High and low activation energy kinetics are different: Implications for hydrogen and protons in condensed matter
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High and low activation energy kinetics are different: Implications for hydrogen and protons in condensed matter Arthur Yelon Department of Engineering Physics, Polytechnique Montreal, PO Box 6079, Station C-V, Montreal, QC H3C 3A7, Canada, and Réseau Québecois sur des Matériaux de Pointe (RQMP) ABSTRACT At high activation energy, kinetic processes require the accumulation of several excitations in order to take place. This accumulation implies an entropy, which we call multi-excitation entropy. This explains the isokinetic rule, which itself explains the existence of isoequilibrium relations. Understanding the implications of these effects helps clarify the microscopic mechanisms in kinetic processes, including hydrogen storage, and transport of protons, and potentially, of other ions, in condensed matter. INTRODUCTION Kinetic processes involving protons and hydrogen in solids are known to involve high activation energies. That is, that they are large compared with the energies of excitations which may provide this energy (phonons, 70 meV and below, and local vibrations, 150 meV and below) and with thermal energies (25 meV at room temperature). For minerals, they are typically 1 eV and higher.1 Even for perovskites which are considered to be promising for energy storage and conversion, they are on the order of 500 meV.2,3 The scientific community is accustomed to thinking about low-activation-energy kinetics. As we show here, high-activation-energy kinetics are quite different. This introduces complications, but also creates opportunities for using observations of kinetic behavior to illuminate physical processes. In what follows we offer suggestions for the mechanism for proton conduction in minerals and ceramics, and propose studies of proton conduction kinetics which may provide further insights. HIGH ACTIVATION ENERGY KINETICS It has long been known that many phenomena are temperature activated. In 1899, Arrhenius proposed a simple law, which describes the majority of such processes: ܺ ൌ ܺ ݁ ିοாΤಳ ் .
(1)
In Eq. (1), ΔE is the activation energy, kB is the Boltzmann constant, and T is the absolute temperature. In a classic paper, Miller and Abrahams4 showed that at modest temperatures, a kinetic process with low activation energy, 'E , in a solid, for example, will take place by the annihilation of a phonon or local (IR or Raman) vibration of almost exactly the energy needed for the process, to that the process takes place at a rate, Q , given by (the Arrhenius equation) ݒൌ ݒ ݁ െ ο ܧΤ݇ ܶ,
(2)
where ݒ is the attempt frequency. There is a strong natural tendency to believe that this is all we need to know. However, there are two important facts which we may tend to forget. First,
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Eyring, in his development, In 1937, of transition state theory, presented an equation f
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