Kinetics of phase transformations for constant heating rate occurring close to the thermodynamic transition
Introduction
First order phase transitions α→β can be detrimental or of outmost importance to many materials due to effects on important applications (e.g. electrical properties) or because they might cause collapse due to excessive lattice mismatch between the β and α phases. For example, the β→α transition in some ion conducting ceramics (e.g. Bi2O3- or La2Mo2O9-based materials) spoils their applicability as solid electrolytes [1], [2], or limits the working temperatures. In addition, repeated cycling between high and low temperatures might lead to failure, even for low differences between the lattice parameters of the high and low temperature phases.
The Johnson–Mehl–Avrami (JMA) theory [3], [4] is often used to describe the fraction transformed for experiments performed at constant temperature. The corresponding Avrami–Nakamura models [5], [6] have been proposed to analyze experiments performed on heating with variable temperature and were successfully used to analyse the crystallisation in glass–ceramic materials. However, the models for non-isothermal conditions are based on the assumption that the relevant kinetic constant is nearly described by a typical Arrhenius dependence, which might be invalidated near the thermodynamic transition, as discussed below.
Detailed models have been proposed mainly for homogeneous nucleation in glass–ceramic systems [7], and also for phase transformations, with inclusion of the effects of strain misfit energy, as described by [8]:where Io is a pre-exponential factor, EN is the energy of migration, and the thermodynamic barrieris dependent on the interfacial energy γ, the free energy of transformation per unit volume ΔGv = ΔG/Vm (Vm being the molar volume), and the strain misfit energy ΔGs. Eqs. (1), (2) thus show that the temperature dependence of the rate of nucleation is far from simple. Most models are thus derived on assuming simplified conditions such as separate stages of nucleation and growth, or nearly instantaneous nucleation at temperatures which are close to the true transition temperature.
The temperature dependence of the growth rate also deviates from the Arrhenius dependence on approaching the phase transition temperature Tt. Though more complex formulae have been proposed for glass–ceramic systems [8], [9], the following formulae is often used to describe the temperature dependence of growth rate:
Uo being a pre-exponential factor, Eg the activation energy and ΔH the enthalpy change. Eq. (3) will be assumed for solid–solid transformations, and will be used to emphasize the deviations from commonly assumed kinetic models.
Section snippets
Temperature dependence of the fraction transformed
Nucleation may be a rather complex process [10], [11], mainly before reaching a steady state regime [12], [13], [14]. However, ready nucleation is likely to occur at internal interfaces or other discontinuities, such as grain boundaries, edges, corners, dislocations, etc. For example, kinks are often active sites for growth and may yield nearly one-dimensional growth of plate-like particles by lateral movement of ledges. In these conditions, nucleation may occur readily at temperatures which
Transformation peak versus rate of change in temperature
A commonly used method to obtain the activation energy of glass crystallisation and similar processes is the Kissinger equation [20] which describes the dependence of crystallisation peak temperature Tp on the rate of change in temperature:
The applicability of Kissinger equation has been discussed in the literature (e.g. [21]), and must also be revised in the present case. Note that the actual temperature dependence of the growth rate deviates strongly from a simple
Dimensionless treatment
In order to minimize the number of relevant parameters one may use dimensionless variables, thus simplifying the relevant models for the dependence of fraction transformed on temperature and rate of change in temperature, and the dependence of the transformation peak temperature on the rate of change in temperature. The chosen dimensionless variables were:
On inserting these variables in Eqs. (14), (15), (16) one thus obtains:
Predictions
Fig. 1 shows finite difference solutions of Eqs. (6), (7) (symbols) and the corresponding predictions of Eq. (14), with ɛ1 and ɛ2 described by Eqs. (15), (16) (solid lines), for the α→β transformation with TtR/Eg = 0.02, = 10−21 and ΔH/RTt = 1, 10 and 102. Note that the actual range of values of dimensionless parameter , in Fig. 1 corresponds to typical values of RTt, Eg, No and U in the order of 10 kJ/mol, 500 kJ/mol, 10−2 μm−3 and 10−2 μm/s, with
Conclusions
Commonly used kinetic models for nucleation and growth do not provide a correct description of phase transformations near the true thermodynamic transformation temperature Tt. This occurs because the growth rate vanishes on approaching Tt. Revised models were thus derived to describe the dependence of fraction transformed on temperature, and the dependence of the peak transformation temperature on the rate of heating or cooling. Dimensionless treatment is suitable to minimize the number of
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