Concentrated Solutions

According to Newman [834], dilute solution theory can be extended to include interactions between all species in an electrolyte. This involves:

electrochemical potentials
activity coefficient variations
variation of ionic diffusion coefficients with concentration

Moderately dilute solution theory accounts for electrochemical potentials and activity coefficients, whereas concentrated solution theory accounts additionally for concentration-dependent diffusion coefficients. The basis of concentrated solution theory is given by Stefan-Maxwell diffusion.

Similar to set-ups that use the electrochemical species model, the Li-Ion Battery Cell model interacts with electric potentials, that are provided here by the Li-Ion Electric Potential model. The species transport is described by the Li-Ion Concentration model, which basically assumes an electroneutral electrolyte—consisting of a solvent that embeds a binary salt that decomposes into a cation and an anion. As a consequence, only a single lithium/salt concentration scalar is solved for, which is understood to be the concentration of either the salt, the cation, the anion, or the intercalated lithium (when in the solid electrodes).

In the solid electrodes and current collectors, the electric current density is defined using Ohm’s law. A simple species diffusion transport equation is solved for lithium (lithium/salt concentration) transport within the electrode phase:

Figure 1. EQUATION_DISPLAY

J = σ E

(4093)

where $σ$ is the effective electrical conductivity and $E = - \nabla ϕ$ .

Within the electrolyte phase, the electric current density $J$ is due to diffusion and migration of ions (salt). Using concentrated solution theory [837], $J$ can be written as:

Figure 2. EQUATION_DISPLAY

J = κ E - \frac{κ}{F} (\frac{s_{+}}{n ν_{+}} + \frac{{t_{+}}^{0}}{z_{+} ν_{+}} - \frac{s_{0} c}{n c_{0}}) \nabla μ_{e}

(4094)

where the first term on the right-hand side corresponds to migration and the second term to diffusion. The effective electrical conductivity $κ$ in the liquid phase accounts for a non-unity porosity $χ$ and tortuosity $τ$ within the separator:

Figure 3. EQUATION_DISPLAY

κ = \frac{χ}{τ} κ_{0}

(4095)

Assuming a binary lithium based electrolyte to parametrize Eqn. (4094) (see [837] and [827]):

Figure 4. EQUATION_DISPLAY

J = κ E + \frac{2 R_{u} T κ}{F} (1 - {t_{+}}^{0}) [1 + \frac{d (ln f_{\pm})}{d (ln c)}] \nabla (ln c)

(4096)

or:

Figure 5. EQUATION_DISPLAY

J = κ E + κ_{d} \nabla (ln c)

(4097)

where:

Figure 6. EQUATION_DISPLAY

κ_{d} = \frac{2 R_{u} T κ}{F} (1 - {t_{+}}^{0}) [1 + \frac{d (ln f_{\pm})}{d (ln c)}]

(4098)

where:

$f_{\pm}$ is the salt activity.
$c$ is the salt concentration.

where:

Simcenter STAR-CCM+ uses this electrical current density definition for the electrolyte that is used in the Li-Ion Battery Cell Model.

Due to charge conservation, the electric current density field must be divergence free:

Figure 7. EQUATION_DISPLAY

\nabla \cdot J = 0

(4099)

Applying Eqn. (4093) and Eqn. (4097) (the electric current density definition) to Eqn. (4099), integration over a control volume, and subsequent application of Gauss’s theorem yields the transport equations for electric potential:

Figure 8. EQUATION_DISPLAY

\oint_{A}^{} σ \nabla ϕ \cdot d a = 0

(4100)

within the (solid) electrode and current collector phases, and:

Figure 9. EQUATION_DISPLAY

\oint_{A} κ \nabla ϕ \cdot d a = \oint_{A}^{} κ_{d} \nabla (ln c) \cdot d a

(4101)

within the (liquid) electrolyte phase.

When using the Li-Ion Battery Cell model (3D-MSE) a simple species diffusion transport equation is solved for lithium (lithium/salt concentration) transport within the electrode phase:

Figure 10. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} c d V = \oint_{A} D \nabla c \cdot d a

(4102)

where $D$ is a constant that is defined by Eqn. (4104).

An additional transport term for lithium cation migration is considered in the (liquid) electrolyte phase, so that:

Figure 11. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} χ c d V = \oint_{A} D \nabla c \cdot d a - \int_{V} \frac{J \cdot \nabla t_{+}^{0}}{F} d V

(4103)

The effective lithium/salt diffusivity coefficient $D$ accounts for non-unity values of porosity $χ$ and tortuosity $τ$ within the separator, and considers the concentrated solution correction to salt diffusion coefficient $D_{0}$ , which is written as:

Figure 12. EQUATION_DISPLAY

D = D_{0} \frac{χ}{τ} [1 - \frac{ⅆ (ln c_{0})}{ⅆ (ln c)}]

(4104)

Convective transport in the electrolyte is not modeled.

When using the concentrated electrolyte model refer to Species Transport.

Electric Potential Interfacing Condition

Conservation of the electric charge that is exchanged across the solid electrolyte interface (SEI) gives the interface condition for the boundary-specific electric current:

Figure 13. EQUATION_DISPLAY

j_{n, s} = - j_{n, l}

(4105)

where $j_{n, s}$ is the boundary-specific electric current on the electrode (solid):

Figure 14. EQUATION_DISPLAY

j_{n, s} = J_{s} \cdot n_{s}

(4106)

and $J_{n, l}$ is the boundary-specific electric current on the electrolyte (liquid):

Figure 15. EQUATION_DISPLAY

j_{n, l} = J_{s} \cdot n_{l}

(4107)

The boundary-specific electric current is the SEI normal component of the electric current density.

The normal vector always points outward of the considered electrode or electrolyte domain:

Figure 16. EQUATION_DISPLAY

n_{s} = - n_{l}

(4108)

Using the definition for the specific electric current ( Eqn. (4105)), the potential boundary condition on the electrode side is:

Figure 17. EQUATION_DISPLAY

\frac{\partial ϕ}{\partial n} |_{SEI} = - \frac{j_{n, s}}{σ}

(4109)

Using Eqn. (4109), the potential boundary condition on the electrolyte side is:

Figure 18. EQUATION_DISPLAY

\frac{\partial ϕ}{\partial n} |_{SEI} = - \frac{j_{n, l}}{κ} + \frac{κ_{d}}{κ} \frac{\partial (ln c_{1})}{\partial n}

(4110)

Concentration Interfacing Condition

By conservation of the flux of lithium $N_{+}$ across the SEI:

Figure 19. EQUATION_DISPLAY

N_{+} \cdot n_{s} = - N_{+} \cdot n_{l}

(4111)

On the electrolyte side, the flux of lithium ions is expressed as:

Figure 20. EQUATION_DISPLAY

N_{+} = \frac{J}{F} = - D Δ c \frac{1}{1 - {t_{+}}^{0}}

(4112)

Taking the inner product of Eqn. (4112) with the SEI normal vector $n_{l}$ , the interfacing condition reads:

Figure 21. EQUATION_DISPLAY

\frac{\partial c}{\partial n} |_{SEI} = - \frac{j_{n, l}}{F D} (1 - {t_{+}}^{0})

(4113)

On the solid electrode side, the lithium non-ionic flux is expressed as:

Figure 22. EQUATION_DISPLAY

N_{+} = \frac{J}{F} = - D Δ c

(4114)

Taking the inner product of Eqn. (4114) with $n_{s}$ , the concentration interfacing condition becomes:

Figure 23. EQUATION_DISPLAY

\frac{\partial c}{\partial n} |_{SEI} = - \frac{j_{n, s}}{F D}

(4115)