**2. Theory**

## *2.1. Transport Regions*

In addition to electrical conduction, ionic conductivity in the electrodes and electrolyte is necessary for the completion of electrochemical reactions. Charged species, including Li-ions can pass through a media under two driving forces: An externally applied electric field and/or a concentration gradient which is described in the Nernst–Planck relation:

$$j\_{\rm ion} = -c\_{\rm ion}v\_{\rm ion} = \frac{u\_{\rm ion}}{z\_{\rm ion}q}kT(\nabla c\_{\rm ion} + \frac{F}{kT}c\_{\rm ion}\nabla \phi) = D\_{\rm ion}(\nabla c\_{\rm ion} + \frac{F}{kT}c\_{\rm ion}\nabla \phi) \tag{1}$$

where *jion* is the ionic current density, *vion* is the drift velocity, *uion* is the electrical mobility of ions, *zion* is the valence number, and *q* is the charge. *K* is the Boltzamn constant, *Dion* is the diffusion coefficient, and ∇*φ* is the gradient of potential. Ohm's law is the relation between current density (*i*), conductivity (*σ*), and the electric field (*ζ*). By substituting the ionic current density (*j*) for *i* into Ohm's relation and including chemical potential in driving forces, as it is needed for the ions, then rewriting the derived formula for *σion* and finally comparing it with Nernst–Plank relation we come to an equation called the Nernst–Einstein:

$$
\sigma\_{ion} = \frac{c\_{ion} D\_{ion} z\_{ion}^2 F^2}{RT} \,\text{.}\tag{2}
$$

During charge and discharge, Li<sup>+</sup> transfers from one side to the other. To study the transport mechanisms in the cell we consider five regions (see Figure 1), which can be defined as follows:


**Figure 1.** Schematic view of the five transport regions considered in a full lithium ion cell.

During the charge period, intercalated Li diffuses from the cathode particle bulk to the interface (E). Particles of the solid active materials at the cathode and anode are assumed to be spherical. At the surface, it donates one electron and crosses the formed SEI layer on particles at the interface (D) to enter the electrolyte. This electrochemical reaction can be explained by the Butler–Volmer equation. Li<sup>+</sup> diffuses further in the electrolyte towards the anode because of the concentration gradient and electric field (C). By reaching the anode particles, the Li<sup>+</sup> ion transfers through the surface layer formed on the particles and receives one electron (B) and enters into the particle. Intercalated lithium diffuses away from the surface due to the concentration gradient (A). Once the cell is fully charged and the discharge starts, the whole transport process takes place in a reverse direction from the anode side to the cathode.

Calculating the ionic conductivity for each of the five regions can give us a simple way to compare the transport performances at different conditions. In this work we calculate ionic conductivity of transport for (1) solid active materials A and E, (2) electrolyte C, and (3) interfaces B and D based on the Nernst–Einstein relation.

### *2.2. Model Description*

The required data for calculation comes from an implemented model which is based on the pseudo two-dimensional (P2D) approach [28]. We used COMSOL Multiphysics® 5.4 software for simulation. A detailed explanation of the governing partial differential equations can be found in the literature [28–31]. The model considers charge and species transport along the electrodes thicknesses direction (*x*) and in the solid particles (*r*) of active materials. Equations governing the *x* and *r* directions are coupled via the electrochemical reactions on the surface of active material particles described by the Butler–Volmer relation. Lithium plating and SEI formation are considered aging mechanisms, so that they are the anode side reactions competing the intercalation reaction during the charge process. This means:

$$j\_{tot} = j\_{int} + j\_{SEI} + j\_{Lip} \tag{3}$$

where *jint* is the intercalation current density, *jSEI* is the current density of the SEI formation, and *jLiP* is the current density of lithium plating. From the Butler–Volmer relation for the intercalation current density we have the definition of:

$$j\_{int} = j\_{0, int} (\exp(\frac{\alpha\_a F \eta\_{int}}{RT}) - \exp(\frac{-\alpha\_c F \eta\_{int}}{RT})) \tag{4}$$

where *j*0,*int* is the intercalation exchange current density, *αa* and *αc* are anodic and cathodic transfer coefficients respectively. Exchange current density for intercalation can be calculated as follows:

$$j\_{0,int} = Fk\_c^{a\_d}k\_a^{a\_c}(c\_{s,max-c\_s})^{a\_d}(c\_s)^{a\_c}(\\\frac{c\_l}{c\_{l,ref}})^{a\_d} \tag{5}$$

*ka* and *kc* are the rate constants of the anodic and cathodic reactions respectively. The maximum possible concentration is *cs*,*max*, and *cs* is the local concentration of solid particles. *cl* represents the electrolyte concentration. SEI formation, which we assumed in this model is the reaction of ethylene carbonate (EC) from electrolyte with lithium ions and electrons. A detailed description about the simulation of the SEI layer can be found in previous works [32,33]. The surface overpotential for each of the reactions (intercalation, SEI, Li-plating) is:

$$\eta\_{(int,SEL,Lip)} = \phi\_{\mathfrak{k}} - \phi\_{l} - E\_{\text{eq}(int,SEL,Lip)} - \Delta\phi\_{fillm} \tag{6}$$

where *φs* and *φl* are the potential of solid and electrolyte phase respectively. <sup>Δ</sup>*φfilm* is the potential drop over the film which is forming because of the SEI and Li-plating side reactions and *Eeq* is the equilibrium potential of the corresponding reaction.

Both side reactions are assumed to be irreversible. The additional oxidation of plated lithium and consequently the formation of the secondary SEI layer on the plated lithium is neglected. Additionally, it is assumed that no partial dissolution of deposited Li during the discharge occurs. The current density of each of the side reactions is calculated, considering only the cathodic part of the Butler–Volmer relation [34]:

$$j\_{(SEI,Lip)} = -j\_{0(SEI,Lip)} \exp(\frac{-\mathfrak{a}^c(\_{SEI,Lip})F\eta\_{(SEI,Lip)}}{RT}) \tag{7}$$

Cell parameters originate from an experimental cell. Key cell parameters and simulation conditions are listed in Table 1. Diffusion coefficient of electrodes are adjusted based on Cabanero's

work [35] to include a degree of lithiation dependency. The cycling is simulated using a constant current/constant voltage (CC/CV) charging strategy. Discharge is simulated using CC only. Since no thermal modeling is included, a 0.5 C charge/discharge rate is applied for the whole simulation so that the temperature variation over charge and discharge can be neglected [36].


**Table 1.** Cell parameters and simulation conditions used in the model.

**Figure 2.** Diffusion coefficient (**A**) of the anode and (**B**) of the cathode as a function of the lithiation degree.

Based on the Nernst–Einstein relation, we defined ionic conductivity for each of the regions as follows:

$$
\sigma\_{l(a,c,s)} = \frac{c\_l D\_{l(a,c,s)}^{eff} F^2}{RT} \tag{8}
$$

*σl* is the ionic conductivity of electrolyte (region C). *Deff l* is the effective diffusion coefficient of electrolyte which is defined depending on its medium i.e., anode, cathode, or separator. It is calculated with the Bruggeman correlation. So that:

$$D\_{l(a,c,s)}^{eff} = D\_l \mathfrak{e}\_{l(a,c,s)}^{\gamma} \tag{9}$$

consisting of the electrolyte diffusion coefficient *Dl*, the electrolyte volume fraction *l*(*<sup>a</sup>*,*c*,*<sup>s</sup>*), and the Bruggeman exponent *γ*.

For the particles of solid active material (region A and E) we have:

$$
\sigma\_{s(a,c)} = \frac{(\mathfrak{c}\_{s,\text{max}(a,c)} - \mathfrak{c}\_{s,\text{ave}(a,c)})D\_{s(a,c)}F^2}{RT} \tag{10}
$$

in which *Ds* is the diffusion coefficient of the solid particles.

To calculate the ionic conductivity for the last two regions (B and D) we assume the interface reactions to consume Li ion from electrolyte at electrode/electrolyte interface. Therefore ionic conductivity is:

$$
\sigma\_{i(a,c)} = \frac{c\_{s(a,c)} \left(\frac{c\_l}{c\_{l-ref}}\right)^{\alpha\_d} F^2}{RT} \tag{11}
$$

including surface and electrolyte concentrations (*cs* and *cl*) coming from the Butler–Volmer relation definitions and effective electrolyte diffusion coefficients.
