# A Chemo-Mechanical Model of Diffusion in Reactive Systems

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Deformation and Diffusion

#### 2.1. Kinematics of Deformation

#### 2.2. Mass Balance

#### 2.3. Helmholtz Free-Energy Density of the System

#### 2.3.1. Elastic Energy Contribution

#### 2.3.2. Configurational Energy Contribution

#### 2.3.3. Interfacial Energy Contribution

#### 2.4. Mechanical Stresses and Chemical Potential

## 3. Chemical Reactions

#### 3.1. Kinetics of Chemical Reactions

#### 3.2. Thermodynamical Model for Reaction-Diffusion Systems

^{3}), ${\rho}_{i}$ denotes the partial density of component i (in units of kg/m

^{3}) and ${M}_{i}$ is the respective molar mass of constituent i (in units of kg/mol). An application of (27) gives the true stoichiometric coefficients ${n}_{kj}$ and the conventional reaction rates ${\mathcal{J}}_{j}$

#### 3.3. Formulation of the Problem

## 4. Computational Studies of Elastic Multicomponent Reaction-Diffusion Systems

#### 4.1. Evolution of a Ternary Reaction-Diffusion System

#### 4.1.1. Modelling the System

#### 4.1.2. Choice of Parameters

#### 4.1.3. Results of the Two-Dimensional Simulations

#### 4.1.4. Results of the Three-Dimensional Simulations

#### 4.1.5. Parametric Studies

#### 4.2. Evolution of Elastic Phase-Separating Systems

#### 4.2.1. A Rod under to Incoming Flux

^{3}[23] (p. 339).

#### 4.2.2. Phase Decomposition in an Elastic Rod

## 5. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Configurational energy density (12) for different Flory–Huggins interaction parameters $\chi $ with ${g}_{1}^{0}=0.1$ and ${g}_{2}^{0}=0.5$. For $\chi =2.5$ the common tangent indicates the equilibrium concentrations ${c}_{\alpha}$ and ${c}_{\beta}$.

**Figure 2.**(

**a**) Contour plot and (

**b**) illustration of the shape of the configurational energy density ${\Psi}^{\mathrm{con}}$. The blue color indicates low energy values and the red color marks high energy values.

**Figure 3.**Morphology evolution in a ternary reaction-diffusion system at early stages for different reaction rates ${\overline{k}}^{\prime}$ and at different stages of evolution $\overline{t}$. The instances of the morphological snapshots correspond to the times marked with circles in Figure 4. Color coding: ${c}_{\mathrm{A}}$ (red), ${c}_{\mathrm{B}}$ (blue), ${c}_{\mathrm{C}}$ (green).

**Figure 4.**Temporal evolution of the averaged concentrations $\u2329{c}_{A}\u232a$ (blue), $\u2329{c}_{B}\u232a$ (red) and $\u2329{c}_{C}\u232a$ (green) of the chemical species A, B and C, respectively, for the two-dimensional simulation (

**a**) and the three-dimensional simulation (

**b**). The maximum deviation between the average 2D and 3D concentrations is less than $6.2\times {10}^{-3}$ for $\overline{t}<{10}^{3}$.

**Figure 5.**Morphology evolution in a ternary reaction-diffusion system at early stages for different reaction rates ${\overline{k}}^{\prime}$ and at different stages of evolution $\overline{t}$. The instances of the morphological snapshots correspond to the times marked with circles in Figure 4. Color coding: ${c}_{\mathrm{A}}$ (red), ${c}_{\mathrm{B}}$ (blue), ${c}_{\mathrm{C}}$ (green).

**Figure 6.**Temporal evolution of the averaged concentrations $\u2329{c}_{A}\u232a$ (blue), $\u2329{c}_{B}\u232a$ (red) and $\u2329{c}_{C}\u232a$ (green) of the chemical species A, B and C, for different values of gradient energy coefficient ${\kappa}_{1}$ using ${\overline{k}}^{\prime}=600$.

**Figure 7.**Evolution of a ternary reaction-diffusion system for locally varying ${\overline{k}}^{\prime}$. The overall concentrations (

**a**) and the corresponding morphology are presented at different stages (

**b**) with the corresponding spatial distribution of ${\overline{k}}^{\prime}$ (

**c**).

**Figure 9.**Phase decomposition in an elastic rod: initial state (

**a**) concentration distribution after 100 time steps (

**b**), and corresponding von Mises stress (

**c**).

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Weinberg, K.; Werner, M.; Anders, D.
A Chemo-Mechanical Model of Diffusion in Reactive Systems. *Entropy* **2018**, *20*, 140.
https://doi.org/10.3390/e20020140

**AMA Style**

Weinberg K, Werner M, Anders D.
A Chemo-Mechanical Model of Diffusion in Reactive Systems. *Entropy*. 2018; 20(2):140.
https://doi.org/10.3390/e20020140

**Chicago/Turabian Style**

Weinberg, Kerstin, Marek Werner, and Denis Anders.
2018. "A Chemo-Mechanical Model of Diffusion in Reactive Systems" *Entropy* 20, no. 2: 140.
https://doi.org/10.3390/e20020140