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In this article, the method of separation of variables (SOV), as illustrated by Subramanian and White (J Power Sources 96:385, 2001), is applied to determine the concentration variations at any point within a three region simplified lithium-ion cell sandwich, undergoing constant current discharge. The primary objective is to obtain an analytical solution that accounts for transient diffusion inside the cell sandwich. The present work involves the application of the SOV method to each region (cathode, separator, and anode) of the lithium-ion cell. This approach can be used as the basis for developing analytical solutions for battery models of greater complexity. This is illustrated here for a case in which nonlinear diffusion is considered, but will be extended to full-order nonlinear pseudo-2D models in later work. The analytical expressions are derived in terms of the relevant system parameters. The system considered for this study has LiCoO 2 -LiC 6 battery chemistry. List of symbolsa Specific interfacial area (m 2 /m 3 ) B, Brugg Bruggeman coefficient c 0 Concentration at initial time t = 0 c i (x, t) Concentration in region i (mol/m 3 ) C i (X, s) Dimensionless concentration in region i D Diffusion coefficient of lithium ions in the electrolyte (cm 2 /s) D eff,i Effective diffusion coefficient of the Li-ion in region i (cm 2 /s) F Faraday's constant (C/mol) i app Applied current density (A/m 2 ) j i Flux density of the Li-ions into the electrode in region i (mol/m 2 s) J i Dimensionless flux density in region i l i Thickness of region i (m) K Ratio of dimensionless flux densities in the electrodes L Total thickness of cell (m) p Dimensionless position of positive electrode/ separator interface q Dimensionless position of separator/negative electrode interface t Time (s) t þ Transference number x Position (m) X Dimensionless position

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Competing fracture in the transfer of thin films from a relatively rigid host substrate to a flexible polymer substrate is studied using finite element simulations with cohesive zone models. Cohesive zone models for delamination based on traction-separation relations with a maximum stress criterion for damage initiation and mode-independent fracture energy for complete separation are explored to identify important parameters that affect transfer printing. Successful transfer of a thin film to a relatively compliant polymer substrate from a stiffer substrate depends on relative crack lengths, interface strengths, and fracture energies. Interface selection occurs where the mode-mix at the crack tip is predominantly due to normal stresses, despite the interface toughness being mode-independent. The observations and the fracture maps developed here predict the interface selection directly with material properties of the interfaces, substrates, and films.

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