Handbook of Thermal Analysis and Calorimetry Vol. 5: Recent Advances, Techniques and Applications M.E. Brown and P.K. Gallagher, editors 2008 Elsevier B.V. 503

Chapter 13

ISOCONVERSIONAL KINETICS

Sergey Vyazovkin

Department of Chemistry, University of Alabama at Birmingham, 901S 14 th

Street, Birmingham, AL 35294, USA, vyazovkin@uab.edu

1. INTRODUCTION

Isoconversional kinetics rest upon evaluating a dependence of the effective activation energy on conversion or temperature and using this dependence for making kinetic predictions and for exploring the mechanisms of thermal processes. This chapter presents the evolution of isoconversional methods and discusses their advantages and limitations. The emphasis is put on the development of new techniques and applications that have taken place over the past decade. The applications covered include both physical and chemical processes. Physical processes include crystallization of polymer melts and glasses, the glass transition, and second-order solid-solid transitions. Chemical processes involve reversible decompositions, degradation of polymers, and crosslinking (curing). The chapter also discusses the methods of making isoconversional (model-free) predictions, as well as the techniques for evaluating the pre-exponential factor and the reaction model.

Thermal analysis methods are not species specific. By measuring the evolution of overall physical properties of a system, these methods provide information on macroscopic kinetics. The macroscopic kinetics are inherently complex because they include information about multiple steps that occur simultaneously. Unscrambling complex kinetics presents a serious challenge that can only be met by kinetic methods that provide means of detecting and treating multi-step processes. As indicated by the results of the ICTAC Kinetics Project [1,2,3], only the methods that use multiple heating programmes can meet this challenge. Isoconversional methods are definitely the most popular of the methods based on the use of multiple heating programmes. The present chapter provides a brief introduction to isoconversional methods and an overview of their application to the kinetic analysis of various physical and chemical processes.

504

2. ISOCONVERSIONAL METHODS

These methods have their origin in the single-step kinetic equation:

dt = A exp f (a) (1)

and make use of the isoconversional principle, which states that, at a constant extent of conversion, the reaction rate is a function only of the temperature so that:

d ln(da / dt) 1 E~ dT-' ' ~ = R (2)

In equation (1) and (2), A and E are the Arrhenius parameters (the pre- exponential factor and the activation energy, respectively), riot) is the reaction model (or conversion function), R is the gas constant, T is the temperature, t is the time, and a is the extent of conversion. Henceforth, the subscript a denotes values related to a constant extent of conversion. A combination of E, A, and J(a) is sometimes called the "kinetic triplet".

In order to obtain data on a variation of the rate at a constant extent of conversion (i.e., the left hand side of equation (2)), isoconversional methods employ multiple temperature programmes (e.g., different heating rates and/or temperatures). Although equation (2) is derived from the single-step kinetic equation (1), the fundamental assumption of isoconversional methods is that a single equation (1) is applicable only to a single extent of conversion and to the temperature region (AT) related to this conversion (Figure 1). That is, isoconversional methods describe the kinetics of the process by simultaneously using multiple single step kinetic equations (Figure 1). This attribute of isoconversional methods allows multi-step processes to be detected via a variation of E~ with a.

Equation (1) is easily rearranged into equation (3):

ln (da i = l n k f ( a ) ] E~ (3) t, dt )~,, RT~,,

505

~3 . . . . .

Ot 2

. . . . . : . . : . : _ . .~m-. . . . . . - - - -_ -_~

! I . I I I

T T T T a,1 a~,2 o%,1 0%,2

Figure 1. Each value of E~ is associated with a narrow temperature region AT that changes with or.

which is the basis of the differential method of Friedman [4]. The subscript i denotes different heating rates. The application of this method to integral data (e.g., TG data) requires numerical differentiation of experimental a vs T curves that tends to yield quite noisy rate data and, therefore, scattered Ea values.

Numerical differentiation can be avoided by using integral methods. Integration of equation (1) for isothermal conditions yields"

,~o>:~ ~o ~exp/~ 1, ,4> o f (a )

where g(a) is the integral reaction model. Rearrangement of equation (4) leads to equation (5):

,n,. :,n[ A- ; g(a) RT~ (5)

Equation (5) allows for evaluating the E~ dependence from a series of runs performed at different temperatures, ~.

506

Nonisothermal runs are most commonly performed at a constant heating rate/3. For such conditions, integration of equation (1) requires solving the temperature integral I(E, T):

g(a) - -fl o -~/(E, T) (6)

which does not have an analytical solution. Its solution is accomplished by using either approximations or numerical integration. For instance, the use of Doyle's approximation [5] yields equation (7):

1.05E~ ln(fl;) = Const - ~ (7)

RT~;

which is the foundation of the most popular isoconversional methods by Flynn and Wall [6] and by Ozawa [7]. A more precise approximation by Coats and Redfern [8] gives rise to equation (8):

l fl-~21= Const E~ In T~., ) RT~., (8)

For even better precision, one can use numerical corrections [9] or numerical integration. The latter, for example, is employed in the method [ 10] that makes use of minimizing the following function:

r ) = ~ ~ I(E~ ,T~,, )]3j (9)

At each particular value of a in equation (9), E~ is determined as a value that minimizes O(E~), and the temperature integral, I(E, T) is solved numerically.

The simpler integral methods (e.g., equations (5), (7) and (8)) assume that the value of E~ is constant in I(E, T) throughout the whole interval of integration, from 0 to a. This assumption causes a systematic error in the value of E~ when E~ varies with a. The error can be as large as 20 - 30% [11 ]. This error does not appear in the differential method of Friedman. Also, it is easily eliminated in the advanced integral methods of Vyazovkin [10,11,12] because they use numerical integration as a part of E~ evaluation. Eliminating the error in equation (9) is accomplished by integration over small temperature segments as follows:

507

I(E,T) = ~ exp dT (10) Ta-Aa

In this case, E~ is assumed constant for only a small interval of conversions, Aa. Integration by segments yields Ea values that are similar to those obtained by the Friedman method [ 11 ]. Integration over segments has also been implemented by Simon et al. [13] in an incremental integral isoconversional method. Budrugeac [14] has proposed a nonlinear differential isoconversional method that uses a numerical algorithm similar to that used in the advanced isoconversional method (equation (9)). He has also demonstrated [14] the asymptotic convergence of the advanced integral method with the differential method.

Although isothermal and constant heating rate temperature programmes are employed most commonly, they do not exhaust all practical needs. Other important programmes include constant rate cooling and distorted linear heating and/or cooling. The former finds wide use in the melt crystallization studies. The latter is frequently encountered as a result of self-heating or self-cooling when studying processes accompanied by significant thermal effects. Such temperature programmes cannot be handled by simpler integral methods (e.g., equations (7) and (8)) because they have been developed under the assumption that/3 is constant and positive. Because these assumptions are not made in the differential method of Friedman, it can be applied to handle data obtained under the aforementioned temperature programmes. However, the advanced integral method (equation (9)) can be adjusted to an arbitrary temperature programme T(t) by replacing integration over the temperature with integration over the time. The resulting advanced isoconversional method of Vyazovkin [11,12] can be used to handle data obtained under arbitrary temperature programmes, Ti(t). In this method, the E~ value is found by minimizing the function:

J[Ea,Ti(ta )1 ~=lj~,J[Ea,rj(ta) ] (11)

where

J[E~ T,(t~)] '~ [-E~ 1 , -- I exp RT,(t)dt (12) ta-Aa

The effect of self-heating/cooling is accounted for in equation (11) by using the so-called "sample temperature" whose variation with the time represents the actual temperature programme (i.e., T(t)) experienced by a sample.

508

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

RTp,~ (13)

looks similar to the isoconversional equation (8). Nevertheless, in equation (13) Tp.i is the peak temperature at different heating rates, and the extent of conversion at the peak is known [9] to vary with the heating rate. In addition, the Kissinger method always produces a single value of E for the whole process. This is an important limitation of the method, because the resulting value is sound only if E~ does not vary with a throughout the process. Yet, variations are very common, and the Kissinger method cannot detect them. Therefore, the E values determined by this method should be considered with care, unless an isoconversional method has been used to prove that E~ is independent of a.

3. CONCEPT OF VARIABLE ACTIVATION ENERGY

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:

logt=Q-F(w) (14) T

where w is the mass decrease in % of the initial value, t is the time to reach the extent of decomposition w at different temperatures, and Q is a "material constant" [ 16], which was determined as the slope of a plot of logt vs T ~. The meaning of Q and F(w) in equation (14) can be easily established by comparing it with equation (5). Clearly, Q is E/2.303R, and F(w) is -log[g(a)/A]. By using the early data of Kujirai and Akahira one can determine that the activation energy for degradation of the insulating materials studied demonstrates a noticeable variation with w (Figure 2). A similar effect was reported for the thermal degradation of a phenolic plastic by Friedman [4] in the first application of his method and for the thermal degradation of Nylon 6 by Ozawa [7] in the first application of his method. Apparently the phenomenon of a variable activation energy has been with isoconversional methods since their first days of existence. Using simulated data for multi-step reactions, Flynn and Wall [6] demonstrated that experimentally observed variations of the activation energy

509

160

140

120

-- ~ - - silk , - - ~ - - m a n i l a paper ', -- ,~-- filter paper ', - -~ - - co t ton

;o ' io ' ;o w / %

Figure 2. Variations in E obtained from the early data [16]

E=E D

T-1

Figure 3. Arrhenius plot for reaction complicated by diffusion

are nothing else but a sign of the process complexity. In spite of this fact, E~ dependencies have not been exploited, but rather ignored as a major nuisance. The potential of isoconversional methods had not been fully appreciated until we [17] brought analysis of the E~-dependencies to the forefront and demonstrated their utility for predicting kinetics and exploring the mechanisms of processes. We also introduced [18] the concept of a variable activation energy. Although this concept does not sit well with the traditionalist's point of view [19], it does provide a reasonable compromise between the actual complexity of condensed phase reactions and oversimplified methods of describing their kinetics [20]. As we argued [18, 21], the expectation that an experimentally determined activation energy should be a single constant value directly related to an energy barrier, originates from kinetic theories of simple gas phase reactions. For such systems, single steps can be easily isolated and their kinetics can be readily measured. As a result, the experimental activation energy can be directly linked to the energy barrier of a reaction. However, isolation of individual steps is not generally possible when applying TG and DSC to study processes that occur in the solid or liquid media. For these types of processes, two factors need to be considered: the simultaneous occurrence of multiple reaction steps and the presence of diffusion.

Diffusion adds an extra step to a chemical reaction so that the overall process rate becomes dependent on the chemical reaction rate as well as on the transport rate of the reactants and products in the reaction medium. In this case, the temperature dependence of the overall rate is described by the effective rate constant, key as follows:

510

1 1 1 = -i

kef kR kD (15)

where kR and kD are the reaction and diffusion rate constants respectively. From equation (15) one can easily derive the effective activation energy for a process that involves both chemical reaction and diffusion:

dlnke/) EDkR + ERk D Eef =-R =

dT -~ k R + k D (16)

Because both kR and kD vary with temperature, the effective activation energy in equation (16) is also temperature dependent. An Arrhenius plot of lnkef against T -1 for such process is nonlinear so that the activation energy derived from its slope changes with increasing temperature from the activation energy of reaction (ER) to the activation energy of diffusion (ED). The activation energy of diffusion of small molecules in the liquid or solid medium is typically a small value, markedly smaller than that of a chemical reaction. In this situation, the Arrhenius diagram shows a characteristic plot that is bent upwards (Figure 3). Only if one of the two steps is much faster than another, will the overall rate be determined by the slowest step and the experimental value of Eef will become the activation energy of this step, i.e., Ee/=ER if kR << kD (so-called kinetic control) o r Eef=ED if kD < < kR (so-called diffusion control).

The dependence of the effective activation energy on both temperature and extent of conversion can be exemplified by a decomposition process that occurs via two parallel pathways as follows"

B~ k, A k~ ;C (17)

If the pathways follow different reaction models, the overall decomposition rate is given by equation (18):

da dt

= k, f, (a) + L (a) (18)

By taking the logarithmic derivative of the reaction rate at a constam a, we can determine the effective activation energy at each conversion as follows:

511

E~ =-R[ dln(da/dt)] --Elkl f ~(a)+E2k2 f2(a) dT -~ ~ k, f I(C~) +k 2 f2 (a)

(19)

A three-dimensional plot of E~ can be visualized by using, for instance, kinetic triplets experimentally determined for parallel channels of decomposition of nickel formate that are as follows: J](a)=(1-a) 2/3, El=200 kJ tool l , A 1 = 1016 min i and f z (a)=a(1-a) and E2=100 kJ mol 1, A2=107 min l [18]. Inserting these triplets in equation (19) yields an effective activation energy, which is a function of the temperature as well as of the extent of conversion (Figure 4).

The two aforementioned examples show that the experimental activation energy is generally a function of the energy barriers of the individual steps of a process. Note that the examples considered are very simplistic, and real processes tend to involve multiple chemical and diffusion steps. In this situation it may be practically impossible to link the experimental values of the activation energy of a process to the energy barriers of the individual steps. To stress this difference, the experimental values of the activation energy are frequently referred to as an "effective", or "apparent", or "global" activation energy. Because of its composite nature the effective activation energy tends to vary throughout the process. That is why the concept of variable effective activation energy has been proposed [18] as a necessary compromise between the actual complexity of condensed phase reactions and oversimplified methods of describing their kinetics

200

......... ............. i1

......... i ............ ! ........... 120I ~ . i i l l ~

3 0 ~ ........ ~ 0 . 4 2 5 0 " - : ~ . . ~ 0 . 6 7'/0C20 15 1.0 O.8 o~

Figure 4. Simulated surface plot of E~ vs a and T (equation (19))

. / /

AG s /' /

/ /./'/"

x

AG v "'.,

Figure 5. Free energy of nucleation as a function of nucleus radius

512

4. KINETICS OF PHYSICAL PROCESSES

4.1. Crystallization

Most melted substances crystallize on cooling. From the thermodynamic standpoint, crystallization should occur spontaneously, just below the equilibrium melting temperature, Tm, because, under these conditions, the Gibbs (free) energy of the solid is lower than that of the liquid phase. The resulting difference in the free energies is a negative value termed the volume free energy, AGv. In reality, no visible crystallization occurs until the melt reaches significant supercooling. The delay in crystallization is caused by a free energy barrier associated with the creation of the solid phase nuclei that is characterized by the surface free energy, AGs. This value represents the difference in the Gibbs energy of the surface and the bulk of the nucleus. Because the free energy of the surface is always larger by the value of the surface energy, o, the value of AGs is positive. The total free energy of the nucleation process is given as"

AG = AG s + AG v (20)

The nuclei are usually assumed to have a spherical shape of the radius, r so that AGs is the surface area of the sphere times the surface energy, and AGv is the volume of the sphere times the volume energy per unit volume, AGv:

4 AG = 4at 2o" + - a'r 3AG,, (21)

3

Equation (21) establishes a dependence of AGs and AGv on the radius of a nucleus (Figure 5). The sum of these two terms (i.e., AG) passes through a maximum that defines the critical size of a stable nucleus. The formation of a nucleus of a larger size would result in its spontaneous growth accompanied by a decrease in AG. The maximum value of AG is the energy barrier for the nucleation process, AG*. The nucleation rate is commonly expressed in the Arrhenius form:

n - n o exp,, kT (22)

The height of the energy barrier can be found from the condition of a maximum for AG (equation 21), which yields:

513

A G * - 16~O "3

3(AGv) 2 (23)

The value AGv is defined as"

- AH f A T AG~ = (24)

7"o

where AT = Tm- T is the supercooling, a n d / ~ f is the heat of fusion. The value of AGv decreases with supercooling (i.e., with decreasing temperature) causing a decrease in the critical nucleus size and the energy barrier (equation (23)).

It follows from equations (23) and (24) that:

AG* "" (T m -T) 2 (25)

Because AG* decreases with decreasing temperature (equation (25)), the nucleation rate increases. This situation gives rise to a temperature dependence that is markedly different from a typical Arrhenius dependence. Figure 6 shows an Arrhenius plot for the nucleation rate. The plot has a positive slope that corresponds to the negative temperature coefficient and/or negative effective energy of activation. Also, the slope varies strongly with the temperature, reaching infinity at T=Tm (equation (25)).

Not far below the melting point, the nucleation rate quickly increases with decreasing temperature. However, the nucleation rate does not increase monotonously, but passes through a distinct maximum at a certain temperature, Tmax (Figure 7). Decreasing the temperature below Tma~ results in decreasing the nucleation rate. This occurs because of a significant increase in the melt viscosity that creates an energy barrier, ED associated with diffusion of molecules across the phase boundary. Introduction of the respective energy term into equation (22) gives [22]:

- AG* exp (26) n = n o exp kT kT

Unlike the AG* term, the ED term represents a typical Arrhenius temperature dependence (Figure 7). Therefore, the product of the two terms (equation (26)) gives rise to a temperature dependence that has a maximum in the nucleation

514

~x0 O . = , ,

1

[ E < 0

T=T E=0 T=Tg , .2 m _ f _ . ~

\ T=T

m a x

Glass: E > 0

I . . . .

T -1

i i""

1 2 �9 \.%

" ' I ' : i I "~ ,

i I "l ,

II :: \,i ~ / '~/ Is ::

: i \ ~ / '\ ! i / \ / , ' , , ! " ' \ I , ~ :

Figure 6. Arrhenius plot for nucleation in temperature range Tg- Tm (1" melt nucleation; 2" glass nucleation)

Figure 7. Temperature dependence of nucleation rate (1" exp(-AG*/kT); 2: exp(-EdkT); 3" product of 1 and 2)

rate. Below the temperature related to the maximum, the process becomes comrolled by diffusion that results in a dramatic decrease of the nucleation rate.

If the maximum nucleation rate is not very large in a particular substance, its cooling may result in semi-crystalline material that contains a substantial fraction of the amorphous (glassy) phase. This fraction can be increased by increasing the rate of cooling. Ultimately, fast cooling results in the formation of an entirely amorphous substance, i.e., a glass. Glasses can be crystallized by heating above their glass transition temperature. The process is frequently called "cold crystallization". The nucleation rate in the glass is largely limited by diffusion and increases with increasing temperature. The corresponding Arrhenius plot has the normal negative slope that represents the positive temperature coefficient and/or positive effective energy of activation (Figure 6). The slope also decreases with increasing temperature.

It follows from the above that crystallization kinetics generally demonstrate non-Arrhenius behaviour (Figure 6). Although for narrow temperature imervals the temperature dependence of the crystallization rate can be approximated by using the Arrhenius equation, for wider intervals one should make an allowance for the temperature variation in E. As seen from Figure 6, the E value should continuously decrease with increasing temperature. It should be positive for the glass crystallization and decrease as the temperature is increased from the glass-, transition temperature, Tg to T~o~. For the melt crystallization it should be negative and increase when the temperature is increased from Tm to Tg.

515

Note that an allowance for variation in E is not made in the Kissinger method that is very frequently used to determine the so-called "activation energy of crystallization" for melts, as well as for glasses. This method is thus inadequate for crystallization of glasses because it masks the temperature dependence of E. Furthermore, it is inapplicable to the processes that occur on cooling, such as the melt crystallization [23]. The use of an advanced isoconversional method (equation (11)) and the method of Friedman (equation (3)) has been recommended [23] as a viable alternative to the Kissinger method. The application of isoconversional methods to the glass and melt crystallization data results in the general types of the E~ dependencies shown in Figure 8. These dependencies reflect the respective temperature dependencies of the slopes of the Arrhenius plots (Figure 6).

Glass: E > 0

T increases .

Melt: E < 0

T decreases

I . I . I i I , I J I

0.0 0.2 0.4 0.6 0.8 1.0

I

Glass crystallization i E > 0 ,1 G

. . . . . . . . . . . . . . . . I ! . . . . . . . . . . . . . . . . .

E = 0 / E- ~ i ~ \

, ' ~ i ' J\ / i ' / ~

,, ~Vlelt crystaltizatio_~

- - - ' ~ < 0

T T ~ T g m

Figure 8. Dependencies of E~ on a Figure 9. Variation in E according to for melt and glass crystallization the Hoffman-Lauritzen theory

Therefore, one must be careful in assigning a physical meaning to the experimental value of E. It is common practice to call an experimentally determined E value an "activation energy", which is not infrequently interpreted as an "energy barrier". Such interpretations are clearly invalid in the case of crystallization. As seen from Figure 6, by performing crystallization in different temperature regions one can obtain practically any value of E from large positive to large negative numbers and this obviously suggests that E is not an energy barrier of crystallization.

516

4.2. Melt and glass crystallization of polymers Negative E~ values (Figure 8) have been obtained experimentally by

Vyazovkin et al. [24] for the melt crystallization of poly(ethylene terephthalate), poly(ethylene oxide) [25], and poly(ethylene 2,6-naphthalate) [26]. Similar dependencies have also been reported by other workers [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42] who applied isoconversional methods to crystallization of the melts. More importantly, it has been demonstrated [43] that the resulting E~-dependencies can be employed for evaluating the parameters of the Hoffman- Lauritzen theory [44]. According to this theory the linear growth rate of a polymer crystal, G depends on temperature, T as follows:

- U exp (27) G = G O exp R(T - -Too ) TATf

where Go is the preexponential factor, U* is the activation energy of the segmental jump, AT=Tm-T is the undercooling, f=2T/(Tm+T) is the correction factor, Too is a hypothetical temperature at which viscous flow ceases (usually taken as Tg- 30K). The parameter Kg is defined as:

2nbcrcreT m Kg = (28)

AhfkB

where b is the surface nucleus thickness, a is the lateral surface free energy, ae is the fold surface free energy, Tm is the equilibrium melting temperature, Ahf is the heat of fusion per unit volume of crystal, k~ is the Boltzmann constant, and n takes the value 2 for crystallization regime I and III, and 1 for regime II. The parameter U" is frequently taken as the "universal" value 1.5 kcal mol ~ [44].

Vyazovkin and Sbirrazzuoli [43] have used equation (27) to determine the theoretical temperature dependence of the effective activation energy (Figure 9) of the growth rate as:

E = -R d In G _ U* T2 T~ - T 2 - TroT (29) dT_-------- T - (T_Too)2 +KgR (Tm_T)2T

Numerous experimental data by Toda et al. [45,46] indicate that the logarithmic derivative of the microscopically measured growth rate is equivalent to the logarithmic derivative of the overall crystallization rate obtained from DSC. For this reason, the temperature dependence of the effective activation energy

517

evaluated from DSC data can be fitted to equation (29) to determine the values of U* and Kg. A dependence of E on T can be easily obtained from an isoconversional dependence of E~ on a. Each value of E~ is linked to a given a, which, in its turn, is related to a narrow temperature region, AT (see Figure 1). Therefore, one can correlate the Ea values with the temperature by replacing a with an average temperatures corresponding to this a at different heating rates.

Equation (29) suggests that the effective activation energy decreases with increasing the temperature throughout both glass and melt crystallization regions [26]. These two regions are usually separated by the maximum crystallization rate. Therefore, the E value typically changes from positive to negative on passing from the glass to melt crystallization region (Figure 9). That is, a change in the temperature region of crystallization would result in changing a value of E. This, as stated earlier, indicates that E is not an "energy barrier" of crystallization, although it is sometimes interpreted as such. In order to stress its difference from the true activation energy, which represents the energy barrier, the E value should be referred to as the effective activation energy.

Figure 10 shows the E~ dependencies for the glass and melt crystallization of poly(ethylene terephthalate). As stated earlier, the glass crystallization data produce positive E~ values that decrease with increasing a (i.e., with increasing T). On the other hand, the melt crystallization data yield negative E~ values that increase with increasing ct (i.e., with decreasing T).

100

0 "7

-~ -100

-200

-300

",E = 0

0 Melt crystallization, E < 0

�9 I �9 i , i . i , i , I

0.0 0.2 0.4 0.6 0.8 1.0

200

100

0 " 7

O

-100

-200

-300

<> Glass crystallization, E > 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E - - 0 ",

O Melt crystallization, E < 0 O

I , I . . , I , I , I ,

400 420 440 460 480 500

T / K

Figure 10. E~ dependencies for the Figure 11. E~ vs T data converted glass and melt crystallization of from E~ dependencies (Figure 10) poly(ethylene terephthalate) and fitted to equation (29) (dashed

line) By converting E~ vs. a for the melt data into an E~ vs T dependence and fitting the latter to equation (29), one obtains the values Kg = 3.2 x 105 K 2 and U* = 4.3

518

kJ mol ~ [43]. Vyazovkin et al. have also used this method to determine the Kg and U* values for the melt crystallization of poly(ethylene oxide) [25] and poly(ethylene 2,6-naphthalate) [26] and found the values to agree with the literature values derived from microscopic measurements. Similarly good agreement has also been reported by other workers [32,35,40,41 ] who applied the aforementioned approach.

Vyazovkin and Dranca [26] have recently demonstrated that equation (29) is suitable for simultaneously fitting combined melt and glass crystallization data. Figure 11 provides an example of such a fit for poly(ethylene terephthalate) that produces the values Kg = 3.6 x 105 K 2 and U*=7.5 kJ mol ~. Compared to the values derived from the melt data alone, the Kg value has not changed much, whereas the U* value has risen closer to the universal value 6.3 kJ mol l . Similar results has been reported [26] for poly(ethylene 2,6-naphthalate) crystallization. The use of the combined data sets [26] improves the precision of the fit as well as the accuracy of the U* value.

4.3. Second-order transitions Second-order transitions do not cause abrupt changes in the volume or enthalpy

but in the derivatives of the respective properties. The resulting effects are quite subtle. Figure 12 displays the ferromagnetic to paramagnetic transition in Ni metal. As seen it has the characteristic appearance of the inverted Greek letter 2.. That is why the transitions of this type are called A-transitions. The critical temperature, T~ determined from DSC is 632 K (359 ~ This value is typically found to be somewhat larger than the Curie temperature (354 ~ determined from magnetic measurements [47]. Increasing the heating rate causes a slight shift of the transition to higher temperatures. This allows one to determine the effective activation energy of the transition by using an isoconversional method. It is seen from Figure 13 that the E~ value changes dramatically throughout the process.

The behaviour of the effective activation energy can be predicted from the Landau theory of phase transitions. According to it, for a second-order transition, the temperature dependence of the relaxation time is given by [48]:

Const

where T~ is the critical temperature at which the two phases would be at equilibrium. From equation (30) one can derive the temperature dependence of the effective activation energy as follows:

0.19

dlnr R T 2 E = R ~ = (31)

d T -1 {T-Tc{

As this equation suggests, the effective activation energy should quickly increase to infinity on approaching the critical temperature (see Figure 12). Obviously, the observed E~ dependence [49] (Figure 13) tracks the theoretical temperature dependence (Figure 12) and simply reflects the phenomenon of passing through the critical point of the transition. Similar behaviour should be expected for other 2-transitions.

0.18

"7

0 E

0.17

~: 0.16

0.15

0.14

0.13

, } 2000

i 1"~176 o 11ooo 3

_,_:_,_~..c ~..~ 0 / 580 600 620 640 660

T / K T /

I , I , 1 ~ I ~ I , I i

560 580 600 620 640 660 680

T / K

1000

1500

o o 0

o 0 o

o 0

0 0

500

519

o

o o

0.0' ' 0'.4'0.6 '1'.0

Figure 12. DSC trace of Ni around the Curie point. Inset: E vs T depen- dence by equation (31)

Figure 13. E~ dependence for the 2- transition in Ni obtained from DSC data (Figure 12)

4.4. Glass transition Many materials can be produced in the glassy state. This is a nonequilibrium

state that is thermodynamically driven to relax to the equilibrium liquid state. The relaxation process can be readily followed by various thermal techniques. The rate of relaxation is characterized by the relaxation time, r, whose temperature dependence is described either by the Arrhenius equation:

r = A exp(~T ) (32)

or, more commonly, by the Williams-Landel-Ferry (WLF) equation"

520

r - C l ( T - T o ) log-- - (33)

r o (C 2 + T - T o)

where To is a reference temperature, r0 is the relaxation time at To, and C1 and C2 are constants. The WLF equation works well for relaxation processes above the glass transition temperature, Tg. The Arrhenius equation is typically applied well below Tg, but may also serve as an approximation above Tg, especially when relaxation is measured over a narrow region of temperatures.

DSC is applied routinely for measuring the glass transition. The transition appears as a step in curve of heat capacity against T. An increase in the heating rate causes this step to shift to higher temperatures. This shift can be used for determining the activation energy of the process according to equation (34) [50]:

E =-R dlnl fl________~f dTg~ (34)

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 from 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, a, can be readily determined from DSC data as the normalized heat capacity [55]:

(Cp -Cp,g)]r C ; = (Cp,l - Cp,g )IT = ~ (3 5)

where Cp is the observed heat capacity, and Cp, g and Cp, t are, respectively, the glassy and liquid heat capacities. Because the values of Cp, g and Cp, t depend on temperature, they are extrapolated into the glass transition region. Figure 14 shows the normalized heat capacities for the glass transition in maltitol measured at different heating rates. The application of an isoconversional method to these data reveals a decreasing dependence of E~ on a (Figure 15).

521

As noted earlier (section 4.2), each value of E~ is related to a given (~ and, therefore, to a respective temperature region, AT (see Figure 1). This allows one to link the E~ values to the average temperature of the AT region. Figure 15 shows the E~ vs T dependence obtained by substituting the average T~ for a. The dependence shows a decrease that is typical of the glass transition [54]. The

1.5

1.0

0.5

0.0

. . . . . . . 10

......... 15

............ 20

- - 2 5 //

i

/ i

I I I , I , I . I

30 40 50 60 70

T / ~

40 45 50 ' ! , i ,

OOoo o n

o 3

o l

150 O . , , , , . . ,. , ,

0.0 0.2 0.4 0 6 0 8 1.0

250

0

E _~ 200

T / ~ 0r

Figure 14. Normalized heat Figure 15. Variation in E with ot and capacities at different heating rates T for the glass transition in maltitol (numbers by the line types)

trend is predicted by the WLF equation that gives rise to a decreasing temperature dependence of the effective activation energy"

d lnr C~C2T 2 E - R ~ = 2 . 3 0 3 R

dT -~ (C 2 + T - T 0)2 (36)

The decrease is explained by the co-operative character of the molecular motion. Co-operativity is very strong in the early stages of the glass transition, when the available free volume is too small to allow for independent motion of individual molecules. As a result, the co-operative motion has a large energy barrier. As the temperature rises, the free volume increases, relieving energetic constraints, so that the E value decreases. A correlation between the co-operativity of the molecular motion and the magnitude of E has been revealed by Vyazovkin and Dranca [56], who compared the glass-transition kinetics of poly(styrene) and

522

poly(styrene)-clay nanocomposite. Because the composite has a brush structure [57], its polymer chains move in a correlated manner that reflects in a markedly larger size of the cooperatively rearranging region and, therefore, in a larger value of the activation energy. A similar correlation has been reported [58] in a comparative study of the glass transition in glucose and maltitol, which is a bulkier derivative of glucose.

The variability in E correlates [54,59] with the dynamic fragility of the glass forming systems. The biggest variation has been observed for polymers, having the largest fragility (e.g., poly(vinyl chloride) and poly(ethylene terephthalate)), whereas the smallest variation is found in the systems, having the lowest fragility (e.g., poly(n-butyl methacrylate) and boron oxide) [59]. This correlation suggests that kinetic models of the glass transition that assume the constancy of the activation energy are conceptually unsuitable for polymer glasses because they tend to have the largest fragility.

5. KINETICS OF CHEMICAL PROCESSES.

5.1. Reversible decompositions A number of solid-state decompositions occur reversibly in accord with the

following general equation:

A(~) r B(,) + C(g~ (37)

Because the activities of pure solid phases are taken to be 1, the equilibrium constant of such a process is:

x = e0 (38)

where Pc is the partial pressure of the gaseous product C and pO is the standard pressure. The temperature dependence of the equilibrium constant:

_AH ~ In K = ~ + Const (39)

RT

allows one to correlate the pressure and temperature at which all three phases can co-exist in equilibrium. It follows from equation (38) and (39) that the logarithm of the equilibrium pressure has a linear dependence on the reciprocal temperature. A good example is the reversible decomposition of calcium carbonate that occurs as follows:

523

CaC03(~) r CaO(~) + C02(g ) (40)

Figure 16 presents a temperature dependence of the equilibrium pressure of carbon dioxide for the decomposition of calcium carbonate [60]. The straight line:

205643 Pc'~: (atm)= 4.67 • 107 exp r (41)

correlates the pressure and temperature at which CaCO3(s), CaO(s) and C O 2 ( g ) a r e

at equilibrium with each other. Below this line, calcium carbonate would decompose to calcium oxide and carbon dioxide. While largely simplified, this thermodynamic model introduces an important notion that a reversible decomposition should start when the equilibrium pressure of the

10 2

10 ~

r

~ ' - 1 0 .2 r 8

10 "4

10 -6

0.4

aGO 3 + CO 2

" ~ 3xlO 4 arm

CaO + CO 2 . . . . . . ,~ ............

O i

018" ' 1.12 ' 1.6

200

"7

150

100

r

o Masuda et al �9 Urbanovici, Segal

q~ * Vyazovkin

_

v V i , i . I | I , . . . I i I

0.0 0.2 0.4 0.6 0.8 1.0

1000 T / K a

Figure 16. Equilibrium pressure of Figure 17. E~ dependencies reported CO2 for decomposition of CaCO3 by several workers for dehydration of

CAC204 "H20

gaseous product rises above its partial pressure at a given temperature. For instance, the partial pressure of carbon dioxide in air under normal conditions is approximately 3x10 4 atm, which is significantly larger than the equilibrium pressure (Figure 16). In order to raise the equilibrium pressure above 3x10 4 atm one needs to raise the temperature above 530~ Then calcium carbonate would start converting to carbon dioxide and calcium oxide. Another obvious way to

524

initiate decomposition would be to decrease the partial pressure of CO2 below its equilibrium value at a given temperature. This is accomplished experimentally by performing reversible decompositions in vacuum.

When a reversible decomposition takes place under a continuously rising temperature, its initial stage occurs not far from equilibrium. For such conditions, the basic rate equation (1) needs to be expanded to include a pressure dependent term [61,62]:

dOt=dt Aexp f (a ) 1 - ~ - (42)

where Pc and Py are the partial and equilibrium pressure of the gaseous product

C in equation (37). According to equation (42), under rising temperature conditions the rate of the initial stage of decomposition will be predominantly determined by a change in the equilibrium pressure with temperature. Once the system is removed far from equilibrium (p(eq >>Pc), the pressure dependent term becomes irrelevant so that the temperature dependence of the rate will follow the basic rate equation (1). Therefore, in a wide temperature region, the temperature dependence of rates of reversible reactions tends to demonstrate significant deviations from Arrhenius behaviour. As a result, the effective activation energy tends to vary with temperature as well as with the extent of reaction.

Pavlyuchenko and Prodan [63,64] have shown that the effective activation energy of a reversible decomposition can be expressed as:

( ecq ) m Eef = E 2 - 2 + mQ (43)

( pcq )m _ (Pc)m

where Ee is the activation energy of the reverse reaction, 3, is the heat of adsorption, m is a constant (0 < m ___ 1), Q is the thermal effect of reaction. Under the rising temperature conditions the value of ~ continuously increases

and causes the last term of the sum in equation (43) to decrease. For the initial stages of the process (i.e., Py ~ Pc), the last term in equation (43) is large so large values of the effective activation energy are obtained. In the later stages (i.e., e~q >> Pc), the last term in equation (43) converges to 1 so that the values of the effective activation energy are close to the activation energy of the forward reaction. The resulting decrease in the effective activation energy is frequently detected when using an isoconversional method for analysis of reversible processes. Figure 17 demonstrates a dependence of the E~ value on the extent of reaction for the thermal dehydration of calcium oxalate

525

monohydrate. The excellent agreement of the E~ dependencies reported in three different papers [11,65,66] should be noted. A similar type of dependence has also been reported for the thermal dehydration of lithium sulfate monohydrate [67,68].

5.2. Thermal and thermo-oxidative degradation of polymers Degradation of polymers involves scission of the bonds between the individual

atoms of a chain. For typical vinyl polymers, that means scission of C-C bonds whose energy is around 350 kJ mol 1. Although this energy is quite large, thermal degradation of vinyl polymers starts rather easily above 200 ~ This is typically explained by the presence of the weak links in the polymer chain. Such links may include head-to-head links, hydroperoxy and peroxy structures that serve as spots where thermal degradation is initiated. The formation of the initial macro radicals is followed by degradation via various radical pathways whose activation energies are markedly smaller than that for the C-C bond scission. Because of the change from the process of radical initiation to propagation, the effective activation energy of thermal degradation tends to change throughout the process. It tends to be lower at earlier stages, which are determined by initiation at the weak links. Once the weak links are exhausted, the effective activation energy rises at the later stages representative of propagation. This variation has been clearly demonstrated by Peterson et al. [69], who applied an advanced isoconversional method (equation (11)) to the TG data on the thermal degradation of poly(styrene), poly(ethylene), and poly(propylene). The E~ dependencies obtained have increasing character [69] as shown in Figure 18 [70]. However, a more complex E~ dependence has been observed by Peterson et al. [71] for the thermal degradation of radically polymerized poly(methyl methacrylate). The dependence demonstrates an increase (60 to 190 kJ mol -~) followed by a quick falling off (190 to 60 kJ mol l ) and another increase (60 to 230 kJ mol~). The complex behaviour is associated with existence various weak links (e.g., the head-to-head linkages, the vinylidene end groups) that give rise to different mechanisms of initiation.

In an oxygen containing atmosphere, polymers undergo thermo-oxidative degradation. As a highly reactive radical, oxygen easily initiates low activation energy pathways for thermal degradation so that thermo-oxidative degradation starts at temperatures about 100 ~ lower than the respective thermal degradation in an inert atmosphere. Thermo-oxidative degradation of vinyl polymers occurs via the formation of the hydroperoxide radical in the propagation step of the process. For this reason, thermo-oxidative degradation tends to yield activation energies not far from 100 kJ mol 1, which is characteristic of the bimolecular decomposition of organic hydroperoxides. The corresponding activation energies are obtained (Figure 18) when applying an

526

isoconversional method to TG data on the thermo-oxidative degradation of vinyl polymers [69].

3 0 0

2 5 0

2 0 0

.7

150

m~' 100

~ 'o ' . ' ' ' 0.0 0 2 0.4 6 0.8 1.0

01 DGEBP+2,4-DNA I "NHi 140 @ DGEBP+3-NPDA f ~ - ~ NO2 (]I O(]I

DGEBP+4-NPDA ~ , J

120 No~

Cli cito 0 o 0t ~ ~ 01 ~ ~@ ~ ~ ~ @ ~

, <1 ,~ <1, �9 <1' ,~, O, <1, <1, <1, '~ - 100 �9 ~,

�9 , NHz ~ NHz

4 0 ~ . . . . , 0.0 0'4 016 0'8 ,'.0

Figure 18. E~ dependencies for Figure 19. Effect of a nitro-group in thermal (squares) and thermo- amines on the E,~ values for the oxidative (pentagons) degradation of epoxy-amine curing reaction. poly(styrene)

5.3. Crosslinking Highly crosslinked polymers are commonly produced from epoxy materials

such as diglycidyl ether of bisphenol A (DGEBA). DGEBA is easily copolymerized with various substances such as amines or anhydrides. Copolymerization of DGEBA with a monoamine yields linear polymer chains. When DGEBA is copolymerized with a diamine, polymer chains crosslink.

The kinetics of crosslinking (curing) are usually complex because of the multitude of steps involved in the process. For example, copolymerization of DGEBA with a diamine involves two steps associated, respectively, with the two hydrogens of a primary amine. Reaction of the first hydrogen with an epoxy group produces a secondary amine as a part of the growing polymer chain. The second hydrogen becomes less accessible and, thus, less reactive with respect to another epoxy group. As a result, the initial stages of copolymerization involve, for the most part, chain extension due to the primary amine reaction. On the other hand, crosslinking associated with the secondary amine reaction tends to take place at later stages. In addition to chemical steps, epoxy-amine curing causes dramatic physical changes of the reaction medium. The changes include an increase in the molecular weight, the viscosity and the glass-transition temperature of the forming polymer. Molecular mobility, however, experiences a dramatic decrease, especially due to crosslinking. The crosslinked chains

527

cannot move one past another so that the reaction medium turns into a glassy or rubbery solid. The glassy solid is formed when the glass-transition temperature of the medium rises above the current actual temperature of the process. A dramatic decrease in the molecular mobility changes the kinetics of curing to a diffusion mode that is controlled by mass transport of the reactants.

The complex kinetic behaviour is revealed by isoconversional methods as a variation of the effective activation energy with the cure progress. A comparison with other kinetic methods reveals [72] the advantage of isoconversional methods in detecting the complexity of the curing kinetics. Yet another benefit is that isoconversional methods yield consistent values of the activation energy for isothermal and nonisothermal conditions [72,73]. Care must, however, be exercised [74] to avoid systematic errors in isoconversional calculations when isothermal curing is carried out below the limiting glass transition temperature, i.e., when the ultimate degree of curing changes with the curing temperature.

The application of isoconversional methods to DSC data on epoxy curing frequently demonstrates a characteristic decrease in the E~ values in the later stages of the process. For instance, Vyazovkin and Sbirrazzuoli [75] have reported a decrease from 60 to 40 kJ mol 1 at o~ > 0.6. The lower values of E~ are characteristic of diffusion of small molecules in a liquid/solid medium, and the effect can be explained by diffusion control associated with vitrification. A correlation of the decrease in E~ with a decrease in molecular mobility has been established experimentally [76] by using temperature modulated DSC. A similar correlation has been reported by other workers [77,78] who also combined an isoconversional method with temperature modulated DSC. The correlation of a decrease in the E~ values with vitrification can also be demonstrated chemically. For instance, Sbirrazzuoli et al. [73] have applied an isoconversional method to curing DGEBA with 1,3-phenylene diamine (m-PDA). They have analyzed two systems: a stoichiometric one that had two moles of DGEBA per mole of the amine and a nonstoichiometric one that had a fivefold excess of the amine. An excess of amine favors chain extension via the primary amine reaction. As a result, the E~ value (--55 kJ mol ~) does not change with the extent of curing, which suggests that a single step determines the rate of the overall process. In the stoichiometric system, the E~ value of the initial stages is similar to that found in the nonstoichiometric system. However, a contribution of crosslinking becomes increasingly important at later stages so that the E~ value drops quickly to low values typical of diffusion. The decrease has been reported [73] to correlate with an increase in the shear modulus and a decrease in the complex heat capacity.

The earliest stages of curing (tx---~0) frequently demonstrate large E~ values that quickly drop to the regular values of the activation energy around 5 0 - 60 mol 1. This phenomenon has been discussed by Vyazovkin and Sbirrazzuoli [79], who

528

attributed it to the temperature dependence of viscosity in the systems of high viscosity. The hypothesis has been experimentally supported by the fact that at (x---~0 the E~ values determined from DSC data are similar to the activation energy of viscous flow estimated from rheological data. Alternatively, the rapid decrease in E~ in the early stages is frequently interpreted as being due to a change from a non-catalyzed to an autocatalytic mode. Although it may be a plausible explanation, one needs to make sure that the values of the activation energy of the non-catalyzed reaction (i.e., E~ at a--~0) are not unreasonably large. A reasonable reference value for the uncatalyzed reaction can be found in paper by Swier et al. [80], who have demonstrated that the cure initiated by the reaction of the primary amine with epoxy-aniline complex has an activation energy of 80 kJ mol 1.

The E~ dependencies can also provide valuable insights into the chemical mechanisms. For example, by applying an isoconversional method to an epoxy- anhydride curing reaction catalyzed by tertiary amine, Vyazovkin and Sbirrazzuoli [81] have obtained an E~ dependence that increases from 20 to 70 kJ mol ~ with increasing a. For the same reaction without anhydride they have found a practically constant value E~-~ 20 kJ mol ~ that provides an estimate of the activation energy of initiation. Consequently, 70 kJ mo1-1 gives an estimate of the activation energy for propagation. Also, the use of an isoconversional method for kinetic analysis of curing of diglycidyl ether of 4,4'-bisphenol (DGEBP) with nitro-substituted phenylenediamine has allowed Zhang and Vyazovkin [82,83] to discover a strong effect on the amine reactivity of the position of the nitro-group. This effect manifests itself as a change in the activation energy from 50 to over 100 kJ mol ~. Figure 19 demonstrates that the presence of a nitro-group next to an amino-group in 2,4-dinitroaniline (2,4- DNA) causes E~ to reach values above 100 kJ mol 1, whereas in the 4-nitro-l,2- phenylenediamine (4-NPDA)/DGEBP system the curing process has E~ of about 50 kJ mol ~, which is a typical value for epoxy-amine reactions. This obviously suggests that a nitro-group separated from an amino-group by more than two carbons has no effect on the curing process. In its turn, a 3-nitro-l,2- phenylenediamine (3-NPDA)/DGEBP system presents an intermediate case, where one of the amino-groups is located next to the nitro-group and another is separated by more than two carbons. As a result, the system demonstrates an increase in E~ from 50 to 100 kJ mol "1, which reflects progress of the reaction from the amine in position 1 to the amine in position 2.

529

6. ISOCONVERSIONAL METHODS AND THE KINETIC TRIPLET

6.1. Is it really needed? It is a commonly-held view that a kinetic description is incomplete until the

whole kinetic triplet has been determined. For this reason, isoconversional methods are commonly criticized as being incapable of directly determining the pre-exponential factor and the reaction model. These two components of the triplet typically appear in isoconversional equations in a conjoint form (cf., equation (5)) so that their separation ultimately requires one to choose the reaction model. Although the pre-exponential factor is usually determined by choosing the reaction model, it should be stressed that a method of its evaluation in a model-independent way also exists [17]. Once the pre-exponential factor is determined, one can also numerically reconstruct the reaction model. The respective methods are illustrated below (section 6.3). Nevertheless, it is our opinion [84] that, compared to the activation energy, the pre-exponential factor and reaction model contribute little extra to understanding the process kinetics. The real need in determining the whole kinetic triplet is that in general it is believed to be necessary for making kinetic predictions. However, this belief is false as can be concluded from the following section that demonstrates that successful kinetic predictions can be accomplished without the pre-exponemial factor and reaction model.

6.2. Isoconversional kinetic predictions Kinetic analysis allows one to solve an important practical task of predicting

thermal stability of materials outside the temperature region of experimental measurements. Thermal stability can be evaluated as the time to reach a certain extent of conversion at a given temperature. Rearrangement of equation (4) gives

t a g(a)

/ where t~ is the time to reach the extent of conversion a at a given temperature, To, in an isothermal run. The major problem of equation (44) is that it is typically used in conjunction with a single heating rate method for evaluating kinetic triplets. Since such methods tend to yield significantly differing kinetic triplets for the same dataset, substitution in equation (44) of the triplets obtained results in highly erratic predictions [85]. The problem is partially addressed in the ASTM E698 method [86] that utilizes the following predictive equation:

530

- In ( l - a ) t a = ( Aexp -

where the value of E is determined by the Kissinger method (equation (13)). The method uses multiple heating rates and yields a model-independent value of E. However, as mentioned earlier, the application of this method is problematic in the case of complex (multi-step processes) that cannot be described by a single value of E. Also, equation (45) assumes explicitly that the process kinetics obey the first-order model, g(a)=-ln(1-a). The same assumption is made [86] to determine the pre-exponential factor as

exp( ) (46

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

~ exp d T

fl RT

t~= ( / e x p -RToE~ (47)

and later modified to employ data from arbitrary heating programmes, as follows

J[E~,T(t~)]

"- ( 1 exp

The respective predictions can be called "model-flee predictions", because they do not require the reaction model to be used explicitly in the numerator of equation (47) and (48). Note, that the pre-exponential factor is also unnecessary for making predictions by equation (47) and (48).

531

It has been experimentally demonstrated [17,75,85] that the model-flee equations give rise to reliable predictions, whereas substitution of the kinetic triplets, obtained from a single heating rate run, into equation (44) yields fundamentally erroneous predictions. It has also been shown that the model-free predictions are superior to the predictions based on the ASTM method (equation

1.0

0.8

0.6

0.4

0.2

0.0

1

1 0 0 ~

t []

1~ 0.0 0.2 0.4 0.6 0.8 1.0

I . i . I , I , I , I a . i ~ I

0 10 20 30 40 50 60 70

t / min

Figure 20. Isoconversional (2) and ASTM (3) predictions against actual data (1). Inset: E~ dependence for the curing process.

110 l l

100

90 O E

~ 80

70

O E (x

~ - l o g A

I i , I i ~ ' ' 7 0.0 01.2 0.4 01.6 01.8 1.0

E

Figure 21. Evaluating logA~ by equation (50) from E~ data on decomposition of ammonium nitrate

(45)). Figure 20 provides an example of using equations (45) and (47) for predicting the curing progress of an epoxy material at To=100 ~ It is seen that the model-free prediction (equation (47)) compares very well with the actual measurement, whereas the ASTM prediction deviates markedly from the experimental data. This example emphasizes the importance of accounting for reaction complexity when making kinetic predictions. In equation (47), this complexity is accounted for by using an isoconversional dependence of E~ on a determined for the curing process (Figure 20). Note that according to equation (47), each value of t~ is predicted by using the respective value of E~ that varies with a from ~90 to ---40 kJ mol ~. On the other hand, ASTM predictions rely on the Kissinger method (equation (13)) that treats any process as a simple single- step reaction, which can be represented by a single value of the activation energy. For the process considered this value is 63 kJ mol 1. This value is

532

consistent with the E~ value related to a ~ 0.45, which is the extent of conversion accomplished at the DSC peak maximum for this process.

6.3. Evaluating the pre-exponential factor and the reaction model. The fundamental flaw of single heating rate methods is that they produce

significantly differing kinetic triplets, most of which provide quite satisfactory description of the same dataset [85]. This occurs because of the mutually compensating correlation of E and A. Known as a compensation effect, this correlation takes the following form

logAj = aE, - b (49)

where a and b are constants, and Ai and E; are Arrhenius parameters associated with a particular reaction model g~(a) orf(a) . Table 1 provides an example of a compensation effect for the thermal decomposition of ammonium nitrate [88]. It is readily noticeable that larger values of E correspond to larger values of logA and the other way around. Note that an increase in A is equivalent to an increase in the rate, whereas an increase in E causes the rate to decrease (see equation (1)). Therefore, almost the same rate can be maintained by simultaneously increasing or decreasing both parameters.

Table 1. Kinetic triplets for the decomposition of ammonium nitrate [88] i Reaction Model gt(o~) EJkJ mol l log(AJmin l )

1/4 i power law a 11.5 -0.2 1/3 2 power law ot 17.7 0.6 1/2 3 power law ot 30.1 2.0 3/2 4 power law ct 104.5 10.2 2 5 one-dimensional diffusion ot 141.6 14.2

6 Mampel (first order) -In(1 - a) 81.5 8.2 7 Avrami-Erofeev [-ln(1 - I~)] 1/4 15.1 0.4

8 Avrami-Erofeev [-ln(1 - (x)]1/3 22.5 1.3

9 Avrami-Erofeev [-ln(1 - o~)]1/2 37.2 3.1

10 three-dimensional diffusion [ 1 - (1 - a)1/312 1 5 6 . 7 15.3

11 contracting sphere 1 - (1 - a)1/3 74.8 6.8

12 contracting cylinder 1 - (1 - (x)1/2 72.4 6.6

One of the useful properties of the compensation effect is that it includes the correct values of the Arrhenius parameters [89]. Therefore, if the value of E is known, one can substitute it into equation (49) to determine logA. For

533

isoconversional methods, the use of the compensation effect allows one to estimate the pre-exponential factor in a model-independent way, i.e., without choosing a suitable reaction model [17]. For instance, the application of an isoconversional method to the thermal decomposition of ammonium nitrate results in the dependence of E~ [88] shown in Figure 21. On the other hand, the logAi and Ei values (Table 1) demonstrate a compensation effect of the following functional form:

log A; = 0.108E; - 1.17 (50)

By substituting the values of E~ (Figure 21) into equation (50), one obtains estimates for the preexponential factor, logA~ (see Figure 21).

Once both E~ and logA~ dependencies are known, it is possible [17] to numerically reconstruct the reaction model in either integral or differential form. The integral form is reconstructed by substituting the values E~ and A~ into equation (51):

Aa g(a) - --F I(E,~ ,T~ ) (51)

where T~ is the experimental value of the temperature corresponding to the conversion a at given heating rate, ft. The differential form can be reconstructed by substituting the values E~ and A ~ in rearranged equation (1):

f (a) = fl -d-T ,~ A,~ exp RTa (52)

The models are reconstructed in the numerical form, i.e., as a set ofg(a) orj(a) values corresponding to different values of a. The analytical form (i.e., equation) can be established by plotting the g(a) or j (a) values against the theoretical dependencies obtained from the model equations (e.g., Table 1). Figure 22 provides an example of reconstructing the integral reaction model for the thermal decomposition of ammonium nitrate. It is seen that no single model from Table 1 fits perfectly the numerical values of g(a) obtained by substituting the values E~ and Aa (Figure 21) into equation (51). The first-order model seems to fit data well at a < 0.6, whereas the Avrami-Erofeev model appears to be the best fit at a > 0.6. This situation reflects the general problem of model-fitting procedures. No matter how large the pool of models (Table 1) is, there is no assurance that the correct model is included. Nevertheless, the use of this

534

procedure has allowed Zhou et al. [90] to determine that the first step of dehydration of the drug, nedocromil sodium trihydrate, follows the single zero- order kinetic model, g(a)=a. This procedure has also been helpful in establishing that crystallization of nifedipine from the glassy state obeys the Avrami-Erofeev model [91 ].

2.0

1.5

,~ 1.0

0.5

0.0

; 9

6 9 / / ~8/

I , I | i . I , I ! [

0.0 0.2 0.4 0.6 0.8 1.0

Figure 22. Experimental values of g(a) obtained by equation (51) (squares) and theoretical

dependencies (lines). Numbers represent models from Table 1

In conclusion, a word of caution must be voiced about reconstructing the reaction model. The procedures described (equations (51) and (52)) are based on the assumption that a single-step kinetic equation (1) or (6) holds for the process under study. This assumption can be valid only if the experimental values of E~ demonstrate no significant systematic variation with c~ (cf., Figure 21). If E~ varies significantly with a, the process involves multiple steps so that establishing a single reaction model for it has little sense.

7. CONCLUSIONS

Isoconversional kinetics is an efficient compromise between the common single-step Arrhenius treatment and the predominantly encountered 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

535

dependence of the effective activation energy on the extent of conversion and/or temperature. Although the resulting activation energies usually are effective or composite values, they can be employed successfully for making reliable kinetic predictions, for learning about complex mechanisms, and, ultimately, for accessing intrinsic kinetic parameters.

8. REFERENCES 1. M.E. Brown, M. Maciejewski, S. Vyazovkin, R. Nomen, J. Sempere, A.

Burnham, J. Opfermann, R. Strey, H.L. Anderson, A. Kemmler, R. Keuleers, J. Janssens, H.O. Desseyn, C.R. Li, T.B. Tang, B. Roduit, J. Malek and T. Mitsuhashi, Thermochim. Acta, 355 (2000) 125.

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