1[1 + (L − 1) × (1 − α)], the reaction order is L = 2, while the activation energy is about 207 KJmol⁻¹. Apparently, both E_k and E_s1 have larger deviations.

Conclusions

For the calculation of activation energy from Kissinger equation, some large errors were displayed when values of α_p which are not constant. Based on the STA technique, a new analytic method, called SKM, was provided. For the degradation of PA6, the calculated results showed that a more accurate value of activation energy was obtained using SKM instead of the Kissinger equation. Moreover, the reaction mechanism function f(α) also can be determined easily from SKM.

Acknowledgements

The author gratefully acknowledges the assistance of Mr. Urs Jörimann, Mettler-Toledo GmbH in obtaining the PA6 test data, and is also grateful to his students, Zuo Jinqiong and Li Yanchun.

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