# 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

- Tikhomirov, V.M. A Study of the Diffusion Equation with Increase in the Amount of Substance, and its Application to a Biological Problem. In Selected Works of A. N. Kolmogorov: Volume I: Mathematics and Mechanics; Tikhomirov, V.M., Ed.; Mathematics and Its Applications (Soviet Series); Springer: Dordrecht, The Netherlands, 1991; pp. 242–270. [Google Scholar]
- Fisher, R.A. The wave of advance of advantageous genes. Ann. Hum. Genet.
**1937**, 7, 355–369. [Google Scholar] [CrossRef] - Cencini, M.; Lopez, C.; Vergni, D. Reaction-Diffusion Systems: Front Propagation and Spatial Structures. In The Kolmogorov Legacy in Physics; Livi, R., Vulpiani, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; Volume 636, pp. 187–210. [Google Scholar]
- Takagi, H.; Kaneko, K. Pattern dynamics of a multi-component reaction diffusion system: Differentiation of replicating spots. Int. J. Bifurc. Chaos
**2002**, 12, 2579–2598. [Google Scholar] [CrossRef] - Mendez, V.; Fedotov, S.; Horsthemke, W. Reaction-Transport Systems. In Mesoscopic Foundations, Fronts, and Spatial Instabilities, 1st ed.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 3–419. [Google Scholar]
- De Groot, S.; Mazur, P. Non-Equilibrium Thermodynamics; Courier Corporation: Amsterdam, The Netherlands, 1962; pp. 197–234. [Google Scholar]
- Larchté, F.C.; Cahn, J.W. The effect of self-stress on diffusion in solids. Acta Metall.
**1982**, 30, 1835–1845. [Google Scholar] [CrossRef] - Eckart, C. The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity. Phys. Rev.
**1948**, 73, 373–380. [Google Scholar] [CrossRef] - Beaulieu, L.Y.; Eberman, K.W.; Turner, R.L.; Krause, L.J.; Dahn, J.R. Colossal reversible volume changes in lithium alloys. Electrochem. Solid-State Lett.
**2001**, 4, 137–140. [Google Scholar] [CrossRef] - Natsiavasa, P.P.; Weinberg, K.; Rosato, D.; Ortiz, M. Effect of Prestress on the Stability of Electrode-Electrolyte Interfaces during Charging in Lithium Batteries. J. Mech. Phys. Solids
**2016**, 95, 92–111. [Google Scholar] [CrossRef] - O’Connell, J.; Haile, J. Thermodynamics: Fundamentals for Applications; Cambridge University Press: New York, NY, USA, 2005; pp. 1–674. [Google Scholar]
- Marsden, J.E.; Ratiu, T.S. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd ed.; Texts in Applied Mathematics; Springer: New York, NY, USA, 2002; pp. 1–517. [Google Scholar]
- Müller, P. Glossary of terms used in physical organic chemistry. (IUPAC recommendations 1994). Pure Appl. Chem.
**1994**, 66, 1077–1184. [Google Scholar] - Cross, M.; Greenside, H. Pattern Formation and Dynamics in Nonequilibrium Systems; Cambridge University Press: Cambridge, UK, 2009; pp. 1–495. [Google Scholar]
- Anders, D.; Weinberg, K. Modeling of multicomponent reactive systems. Tech. Mech.
**2012**, 32, 105–112. [Google Scholar] - Anders, D.; Dittmann, M.; Weinberg, K. A higher-order finite element approach to the Kuramoto-Sivashinsky equation. J. Appl. Math. Mech.
**2012**, 92, 599–607. [Google Scholar] [CrossRef] - Hesch, C.; Schuß, S.; Dittmann, M.; Franke, M.; Weinberg, K. Isogeometric analysis and hierarchical refinement for higher-order phase-field models. Comput. Methods Appl. Mech. Eng.
**2016**, 303, 185–207. [Google Scholar] [CrossRef] - Anders, D.; Weinberg, K. A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation. J. Appl. Math. Mech.
**2011**, 91, 609–629. [Google Scholar] [CrossRef] - Anders, D.; Weinberg, K. Numerical simulation of diffusion induced phase separation and coarsening in binary alloys. Comput. Mater. Sci.
**2011**, 50, 1359–1364. [Google Scholar] [CrossRef] - Glotzer, S.; Di Marzio, E.; Muthukumar, M. Reaction-controlled morphology of phase-separating mixtures. Phys. Rev. Lett.
**1995**, 74, 2034–2037. [Google Scholar] [CrossRef] [PubMed] - Horiuchi, S.; Matchariyakul, N.; Yase, K.; Kitano, T. Morphology development through an interfacial reaction in ternary immiscible polymer blends. Macromolecules
**1997**, 30, 3664–3670. [Google Scholar] [CrossRef] - Anders, D.; Weinberg, K. An extended stochastic diffusion model for ternary mixtures. Mech. Mater.
**2013**, 56, 122–130. [Google Scholar] [CrossRef] - Zhao, Y.; Stein, P.; Xu, B.-X. Isogeometric analysis of mechanically coupled Cahn-Hilliard phase segregation in hyperelastic electrodes of Li-ion batteries. Comput. Methods Appl. Mech. Eng.
**2015**, 297, 325–347. [Google Scholar] [CrossRef]

**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