Cracking in nanostructured Li-batteries

During the charge/discharge cycle of rechargeable Li-batteries, Li-ions are attracted to the negative and positive electrodes interchangeably, and therefore the electrodes expand and contract repeatedly. In order to obtain complete knowledge of rechargeable Li-batteries, the mechanical behavior of the electrodes during each cycle needs to be analysed. A first attempt in this direction was by Aifantis and Hackney [1], who modelled the stress that is developed in the electrodes during the charge/discharge cycle. The most important effect, however, that this Li-insertion/de-insertion has for the electrodes is the formation of cracks, since this affects the electrochemical properties of the battery cell [1]. In the present analysis the geometry considered for both electrodes is that of long cylindrical Li-insertion sites (Sn - in the negative electrode, and LiMn₂O₄ - in the positive electrode) embedded in a glass matrix. A unit cell consists of a Li-insertion site, whose radius is taken to be 100nm, surrounded by a cylindrical glass area, with radius 1000nm. The cracks that develop at the Sn or LiMn₂O₄-glass interface are treated as being radial, in accordance with experimental evidence [1]. By choosing an appropriate displacement condition at the Li-insertion site/glass matrix interface, and by considering three different cases at the glass/glass outer boundary (i.e. stress free surface, clamped surface, self-equilibrated loading) the energy release rate is computed, which allows for the determination of the maximum crack length that can be attained during stable cracking. Sample numerical calculations are presented for the negative electrode.