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3.2. OPTIMIZATION AND KINETICS
Theory
The noticeable weakness of the ‘single curve’ methods (determination of the
kinetic parameters from a single run recorded with one heating rate only) has led
to the introduction of the ‘multi-curve’ methods over the past few years [1-3] -
International ICTAC Kinetics project. Only a series of non-isothermal measurements
carried out at different heating rates can give a data set which generally
contains the necessary amount of information required for full identification of
the complexity of a process. This data set usually contains:
- the relationship between specific conversion, xi, and temperatures for
different heating rates (non-isothermal mode).
- the relationship between specific conversion, xi, and time for different
temperatures (isothermal mode).
Commonly applied are the following three isoconversional methods known as: Friedman
[5], Ozawa-Flynn-Wall [6-7] and the ASTM E698 analysis [8]. A detailed analysis
of the various isoconversional methods (i.e. the isonconversional differential
and integral methods) for the determination of the activation energy has been
reported in the literature by Budrugeac [9]. The convergence of the activation
energy values obtained by means of a differential method like the Friedman method
[5] with those resulting from using integral methods with integration over small
ranges of reaction progress x, comes from the fundamentals of the differential
and integral calculus. In other words, it can be mathematically demonstrated
that the use of isoconversional integral methods (for example: Ozawa- Flynn-Wall
[6-7]) can yield systematic errors when determining the activation energies.
These errors depend directly on the size of the small ranges of reaction
progress Dx over which the integration is
performed. These errors can be avoided by using infinitesimal ranges of reaction
progress Dx. As a result, isoconversional
integral methods revert to the differential isoconversional methods formerly
proposed by Friedman [5].
The differential methods for the calculation of the kinetic parameters are based
on the use of the well-known reaction rate equation:
where b is the heating rate, T the temperature, A the preexponential factor and
f(x) is the differential conversion function.
As far as isoconversional integral methods are concerned, these techniques are
based on the equation:
where g(x) is the integral conversion function.
The isoconversional integral methods with the integration over low ranges of the
degree of conversion and temperature respectively, are based on the equation:
which is derived by supposing that in the range of the variation of the
conversion degree Dx, the activation energy
E can be assumed constant. Obviously, the use of such an approach leads to a
plot of E versus the degree of conversion x. However, the
activation energy as a function of the conversion progress looks like a stair
function in which the low ranges of Dx
where E keeps a constant value are clearly marked. The number of stairs
depends directly on the size of Dx.
In order to evaluate the integrals from the previous equation, one can use the
theorem of the average value; we obtain:
where
Since the number of stairs (where the activation energy E is assumed
constant in the isoconversional integral methods) depends directly on the range
of chosen Dx, then an unlimited number of
stairs can be reached by making Dx
infinitesimal. For Dx -> 0, we have Tx
-> T and f(xx)->f(x). As a
consequence, the previous equation returns to its differential form:
that grounds the isoconversional differential methods which correspond to the
Friedman approach. More generally, the conversion rate expression can be adapted
to an arbitrary variation of temperature (as well as to isothermal conditions)
by replacing b(dx/dT) with dx/dt.
Friedman analysis, based on the Arrhenius equation, applies the logarithm of the
conversion rate dx/dt as a function of the reciprocal temperature at
different degrees of the conversion.
with i: index of conversion, j: index of heating rate, and f(x) the function
dependent on the decomposition mechanism.
Table: The forms of the f(x) function dependent on the reaction model.
Autocatalytic : (1-x)^n x^m
F1 : 1-x
F2 : (1-x)^2
F3 : (1-x)^3
Fn : (1-x)^n
P1 : x^0
P2 : 2 x^(1/2)
P3 : 3 x^(2/3)
P4 : 4 x^(3/4)
Pn : n x^(1-1/n)
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A1.5 : 1.5 (1-x) [-ln(1-x)]^(1/3)
A2 : 2 (1-x) [-ln(1-x)]^(1/2)
An : n (1-x) [-ln(1-x)]^(1-1/n)
R2 : 2 (1-x)^(1/2)
R3 : 3 (1-x)^(2/3)
Rn : n (1-x)^(1-1/n)
D1 : 1/(2x)
D2 : [-ln(1-x)]^-1
D3 : 1.5 [1-(1-x)^(1/3)]^-1 (1-x)^(2/3)
D4 : 1.5 [(1-x)^(-1/3)-1]^-1 |
As f(x) is constant at each conversion degree xi, the
dependence of the logarithm of the reaction rate over 1/T is linear with
the slope of m = E/R as presented in the next figure.
Figure: Friedman analysis
If the decomposition follows a single mechanism then the reaction can be
described in terms of a single pair of Arrhenius parameters and the commonly
used set of reaction models. In such cases the dependence of the logarithm of
the reaction rate over 1/T is linear with the same slope of m = E/R
for all conversion degrees xi. The reaction rate can be
described by only one value of the activation energy E and one value of the pre-exponential factor A by the following expression:
However, this approach is not acceptable for most of the decomposition reactions
because, as presented in the Friedman analysis plot and in the next figure for the examined samples, the activation energy is often strongly dependent on the
reaction progress.
Figure: Activation energy as a function of the reaction progress for
decomposition of the organic substance (DSC closed crucibles).
Decomposition reactions are often too complex to be described in terms of a
single pair of Arrhenius parameters and the commonly applied set of reaction
models. As a general rule, these reactions demonstrate profoundly multi-step
characteristics. They can involve several processes with different activation
energies and mechanisms. In such situations the reaction rate can be described
only by complex equations, where the activation energy term is no longer constant
but is dependent on the reaction progress x (E not constant but E=E(x)).
Thus a simplified kinetic analysis can not lead to an accurate description
of the experimental data. For multistage overlapped reactions the prediction of
the thermal behavior under any new temperature profile, without taking into
account the dependence of the activation energies E(x) on the conversion
degree x, is of little value.
The accurate determination of the kinetic parameters under experimental
conditions applied which enables the correct fit of the experimental data is a
prerequisite for prediction of the reaction progress under any new temperature
profile. When solving the complicated interrelation between the baseline, the
kinetic parameters of the reaction and reaction progress, two important points
have to be considered:
(1) The reaction rate must be of Arrhenius type.
(2) When measuring the progress of a reaction one tries to eliminate the
systematic errors, so that only accidental errors have to be taken into account.
In that case the measured values will spread around the average value for each
heating rate, in a form of the Gaussian-type curve. The Gaussian distribution
results from a summation of several events e.g. overlapping reactions, noise,
drift, artifact, and uncertainties in the baseline construction. During the
optimization, the true information has to be extracted with respect of the
point 1 and the whole optimization becomes more and more stable. Once the
stability is reached, the optimization is finished and the reliability of the
predictions depends on how big was the sum of all possible sources of errors.
Therefore, under consideration of all heating rates the mathematical approach
has to determine for each heating rate the 'best value' or 'central tendency' of
the signal, for which the chance of the good reproducibility on subsequent
measurements is maximal.
Fulfilling both above conditions (I: Arrhenius type reaction rate and II:
Gaussian-type distributed errors) makes possible the iterative calculation and
objective determination of the correct baseline for each signal measured under
different heating rates. This objective determination is done by the iterative
calculation of all tangent parameters for each heating rate:
ai,k, bi,k, aj,k, bj,k for
each heating rate k
with
i = indices of the slope and intercept of the tangent at the beginning of the
signal S(T) with the heating rate 'k'
j = indices of the slope and intercept of the tangent at the end of the signal
S(T) with the heating rate 'k'.
The baselines are no longer arbitrarily chosen by the users but objectively
optimized taking into account:
- statistics, for the consideration of the experimental noise and shape of the
signals.
- the kinetic parameters, for the consideration of reaction rates following
Arrhenius relationship.
Determination of the best baselines and calculation of the kinetic parameters
By clicking on the 'Run' button, the software
proceeds with a mathematical analysis of the signal.
This analysis is important because it
will set the basics and initial guess variables for expressing the signal in
terms of Arrhenius equations.
This analysis has to be done for each curve. The left button 'Run' enables
to make the analysis of each curve step by step, whereas the right button 'Run'
proceeds with an automatic screening of all curves.
The strength factor is useful when we
are facing a curve with big noise (e.g. MS signals). It enables you
to go through
the noise and conclude on the form of the signal which is essential for any
kinetic consideration.
- The strength factor can vary between 0 and 1000 per mil, in other words
between 0 and 1.
- The strength factor must not be too high because it does not allow the
accurate description of the signal accurately (loss of information).
- The strength factor must not be too small for noisy signals because it will
reflect the noise too strongly in the subsequent calculations (information on
the shape of the real signal is not clear here).
Recommendations for the strength factor:
Start usually from 1, click on ‘Run’ and zoom the signal. If you have a suitable description of the signal then it is OK. If you do not have a suitable description of
the signal (noisy), increase slowly but progressively the strength factor
towards 1000, click on ‘Run’, and try to get a suitable description.
The ‘boundary’ term is combined with the selection of the temperature (or time)
range for the evaluation of the recorded signals. It is similar to the boundary
conditions of a system of equations. We are facing two possibilities: we can
decide to have strong boundary conditions by setting this factor at 100 percent
or we can allow freedom to these boundary conditions
by setting this factor at 0.
The problem of the boundary conditions is directly related to the second
questions:
- where does the signal start?
- where does the signal stop?
In fact nobody really knows where the signal starts or ends. For example under
non-isothermal conditions, there is always a small temperature window where the
signal is starting or ending. This boundary factor is related to the degree of
freedom the user wants to set for the determination of the boundaries.
Recommendations for the boundary factor:
Start with a boundary factor that is automatically set, click on ‘Run’, and zoom
the signal. If you have an acceptable
description of the signal at the boundaries then
it is OK. If you do not have a suitable description of the signal at the boundaries,
progressively increase the trackbar toward 100 percent, click on ‘Run’,
and try to get a suitable description.
The smooth feature contains two options. It
gives you the
possibility to act directly on the experimental noise and robustness of the
optimization procedure. Always start with the ‘No smoothing’ option.
Recommendations for setting the ‘Smoothing’:
Start always with the ‘No smoothing’ option.
The 'Smoothing' option should be used only occasionally .
The baseline optimization and kinetics
analysis are performed by clicking on the button 'Run analysis'
under the tabbar 'Optimize':
The miniature plot at the top right of the form gives the user an indication
of how fast the optimization goes. The software always continues to search for the
best residuals by using different optimization methods.
Once the optimization is done, the software displays the reaction rates
(normalized signals after correctly calculated baselines and kinetics) for the
examined reaction as a function of the time. Experimental data are represented as
symbols; solid lines represent the calculated signals. The values of the heating
rates or temperatures used for calculation are marked on the curves. The results
are displayed under the tabbar ‘Plot analysis results’.
With the ‘No smoothing’ option the optimization usually converges after
a few loops (three or four loops). If the optimization failed after
clicking on the ‘Run analysis’ button, then try first to act on the baselines by
reconstructing them. For this just click again on the tabbar ‘Kinetics’ and draw
again the tangents or modify slightly their slopes by using the buttons
set for this task:
It is important to construct the baselines as precisely as possible to be
successful with the optimization. For this occurrence, make large use of the zoom feature.
The better the determination of the initial baselines
is the more robust will be the subsequent optimization
and the more
reliable the predictions. The optimization might fail for example if the reaction progress of each heating
rate under non-isothermal conditions is crossing with another heating rate when
you click on the button ‘view reaction progress’. This may arise with data
containing a large number of experimental errors or if the baselines are not
constructed correctly. You can examine the way the baselines are constructed by
displaying the reaction progress of each curve as a function of the temperature
under the tabbar 'Optimize'.
Additional remarks concerning the ‘Smoothing’ option:
If the optimization still fails after the reconstruction of the baselines then
click on the radiobutton ‘Smoothing’ and rerun the optimization by clicking
again on the ‘Run analysis’ button. Start with 10 loops and click on plot
analysis results. If you do not have a nice fitting just come back to the tabbar
‘Optimize’ and click again on ‘Run analysis’ for five additional loops. And
continue like this… Generally 10 or 15 loops are necessary. Do not usually
perform more than 30 loops. If you need more loops it might be because:
- the signals are not properly measured (measure again the curve which presents
some deviation),
- the experimental noise might be substantial,
- the baselines are still not correctly constructed
- or the signals are not correctly cut.
In any case go back to the corresponding tabbar and reconstruct or modify
slightly the baseline or go back to ‘Data Input’ and cut again the signal.
Once the optimization is finished,
the software displays the conversion
rates as a function of time:
Figure:
Conversion rates (normalized
DSC-signals after correctly calculated baselines and kinetics) as a function of
time for the decomposition of an organic substance. Experimental data are
represented as symbols, solid lines represent the calculated signals. The values
of the heating rates are marked on the curves.
=> (units of S(T)Rate = [1/s])
= conversion rate
Clicking on the button:
displays the conversion rates as a function of the temperature.
Figure: Conversion rates (normalized DSC-signals after correctly calculated
baselines and kinetics) as a function of the temperature for the decomposition
of an organic substance. Experimental data are represented as symbols, solid
lines represent the calculated signals. The values of the heating rates are
marked on the curves.
=> (units of S(T)Rate = [1/s])
= conversion rate
The kinetic parameters calculated from the non-isothermal experiments then make
possible the prediction of the reaction progress for any other heating rate and
more generally for any temperature mode such as:
· isothermal
· non-isothermal
· stepwise
· modulated temperature or periodic temperature variations
· rapid temperature increase (temperature shock)
· real atmospheric temperature profiles for investigating properties of
low-temperature decomposed solids under different climates (yearly temperature
profiles with daily minimal and maximal fluctuations)
· adiabatic (safety analysis, storage, scale-up).
To predict the reaction progress under any new temperature profile just click on
one of the tabbars:
The kinetic parameters of the reaction are displayed under the tabbar:
Under this tabbar the results of the different types of kinetic analysis are
displayed:
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