I l z = − f ( ∈ l ) ( σ l, z x ∂ ϕ l ∂ x + σ l, z y ∂ ϕ l ∂ y + σ l, z z ∂ ϕ l ∂ z ) Y component of the electrolyte current density due to migration I l y = − f ( ∈ l ) ( σ l, y x ∂ ϕ l ∂ x + σ l, y y ∂ ϕ l ∂ y + σ l, y z ∂ ϕ l ∂ z ) X component of the electrolyte current density due to migration I l x = − f ( ∈ l ) ( σ l, x x ∂ ϕ l ∂ x + σ l, x y ∂ ϕ l ∂ y + σ l, x z ∂ ϕ l ∂ z ) Z component of the electrolyte current density I z = i l z + 2 R T ( 1 + ∂ l n ( f ± ) ∂ l n ( c l ) ) ( 1 − t + ) f ( ∈ l ) ( σ l, z x ∂ c l ∂ x + σ l, z y ∂ c l ∂ y + σ l, z z ∂ c l ∂ z ) F ( m a x ( c l, 0.1 ) ) Y component of the electrolyte current density I y = i l y + 2 R T ( 1 + ∂ l n ( f ± ) ∂ l n ( c l ) ) ( 1 − t + ) f ( ∈ l ) ( σ l, y x ∂ c l ∂ x + σ l, y y ∂ c l ∂ y + σ l, y z ∂ c l ∂ z ) F ( m a x ( c l, 0.1 ) ) X component of the electrolyte current density I x = i l x + 2 R T ( 1 + ∂ l n ( f ± ) ∂ l n ( c l ) ) ( 1 − t + ) f ( ∈ l ) ( σ l, x x ∂ c l ∂ x + σ l, x y ∂ c l ∂ y + σ l, x z ∂ c l ∂ z ) F ( m a x ( c l, 0.1 ) ) Our results highlight the importance of considering the effect of anisotropic tortuosity in such electrodes. Furthermore, we used the modified implementation to simulate the performance of HOLE graphite anodes with and without anisotropic tortuosity. To overcome this limitation, we propose a modification to the COMSOL software. Despite COMSOL's capability to solve the model equations in three dimensions, the current implementation of the model equations (specifically in COMSOL 5.4) cannot be directly applied to simulate the electrochemical performance of electrodes with anisotropic tortuosity. In such architectures, the gradients in the electrolyte concentration and potential are three dimensional, and thus provide driving force for transport in all three directions.ĬOMSOL Multiphysics Ⓡ is a widely used tool by the Li-ion battery modeling community for solving the coupled partial differential equations of the Doyle model in conventional electrodes. On the other hand, when advanced, three-dimensional electrode architectures like highly ordered laser-patterned electrodes (HOLE) are considered, it becomes necessary to account for the anisotropic tortuosity in the electrochemical simulations.
Thus, it is sufficient to set tortuosity, (or alternatively the effective diffusivity), to correspond to that for the through-thickness direction. However, models typically do not need to account for the difference in tortuosity because the macroscopic ion-transport occurs only along the thickness in conventional electrodes if the edge effects are insignificant. For example, a graphite anode was reported to have a through-plane (along the electrode thickness) tortuosity factor that is four times the in-plane value, for an electrode porosity of 40%. Standard Li-ion battery electrodes exhibit anisotropic tortuosity in the electrolyte phase, owing to the non-equiaxed shape of the active material particles and the manufacturing technique, post-casting calendaring, used to make them. ,, and its variants, which are built on the porous electrode theory (PET). These simulations are based on the model developed by Doyle at al. To gain further insights into the underlying mechanisms and to further improve the performance of these devices, electrochemical simulations are widely used. Li-ion batteries have become immensely popular mobile energy storage devices in our modern lives because of their high energy density and moderately long life.
COMSOL Multiphysics Ⓡ is one of the leading tools used by the Li-ion battery modeling community to simulate the electrochemical dynamics of the Li-ion batteries. However, the anisotropy becomes important to consider with three-dimensional architectures, such as those generated by laser-patterning. Such anisotropy can be ignored in conventional electrodes because all the macroscopic ion transport occurs along the electrode thickness, making the ion transport effectively one dimensional.
Li-ion battery electrodes, such as widely used graphite anodes, may have anisotropic tortuosity due to the non-equiaxed shape of the active material particles and the post-casting calendaring process.