Imperfections of Kissinger Evaluation Method and the Explanation of Crystallization Kinetics of Glasses and Melts

Abstract

The famous Kissinger’s kinetic evaluation method (see Anal. Chem. 1957) is examined with respect to both the relation between the DTA signal θ(t) and the reaction rate r(t) ≡ dα/dt, the requirements on reaction mechanism model f(α), and the relation of starting kinetic equation to the equilibrium behavior of sample under study. Distorting effect of heat inertia and difference between the temperature T _p of extreme DTA deviation and the temperature T _m at which the reaction rate is maximal are revealed. DTA equation of Borchard and Daniels is criticized regarding the neglection of heat inertia correction. The kinetic equations respecting the influence of equilibrium temperature T _eq , especially fusion/melting temperature T _f , are tested as bases for a modified Kissinger-like evaluation of kinetics. Crystallization kinetics on melt solidification is examined under integration of undercooling and needed Gibbs approximations are explored. This chapter provides a new insight into the routine practice of nonisothermal kinetics showing forward-looking outlook and encompasses hundreds of references.

Pavel Holba—Deceased

Appendices

Appendix 1: Gibbs Energy Approximations

For theoretical study on nucleation and crystal growth during solidification , the difference in Gibbs free energy between liquid and solid is usually expressed by various authors as ΔG, thus emerging in several theoretical formulations in phase transfer phenomena [4, 60, 94–96]. By definition, ΔG, is given by ΔG = ΔH − TΔS which can be expanded using the popularly known expressions for ΔG and ΔS as

$$\Delta G = \Delta H_{m} \frac{\Delta T}{{T_{m} }} - \int\limits_{T}^{{T_{m} }} {\Delta C_{p} } dT + T\int\limits_{T}^{{T_{m} }} {\Delta C_{p} \frac{dT}{T}}$$

leading to ΔG = ΔH ΔT/T _eq  + ΔCp (ΔT)²/2T (1 − ΔT/6T).

While the enthalpy of melting, ΔH _m , is readily available for most materials, the difference in solid–liquid specific heats, ΔCp, is not, because its experimental measurements is difficult due to the liquid metastability. Hence, having ΔCp neglected and after approximating a meaningful value of ΔG, it gives simple ΔS _m ΔT which is the most convenient and widely used expressions attributed to Turnbull [90] and valid over small undercooling (conventional solidification processes like casting). The value of ΔG is therefore overestimated which was approved by Thompson and Spaepen [97] for a linear ΔCp approximation yielding after simplification the more workable ΔG = ΔS _m T ln (T _m /T) which after expanding the logarithmic term can be extended to feasible ΔG = ΔS _m ΔT (2T/(T _m  + T).

More recent attempts involve expansions involving higher powers of undercooling and incorporating theoretical quantities like Kauzmann temperature [60]. A noteworthy example and one that is applied in this work is a simple looking expression obtained by Lad et al. [98] upon using the series expansion, $\ln \left( {1 - \frac{\Delta T}{{T_{m} }}} \right) = \frac{\Delta T}{{T_{m} }} - \frac{{\Delta T^{2} }}{{2T_{m}^{2} }} + \frac{{\Delta T^{3} }}{{3T_{m}^{3} }} \ldots$ as to obtain a parabolic approximation for ΔG, leading to [98]

$$\Delta G = \Delta S_{m} \Delta T\left( {1 - \frac{\Delta T}{{2T_{m} }}} \right)$$

(10.49)

In the classical study by Hoffmann [60, 99], the constant ΔC _p was also assumed arriving at a simple approximation ΔG = ΔH ΔT/T _eq (T/T _eq ) while Dubay and Ramanchadro [100] proposed another specification as ΔG = ΔH ΔT/T _eq (2T)/(T _eq  + T). By incorporating the third-order term in the TS-approximation, Ji and Pan [101] obtained more recently a new relation

$$\Delta G = 2\Delta S_{m} \Delta T\left( {\frac{T}{{T_{m} + T}} - \frac{{\Delta T^{2} T_{m} }}{{3(T_{m} + T)^{3} }}} \right)$$

(10.50)

Pillai and Málek [96] used the arithmetic averaging to obtain an innovative approximation as

$$\Delta G = \Delta S_{m} \Delta T\left( {\frac{T}{{T_{m} + T}} + \frac{T}{{2T_{m} }}} \right)$$

(10.51)

In conclusion, it follows that for small undercoolings , the simplest form of the Turnbull [90] (see Paragraph 4) or Hoffmann [99] approximations can be readily used [60, 96]. However, when the liquid is deeply undercooled the other approximations should be preferred in conjunction with testing [60, 96]. In extreme when the liquid is undercooled down to the glass transition temperature (or even lower), the simple utilization of Turnbull becomes incorrect, thus recommending the arithmetic mean values such as Eq. (10.51).

Appendix 2: Experimental

Numerical evaluation and analysis of transformation kinetics is just an end consequence of specific experimental study often related to the so-called macro-system, i.e., giving way to analysis of bulk samples of certain property and treated under ordinary heat conditions often provided by commercial instruments. ICTAC Kinetic committee [50] presented a manual for kinetic analysis staying, however, on the surface of the problem citing. “The temperature used in the kinetic analysis must be that of the sample. Thermal analysis instruments control precisely the so-called reference temperature (T), whereas that of sample can deviates from it due to the limited thermal conductivity of the sample or due to the thermal effect of the process that may lead to self-heating/cooling.” There is no comment that such a T-deviation is a factual instrumental response which is exploited for kinetic data determination [27, 28] under appeal of proper analyzing alongside with the impact of heat inertia [29]. Further “Typical approaches to diminishing the deviation of the sample temperature from the reference temperature are decreasing the sample mass as well as the heating rate” brining no clarification on the sample factual temperature due to both the gradients and the grain size structure. Neglecting the customary quandary towards the sample makeup (sample holder, mutual thermal contacts, in-weight, averaging, powdering, etc.) most important is the upcoming prospect of thermal analysis going down to the nano-dimensions and, in limit, to quantum world [102–105]. The experimentation may extend to the novel features touching the significance of temperature [93] during the ultrafast T-changes (see Chap. 3). Accordingly, we would need to recognized specialized fields of heat micro-transfer [104] exhibiting unusual assets still regarding validity of constitutive equations [105] but paying attention to the novel techniques based on highly sensitive thin microchips enabling rapid scanning microcalorimetry [67, 106, 107]. Particularity of the emerging area of nanomaterials can be studied by alteration calorimetry [108–112] forcing us to learn about its specificity (see Chap. 18) providing extra dimension in thermodynamic description [113, 114]. The overall examination was the subject of our invited lectures [115, 116]. Moreover, it brought some unexpected consequences of non-constancy of certain thermodynamic values when severally diminishing particle curvature [117, 118]. It is just a short inventory of material basis wherein the kinetic analysis is employed and which should be kept in the researchers’ attention in order to carry on a respectable kinetic study [46].