Theoretical analysis of transient solution phase concentration field in a porous composite electrode with time-dependent flux boundary condition

Abstract

Energy conversion and storage in a Li-ion cell involves multiple closely coupled transport processes, such as species diffusion through solid and solution phases in the electrode. Mathematical modeling of these processes is critical for fully understanding and optimizing the performance of a Li-ion cell. While a number of analytical and numerical models have been presented for solution phase diffusion, most of such work is based on the assumption of a constant current density. This paper presents analytical modeling of solution phase diffusion in a separate–electrode composite for a generalized, time-dependent current density. An analytical solution for the concentration field in a separator–electrode composite in such conditions is derived using the method of eigenfunction expansion. Good agreement with past work as well as numerical simulations is shown. Results for linear, periodic and step-function boundary conditions are discussed. The theoretical analysis presented here may help accurately model realistic processes where the applied current changes over time, for example, cyclic charge and discharge in an electric vehicle, or sudden changes in the battery load. Results presented here contribute towards the fundamental understanding of solution phase diffusion in Li-ion cells, and provide a basis for improving electrochemical energy conversion and storage processes.

Graphic abstract

Appendix A: Dimensional equations and non-dimensionalization scheme

Based on the assumptions listed in Sect. 2, the concentration conservation equations for separator and electrode layers may be written in dimensional form as follows:

$$ \frac{{\partial c_{1} }}{\partial t} = D\frac{{\partial^{2} c_{1} }}{{\partial x^{2} }} $$

(37)

and

$$ \varepsilon \frac{{\partial c_{2} }}{\partial t} = D\varepsilon^{3/2} \frac{{\partial^{2} c_{2} }}{{\partial x^{2} }} - \frac{{I\left( t \right)\left( {1 - t_{ + } } \right)}}{{F \cdot L_{2} }} $$

(38)

Note that the migration term is zero based on the common assumption of constant transference number [42]. The term $- \frac{{I\left( t \right)\left( {1 - t_{ + } } \right)}}{{F \cdot L_{c} }}$ appearing in Eq. (38) represents the pore wall flux, assumed to be uniform and constant [42]. The dimensional boundary conditions are

$$ \frac{{\partial c_{1} }}{\partial x} = - \frac{{I\left( t \right)\left( {1 - t_{ + } } \right)}}{F \cdot D}\quad {\text{at}}\,x = 0 $$

(39)

$$ c_{1} = c_{2} \quad {\text{at}}\,x = L_{1} $$

(40)

$$ \frac{{\partial c_{1} }}{\partial x} = \varepsilon^{3/2} \frac{{\partial c_{2} }}{\partial x}\quad {\text{at}}\,x = L_{1} $$

(41)

$$ \frac{{\partial c_{2} }}{\partial x} = 0\quad {\text{at}}\,x = L_{1} + L_{2} $$

(42)

The initial condition is

$$ c_{1} = c_{2} = c_{0} \quad {\text{at}}\,t = 0 $$

(43)

The non-dimensional variables are defined as follows

$$ \tau = \frac{Dt}{{L_{1}^{2} }};\quad \xi = \frac{x}{{L_{1} }};\quad C_{1} = \frac{{c_{1} }}{{c_{0} }};\quad C_{2} = \frac{{c_{2} }}{{c_{0} }} $$

(44)

Substituting in Eqs. (37)–(38), one may obtain the following non-dimensional governing equations:

$$ \frac{{\partial C_{1} }}{\partial \tau } = \frac{{\partial^{2} C_{1} }}{{\partial \xi^{2} }} $$

(45)

$$ \frac{{\partial C_{2} }}{\partial \tau } = \sqrt \varepsilon \frac{{\partial^{2} C_{2} }}{{\partial \xi^{2} }} + J\left( \tau \right) $$

(46)

where

$$ J\left( \tau \right) = - I\left( \tau \right)\frac{{\left( {1 - t_{ + } } \right)L_{1}^{2} }}{{FDL_{2} c_{0} \varepsilon }} $$

(47)

Similarly, substituting in the boundary conditions, given by Eqs. (39)–(42), one may obtain

$$ \frac{{\partial C_{1} }}{\partial \xi } = \varepsilon rJ\quad {\text{at}}\,\xi = 0 $$

(48)

$$ C_{1} = C_{2} \quad {\text{at}}\,\xi = 1 $$

(49)

$$ \frac{{\partial C_{1} }}{\partial \xi } = \varepsilon^{3/2} \frac{{\partial C_{2} }}{\partial \xi }\quad {\text{at}}\,\xi = 0 $$

(50)

$$ \frac{{\partial C_{2} }}{\partial \xi } = 0\quad {\text{at}}\,\xi = 1 + r $$

(51)

where r = L₂/L₁. The initial condition in non-dimensional form is

$$ C_{1} = C_{2} = 1\quad {\text{at}}\,\tau = 0 $$

(52)

Equations (45)–(52) constitute the set of non-dimensional equations solved in this work (Eqs. (1)–(4)).