Strain derivatives for practical charge rate characterization of lithium ion electrodes

During battery use, electrode materials are known to expand and contract in repeatable patterns, and this strain has been previously correlated with battery properties such as state of charge and state of health. In this study, we show that the second derivative of strain, d²ε/dQ², is mathematically proportional to dV/dQ within an electrode stage. We also experimentally quantify peaks in the strain curves for electrode stage transitions at practical charge rates of up to C/2 and confirm that transitions are visible in the practical scenario of discharging at the higher rate of 1C. Moreover, the location of the transition measured by d²ε/dQ² changes by less than 10% from 0.05 C to 0.5 C, but the transition measured with dV/dQ decreases by more than 15% from 0.05 C to 0.3 C, demonstrating the reliability of strain to measure electrode transitions at moderate charge rates. We also note that d²ε/dQ² exhibits similar peak shifts as those expected in dV/dQ as the cell ages. Our derivations for the model system of graphite and lithium cobalt oxide can be generalized to other battery systems and used to characterize materials at practical charge rates impossible with only voltage.