Governing Equations on the Micro Scale

Continuum models encompass both micro and macro scales and in li-ion models the microscale is governed by the solid phase diffusion equation. The coupling of the microscale and the macroscale variables pose computational limitations. 

The concentration of lithium at the solid/solution interface, , is obtained by solving the Pick s law in spherical coor- 

The porewall flux occurs as a boundary condition for the micro-scale as opposed to being a source term for the macroscale. For certain battery chemistries rectangular or cylindrical diffusion might be more relevant. 

the diffusion coefficient, which is assumed to be a constant for most modehng purposes, sometimes needs to be functionalized. In cases where the diffusion coefficient depends non-linearly on concentration the coupling becomes much harder. 

When the diffusivity changes as a function of concentration (SOC), the governing equation is 

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