Some mathematical aspects of nonisothermal kinetics

The results of an attempt to derive correct nonisothermal kinetic equations from isothermal ones through the classical nonisothermal change (CNC) of the postulated primary kinetic equations are presented. An alternative possibility through use of the model of infinitesimal isothermal portions (MIIP) is discussed.

---

Zusammenfassung Die Autoren stellen die Ergebnisse eines Versuchs vor, korrekte Gleichungen für die nicht-isotherme Kinetik durch den klassischen nichtisothermen Übergang aus den als primär postulierten isothermen kinetischen Gleichungen abzuleiten. Als alternative Möglichkeit wird das Modell der infinitesimalen isothermen Abschnitte diskutiert.

---

. .

     

On the nonlinear isoconversional procedures to evaluate the activation energy of nonisothermal reactions in solids

A general procedure for deriving the equations that underlie various isoconversional nonlinear methods for evaluating the activation energy is presented. A new integral isoconversional nonlinear method with integration over a given range of conversion is suggested. This method was applied to simulated nonisothermal data as well as to data for the nonisothermal decomposition of ammonium perchlorate. The obtained dependencies of the activation energy on the degree of conversion were compared with those resulting from other nonlinear and linear methods of analysis. © 2003 Wiley Periodicals, Inc. Int J Chem Kinet 36: 87–93 2004     

Isothermal and nonisothermal crystallization kinetics of irradiated nylon 1212

The crystallization behavior of nylon 1212, irradiated at ⁶⁰Co γ‐rays (50 kGy), was studied by a rheometer, polarized optical microscopy (POM), and differential scanning calorimeter (DSC). The results showed that irradiated nylon 1212 samples exhibited abnormal crystallization behavior during the crystallization process: The Avrami exponent n was calculated and was found to be in the range from 2.06–2.41 for isothermal crystallization, and from 2.67–4.91 for nonisothermal crystallization; the spherulite morphology also changed largely by polarized optical microscopy (POM); the crystallization activation energy ΔE for isothermal and nonisothermal crystallization process of irradiated nylon 1212 are determined to be 57.4 kJ/mol and 78.65 kJ/mol, respectively, which are lower than that of nonirradiated nylon 1212. At the same time, a new method by a combination of the Avrami and Ozawa equations was successfully applied to analyze the noncrystallization process of irradiated nylon 1212. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2326–2333, 2005     

The heating rate as a variable in non-isothermal kinetics. A new method to evaluate the nonisothermal kinetic parameters

The heat exchange calorimetry, hitherto developed for ordinary size samples, was modified for diminishing the sample size to one tenth or less of that commonly used in previous reports. Improvements were made with respect to the vessels for sample and reference, the stirrer of the sample solution, the thermistor and the calibration heater. The value of α, a constant relating to the heat transfer and critically affecting the sensitivity to smaller heat effects, was given an appropriately small value. The improved version of the calorimeter employed a sample vessel of a capacity of 6 cm³ and was suited to accomodate about 3 cm³ of a solution. The calorimeter proved to give sufficiently precise results when total heats ranging from 0.05 to 0.4 J were evolved.     

Extension of the diffusional kinetics models involving changes in activation energy

On the basis of previous modifications of the Zhuravlev and Ginstling-Brounshtein models, a generalization of kinetic diffusional models is proposed. With the assumption that the rate of the activation energy change during the reaction is inversely proportional to the reaction time, it has been shown that all diffusional kinetic equations in heterogeneous systems take the formF()=KT ⁿ , whereF() is a function of the degree of conversion andK andn are constants related to the rate constant.

---

Zusammenfassung Auf Grund vorangegangener Modifikationen der Modelle von Zhuravlev und Ginstling-Brounshtein wird eine Verallgemeinerung der kinetischen Diffusionsmodelle vorgeschlagen. Mit der Annahme, daß die Geschwindigkeit der Änderung der Aktivierungsenergie während der Reaktion umgekehrt proportional der Reaktionszeit ist, wird gezeigt, daß alle kinetischen Diffusionsgleichungen für heterogene Systeme die FormF()=KT ⁿ haben, woF() eine Funktion des Konversionsgrades undK undn mit der Geschwindigkeitskonstante in Beziehung stehende Konstanten sind.

---

- . , , , F()=KT ⁿ , F() — , K n — .

     

Actual activation energy of electrode process under mixed kinetics conditions

In the case of a single-electron reaction with account for slow diffusion of reagents, equations for actual (experimentally determined) activation energies of two types were derived and analyzed: real energy A f, i.e., the energy measured at a constant electrode polarization value η = const) and formal energy (Ω_f, i.e., the value measured at a constant value of potential vs. an ambiguously chosen reference electrode E = const). It is found that under the conditions of a sufficiently significant deviation from equilibrium, the actual activation energy A _f is the weighted arithmetic mean of the diffusion activation energy and the sum of A ₀ + αFη (where A ₀ is the real activation energy of the discharge stage at polarization of η = 0); herewith, the weighting coefficients are the corresponding values of the current of the discharge stage and the limiting diffusion current. A similar relationship is also obtained for Ω_f. It is found that the A _f, η- and Ω_f, E-curves can in a number of cases feature regions with the negative A _f and Ω_f values in the mixed kinetics range.     

Isothermal and nonisothermal crystallization kinetics of nylon‐46

Isothermal and nonisothermal crystallization kinetics of nylon‐46 were investigated with differential scanning calorimetry. The equilibrium melting enthalpy and the equilibrium melting temperature of nylon‐46 were determined to be 155.58 J/g and 307.10 °C, respectively. The isothermal crystallization process was described by the Avrami equation. The lateral surface free energy and the end surface free energy of nylon‐46 were calculated to be 8.28 and 138.54 erg/cm², respectively. The work of chain folding was determined to be 7.12 kcal/mol. The activation energies were determined to be 568.25 and 337.80 kJ/mol for isothermal and nonisothermal crystallization, respectively. A convenient method was applied to describe the nonisothermal crystallization kinetics of nylon‐46 by a combination of the Avrami and Ozawa equations. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1784–1793, 2002     

New approximations of the temperature integral for nonisothermal kinetics

The accuracy and scope of application of previously reported approximations of the temperature integral were evaluated. The exact solution was obtained independently by solving the temperature integral numerically be Simpson's rule, the trapezoidal rule and the Gaussian rule. Two new approximations have been proposed: $$\begin{gathered} P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 5.2/X^2 ) \hfill \\ P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 4.6/X^2 ) \hfill \\ \end{gathered} $$ whereX=E/RT. The first equation gives higher accuracy, with a deviation of less than 1% and 0.1% from the exact solution forX≥7 andX≥10, respectively. The second equation has a wider scope of application, with a deviation of less than 1% forX≥4 and of less than 0.1% forX≥35.     

Isothermal and nonisothermal crystallization kinetics of novel odd-odd polyamide 9 11

A study on isothermal and nonisothermal crystallization kinetics of odd-odd polyamide 9 11 was carried out by differential scanning calorimetry (DSC). The equilibrium melting temperature of polyamide 9 11 was determined to be 199.1 °C. The Avrami equation was adopted to describe isothermal crystallization of polyamide 9 11. Nonisothermal crystallization was analyzed using both the Avrami relation modified by Jeziorny and the equation suggested by Mo. The isothermal and nonisothermal crystallization activation energies of polyamide 9 11 were determined to be −310.9 and −269.0 kJ/mol using the Arrhenius equation and the Kissinger method, respectively.     

On the linearity of the energy of activation in theEyring theory of kinetics

Summary A result obtained earlier implies that — if theEyring theory is correct — the energy of the transition state is less thanU+E _a whereU is the sum of the internal energies of the chemical reactants andE _a is the energy of activation. The energy of the transition state for the forward reaction is not equal to that of the inverse reaction; it varies linearly with the temperature.

---

Zur Linearität der Aktivierungsenergie in derEyringschen Theorie

Zusammenfassung Aus einem früher erhaltenen Ergebnis folgt, daß — Richtigkeit derEyring-Theorie vorausgesetzt — die Energie des Übergangszustands geringer ist alsU+E _a (U: Summe der inneren Energien der Reaktanden;E _a: Aktivierungsenergie). Die Energie des Übergangszustands ist für Hin- und Rückreaktion unterschiedlich und hängt linear von der Temperatur ab.

     

Isothermal and nonisothermal crystallization kinetics of poly(propylene terephthalate)

The kinetics of crystallization of poly(propylene terephthalate) (PPT) samples of different molecular weights were studied under both isothermal and nonisothermal conditions. The Avrami and Lauritzen–Hoffmann treatments were applied to evaluate kinetic parameters of PPT isothermal crystallization. It was found that crystallization is faster for low‐molecular‐weight samples. The modified Avrami equation, and the combined Avrami–Ozawa method were found to successfully describe the nonisothermal crystallization process. Also, the analysis of Lauritzen–Hoffmmann was tested and it resulted in values close to those obtained with isothermal crystallization data. The nonisothermal kinetic data were corrected for the effect of the temperature lag and shifted alone with the isothermal kinetic data to obtain a single master curve, according to the method of Chan and Isayev, testifying to the consistency between the isothermal and corrected nonisothermal data. A new method for ranking of polymers, referring to the crystallization rates, was also introduced. This involved a new index that combines the maximum crystallization rate observed during cooling with the average crystallization rates over the temperature range of the crystallization peak. Furthermore, the effective energy barrier of the dynamic process was evaluated with the isoconversional methods of Flynn and Friedmann. It was found that the energy barrier is lower for the low‐molecular‐weight PPT. The effect of the catalyst remnants on the crystallization kinetics was also investigated and it was found that this is significant only for low‐molecular‐weight samples. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3775–3796, 2004     

Some aspects of mathematical statistics as applied to nonisothermal kinetics

Restrictions of the simplest use of correlation and regression analysis to obtain a single-valued solution to the inverse kinetic problem are considered. The Coats-Redfern method is suggested as a version of nonlinear regression analysis to increase the unambiguity of the solution.

---

Zusammenfassung Die Beschränkungen der einfachsten Anwendung der Korrelations- und Regressionsanalyse zur eindeutigen Lösung des inversen kinetischen Problems werden erörtert. Die Coats-Redfern-Methode wird als eine Version der nichtlinearen Regressionsanalyse zur Erhöhung der Eindeutigkeit der Lösung vorgeschlagen.

---

. - .

     

Some aspects of mathematical statistics as applied to nonisothermal kinetics

The suggested unified approach is based on the consideration of the entire set of kinetic functions as some space. The analysis of this space provides four classes of kinetic functions, characterized by similar properties and close values of kinetic parameters. The solution of the inverse kinetic problem within the unified approach reduces to finding unambiguously the class of kinetic functions.

---

Zusammenfassung Die vorgeschlagene vereinheitlichte NÄherung geht davon aus, da\ das gesamte System der kinetischen Gleichungen als Raum angesehen wird. Die Analyse dieses Raumes ergibt vier Klassen von kinetischen Funktionen, die durch Ähnliche Eigenschaften und geschlossene Werte der kinetischen Parameter charakterisiert sind. Die Lösung des inversen kinetischen Problems reduziert sich in der vereinheitlichten NÄherung auf das eindeutige Auffinden der Klasse der kinetischen Funktionen.

---

. , , , .

     

Analysis of the nonisothermal crystallization kinetics in three linear aromatic polyester systems

Melt or cold crystallization kinetics has a strong bearing on morphology and the extent of crystallization, which significantly affects the physical properties of polymeric materials. Nonisothermal crystallization kinetics are often analyzed by the classical Johnson–Mehl–Avrami–Kolmogorov (JMAK) model or one of its variants, even though they are based on an isothermal assumption. As a result, during the nonisothermal (e.g. constant heating or cooling rate) crystallization of polymeric material, different sets of model parameters are required to describe crystallization at different rates, thereby increasing the total number of model parameters. In addition, due to the uncorrelated nature of these model parameters with the cooling or heating rate, accurate modeling at any intermediate condition is not possible. In the present work, these two limitations of the conventional approach have been eliminated by exhibiting the existence of a functional relationship between cooling or heating rate and effective activation energy during nonisothermal melt or cold crystallization in three linear aromatic polyesters. Furthermore, it has been shown that when the JMAK model is used in conjunction with this functional relationship, it is possible to precisely predict the experimental nonisothermal melt or cold crystallization kinetics at any linear cooling or heating rate with a single set of model parameters.     

Random activation energy model and disordered kinetics, from static to dynamic disorder

We suggest a unified path integral approach for random rate processes with random energy barriers, which includes systems with static and dynamic disorder as particular cases. We assume that the random component of the activation energy barrier can be described by a generalized Zubarev-McLennan nonequilibrum statistical ensemble that can be derived from the maximum information entropy approach by assuming that the time history of the fluctuations of the random components of the energy barrier are known. We show that the average survival function, which is an experimental observable in disorderd kinetics, can be computed exactly in terms of the characteristic functional of this generalized Zubarev-McLennan nonequilibrium statistical ensemble. We investigate different types of disorder described by our approach, ranging from static disorder with infinite memory to random processes with long or short memory, and finally to rapidly fluctuating independent random processes with no memory. We derive expressions of the average survival function for all these types of disorder and discuss their implications in the evaluation of kinetic parameters from experimental data. We illustrate our approach by studying a simple model of dynamic disorder of the renewal type. Finally we discuss briefly the implications of our approach in molecular biology and genetics.     

On the changes of the activation energy with the conversion degree in non-isothermal kinetics

The authors present their results concerning the changes of activation energy determined under non-isothermal conditions with the conversion degree. An interpretation of this effect is suggested.     

A new method to explain the model dependence of apparent activation energy derived from a single nonisothermal dynamic curve

A single nonisothermal dynamic curve can be relatively correctly described by several reaction models in model-fitting approach, which yields model-dependent kinetic parameters. The model dependence of the apparent activation energy is explained mathematically by a new method which is developed from the peak property method in this paper. It has been found that the apparent activation energy rises linearly with the increase of the order of nth-order and Avrami-Erofeev reaction models, which can be verified by data from the literature and can explain some phenomena appeared in some articles. Moreover, the apparent activation energies derived from fitting a single nonisothermal dynamic curve to nth-order and Avrami-Erofeev reaction models can be correlated through the activation energy of 1st-order model.     

Some aspects of mathematical statistics as applied to nonisothermal kinetics V

The paper gives a quantitative comparison of two methodological approaches to the solution of the inverse kinetic problem: the traditional approach and the nontraditional approach suggested by the authors. It is shown that the amount of information (in the sense of Shannon) obtained within the scope of the nontraditional approach is always greater than that obtained with the use of the traditional approach.

---

Zusammenfassung Zwei methodologische Näherungen der Lösung des inversen kinetischen Problems werden quantitativ verglichen, nämlich die traditionelle und die von den Autoren vorgeschlagene nicht-traditionelle Näherung. Es wird gezeigt, daß mit der nicht-traditionellen Näherung erhaltene (im Sinne von Shannon verstandene) Informationsmenge immer größer als die durch Anwending der traditionellen Näherung erhaltene ist.

---

: . , ( ), , , .