Isoconversional Methodology - Isoconversional Kinetics of Thermally Stimulated Processes - page 32

2.1.2 Modern Methods

Any further increase in the accuracy can be reached by means of isoconversional
methods that compute the temperature integral as a part of evaluating E ʱ . These
methods can be referred to as flexible integral methods. By way of contrast, the
traditional integral methods can be referred to as rigid integral methods. This is be-
cause in the latter integration was carried out as a part of deriving the final equation
and thus cannot be modified. Several flexible isoconversional methods have been
developed by Vyazovkin [ 34 - 36 ]. The methods employ the same numerical algo-
rithm [ 34 ] that was developed under the basic isoconversional assumption that for
any given ʱ, g ( ʱ ) remains unchanged when changing the temperature program, e.g.,
the heating rate. Then, for n different heating rates, one can use the basic integral
rate Eq. 2.8 to write the following equality:

A IE T A IE T A IE T

n

α α α

g () (, α = ) = ( , ) =…= (, ).

(2.15) αα , 1 αα , 2 αα , n

β β β

1 2

When the equality holds strictly, Eq. 2.15 is equivalent to

n (, n

IE T
IE T

) β

β =−

∑ ∑

αα , i j

nn ( 1 ).

(2.16)

(, )

i = 1 ji ≠ αα , j i

Since the values of T ʱ are unavoidably measured with some error, the strict equality
cannot be reached and the difference between left- and right-hand sides of Eq. 2.16
has to be minimized. The difference converges to a minimum when the left-hand
side reaches a minimum. That is, E ʱ is found as the value that secures a minimum
of the following function:

n n

IE T
IE T

(, ) β

β

αα , i j

∑ 1 ∑

Φ() E = .

(2.17)

α

(, )

i = ji ≠ αα , j i

The temperature integral in this equation is solved numerically or, for faster results,
can be replaced with one of the highly accurate approximating functions [ 17 ] in ac-
cord with Eq. 2.10, i.e., without setting S ( T 0 ) to 0. The minimization procedure is
repeated for each value of ʱ to obtain a dependence of E ʱ on ʱ .
A comparison of the relative errors in the activation energies estimated by
Eqs. 2.17 and 2.13 (the method of Kissinger-Akahira-Sunose) indicates (Fig. 2.5 )
that the former is virtually error-free. On the other hand, the errors associated with
the latter are practically negligible except when x = E / RT is very small, i.e., when a
process occurs at high temperature and has unusually small activation energy. How-
ever, the major advantage of the proposed flexible method is that the user has total
control over the process of integration and thus can modify it. In order to appreciate
the value of such approach, one needs to realize that when using the traditional (i.e.,