#
Electrochemical Model-Based Investigation of Thick LiFePO_{4} Electrode Design Parameters

^{*}

## Abstract

**:**

_{4}electrode based on the P2D model and discusses it with common literature values. With a superior macrostructure providing a vertical transport channel for lithium ions, a simple approach could be developed to find the best electrode structure in terms of macro- and microstructure for currents up to 4C. The thicker the electrode, the more important are the direct and valid transport paths within the entire porous electrode structure. On a smaller scale, particle size, binder content, porosity and tortuosity were identified as very impactful parameters, and they can all be attributed to the microstructure. Both in modelling and electrode optimisation of lithium-ion batteries, knowledge of the real microstructure is essential as the cross-validation of a cellular and lamellar freeze-casted electrode has shown. A procedure was presented that uses the parametric study when few model parameters are known.

## 1. Introduction

_{4}), shortform LFP. The anode is typically graphite. Lithium-ion accumulators are characterised by a high efficiency and energy density, a good cycle stability and a satisfying operating life. They can be used in a wide temperature range (243–333 K) and have a low self-discharge [1]. However, the electrification of the powertrain is limited by the relatively low energy and power density of lithium-ion batteries compared to fossil fuels. In addition to a high specific capacity, the electrodes must provide a fast charging capability, a low price and excellent safety. On the material level of the cathode, LFP could be one of the promising candidates [2]. Besides the continuous improvement of cathode materials, electrolytes [3] and separators [4,5] or the replacement of graphite by metal anodes [6], the cathode structure could be an essential factor to achieve these goals. In general, porous electrodes with low tortuosity provide efficient ion flow and ensure uniform replenishment of the lithium supply, especially at high discharge rates [7]. The charge and discharge cycles can also be stabilised by a porous structure with small particles [8]. In order to gain a deep scientific understanding of physical and electrochemical processes associated with the material structure, extensive data or simulations are required [9]. Without this prior knowledge, the search for a suitable material composition and structure (e.g., porosity, tortuosity, thickness or mass loading) is like the fabled search for the needle in a haystack. Simulations can provide a quick insight into problems or processes and save resources, time and effort.

## 2. Model Development

_{6}) electrolyte in a mixture of propylene carbonate, ethylene carbonate and dimethyl carbonate (PC/EC/DMC, 10:27:63 by volume). The lithium zone is modelled as a boundary condition. During the discharge process, shown in Figure 1, lithium ions diffuse from the lithium electrode through the separator into the electrolyte and finally the LiFePO

_{4}electrode.

#### 2.1. Electrochemical Model

#### 2.2. Electrolyte Equations

#### 2.3. Electrode Equations

_{4}at room temperature, the standard equilibrium potential ${E}_{0,ref}$ and the entropy change $\frac{\partial {E}_{eq}}{\partial T}$ were found to be approximations of the State-of-Lithiation (SoL) [28]:

^{−1}and the mass of the active material:

#### 2.4. Material and Geometric Properties

^{−1}[48].

## 3. Results and Discussion

#### 3.1. Investigation of Electrode Design Parameters

#### 3.1.1. Influence of the Electrode Volume Fraction

#### 3.1.2. Influence of the Electrode Conductivity

^{−1}) [51]. Delacourt et al. proposed LFP to be coated with carbon, which forms a conductive network that improves electrical conductivity depending on the carbon amount, especially at high current rates [51]. A correlation between electrical and ionic conductivity was observed, with higher electrical conductivity possibly creating a space charge region leading to better ionic conductivity. Byles et al. emphasised the importance of the relationship between the two conductivities and, contrary to the prevailing opinion, showed the importance of the ionic conductivity using the example of manganese oxide [52]. Nevertheless, the ionic conductivity is usually expressed only in terms of the Li-ion diffusion coefficient [52], as was also assumed in this modelling approach and only the electrical conductivity was examined.

^{−1}by Srinivasan et al. [34] in combination with the chosen default values was not applicable to the model and led to an insufficient gravimetric capacity. Thorat et al. pointed out the difference between wet (electrolyte-flooded) and dry (bulk) electrical conductivity, with dry electrical conductivity being ten times higher [26]. A more precise specification was nowhere to be found in any further modelling approach. The comparatively low values of 0.03 S m

^{−1}determined by Kashkooli et al. [30] and 0.04 S m

^{−1}determined by Saw et al. [24] also hindered the full activation of the LFP electrode. It is noticeable that the lithiation for low conductivities (<10

^{−1}S m

^{−1}) is strongly inhomogeneous and depends on the height of the electrode (cathode thickness). The value of Li et al., i.e., 0.5 S m

^{−1}, was already sufficient for activating the entire thickness of the electrode [27]. However, this value still increases the ohmic overpotential. Only values above >10

^{−0}S m

^{−1}no longer show any noticeable changes. The value of 16 S m

^{−1}, which corresponds to a carbon content of 4.75% within the LFP [51], was chosen to be the best because the investigated freeze-casted electrodes have a similar carbon content of 5%. Higher values seem to be implausible when comparing all the examined values. Since the majority of the most models assume an electrode thickness between 50 and 75 µm, the massive deviations in the values are obviously not significant. It is thus questionable to use this parameter for model fitting.

#### 3.1.3. Influence of the Electrode Diffusion Coefficient

^{−1}by approx. 25%. Increasing the prefactor 4.66 times ($~5.5\times {10}^{-18}$ from [31]) leads the limitation to decrease to approx. 6%. The effect of the prefactor is most pronounced in this range. Maheshwari et al. [29] and Saw et al. [24], presenting values situated four orders of magnitude above the previously considered values, utilise the full potential of the electrode. For further consideration in this work, the prefactor of $2.2\times {10}^{-14}$ was chosen, since this value is sufficient for completely activating the electrode and higher values have no additional influence on the performance. Based on the problem of determining the diffusion coefficient [56], this value is a reasonable fitting parameter for the capacity of a model, as there is no significant impact on other parameters.

#### 3.1.4. Influence of the Particle Radius

#### 3.1.5. Influence of the Filler Content

#### 3.1.6. Influence of the Electrolyte

_{6}in a mixture of PC/EC/DMC, in a temperature range between 263 and 333 K and in a concentration range between of $7.7\times {10}^{-6}$ M and 3.9 M [41]. The empirical-analytical form for the diffusivity and for the ionic conductivity has been successfully used in many models [27,28,31,47]. Valøen and Reimers have also pointed out that the magnitude of the diffusion coefficient for their composition are similar to those of EC/EMC and EC/DEC, which makes an adaptation evident [41]. The results of Berhaut et al. confirm and extend this statement to the mixture of EC/DMC [63]. However, pure EC and DMC have very different properties in combination with Li ions [64], meaning that the formula cannot cover all combinations of solvents and an adaptation is necessary. This work uses the formulas given by the Equations (3)–(5), which were modified to obtain the reported values at 1 M and 298.15 K.

^{2}s

^{−1}at 298.15 K and 1 M from [27]) allows full activation of the electrode. Halving the prefactor, which corresponds to an electrolyte diffusion coefficient of about $1.5\times {10}^{-10}$ m

^{2}s

^{−1}reported in [26], shows a considerable loss of energy in which the entire length of the electrode is not sufficiently supplied with lithium ions. Even lower values intensify this effect and, in addition to a massive loss of specific capacity, also lead to a drop in cell potential. The electrolyte diffusion coefficient influences the diffusion term in Equation (11) and thus the Li-ion transport in the liquid phase. The already mentioned filler content scales Equation (11) not only in terms of diffusivity. Since the simulation of both parameters produces similar results, this implies that transport by diffusion is the dominant process. As already mentioned in the subsection on electrode conductivity, this effect may not be noticeable in most models, as it only occurs at a penetration depth of 100 µm, even with a low diffusion coefficient, as depicted in Figure 8b. However, it should be critically noted that the electrode has already been optimised for pore volume at this point.

^{−1}have no influence on the lithiation and almost none on the voltage curve, as depicted in Figure 9. In contrast, the lowest value of 0.25 S m

^{−1}strongly flattens the voltage plateau with increasing discharge capacity, but practically no capacity loss occurs. This effect could be an interesting fitting parameter due to the uncertainty of using different electrolyte compositions.

#### 3.1.7. Influence of the Geometry

^{−1}and a diffusion coefficient of $1.18\times {10}^{-18}$ of LFP, as well as the lowest electrolyte diffusion peak of $7.5\times {10}^{-11}$, have practically no effect on the cell performance for thin electrodes as long as the particle radius is small. Figure 11b shows that with bigger particle radius the associated type of ion transport (solid or liquid phase) has a tremendous influence, even on thin electrodes. It can be deduced that in the investigated models, the particle size distribution (PSD) in combination with the diffusion coefficient of the electrode is decisive for the achievable capacities.

#### 3.1.8. Influence of the Tortuosity

#### 3.2. Performance Tests

^{−1}. With a nominal cell voltage of 3.2 V, this results in a formal energy density of 544 Wh kg

^{−1}and at 4C in a power density of 2176 W kg

^{−1}. In the following, the time average of the voltage curve was used to calculate the power and energy density instead of considering the nominal cell voltage.

^{−1}) with increasing current up to 4C, provided there is sufficient solid diffusion. As already discussed, a high content of active material leads to a reduction in power and energy density. The electrolyte channel can counteract this in a similar way, e.g., through increasing the porosity in the single structure. Thus, a lamellar structure offers more adjustment parameters for designing an optimised electrode especially intended for high currents. For small currents up to 1C, it is rather unimportant whether the necessary increase in porosity for a good performance with thick electrodes occurs through the macrostructure or through the microstructure, as long as the overall porosity remains approximately the same (as depicted in Figure 17a,c). This statement could be relativised in that sense as real tortuosities are used for the microstructure, and they are often significantly larger [25]. The validation in the next subsection also shows the limitation of the idealised P2D approach, since although the cellular structure presented, i.e., comparable to a highly porous electrode, had a similar porosity to the lamellar structure, the performance of the lamellar structure was significantly better.

#### 3.3. Validation and Usability

- All physical parameters that are verified, e.g., by the manufacturer’s specifications or experiments, must be taken into account.
- The best parameter set related to the theoretical study should be chosen for the undetermined parameters in order to start without any limitations.
- The external circumstances such as the discharge or charging current and the temperature must be clearly defined.
- The electrode diffusion coefficient should be determined to scale the achievable capacity (see Figure 5).
- An adjustment of the inhomogeneities can be done by the diffusivity as well as the conductivity of the electrolyte. Since in the porous electrode these two parameters are influenced by the microstructure (see Equations (8) and (9)), an adjustment of the microstructure would likewise be target-oriented (see Figure 7, Figure 8 and Figure 9).

^{−1}. Hence, a new solid diffusion prefactor was determined with ${D}_{LFP}=4.5\times {10}^{-19}$. Nevertheless, the measurement compared to the simulation reveals that the mass transport is hindered with decreasing State-of-Charge. According to the procedure, it is obvious to fit the curve next by the electrolyte diffusion coefficient. However, this would counteract the idea of the proposed procedure, as the lamellar and cellular parameter set would no longer be validated among each other. Therefore, it is more obvious to make an adjustment via the microstructural parameters, which are different for the lamellar structure as shown in [39]. The obtained parameter set A is shown in Table 4. Due to the measured porosities, an even higher solid content in the lamella is assumed (see Table 4 parameter set B), and the solid diffusion prefactor had to be modified. This in turn reiterates the importance of microstructure, which has already been emphasised several times, as it has a significant influence on validation.

## 4. Conclusions

_{4}electrode to study the design parameters of the electrode. For this purpose, the discharge curves at different parameters and the lithiation of the electrode were examined in more detail in order to understand more about the underlying processes in a visual way. The parameters were taken from a variety of other studies and discussed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LFP | Lithium iron phosphate |

SoL | State-of-Lithiation |

P2D | Pseudo two-dimensional model |

DoD | Depth-of-Discharge |

## Appendix A

Name Identifier | Parameter | Value | Source |
---|---|---|---|

Haverkort | ${\epsilon}_{s}$ | 0.4 | [36] |

Saw et al. | ${\sigma}_{LFP}$ | 0.04 S m^{−1} | [24] |

${D}_{LFP}$ | $8\times {10}^{-14}$ | ||

${r}_{p}$ | 1 µm | ||

Li et al. | ${\sigma}_{LFP}$ | 0.5 S m^{−1} | [27] |

${D}_{LFP}$ | $1.18\times {10}^{-18}$ | ||

${r}_{p}$ | 3.5 µm | ||

${D}_{l}$ | $3.2\times {10}^{-10}$ m^{2} s^{−1} | ||

Mastali et al. | ${\sigma}_{LFP}$ | 6.75 S m^{−1} | [31] |

${D}_{LFP}$ | $5.5\times {10}^{-18}$ | ||

${r}_{p}$ | 320 nm | ||

Yu et al. | ${\sigma}_{LFP}$ | 11.8 S m^{−1} | [23] |

${D}_{LFP}$ | $6\times {10}^{-18}$ | ||

${D}_{l}$ | $3\times {10}^{-10}$ m^{2} s^{−1} | ||

Huang et al. | ${\sigma}_{LFP}$ | 91 S m^{−1} | [49] |

${r}_{p}$ | 8 µm | ||

${D}_{l}$ | $7.5\times {10}^{-11}$ m^{2} s^{−1} | ||

Delacourt et al. | ${\sigma}_{LFP}$ | 16 S m^{−1} | [51] |

Kashkooli et al. | ${D}_{LFP}$ | $7\times {10}^{-18}$ | [30] |

Srinivasan et al. | ${D}_{LFP}$ | $8\times {10}^{-18}$ | [34] |

${r}_{p}$ | 52 nm | ||

Maheshwari et al. | ${D}_{LFP}$ | $2.2\times {10}^{-14}$ | [28] |

${D}_{l}$ | $1.3\times {10}^{-10}$ m^{2} s^{−1} | ||

${\sigma}_{l}$ | 0.77 S m^{−1} | ||

Zavareh et al. | ${r}_{p}$ | 125 nm | [39] |

Thorat et al. | ${D}_{l}$ | $1.5\times {10}^{-10}$ m^{2} s^{−1} | [26] |

Prada et al. | ${\sigma}_{l}$ | 0.25 S m^{−1} | [65] |

Valøen et al. | ${\sigma}_{l}$ | 1.2 S m^{−1} | [41] |

Wu et al. | ${\sigma}_{l}$ | 1.4 S m^{−1} | [57] |

**Figure A1.**Time evolution of the normalised concentration of Li ions ${c}_{Li}$ (lithiation) within the single structure LFP electrode. The volume fraction is ${\epsilon}_{s}=0.4$ with an electrode conductivity of ${\sigma}_{LFP}=16$ S m

^{−1}and a diffusion prefactor of ${D}_{LFP}=1.18\times {10}^{-18}$ at 1C discharge current and 298.15 K. The illustration of the cell (top—separator, bottom—current—collector) is not to scale.

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**Figure 1.**Schematic view of the electrochemical cell consisting of a porous LFP electrode and an ideal lithium electrode separated by an electrolyte zone and a separator. The figure shows the lithium diffusion through the cell and the lithiation of the LFP electrode during discharge.

**Figure 2.**Schematic view of the geometric cell development with (

**a**) a single structure and (

**b**) an ideal lamellar structure with an electrolyte channel width ${w}_{l}$. The single structure is comparable to most model designs and represents a solid electrode with an inner porous structure as shown in Figure 1.

**Figure 3.**Parameter variation of the solid volume fraction. The discharge performance at 1C is shown in (

**a**), and the lithiation along the electrode at a DoD = 1 is illustrated in (

**b**), where the height of 0 mm corresponds to a position close to the current collector, and 0.5 mm is situated near the electrolyte reservoir.

**Figure 4.**Parameter variation of the LFP electrode conductivity. The discharge performance at 1C with an optimised ${\epsilon}_{s}=0.4$ is shown in (

**a**), and the lithiation along the electrode height at an DoD = 1 is illustrated in (

**b**).

**Figure 5.**Parameter variation of the LFP electrode diffusion prefactor. The discharge performance at 1C with an optimised ${\epsilon}_{s}=0.4$ and ${\sigma}_{LFP}=16$ S m

^{−1}is depicted in (

**a**), and the lithiation along the electrode height at an DoD = 1 is shown in (

**b**).

**Figure 6.**Parameter variation of the particle radius of the LFP electrode: (

**a**) the discharge performances with ${D}_{LFP}=2.2\times {10}^{-14}$at 1C with an optimised ${\epsilon}_{s}=0.4$ and ${\sigma}_{LFP}=16$ S m

^{−1}and (

**b**) with ${D}_{LFP}=1.18\times {10}^{-18}$.

**Figure 7.**Parameter variation of the filler volume fraction within the LFP electrode. The discharge performance is depicted in (

**a**) for the best and the worst solid diffusion at 1C with an optimised ${\epsilon}_{s}=0.4$ and ${\sigma}_{LFP}=16$ S m

^{−1}and in (

**b**) the lithiation at reaching the cut-off voltage of 2.5 V.

**Figure 8.**Parameter variation of the electrolyte diffusion prefactor ${D}_{LiPF6}$ correlated to the reported values of the electrolyte diffusion coefficient ${D}_{l}$ at 1 M and 298 K; (

**a**) shows the discharge performance and (

**b**) the lithiation at a DoD = 1.

**Figure 9.**Parameter variation of the electrolyte conductivity prefactor ${\sigma}_{LiPF6}$ correlated to the reported values of ${\sigma}_{l}$ at 1 M and 298 K as well as the variation of the transport number ${t}_{+}$; (

**a**) shows the discharge performance and (

**b**) the lithiation at a DoD = 1.

**Figure 10.**Parameter variation of the basic geometric variables of the LFP electrode. (

**a**) shows the discharge performance related to the variation of the height (thickness). The discharge performance for the variation of the electrode width is depicted in (

**b**).

**Figure 11.**Discharge performance of a thin 50 µm electrode at 1C, ${\epsilon}_{s}=0.4$, ${\sigma}_{LFP}=0.04$ S m

^{−1}and ${\sigma}_{LFP}=16$ S m

^{−1}as well as peak electrolyte diffusivity of ${D}_{l}=7.5\times {10}^{-11}$ m

^{2}s

^{−1}and ${D}_{l}=3.2\times {10}^{-10}$ m

^{2}s

^{−1}at 1 M. In (

**a**) the particle size is ${r}_{p}=52$ nm and in (

**b**) ${r}_{p}=8$ µm. The black curve represents a diffusion coefficient of $1.18\times {10}^{-18}$, and the blue curve stands for a diffusion coefficient of $2.2\times {10}^{-14}$.

**Figure 12.**Changing the tortuosity by building a macrostructure that creates a transport channel with an adjustable width ${w}_{l}$ and $\tau =1$. The discharge performances are shown in (

**a**) with ${D}_{LFP}=2.2\times {10}^{-14}$ at 1C with an optimised ${\epsilon}_{s}=0.4$ and ${\sigma}_{LFP}=16$ S m

^{−1}and in (

**b**) with ${D}_{LFP}=1.18\times {10}^{-18}$.

**Figure 13.**Simulation of a macrostructure with an additional transport channel with an adjustable width ${w}_{l}$ and $\tau =1$; (

**a**) shows the discharge performance at 4C with ${D}_{LFP}=2.2\times {10}^{-14}$ and in (

**b**) with ${D}_{LFP}=1.18\times {10}^{-18}$ at 4C.

**Figure 14.**Performance test of the single-structured electrode; (

**a**) shows the discharge performance from C/4 up to 4C and (

**b**) the temperature dependency from 273.15 up to 333.15 K at 1C.

**Figure 15.**Discharge Performance of the lamella structure at 1C related to the gravimetric (

**a**,

**b**) and volumetric capacity (

**c**,

**d**) at D

_{LFP}= 2.2 × 10

^{−14}(

**a**,

**c**) and D

_{LFP}= 1.18 × 10

^{−18}(

**b**,

**d**) depending on the solid content ε

_{S}. The circle markers identify the specific time points shown in the corresponding 2D plots. The lamella shows the normalized concentration of Li ions C

_{Li}, and the transport channel shows the electrolyte potential ϕ

_{l}Points 2 and 5 represent the time after 30 min discharge.

**Figure 16.**Discharge performance at a constant current density of ${j}_{DC}=10$ mA cm

^{−2}at ${D}_{LFP}=2.2\times {10}^{-14}$ for the lamella structure (

**a**,

**c**) and for the single structure (

**b**,

**d**) (see Figure 2)

**.**In (

**c**,

**d**), the capacity is related to the current collector area and in (

**a**,

**b**) to the active mass.

**Figure 17.**Ragone plot comparing the gravimetric energy vs. power density for different macro- and microstructure combinations. The energy density is obtained by multiplying the specific capacity with the average discharge voltage, and the power density is calculated by dividing the energy density by the discharge time. The variation of the electrolyte channel width is shown with a dense (

**a**) and with a porous (

**b**) lamella at ${D}_{LFP}=1.18\times {10}^{-18}$and with ${D}_{LFP}=2.2\times {10}^{-14}$ (

**d**,

**e**) as long with the variation of the solid content in a single structure porous electrode (

**c**,

**f**).

**Figure 18.**Determination of the model parameters using the proposed procedure for simulating the cellular structure.

**Table 1.**Parameter set for the standard equilibrium potential taken from [28].

$\mathit{i}$ | ${\mathit{k}}_{\mathit{i}}$ | ${\mathit{l}}_{\mathit{i}}$ | ${\mathit{m}}_{\mathit{i}}$ |
---|---|---|---|

1 | −1.239 | −7.903 | 0.3821 |

2 | $3.644\times {10}^{-10}$ | 21.12 | 30.37 |

3 | $8.249\times {10}^{-12}$ | 22.39 | 1.56 |

**Table 2.**Parameter set of the entropy change taken from [28].

${\mathit{g}}_{0}$ | ${\mathit{g}}_{1}$ | ${\mathit{g}}_{2}$ | ${\mathit{g}}_{3}$ | ${\mathit{g}}_{4}$ | ${\mathit{g}}_{5}$ | ${\mathit{g}}_{6}$ | ${\mathit{g}}_{7}$ | ${\mathit{g}}_{8}$ |
---|---|---|---|---|---|---|---|---|

$1.9186\times {10}^{-5}$ | 0.0032158 | −0.046272 | 0.28857 | −0.98716 | 1.9635 | −2.2585 | 1.3902 | −0.35376 |

Parameter | Value | Unit | Reference |
---|---|---|---|

electrolyte | |||

${D}_{l}$ | (5) | cm^{2} s^{−1} | [41] |

${D}_{LiPF6}$ | 1 | - | [41] |

${\sigma}_{l}$ | (3) | mS cm^{−1} | [41] |

${\mathsf{\sigma}}_{LiPF6}$ | 1 | - | [41] |

$f$ | (4) | - | [41] |

${t}^{+}$ | 0.38 | - | [41] |

${c}_{l,ref}$ | 1000 | mol m^{−3} | [47] |

electrode | |||

${D}_{s}$ | (27) | m^{2} s^{−1} | [27] |

${D}_{LFP}$ | $1.18\times {10}^{-18}$ | - | [32] |

${\sigma}_{LFP}$ | 0.5 | S m^{−1} | [27] |

${E}_{eq}$ | (12) | V | [31] |

${c}_{s,max}$ | 16,481 | mol m^{−3} | calculated |

${c}_{ref,Li}$ | 1 | mol m^{−3} | assumed |

$So{L}_{min}$ | 0.01 | - | assumed |

$So{L}_{\mathrm{max}}$ | 0.99 | - | assumed |

${r}_{p}$ | $1.25\times {10}^{-7}$ | m | [39] |

${\epsilon}_{s}$ | 0.4 | - | calculated |

${\epsilon}_{l}$ | 0.6 | - | [36] |

$\alpha $ | 1.5 (spherical) | - | [40] |

${\rho}_{LFP}$ | 2.6 | g cm^{−3} | [39] |

electrode kinetics | |||

${\alpha}_{a}$ | 0.5 | - | [32] |

${\alpha}_{c}$ | 0.5 | - | [32] |

${k}_{a}$ | (17) | m s^{−1} | [27] |

${k}_{c}$ | (18) | m s^{−1} | [27] |

${k}_{Li}$ | $1\times {10}^{-4}$ | m s^{−1} | assumed |

${E}_{a}$ | 30 | kJ mol^{−1} | [27] |

${E}_{c}$ | 20 | kJ mol^{−1} | [27] |

${E}_{s}$ | 35 | kJ mol^{−1} | [27] |

z | 1 | - | assumed |

m | spherical: = 3 | - | assumed |

separator | |||

${\epsilon}_{l,sep}$ | 0.724 | - | [24] |

current collector | |||

${\sigma}_{alu}$ | $-0.0325{T}^{3}+37.07{T}^{2}-1.5\times {10}^{4}T+2.408\times {10}^{6}$ | S cm^{−1} | [29] |

**Table 4.**Set of parameters for sets A and B in Figure 19.

Set | ${\mathit{D}}_{\mathit{L}\mathit{F}\mathit{P}}\text{}$ | ${\mathit{D}}_{\mathit{L}\mathit{i}\mathit{P}\mathit{F}6}\text{}$ | ${\mathit{\epsilon}}_{\mathit{s}}$ | ${\mathit{\epsilon}}_{\mathit{l}}$ |
---|---|---|---|---|

A | $4.5\times {10}^{-19}$ | 0.162 | 0.4 | 0.35 |

B | $4.2\times {10}^{-19}$ | 0.3 | 0.6 | 0.35 |

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**MDPI and ACS Style**

Franke-Lang, R.; Kowal, J.
Electrochemical Model-Based Investigation of Thick LiFePO_{4} Electrode Design Parameters. *Modelling* **2021**, *2*, 259-287.
https://doi.org/10.3390/modelling2020014

**AMA Style**

Franke-Lang R, Kowal J.
Electrochemical Model-Based Investigation of Thick LiFePO_{4} Electrode Design Parameters. *Modelling*. 2021; 2(2):259-287.
https://doi.org/10.3390/modelling2020014

**Chicago/Turabian Style**

Franke-Lang, Robert, and Julia Kowal.
2021. "Electrochemical Model-Based Investigation of Thick LiFePO_{4} Electrode Design Parameters" *Modelling* 2, no. 2: 259-287.
https://doi.org/10.3390/modelling2020014