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Storage particles of lithium ion batteries undergo significant mechanical stress during charging and discharging due to the inhomogeneous volume change within the particles when lithium is inserted and extracted. This stress potentially leads to fracture of the particles resulting in detrimental effects for the capacity and internal resistance of a lithium ion battery, such as the growth of additional solid electrolyte interface, loss of contact in conductive pathways or complete disintegration of the electrode. Here, we tackle the problem of fracture in storage particles by merging a coupled model of mechanical stress and diffusion of lithium ions with a phase field description of an evolving crack. This approach allows the simultaneous study of the evolution of the lithium concentration together with the growth of a crack without restrictions to specific geometries, simulation dimensions and presuppositions regarding the crack path. The model was successfully applied to study crack growth during lithium insertion in an earlier work. It was shown that inertia effects play a crucial role with respect to a possible fragmentation of the storage particles. Here, we focus on the opposite charging condition and examine the circumstances under which unstable crack growth and particle breakage can occur during lithium extraction.

SUMMARYThis paper presents a general framework for the macroscopic, continuum-based formulation and numerical implementation of dissipative functional materials with electro-magneto-mechanical couplings based on incremental variational principles. We focus on quasi-static problems, where mechanical inertia effects and time-dependent electro-magnetic couplings are a priori neglected and a time-dependence enters the formulation only through a possible rate-dependent dissipative material response. The underlying variational structure of non-reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so-called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum-based modeling in macroscopic electro-magnetomechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle-point principles. Next, the implications on the formulation of boundary-value problems are addressed, which appear in energy-based formulations as minimization principles and in enthalpy-based formulations in the form of saddle-point principles. Furthermore, the material stability of dissipative electro-magneto-mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro-magneto-mechanics, an energy-based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy-based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials.

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Electrochemical strain microscopy (ESM) can provide useful information on the ionic processes in materials at the local scale. This is especially important for ever growing applications of Li-batteries whose performance is limited by the intrinsic and extrinsic degradation. However, the ESM method used so far has been only qualitative due to multiple contributions to the apparent ESM signal. In this work, we provide a viable approach for the local probing of ionic concentration and diffusion coefficients based on the frequency dependence of the ESM signal. A theoretical basis considering the dynamic behavior of ion migration and relaxation and change of ion concentration profiles under the action of the electric field of the ESM tip is developed. We argue that several parasitic contributions to the ESM signal discussed in the literature can be thus eliminated. The analysis of ESM images using the proposed approach allows a quantitative mapping of the ionic diffusion coefficients and concentration in ionic conductors. The results are validated on Li-battery cathodes (LiMnO) extracted from commercial Li-batteries and can provide novel possibilities for their development and further insight into the mechanisms of their degradation.