Lithium-battery electrodes made from silicon  increase volume by a factor of four when charging from a de-lithiated to fully lithiated state, but by charging and discharging slowly enough, stresses can be reduced, which avoids stress damage to the electrode. We provide a mathematical description comprehending the relevant solid mechanics (i.e., stress and strain) and diffusion processes so as to describe highly expanding battery electrode materials and determine the maximum current at which lithium-silicon can be charged and avoid exceeding the material yield stress. An approximate metric for acceptable stress levels is that they are less than 1% of the elastic modulus. At these low stress levels, infinitesimal strain theory can be used for the computations. The rate of charge and discharge that keeps stress below this level depends on values for the elastic modulus, the Poisson ratio, the lithium diffusion coefficient, the particle size, the open-circuit potential of the material, and several other parameters whose values are generally concentration dependent. The derived formulas are analytic and straightforward to apply (e.g., they can be employed in conventional spreadsheet software) and can be used to assess potential new materials, design better electrodes, and improve operating strategies.

Example outputs from the mathematical model can be seen in the two figures below.  The upper figure provides calculated results for the open-circuit potential, modulus, particle radius (left ordinate) and perturbation parameter (right ordinate) as a function of time (or average fraction of filled lithium sites y₀).  During charge, the potential and the modulus (stiffness) decline monotonically, the radius increases monotonically, and the perturbation parameter is a complicated function of y₀.  Histories of the radial, hydrostatic, and tangential stresses as a function of time (or average fraction of filled lithium sites y₀) and at radial positions r/a = 0, 0.5, and 1 are provided in the lower figure.  Note that the abscissas are identical.