# Stress-Affected Lithiation Reactions in Elasto-Viscoplastic Si Particles with Hyperelastic Polymer Coatings: A Nonlinear Chemo-Mechanical Finite-Element Study

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Chemo-Mechanical Model for a Polymer-Coated Spherical Si Particle

#### 2.1. Model Background

_{−}(a-Si) + ${n}_{\ast}$Li → ${\mathrm{Li}}_{{n}_{\ast}}{\left(\mathrm{a}-\mathrm{Si}\right)}_{{n}_{-}}$, where it is assumed that the transformed material has 3.75 lithium atoms per one Si atom (i.e., Li

_{3.75}(a-Si))—this in line with the previously published papers (see e.g., [27]), where the ratio of stoichiometric coefficients ${n}_{\ast}$/n

_{−}varies from 2.5 to 3.75. In addition, the maximum achievable concentration of Li ions in a-Si, taken here as 4.4 (Li) to 1 (a-Si) ratio, is used to calculate solubility (see [30])–the latter also signifies that there are 0.65 free Li atoms (per 1 a-Si atom) available for the diffusion process.

#### 2.2. Governing Equations for a Coated Spherical Si Particle With a Non-Stationary Reaction Front

_{p}, respectively, which are given as

_{1}+ C

_{2}r, where the coefficients C

_{1}and C

_{2}are determined from the boundary conditions (cf. Equations (7) and (8)).

#### 2.3. Constitutive Relations

#### 2.3.1. Transformed and Untransformed Components of Si Particle

#### 2.3.2. Hyperelastic Polymer Coating

## 3. Finite Element solution of the Problem

_{Γ}, and either stress-free or fully-constrained conditions at R = R

_{T}), follows a 1D numerical solution procedure due to the spherical symmetry of the problem. Thus, the problem is discretised by dividing the entire particle domain with the coating ${R}_{\mathrm{T}}$ into $\left(N+1\right)$ equally spaced nodes, each separated by the fixed distance $\Delta R$ = R

_{0}/N, where R

_{0}denotes the centre of the particle. The corresponding 1D FE mesh is shown in Figure 2, with the reaction front moving by one node towards the centre in a single timestep between the two time instances t = t

_{n}and t

_{n}

_{+1}, transforming previously chemically untransformed finite-element domains in the process. Thus, the position of the reaction front can only be at nodal points of the mesh.

_{i}and timestep $\Delta t$. The entire numerical procedure was implemented via MATLAB scripting.

## 4. Results and Discussion

#### 4.1. Model Parameters

^{−1}nm to h = 10

^{2}nm, which is in agreement with some coating thicknesses reported in the literature. For example, a polymer-based particle coating made of polypyrrole (PPy) with a thickness of 5–10 nm was applied to Si nanoparticles with diameters of 150 nm and 380 nm [19]. Shear moduli of typical polymers at room temperature range from a few tenths of MPa (cross-linked rubbers) to several GPa (thermosetting resins). However, here a wider range of the shear moduli from G

_{T}= 10

^{−2}GPa to G

_{T}= 10

^{1.5}GPa is investigated, which is well beyond currently existing polymers. Nevertheless, this extended range enables a more systematic parametric study, and it also provides an incentive for developing new polymeric materials with enhanced mechanical properties for energy storage-related applications. A similarly wide range of bulk moduli is studied from K

_{T}= 10

^{−2}GPa to K

_{T}= 10

^{1.5}GPa, where a typical value of bulk modulus for polymers is at the order of a few GPa–again, that extended range allows for a more systematic study of the influence of polymer coating properties on the lithiation process. A radius of R

_{P}=150 nm was chosen for Si particles, based on the previous research [34]. All other parameters used are given in Table A1 in Appendix A.

#### 4.2. Effects of Coating Moduli on Stresses During Lithiation Process

_{p}). Those complicated stress profiles result from the accumulation of the plastic deformation in the core, and were discussed in detail for the case of Si particles without the coating layer (see [30]).

_{p}), those stresses were investigated further at different levels of lithiation. The results are plotted in Figure 5a for six different combinations of shear and bulk moduli. Compressive circumferential stresses are found to be dominant in the initial stages of lithiation (up to ${R}_{\mathsf{\Gamma}}=0.6{R}_{\mathrm{P}}$). As the reaction front moves, the stresses become tensile for three different combinations of coating properties (${G}_{\mathrm{T}}={K}_{\mathrm{T}}=1\mathrm{GPa}$; ${G}_{\mathrm{T}}=1\mathrm{GPa}$ and ${K}_{\mathrm{T}}=10\mathrm{GPa}$; ${G}_{\mathrm{T}}=10\mathrm{GPa}$ and ${K}_{\mathrm{T}}=1\mathrm{GPa}$)—this in principle indicates that those combinations can lead to particle cracking (in opening mode I) at advanced stages of the lithiation process. The circumferential stresses never become tensile for the moduli combination of ${G}_{\mathrm{T}}={K}_{\mathrm{T}}=10\mathrm{GPa}$–coatings with those moduli values would prevent from opening mode I particle fracture from the particle edge. Figure 5b shows pressure, $p=-R$ at the reaction front, plotted as a function of the reaction front position for different moduli combinations. Similarly, as for the circumferential stresses at the edge of the particle, there is a large discrepancy in the results when the moduli combination is ${G}_{\mathrm{T}}=$ 10 GPa and ${K}_{\mathrm{T}}=$10 GPa. In particular, this moduli combination results in a constant rise in pressure at the reaction front during the lithiation, while in the case of the other moduli combinations the pressure becomes negative up to ${R}_{\mathsf{\Gamma}}=0.8{R}_{\mathrm{P}}$, and only then it starts to increase.

#### 4.3. Effect of Coating Moduli on Reaction Front Velocity

_{T}=10

^{1.5}GPa, the reaction did not start as the constraints from the volumetrically stiff coating generated too large stresses in Si particle for the reaction front to propagate. Irrespectively of the value of the coating shear modulus, the lowest value of the bulk modulus, K

_{T}= 0.1 GPa, resulted in similar results for the reaction front velocity as when no coating was present.

#### 4.4. Effect of Coating Thickness on Stresses and Reaction Front Velocity

#### 4.5. Sensitivity Analysis of Reaction Front Velocity to Coating Properties

_{P}), and their absolute values increase with the movement of the reaction front. The significant increase in sensitivity values is caused by the normalisation used here, which requires division by the velocity—the latter decreases to zero towards the end of lithiation process, as shown in Figure 9d. In Figure 10, it is shown that the coating thickness is the most significant parameter influencing the velocity of the lithiation process in Si particles for all investigated combinations of the moduli.

#### 4.6. Combined Effects of Coating Parameters and Chemical Energy

^{3}. It was also found that the increase in the bulk and shear moduli of the coating reduces the degree of lithiation for the given value of chemical energy. This results from the fact that the coating parameters lead to changes in stresses within the particle, as discussed in Section 4.2. In turn, those stresses affect the kinetics of the reaction (cf. Equations (4) and (5)), where it can be seen that at low values of $\mathsf{\gamma}$, the velocity becomes much more sensitive to the stresses, and consequently to the values of coating moduli. Qualitatively, similar trends can be observed when varying the coating thickness (see Figure 11b)—no lithiation is possible below the threshold value of chemical energy parameter $\mathsf{\gamma}=$ 4 J/mm

^{3}, and the degree of lithiation is reduced with an increase of coating thickness. The latter results from the constraint effect imposed on the particle (for the particle subject to stress-free boundary conditions), which increases with the increasing thickness of the coating, and thus leading to the increase in stress value that in turn affect the reaction front velocity.

#### 4.7. Effect of Particle Constraints (Non-Stress-Free Boundary Conditions)

_{T}= ${10}^{2}$ nm was used in this particular study, as it was found that for a fully constrained particle only a sufficiently thick coating can facilitate the expansion of the particle, while thinner coatings immediately led to the arrest of the reaction (as the coating is subject to radial constraints). As shown in Figure 12a, for the fully constrained particle to achieve a higher degree of lithiation, coatings with lower shear moduli are favored. Moreover, for the given value of coating shear modulus (G

_{T}= 1 GPa) and coating thickness (h

_{T}= 10

^{2}nm), the reaction is blocked at early lithiation stages, between ${R}_{\mathsf{\Gamma}}=0.035{R}_{\mathrm{P}}$ and ${R}_{\mathsf{\Gamma}}=0.045{R}_{\mathrm{P}}$, for coating bulk moduli from K

_{T}= 10

^{−2}to 10

^{0}GPa, respectively of the value of the bulk modulus.

## 5. Conclusions

^{1}GPa, which led to an increase of the absolute values of the radial and circumferential stress components in the particle. Due to the stress-diffusion–velocity coupling, the increase in stresses reduced the velocity of the reaction front, and also the degree of lithiation. A simple deterministic sensitivity analysis of the reaction front velocity with respect to coating parameters showed that the coating thickness is the most influential design parameter that affects the velocity of the reaction front.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|---|---|

${K}_{+}$ [$\mathrm{GPa}$] | 28.5 | ${\mathsf{\rho}}_{-}$ [$\frac{g}{c{m}^{3}}$] | 2.285 | ${M}_{-}$ [$\frac{g}{mol}$] | 28.0855 | ${n}_{+}$ [None] | 1/15 |

${G}_{h}$ [$\mathrm{GPa}$] | 0.5 | $q$ [None] | 24 | $T$ [$\mathrm{K}$] | 293.15 | $\mathsf{\gamma}$ [$\frac{J}{m{m}^{3}}$] | 5 |

$D$ [$\frac{{m}^{2}}{s}$] | ${10}^{-12}$ | $\mathsf{\alpha}$ [$\frac{nm}{s}$] | 2000 | $g$ [None] | 4^{1/3} | ${E}_{-}$ [$\mathrm{GPa}$] | 80 |

${\mathsf{\tau}}_{0}$ [$ns$] | 2 | ${c}_{\ast}$ [$\frac{mol}{c{m}^{3}}$] | 0.053 | ${n}_{-}$ [None] | 4/15 | ${v}_{-}$ [None] | 0.22 |

${\mathsf{\sigma}}_{0}$ [$\mathrm{GPa}$] | 1 | ${k}_{\ast}$ [$\frac{nm}{s}$] | 86 | ${n}_{\ast}$ [None] | 1 | ${E}_{+}$ [$\mathrm{GPa}$] | 41 |

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**Figure 1.**Spherical polymer-coated Si particle undergoing a two-phase lithiation process (the lithiated particle is shown as mapped onto the reference (undeformed) configuration); coating thickness $h={R}_{T}-{R}_{P}$; ${R}_{\Gamma}$ denotes the position of the reaction front mapped onto the reference (undeformed) configuration.

**Figure 2.**Schematic representation of the FE mesh and the reaction front movement from the edge (R

_{N}) of the particle to its centre (R

_{0}).

**Figure 3.**(

**a**) Radial stress component as a function of normalised particle radius for different shear moduli of the coating, and h = 10 nm and K

_{T}= 1 GPa. (

**b**) Zoom-in on the radial stresses close to the edge of the particle and within the coating.

**Figure 4.**(

**a**) Circumferential stress component as a function of normalised particle radius for different shear moduli, and h = 10 nm and K

_{T}= 1 GPa. (

**b**) Zoom-in on the circumferential stresses close to the edge of the particle and within the coating.

**Figure 5.**Circumferential stresses and pressure as functions of the reaction front position for different combinations of the coating moduli: (

**a**) circumferential component; (

**b**) pressure; h = 10 nm.

**Figure 6.**Profiles of reaction front velocity for different coating shear moduli. (

**a**) K

_{T}= 1 GPa and h = 10 nm. (

**b**) K

_{T}= 10 GPa and h = 10 nm thickness.

**Figure 7.**Profiles of reaction front velocity for different coating bulk moduli. (

**a**) G

_{T}= 1 GPa, h = 10 nm. (

**b**) G

_{T}= 10 GPa and h = 10 nm.

**Figure 9.**Stresses for different coating thicknesses h = 10

^{−1}÷ 10

^{1.5}nm, and G

_{T}= K

_{T}= 1 GPa; (

**a**) radial stresses; (

**b**) zoom-on of radial stresses; (

**c**) circumferential stresses; (

**d**) reaction front velocity.

**Figure 10.**Sensitivity profiles of the reaction front velocity with respect to shear modulus (G

_{T}), bulk modulus (K

_{T}), and thickness (h

_{T}) of the coating for different combinations of shear and bulk moduli; nominal coating thickness of 10 nm is used in all simulations. (

**a**) G

_{T}= K

_{T}=1 GPa; (

**b**) G

_{T}= 1 GPa, K

_{T}= 10 GPa; (

**c**) G

_{T}= 10 GPa, K

_{T}= 1 GPa; (

**d**) G

_{T}= K

_{T}= 10 GPa.

**Figure 11.**Degree of lithiation achieved as a function of the chemical energy parameter γ; (

**a**) effect of coating moduli G

_{T}and K

_{T}when coating thickness h

_{T}=10 nm. (

**b**) Effect of h

_{T}when G

_{T}= K

_{T}= 1 GPa.

**Figure 12.**Reaction front velocity profiles of the fully-constrained particle. (

**a**) For different coating shear moduli and coating bulk modulus K

_{T}= 1 GPa and thickness of h

_{T}= 10

^{2}nm; (

**b**) For different coating bulk moduli and coating shear modulus G

_{T}= 1 GPa and h

_{T}= 10

^{2}nm.

**Figure 13.**Reaction front velocity profiles for particles with different coating thicknesses and subject to different boundary conditions (fully-constrained (FC) and stress-free (SF)); coating shear modulus of 10

^{−2}GPa and bulk modulus of 10

^{0}GPa; (

**a**) full range; (

**b**) limited range—zoomed-in velocity profiles for SF boundary conditions.

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**MDPI and ACS Style**

Lippmann, P.; Poluektov, M.; Figiel, Ł.
Stress-Affected Lithiation Reactions in Elasto-Viscoplastic Si Particles with Hyperelastic Polymer Coatings: A Nonlinear Chemo-Mechanical Finite-Element Study. *Coatings* **2018**, *8*, 455.
https://doi.org/10.3390/coatings8120455

**AMA Style**

Lippmann P, Poluektov M, Figiel Ł.
Stress-Affected Lithiation Reactions in Elasto-Viscoplastic Si Particles with Hyperelastic Polymer Coatings: A Nonlinear Chemo-Mechanical Finite-Element Study. *Coatings*. 2018; 8(12):455.
https://doi.org/10.3390/coatings8120455

**Chicago/Turabian Style**

Lippmann, Philip, Michael Poluektov, and Łukasz Figiel.
2018. "Stress-Affected Lithiation Reactions in Elasto-Viscoplastic Si Particles with Hyperelastic Polymer Coatings: A Nonlinear Chemo-Mechanical Finite-Element Study" *Coatings* 8, no. 12: 455.
https://doi.org/10.3390/coatings8120455