Kinetic analysis z(a), y(a)

The activation energy of the process and its determination

The methods how to determine the activation energy of observed process can be generally divided into two groups. The first group of methods analysed only one DSC curve (only one heating rate). This kind of method is relatively often used, but the obtained results can be significantly different than reality.

Table I.

The second group of methods requires several DSC curves (for different heating rates). Ther most frequently used method from this group is Kissinger method […] or Ozawa method […] which evaluate shift of temperature corresponding to the maximum of DSC peak with heating rate. The Kissinger method is given by equation 4 and Ozawa method by equation 6. In both these methods the linear dependence should be obtained where the slope of these equations is used to calculate E. In the case that the linear dependence is not observed, the process can be complex and kinetic equation may not be valid.

Equation 1
Equation 2

Another method is Friedman method […], so called isoconversional method, which can be used for non-isothermal as well as for isothermal data. Activation energy is evaluated from the slope of dependence of logarithm of heat flow on reciprocal temperature (equation 5); both determined for selected value of conversion from several curves. These dependences should be linear and obviously the data in the interval of conversion <0.2; 0.8> are evaluated.  Isoconversional method provides information about dependence of E on conversion which can influence interpretation of studied process – in the case that E is not dependent on conversion the process is a simple one, but when E strongly depends on conversion then the process is more complicated (such as overlapped peak covering more than one process).

The determination of kinetic model based on z(a) and y(a) functions

Equation 7a
Equation 7b

The appropriate kinetic model describing experimental data can be selected using functions z(a) and y(a) obtained by transformation of experimental data and normalised within <0;1> interval. The normalisation of data enables easy comparison of several experiments. The function z(a) can be calculated using equation 7a for non-isothermal conditions and using equation 7b for isothermal conditions. The advantage of z(a) function is that it can be calculated without knowledge of the value of E.

Table II

Conversion corresponding to the maximum of function z(a) is characteristic for some of kinetic models; usually it is not only specific value of a but it is in the form of narrow interval which covers inaccuracy arising from experimental reproducibility and data treatment of original DSC curves. However, the more precise determination of appropriate kinetic model needs performing of other steps of kinetic analyses.

Equation 8a
Equation 8b

Another step of kinetic analysis is evaluation of y(a) function, which can be calculated according to equation 8a for non-isothermal conditions and equation 8b for isothermal conditions. In the case of non-isothermal data the value of E is necessary. The shape of y(a) dependence on conversion is characteristic for given kinetic model.

Table III

The shape of both functions z(a) and y(a) is independent on sample mass and heating rate used. In the case that the shape of z(a) or y(a) function is changing with the heating rate then the other factors influencing the kinetic measurement should be analysed. The shape of y(a) functions for selected kinetic models are summarised in Table III.

Kinetic parameters

In the case that the previously mentioned analyses reveals that the most suitable kinetic model is RO, JMA or SB then the kinetic parameters of these models should be determined.

RO(n) model

Equation 9

Kinetic parameter of this model can be determined by iterative procedure according to equation 9 for conversion corresponding to the maximum of DSC peak, where n(x) is approximation of so called temperature integral.

JMA(m) model

The way of determination of JMA(m) model parameter depends on the shape of y(a) function. If the shape of y(a) function is monotonous then the parameter m can be calculated using equation 10.

Equation 10

When the function y(a) shows maximum then the parameter is m>1 and can be calculated using equation 11 where alpha(m) corresponds to the maximum of y(a) function.

Equation 11
Equation 12

SB(M,N) model

The evaluation of parameters of this model consists of two following steps […]. Firstly, the ratio of kinetic parameters M/N is determined according to the equation 12. Secondly, the parameter N is determined from the slope of dependence given as equation 13. The value of parameter M is then calculated using the M/N ratio.

Equation 13

Pre-exponential factor

The last step of kinetic analysis is the determination of value of pre-exponential factor according to the equation 14, where xp is the reduce activation energy in the maximum of DSC peak and the function in denominator corresponds to differential form of appropriate kinetic model. However, this way of A determination is not very common. The value of A is mostly determined using the numerical fitting of experimental data.

Equation 14

The area below the peak or ∆H, respectively

The area below the DSC peak or ∆H can be calculated according to the equation 15. This procedure is done only as a control and the results should correspond to experimental value.

Equation 15

Thanks to Pavla Honcová for this text.

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