Despite its continued use as a tool for rough rate capability estimates, it is immediately evident that Peukert’s classical power-law type correlation between capacity and applied current for lead acid batteries yields a qualitatively erroneous functional form of the rate-dependence of the available capacity in batteries relying on insertion-type electrodes, where the relevant rate-limiting processes are diffusion relaxation processes. To introduce a more rational way of quantifying rate performances, we can draw inspiration from the theory of frequency dependent relaxation processes that is well established in the context of diffusion controlled impedance, as well as to explain diffusion effects on NMR or inelastic neutron spectroscopy. It will be shown that as long as electronic conductivity can be ensured (e.g. by carbon coating) a stretched exponential, i.e. Kohlrausch-Williams-Watts-like, function between relative capacity Qrel and C-rate is found, which may be understood as the consequence of two (or more) competing relaxation processes. This description not only yields a rational quantitative measure of rate capabilities for insertion-type cathode materials; it also allows for a rational assessment of the relative impact of migration barrier height for the inserted or extracted Li+, pathway dimensionality, the fraction of unoccupied low energy sites in the pathways as well as particle size and operation temperature on the achievable rate performance.
In the aim to proceed beyond a qualitative treatment by establishing a quantitative figure of merit that allows us to directly predict the achievable rate capability for a given structure type we utilize the characteristic C-rate (i.e. the C-rate where the capacity drops to 1/e of its low rate value) and the stretching exponent b as key features. C0 can be empirically correlated to an exponential activation term involving the ratio between activation energy and effective pathway dimensionality to the power of 1.5. On the other hand, C0 is also inversely proportional to the length of the diffusion path inside the particles of the active material, which can be approximated by the particle size to the power of the pathway tortuosity t (where t is typically only slightly larger than 1). In detail the necessary diffusion length for completion of the (de)lithiation reaction will of course also depend on the particle shape as well as (for compounds with low-dimensional pathways) on the orientation of the transport pathways relative to the long and short particle extensions. Thus downsizing the cathode material particles will in general increase the rate capability only slightly more than linearly, while the stretched-exponential dependence of C0 on the hight of the crucial migration barrier along the pathways for the mobile ion and the even stronger dimensionality of the ion migration pathways will control the order of magnitude of the rate performance. The value of the stretching exponent in the KWW-like relationship essentially relates to how complex the transport process will be. In more ordered compounds where defect formation and defect migration have to be considered and most attempted hops will be unsuccessful, a stretching exponent of 0.5 is to be expected, while in “fast-ion conductors” with a high concentration of mobile defects the exponent will approach 1 as the creation of additional defects plays no significant role for the anyways high success rate of elementary migration steps for mobile ions with sufficient kinetic energy.
1. (a) Adams S., in “Bond Valences” Structure and Bonding 158 (2014) pp. 91-128, (b) Adams S, Prasada Rao R.; ibd. pp.129-159.