Effective Diffusivity of Electrolyte in Porous Structure Using a Three-Dimensional Random Pore Network Model

In Li-ion cell, electrochemical discharge and intercalation/deintercalation occur at length scales that differ by an order of magnitude¹. Analysis of these processes has been conventionally done using the classical pseudo 2D-model² that is based on the porous-electrode theory, which solves the volume-averaged species conservation in the electrolyte.  However, the species conservation in solid active material is solved by using the intrinsic diffusion coefficient since the model uses a microstructure that is composed of a single spherical particle placed at each grid location in the electrode. Hence, this model is unable to account for the percolation network in real particle clusters. Therefore, to make volume averaging process applicable at each grid location in the electrode, it is necessary to estimate the effective diffusivity in both particle and electrolyte phases.

In this work, a model has been used to predict the effective diffusivity in the porous structure. It is assumed that the micro-porous structure of an electrode in Li-ion battery can be approximate by 3-D random pore network model as shown in Fig. 1. In this model, a pore is assumed to be circular bond segment which connects any two nodes. To make the model more realistic, length and diameter of each pore segment are chosen with the circular diameter of each bond segment distributed using a Gaussian distribution. The length of the bond segment, which is the distance between the nodes placed randomly is also prescribed using a second Gaussian. The lengths and diameters of each pore segment are iteratively adjusted such that the total pore volume fraction generated is equal to the porosity.

When a pore geometry of required porosity is attained, fixed concentration boundary conditions are prescribed at two faces (with zero-flux imposed on all other faces) and the steady-state Fick’s diffusion equation is solved in the electrolyte phase. By applying mass balance equation at each node of the lattice, the effective mass diffusivity can be calculated. The sensitivity of number of nodes (from 3 ᵡ 3 ᵡ 3 nodes to 50 ᵡ 50 ᵡ 50 nodes or more if necessary) over  effective diffusivity for a range of electrode porosities will be studied to determine the optimum number of nodes over which the transport parameter becomes independent of the node number. At optimum node network, the calculated effective diffusivity will be compared with the existing theoretical models based on idealizations and simplifying assumptions such as Bruggeman, Weissberg and Maxwell correlation.

So far, no electrochemical reaction or transport is assumed at solid-liquid interface to evaluate the effective diffusivity in electrolyte phase. Apart from effective diffusivity and conductivity in electrode, the interfacial electrochemical reaction rate is another homogenized quantity based on idealized spherical particle in pseudo 2D model. Now to estimation effective diffusivity in solid phase and electrochemical reaction flux, the corresponding microscopic transport equations, for ion concentration and electric potential in solid and liquid phase in steady state is to be solve. To solve the above transport equations, specifying the state variables value at opposite end of the cube as boundary condition (Dirichlet boundary condition). Finally, the solution of transport equations is used to calculate the localized electrochemical reaction flux along the interface based on Butler-Volmer equation. To evaluate the normalized overall reaction current density, the flux integrate over the entire interface and is divided by total interfacial area. All the transport equations are solved in steady state, because diffusion and conduction occur over a much smaller distance in cubic geometry (placed at each node) than in electrode.

References

1. A. Gupta, J. H. Seo, X. Zhang, W. Du, A. M. Sastry, and W. Shyy , J. Electrochem. Soc.,  158, 487 (2011).

2. M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc., 140, 1526 (1993).

3. D. Mu, Z. S. Liu, and C. Haung, Microfluid Nanofluid, 4, 257 (2008).