# LOGISTIC MIXTURE MODEL VS ARRHENIUS FOR KINETIC STUDY OF

```LOGISTIC MIXTURE MODEL VS ARRHENIUS FOR KINETIC STUDY OF
ANALYSIS
SHORT TITLE:
LOGISTIC MIXTURE MODEL VS ARRHENIUS FOR KINETIC STUDY OF
MATERIALS
Salvador Naya1, Ricardo Cao1, Ignacio López de Ullibarri1, Ramón Artiaga2, Fernando
1
Department of Mathematics, University of A Coruña, Spain
Department of Industrial Engineering II, University of A Coruña, Spain
2
Ricardo Cao
Campus de Elviña
15071 A Coruña. -SPAIN
Tel: +34 981 167 000
Fax: +34 981 167 160
E-mail: [email protected]
LOGISTIC MIXTURE MODEL VS ARRHENIUS FOR KINETIC STUDY OF
ANALYSIS
Keywords: logistic regression, TGA curve, thermogravimetric experiments.
Abstract
In this work, an alternative method to the Arrhenius equation for thermogravimetric
analysis is presented. It is based in performing a logistic regression of the raw TGA
data. This model assumes that more than one physical process may be involved in each
mass loss step and that each physical process may extend along all the experiment. The
logistic mixture obtained explains the complete TGA trace, including as many mass loss
steps as the experiment has. The typical asymptotic tendency of the mass loss steps is
perfectly reproduced by the model. A discussion of the model from the statistical point
of view is presented as well as a comparison with other classical models.
INTRODUCTION
Thermogravimetric analysis (TGA) is widely used to determine kinetic parameters for
polymer decomposition. Both isothermal and dynamic heating experiments can be used
to evaluate kinetic parameters. Each has advantages and disadvantages [1]. In dynamic
thermogravimetric analysis, the mass of the sample is continuously monitored while the
sample is subjected, in a controlled atmosphere, to a thermal program, where the
temperature is ramped at a constant heating rate. Ideally, a single thermogram has been
said to be equivalent to a very large family of comparable isothermal volatilization
curves and, as such, it constitutes a rich source of kinetic data for volatilization [2].
The classical way to study the kinetics of these processes by TGA starts from the
assumption that the mass loss follows the Arrhenius equation:
⎛ E ⎞
k (T ) = A ⋅ exp⎜ −
(1)
⎟
⎝ RT ⎠
where k, the reaction rate, depends on the temperature, T, and the activation energy, E,
may be considered constant in each degradation process (that appears as a clear step in
the mass trace) since the degradation mechanism is supposed not to change in a narrow
range of temperatures. The constant A may be calculated from A = qts , where s is the
reaction order, in the case that the kinetics follow a reaction order model.
OTHER MODELS
Many other models start from the Arrhenius equation, modified by Sesták-Berggren [3]:
dα
n
p
= k (α ) m (1 − α ) [− ln (1 − α )]
(2)
dt
where n, m and p are constants. Two of the most used derivative models based on this
equation are Freeman and Carroll [5] and Friedman [6].
There are also some integrate models, like Ozawa [7], Flyn [8] and the one proposed by
Popescu [9], that allows for calculation of n and A from TGA data obtained at several
heating rates. The method proposed by Conesa considers that some organic fractions of
the sample decompose independently giving an organic residue and an inorganic
fraction. This model gave good correlation with the mass loss derivative data for
different rubbers [10]. The method proposed by Carrasco and Costa [11] has been
successfully applied to the thermal degradation of polystyron. Although the application
of these models to specific cases has been checked by detailed statistical studies, all of
them are based on the Arrhenius equation and can not be generally applied to material
degradations following very different kinetics. Moreover, its methodology is sometimes
unease.
It has been said for methods based on one simple heating rate that quite different
reaction models fit the data equally well (from the statistical point of view) whereas the
numerical values of the corresponding Arrhenius parameters crucially differ (Vyazovkin
[12]). Its physical meaning is obscure and no predictions can be done outside the range
of experimental temperature. Other authors deemed the Arrhenius model inappropriate
for the calculation of kinetic parameters from non-isothermal thermogravimetric curves
[13]. Moreover, arising from the Kinetics Workshop, held during the 11th International
Congress on Thermal Analysis and Calorimetry (ICTAC) in Philadelphia, USA, in
1996, sets of kinetic data were prepared and distributed to volunteer participants for
their analysis using any, or several, methods they wished. The results obtained by each
researcher were different than the ones obtained by the others, Brown et al. [14]. All of
this confirms our believing that the existing models cannot be generally applied and
sometimes it is not clear which one is the best suitable to each case. That is the reason to
propose an alternative model that will be described in the following sections.
LOGISTIC MIXTURE MODEL PROPOSED
The logistic mixture model (see Naya et al. [15] and Artiaga et al. [16]) proposes to
decompose the TGA trace in several logistic functions, assuming that each of the
functions represents the degradation kinetics of each component of the sample. Even in
the case of homogeneous materials, it is supposed that several different structures may
exist, each one following its specific kinetics that may be different from the others.
In this model, it is assumed that a TGA trace may be fitted by a combination of logistic
functions:
k
Y (t ) = ∑ wi f (a i + bi t )
i =1
et
.
f (t ) =
1 + et
(3)
where i = 1,2,..., k represent different components from the mass loss process point of
view, not necessarily different chemical compounds.
In order to model the mass loss along time, it is assumed that the candidate functions to
estimate this quantity (t , Yi (t ) ) have to satisfy that the response, Yi (t ) , should tend to 0
as t → ∞ . It implies that the parameters bi have to be negative. When t = 0 , the
function Yi (t ) has to tend to the mass of the original sample. This means that, the
constants wi correspond approximately, when ai is large enough, to the mass loss of
the sample in each mass loss process. These processes generally appear as clear steps of
the TGA trace.
The function Y (t ), that represents the overall TGA trace, may be expressed as a sum of
components of the form Yi (t ) = wi f (a i + bi t ) . The constants ai and bi can be
interpreted in the following manner. The values bi represent the slope of the mass steps
while the ratio − ai / bi denotes the step location. The values wi account for the weight
of each component in the sample.
Figure 1 shows an example of a logistic mixture with k = 4 , w1 = 5, a1 = 12, b1 = −4,
w2 = 4, a 2 = 14, b2 = −2, w3 = 7, a 3 = 43, b3 = −5, w4 = 1, a 4 = 16, b4 = −1.
Once the regression function of the TGA trace is obtained, it is immediate to compute
its derivatives. Thus, for example, the first derivative of the TGA trace (DTG), which is
used by many kinetic models, since it represents the mass loss rate along time, may be
expressed by the following equation:
k
dTGA(t ) = ∑ wi bi f ' (a i + bi t )
i =1
f ' (t ) =
et
(1 + e )
t 2
k
⎛ t − a' i ⎞
a
⎟⎟ with a ' i = − i and
A reparametrization of equation (3) gives Y (t ) = ∑ wi f ⎜⎜
bi
i =1
⎝ b' i ⎠
1
b' i = . The value b'i represents the mass loss rate and a'i corresponds to the half
bi
mass loss location of the i-th step.
LOGISTIC MIXTURE PARAMETRIC FITTING
For fitting the data to a logistic mixture model some estimation of the parameter values
in (3) is needed. This task is usually performed by using statistical software. In this
case, we have used the non linear regression and derivatives packages of S-plus.
We considered a non linear regression model:
y i = m( x i , θ ) + ε i , i = 1,2,..., n.
where the response variable and the independent variable values are represented by yi
and xi , respectively, θ is the parameter vector, that will be estimated by least squares
and ε i are the errors, assumed to have normal distribution, with zero mean and constant
variance.
The residuals of the model are defined as:
ei (θ ) = y i − m( x i ; θ ), i = 1,2,..., n
k
The parameters of the model were estimated by minimizing
∑ e (θ )
i =1
i
2
a non linear least
squares methods. The fundamentals of this method were described by Gay [17]
The Levenberg-Marquardt method routine for generation of the approximation sequence
to the minimum point, based in the “trust region” algorithm, was used for the
computation of the parameter values that minimize the residuals squared sum. This
algorithm was discussed by Chambers and Hastie [18]. Details about its implementation
in S-plus are given in Dennis et al. [19].
One of the problems that appear when using this fit is to choose some starting points for
the different parameters to estimate. To do this, one possibility consists in trying to
estimate the inflexion point by direct observation of the TGA trace. Since this method is
not easy and requires previous expertise, we propose a method based in the idea of
assuming that the data follow locally a logistic regression. So it is possible to fit the
function logit Yi (t ) / wi to a straight line with intercept ai and slope bi ,
where logit (u ) = log(u / 1 − u ) . The reason for this linear fitting is explained as follows:
Yi (t )
wi
exp(a i + bi t )
Yi (t ) = wi
⇒
= exp(a i + bi t )
Yi (t )
1 + exp(a i + bi t )
1−
wi
⎛ Yi (t ) ⎞
⎟
⎜
⎛ Yi (t ) ⎞
wi ⎟
⎜
⎟⎟ = log
So logit⎜⎜
⎜ Yi (t ) ⎟ = a i + bi t
⎝ wi ⎠
⎟
⎜⎜ 1 −
wi ⎟⎠
⎝
PARAMETER ESTIMATION
An algorithm has been implemented for automatic calculation of the model parameters.
It consists of the following steps:
1. The values of those inflection points where the second derivative changes from
negative to positive are estimated. The derivative of the TGA curve is obtained by
approximating, with a small enough h, the limit:
lim
h →0
y (t + h ) − y (t )
h
The second derivative is obtained, in a similar way, starting from the first derivative.
To identify the points where y ′′(t ) = 0 , a narrow band [− ε ,+ε ] is considered along
the y axis, preventing to identify as inflection points values where the second
derivative is zero as a consequence of the derivative estimation errors. The middle
points of the range where the second derivative crosses the band [− ε ,+ε ] are finally
selected.
2. The values w1, 0 , w2,0 ,..., wk , 0 are computed using the difference between the mean
values on the y-axis of consecutive inflection points, previously calculated in Step 1.
These values visually reproduce the fall steps of the TGA curve. Denoting by m j
with j = 1,2,..., k − 1 , these middle points, the w j , 0 can be obtained as:
w1,0 = max{Yi / i = 1,2,..., n}− m1
w j ,0 = m j −1 − m j , for j = 2,3,..., k − 1
wk ,0 = m k −1 − min{Yi / i = 1,2,..., n}
3. Each mass loss step is detected by defining
y j (t ) = y (t ) − m j
for j = 1,2,..., k , in the pertaining range on the horizontal axis.
4. In order to find the values a j , 0 and b j , 0 , a straight line is fitted to the points
(t i , logit ( y j (t i ) w j )) in each fragment of the TGA curve corresponding to the fall
steps previously identified:
⎛ y j (t ) ⎞
⎟ = a j + bjt
logit ⎜
⎜ w ⎟
j
⎝
⎠
Once the initial values of the parameters have been obtained, they will be optimized
using the Levenberg-Marquardt algorithm.
HYPOTHESIS TEST
An important issue when fitting a logistic mixture to the TGA curve is to determine the
number of components in the model. To answer this questions a hypothesis test view is
adopted. Starting from a k+1 components model
Yi = y (t i ) + ε i
k +1
y (t i ) = ∑ w j f (a j + b j t i ).
j =1
we consider the null hypothesis:
H 0 : wk +1 = 0 ,
which means that the maximum number of components of the process is k, since the
weight of a (k+1)-th component would be wk +1 = 0 .
The alternative hypothesis is:
H 1 : wk +1 ≠ 0 ,
i.e., the process needs the (k+1)-th component to be explained. We fixed the level
α = 0.01 and have chosen the Average Squared Error (ASE) as a test statistic.
The test will be performed through these steps:
1. Initialize the number of logistic components k = 1 .
2. Use
the
data
(t i , Yi )
to
estimate
the
model
parameters
θ = ( w1 , a1 , b1 , w2 , a 2 , b2 ,..., wk , a k , bk ) by means of the Levenberg-Marquardt
algorithm.
3. Calculate the ASE for the model with k components:
ASE ( k )
(
k
1 n ⎛
= ∑ ⎜⎜ Yi − ∑ wˆ j f aˆ j + bˆ j t i
n i =1 ⎝
j =1
)
⎞
⎟
⎟
⎠
2
where θˆ = ( wˆ 1 , aˆ1 , bˆ1 , wˆ 2 , aˆ 2 , bˆ2 ,..., wˆ k , aˆ k , bˆk ) is the parameter values estimated in
the previous step.
4. Repeat B = 500 times the following mechanism:
(
)
4.1. Draw bootstrap resamples t i , Yi* , for i = 1,2,..., n, as it will be detailed later
on.
4.2. Use the bootstrap resamples to estimate the parameters replication
θ * = ( w1* , a1* , b1* , w2* , a 2* , b2* ,..., wk* , a k* , bk* ) by Levenberg-Marquardt.
4.3. Calculate the bootstrap version of ASE:
ASE(*k ) =
1 n ⎛ * k *
w j f a *j + b *j t i
∑ ⎜ Yi − ∑
n i =1 ⎜⎝
j =1
(
)⎞⎟⎟
⎠
5. The p-value is approximated by bootstrap using the proportion:
p − value =
{
( j)
1
# j / ASE * ( k ) > ASE ( k )
B
}
5.1. If p − value > 0.10 , we accept H 0 , that the number of logistic components of
the process is k.
5.2. If p − value < 0.01 , the number of components, k, is increased in one unit (we
reject H 0 ) and come back to Step 2.
5.3. If 0.01 < p − value < 0.10 , the result is shown and let the user to choose
between accepting the model or looking for more complex one.
Obtaining bootstrap resamples
In order to simulate the bootstrap resamples t i , Yi* , for i = 1,2,..., n , a random error will
be added to the fitting of the experimental trace obtained in Step 2:
(
k
(
)
)
Yi * = ∑ wˆ j f aˆ j + bˆ j t i + ε i*
j =1
The random errors will be generated following these steps in order to incorporate the
sample autocorrelation:
1. Denoting by Ei the difference between consecutive Yi for i = 1,2,..., n − 1 :
Ei = Yi +1 − Yi
The quantities E and σˆ E2 are computed:
E=
1 n −1
∑ Ei
n − 1 i =1
σˆ E2 =
1 n 2
∑ Ei − E
n − 1 i =1
2. Next, the values ε i* will be obtained for i = 1,2,..., n by ε i* = ρˆ E ε i*−1 + ai where
1 n−2
∑ Ei Ei+1
n − 2 i =1
ρˆ E =
1 n −1 2
∑ Ei
n − 1 i =1
and ai is a sequence of independent and identically distributed random variables
d
normal ( ai = N (0, σˆ a2 ) ) with σˆ a2 = σˆ E2 (1 + ρˆ E ) . This dependence structure assumes
that the instrument measurement error follows a first order autoregressive process.
See Figure 2 to observe the correlation of the measurement error.
Once the test was performed the required number of times, the number, k, of
components needed for a correct fitting is found. Moreover, an estimation of the
parameters is obtained θˆ = ( wˆ 1 , aˆ1 , bˆ1 , wˆ 2 , aˆ 2 , bˆ2 ,..., wˆ k , aˆ k , bˆk ) .
PHYSICAL MEANING OF THE PARAMETERS
Consider a global logistic mixture fit to a TGA curve. The amount of mass evolved in
each degradation step is approximately wi . In the time axis, each process is centred at
− ai bi . The value of the derivative of the i-th component at that point is 1 4 wi bi ,
which means that bi measures the mass loss rate, i.e. the loss speed per unit of mass
with respect to time (or temperature, in experiments with a constant heating rate).
CASE STUDIES
In order to validate the model in extreme situations, some TGA experiments exhibiting
very different behaviours were considered. The first one corresponds to the analysis of
wood from Eucalyptus globulus. Wood is a very complex material, where the main
components are cellulose and lignin. Its thermal behaviour is not simple and
overlapping processes seem to be involved. Apparently, it decomposes into three main
steps. Other complex case considered was an Epoxy-Araldite sample.
Eucalyptus wood experiment
In this case three logistic components were assumed. The fitting to obtain the starting
values was performed in three ranges. The final values for the parameter estimates are
collected in Table 1:
wi
i =1
13.0479
i=2
41.094
i=3
22.534
ai
5.06769
15.4589
162.17
bi -0.0113 -0.0085 -0.0856
Table 1: Parameters of the logistic mixture model for the Eucalyptus sample.
Epoxy-Araldite experiment
A constant was assumed in this case to represent the asymptotic value at the end of the
TGA curve. Four components have been assumed. After iterating the algorithm, the
fitted values of the parameters are given in Table 2, with a constant value of 0.705.
wi
i =1
2.404
i=2
6.971
i=3
3.075
i=4
3.448
ai
11.333 35.874 24.239 19.481
bi 0.010 0.014 0.019 0.010
Table 2: Parameters of the logistic mixture model for the Epoxy-Araldite sample.
Two plots of the original TGA traces compared to the estimated functions via the
logistic mixture model are given in Figures 3 and 4.
PUT FIGURE 3 AND 4 ABOUT HERE
COMPARISON BETWEEN THE LOGISTIC MIXTURE MODEL AND THE
ARRHENIUS MODEL
Since the Arrhenius model is usually applied to apparently single processes, in order to
compare it with the logistic mixture model, separate single mass loss steps of a
polyether-polyurethane TGA trace (Figure 5) were used for Arrhenius, while the logistic
mixture was applied to the overall TGA curve. The TGA test was performed at a
constant heating rate of 10 ºC/min, using 50 ml/min of Argon as purge. Three Arrhenius
based methods were applied: linear regression with the standard Arrhenius equation,
Freeman and Carroll and Sestak-Berggren. Table 3 shows the fitted parameters as well
as the correlation coefficient, r, obtained with the three Arrhenius methods in the range
from 2250 to 2808 s. Since the best correlation coefficient resulted from the linear
regression Arrhenius, this method was applied to all the apparently single steps of the
curve. Table 3 shows the p-values and ASE obtained with the logistic mixture for 1, 2,
3, 4 and 5 components. The later resulting to be the optimal fitting. Table 4 shows the
ASE obtained in the Arrhenius and the logistic mixture cases in the time ranges where
Arrhenius model was fitted and Table 5 shows the p-value and the ASE obtained in the
hypothesis test of the polyether-polyurethane case.
As it can be seen in Table 4 the performance of the logistic mixture fit is good
independently of the range considered, which is not the case of the Arrhenius method.
Arrhenius-linear regression Freeman and Carroll Sestak-Berggren
-16 +/- 821 kJ/mol
494 +/- 115 kJ/mol
Ea -497 +/- 38.2 kJ/mol
1.475 +/- 0.025
4.310 +/- 004
-43.516 +/- 11.159
n
-194.570 +/- 46.678
M
393.53 +/- 90.573
P
1.4E-99
A 2.8E-34
0.9781
0.0011
0.977
r
Table 3. Parameter values and correlation coefficient obtained from a polyetherpolyurethane TGA trace for three Arrhenius based models.
Time range/s Arrhenius
Logistic mixture
1405.5-1685.5 0.00058262 0.00039103
1681.5-1921.5 0.246024348 0.000442474
1405.5-1861.5 0.00806295 0.00043081
1801.5-2133.5 0.00026211 0.00020021
Table 4. ASE values obtained, in the specified ranges, with the Arrhenius and logistic
mixture models.
ASE
Number
of p-value
components
1
0
0.202061
2
0.01
0.011231
3
0.004
0.004747
4
0.004
0.004570
5
1
0.000256
Table 5. p-values and ASE obtained in the hypothesis test for the logistic mixture model
in the polyether-polyurethane case.
wi
i =1
3.158
i=2
0.686
i=3
1.296
i=4
1.714
i=5
0.490
ai
51.812 207.606 47.998 24.057 8.208
-0.026 -0.010 -0.003
bi -0.031 -0.124
Table 6: Parameters of the logistic mixture model for the polyether-polyurethane
sample.
CONCLUSIONS
1. The logistic mixture model allows for including at once the overall trace from a
TGA experiment, while the classical methods can only be applied to a single step
each time.
2. Overlapping degradation processes can be explained by the new method. Since the
existing models were proposed to explain single processes, they generally fit very
poorly to overlapped processes.
3. The thermal degradation of each component of the sample can be explained, through
the logistic mixture model, by a single function that may be easily understood from
the physical point of view.
4. This model shows the contribution of each single degradation process to the overall
curve. It is very useful in order to improve thermal stability of materials.
5. It allows for measuring the statistical goodness of fit by signification tests.
6. The classical kinetic models are easier to apply on the estimated functions obtained
by the new method than on the raw TGA data, since the raw data content noise that
affect derivative estimation.
7. The asymptotics are perfectly reproduced at the beginning and end of each
ACKNOWLEDGEMENTS
The authors gratefully acknowledge for the MEC Grant MTM2005-00429 (European
FEDER support included), for the first four authors and XUGA Grant
PGIDT03PXIC10505PN, for the first three authors.
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FIGURES:
18
16
14
Mass /mg
12
LC1
LC2
LC3
LC4
Mixture
10
8
6
4
2
0
0
5
10
15
20
25
Time /min
Figure 1: A logistic mixture model with four components.
7.32
7.315
Mass /mg
7.31
Data
Fitting
7.305
7.3
7.295
7.29
1500
1505
1510
1515
1520
Time /s
Figure 2: Autocorrelation data with fitted logistic mixture model.
1525
1530
120
100
Mass /%
80
Measured
LC1
LC2
LC3
fitting
60
40
20
0
0
500
1000
1500
2000
Time /s
Figure 3: Plot of the original TGA trace compared to the fitted logistic mixture model for the
Eucalyptus sample.
18
16
14
Measured
LC1
LC2
LC3
LC4
fitting
Mass /mg
12
10
8
6
4
2
0
0
500
1000
1500
2000
2500
3000
Time /s
Figure 4: Plot of the original TGA trace compared to the fitted logistic mixture model for an epoxyAraldite sample.
8
7
Mass /mg
6
Measured
LC1
LC2
LC3
LC4
fitting
5
4
3
2
1
0
0
500
1000
1500
2000
2500
3000
Time /s
Figure 5: Plot of the original TGA trace compared to the fitted logistic mixture model for a
polyether-polyurethane sample.
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