Isoconversional methods

For the very restricted conditions where Eq. (5.2) provides a rigorous description of the reaction kinetics, the activation energy, E, is a constant independent of conversion. But in most cases it is found that E is indeed a function of conversion, E (x). This is usually attributed to the presence of two or more mechanisms to obtain the reaction products e.g., a catalytic and a noncatalytic mechanism. However, the problem is in general associated to the fact that the statement in which the isoconversional method is based, the validity of Eq. (5.1), is not true. Therefore, isoconversional methods must be only used to infer the validity of Eq. (5.2) to provide a rigorous description of the polymerization kinetics. If a unique value of the activation energy is found for all the conversion range, Eq. (5.2) may be considered valid. If this is not true, a different set of rate equations must be selected. 

For this particular case, both a, and a2 are unique functions of conversion, meaning that dx/dt depends only on conversion and temperature i.e., the polymerization kinetics may be described by the phenomenological Eq. (5.1). Moreover, if one of the mechanisms (e.g., the catalytic) predominates over the other one (e.g., the noncatalytic), Eq (5.2) may be used to correlate experimental results and the activation energy may be obtained using isoconversional methods. 

Therefore, for particular values of the conversion of functional groups and temperature, the rate of chainwise polymerizations depends on the concentration of active species which, in turn, depends on the particular thermal history. Thus, phenomenological equations derived from Eq. (5.1), or isoconversional methods of kinetic analysis, should not be applied for this case. 

When the isoconversional method is applied to the set of dynamic runs, an activation energy lying in the 69-73 kJ mol-1 range is obtained, without any definite trend with conversion. The value is very close to the one reported by Montserrat and Malek (1993) using this method again, this is an apparent value without any physical meaning. 

These methods were presented in the previous chapter (Section 11.4.3.2). In their principles, isoconversional methods use a mathematical description of the conver-... 

Vyazovkin and Liimert [56] argue that kinetic data, A and E, values obtained on the assumption of a one-step reaction may be incorrect because the possibility that thermal decompositions proceed by multistep processes has been ignored. This potential error can be avoided by using isoconversional methods to calculate Arrhenius parameters as a fimction of a. A real isokinetic relationship in a multistep process can be identified fi om the dependence of and its confidence limits, on a. The contribution fi om the second reaction step is negligible at the start of chemical change and thereafter rises as ar increases. 

When more than one set of experimental results is available, the unknown form of the conversion function f(a) or g(ar) may be eliminated by comparing measurements made at a common value of a under the two (or more) sets of different conditions. These isoconversional methods are thus model independent, or nondiscriminating methods of estimating the Arrhenius parameters [14,42,43]. 

Isoconversional methods [43] rely on the general equation (see equation (5.2)) ... 

To avoid discarding potentially significant information, the parameters obtained fi om isoconversional methods may be used with the original data to determine the kinetic model [49], although this is often not done. Activation energies determined from isoconversional methods [43] are in good agreement with values from isothermal experiments. 

Vyazovkin and Lesnikovich [93] proposed that the type of complex process encountered in non-isothermal experiments could be identified by analysis of the shape of the curve of the dependence of the apparent on a, foimd by isoconversional methods. Concurrent competitive reactions are characterized by an increasing dependence of the apparent value of E on a, but detailed shapes are dependent on the ratios of the contributing rates. A decreasing dependence of on a, was found for intermediate reversible processes [93]. 

An advantage of the advanced isoconversional method is that it can be applied to study the kinetics under arbitrary temperature programs such as distorted linear (e.g., self-heating/cooling) or purposely nonlinear heating e.g, temperature modulation). To more adequately account for a strong variation of... 

Sometimes the popular Kissinger method [15] is erroneously classified as an isoconversional method. The confusion appears to stem fi om the fact that the basic equation of die method ... 

To our knowledge the first application of an isoconversional method is due to Kujirai and Akahira [16], who studied the decomposition rates of insulating materials. In their work, they used an empirical equation ... 

The value of Tg can be defined in several ways. For instance, Moynihan et al. [50] have used Tg determined as the extrapolated onset, the inflection point, and the position of a DSC peak obtained on heating. The differently defined values of Tg correspond to different stages of the glass transition. It has been reported in several papers [51,52,53] that the E value estimated fi-om equation (34), decreases with increases in the Tg value. It is markedly bigger when Tg is taken as an onset temperature and smaller when it is taken as the midpoint and/or peak temperature. To explore this phenomenon more closely, Vyazovkin et al. [54] have employed an isoconversional method that allowed them to reveal a variation in E throughout the glass transition. The conversion, or, can be readily determined fi-om DSC data as the normalized heat capacity [55] ... 

On the other hand, reliable kinetic predictions can be accomplished in entirely model-free way by using the dependence of Ea on a determined by an isoconversional method. The relevant predictive equation [17,87] was originally obtained in the following form... 

Isoconversional kinetics is an efficient compromise between the common single-step Arrhenius treatment and the predominantly encoxmtered processes whose kinetics are multi-step and/or non-Arrhenius. Isoconversional methods are capable of detecting and handling such processes in the form of a... 

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