# (p.419) Appendix D Relaxation Kinetics

# (p.419) Appendix D Relaxation Kinetics

As is mentioned in the main text, the time and frequency behavior of a dielectric depend on many experimental parameters. One of the interesting aspects of relaxation in this regard is the temperature dependence of the characteristic relaxation times, i.e., the *relaxation kinetics*. Historically, the term “kinetics” was introduced in the field of chemistry to describe the temperature dependence of chemical reaction rates. The simplest model which describes the dependence of the reaction rate (*k*) on temperature (*T*) is the so-called Arrhenius law [1]:

where ${E}_{a}$ is the activation energy and ${k}_{0}$ is the pre-exponential factor corresponding to the fastest reaction rate in the limit $T\to \mathrm{\infty}$. In his original paper [1], Arrhenius deduced this kinetic law from transition state theory. The basic idea behind Eq. (D1) addressed the single particle transition process between two states separated by the potential barrier of height ${E}_{a}$.

The next development in the chemical reaction rate theory was provided by Eyring [2–4] who suggested a more advanced model:

where $\mathrm{\Delta}G=\mathrm{\Delta}H-T\mathrm{\Delta}S$ is the Gibbs free energy, with $\mathrm{\Delta}S$ being the activation entropy, $\mathrm{\Delta}H$ the activation enthalpy, and $\mathrm{\hslash}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}6.626\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}{10}^{-34}\phantom{\rule{thinmathspace}{0ex}}\mathrm{J}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}$. As with the Arrhenius law, the Eyring law (Eq. (D2)) is based on the idea of a transition state, according to which the reaction rate (or the velocity) represents the change in concentration of a particular species. However, in contrast to the Arrhenius equation, the Eyring equation is based on more accurate evaluations of the equilibrium reaction rate constant, producing the extra factor proportional to temperature.

An interpretation of the Arrhenius and Eyring laws that is more relevant to the concept of relaxation time in dielectric spectroscopy is that the reaction rate represents the number of molecules per unit time crossing a barrier of energy from a state A (excited) to state B (relaxed). With this interpretation (Figure D1) in mind, it may be said that the relaxation time is inversely proportional to the rate constant (i.e., $\mathrm{\tau}\sim 1/k$) and that Eqs. (D1) and (D2) describe the temperature dependence of the relaxation time $\mathrm{\tau}$ for dielectric or mechanical relaxation provided by the transition between the initial and final states separated by an energy barrier.

The relaxation kinetics of the Arrhenius and Eyring types was found to be valid for an extremely wide class of systems in various aggregation states [5–7]. Nevertheless, these laws cannot explain the experimentally observed temperature dependences of relaxation rates observed in several experiments. Thus, to describe the relaxation kinetics, especially for amorphous and glass-forming substances [8–11], several authors have used the Vogel–Fulcher–Tammann (VFT) law:

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where *T*_{VFT} is the VFT temperature and *E*_{VFT} is the VFT energy. This model was first proposed in 1921 by Vogel [12]. Shortly after it was independently discovered by Fulcher [13] and then utilized by Tammann and Hesse [14] to describe their viscosimetric experiments. It is widely held now that VFT relaxation kinetics found its explanation in the framework of Adam and Gibbs model [15]. This model is based on the Kauzmann concept of configurational entropy [16, 17], which is supposed to disappear for an amorphous substance at temperature ${T}_{k}$. Thus, based on this configurational entropy concept, the coincidence between the experimental data and VFT law is usually interpreted as a sign of cooperative behaviour in a disordered glass-like state.

An alternative explanation of the VFT model (D3) is based on the free volume concept introduced by Fox and Floury [18–20] to describe the relaxation kinetics of polystyrene. The main statement of this concept is that the probability of moving a polymer molecule segment is related to the free volume availability in a system. Later the concept of free volume was applied to the wider class of disordered solids by Doolittle [21] and Turnbull and Cohen [22], who suggested similar relationships that in the terms of relaxation times could be rewritten in the form:

where ${v}_{0}$ is the volume of a molecule (a mobile unit) and ${v}_{f}$ is the free volume per molecule (per mobile unit). Thus, if the free volume grows linearly with temperature ${v}_{f}\phantom{\rule{thinmathspace}{0ex}}\sim \phantom{\rule{thinmathspace}{0ex}}T-{T}_{k}$ then from (D4) the VFT law (D3) is immediately obtained.

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Later, the VFT kinetic model was generalized by Bendler and Shlesinger [23]. Starting from the assumption that the relaxation of an amorphous solid provided by some mobile defects, they deduced the relationship between *τ* and *T* in the form:

where *B* is a constant dependent of the defect concentration and the characteristic correlation length of the defects space distribution [23]. The model (D5) is not as popular as the VFT law; however, it has been found to be really useful for some particular substances [24, 25].