# Influence of Interface Proximity on Precipitation Thermodynamics

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## Abstract

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## 1. Introduction

## 2. Precipitation Near Free Surfaces and Rigid Interfaces

## 3. Precipitation Close to Shear-Coupled Grain Boundaries

## 4. Effective Scale Bridging Description of Precipitation Near Interfaces

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Barrilao, J.L.; Kuhn, B.; Wessel, E. Microstructure evolution and dislocation behaviour in high chromium, fully ferritic steels strengthened by intermetallic Laves phases. Micron
**2018**, 108, 11–18. [Google Scholar] [CrossRef] [PubMed] - Bhadeshia, H.K.D.H. Bainite in Steels; IOM: London, UK, 2001. [Google Scholar]
- Fielding, L.C.D. The Bainite Controversy. Mater. Sci. Technol.
**2013**, 29, 383–399. [Google Scholar] [CrossRef] - Swallow, E.; Bhadeshia, H.K.D.H. High resolution observations of displacements caused by bainitic transformation. Mater. Sci. Technol.
**1996**, 12, 121–125. [Google Scholar] [CrossRef] - Cahn, J.W. Coherent fluctuations and nucleation in isotropic solids. Acta Metall.
**1962**, 10, 907–913. [Google Scholar] [CrossRef] - Cahn, J.W.; Larché, F. A simple model for coherent equilibrium. Acta Metall.
**1984**, 32, 1915–1923. [Google Scholar] [CrossRef] - Williams, R.O. The calculation of coherent phase equilibria. Calphad
**1984**, 8, 1–14. [Google Scholar] [CrossRef] - Johnson, W.C.; Vorhees, P.W. Phase equilibrium in two-phase coherent solids. Metall. Trans. A
**1987**, 18A, 1213–1228. [Google Scholar] [CrossRef] - Schwarz, R.B.; Khachaturyan, A.G. Thermodynamics of open two-phase systems with coherent interfaces. Phys. Rev. Lett.
**1995**, 74, 2523. [Google Scholar] [CrossRef] [PubMed] - Schwarz, R.B.; Khachaturyan, A.G. Thermodynamics of open two-phase systems with coherent interfaces: Application to metal–hydrogen systems. Acta Mat.
**2006**, 54, 313–323. [Google Scholar] [CrossRef] - Roytburd, A.L. Equilibrium and phase diagrams of coherent phases in solids. Sov. Phys. Solid State
**1984**, 26, 1229–1233. [Google Scholar] - Spatschek, R.; Gobbi, G.; Hüter, C.; Chakrabarty, A.; Aydin, U.; Brinckmann, S.; Neugebauer, J. Scale bridging description of coherent phase equilibria in the presence of surfaces and interfaces. Phys. Rev. B
**2016**, 94, 134106. [Google Scholar] [CrossRef][Green Version] - Weikamp, M.; Hüter, C.; Spatschek, R. Linking Ab Initio Data on Hydrogen and Carbon in Steel to Statistical and Continuum Descriptions. Metals
**2018**, 8, 219. [Google Scholar] [CrossRef][Green Version] - Weikamp, M.; Spatschek, R. Effect of shear-coupled grain boundary motion on coherent precipitation. Phys. Rev. B
**2019**, 100, 054103. [Google Scholar] [CrossRef][Green Version] - Fratzl, P.; Penrose, O.; Lebowitz, J.L. Modeling of phase separation in alloys with coherent elastic misfit. J. Stat. Phys.
**1999**, 95, 1429–1503. [Google Scholar] [CrossRef][Green Version] - Neugebauer, J.; Scheffler, M. Mechanisms of island formation of alkali-metal adsorbates on Al (111). Phys. Rev. Lett.
**1993**, 71, 577. [Google Scholar] [CrossRef][Green Version] - Buranova, Y.; Kulitskiy, V.; Peterlechner, M.; Mogucheva, A.; Kaibyshev, R.; Divinski, S.V.; Wilde, G. Al
_{3}(Sc,Zr)-based precipitates in Al–Mg alloy: Effect of severe deformation. Acta Mater.**2017**, 124, 210–224. [Google Scholar] [CrossRef] - Spatschek, R.; Karma, A. Amplitude equations for polycrystalline materials with interaction between composition and stress. Phys. Rev. B
**2010**, 81, 214201. [Google Scholar] [CrossRef][Green Version] - Xu, Y.C.; Geslin, P.A.; Karma, A. Elastically-mediated interactions between grain boundaries and precipitates in two-phase coherent solids. Phys. Rev. B
**2016**, 94, 144106. [Google Scholar] [CrossRef][Green Version] - Geslin, P.A.; Xu, Y.C.; Karma, A. Morphological instability of grain boundaries in two-phase coherent solids. Phys. Rev. Lett.
**2015**, 114, 105501. [Google Scholar] [CrossRef] - Brener, E.A.; Weikamp, M.; Spatschek, R.; Bar-Sinai, Y.; Bouchbinder, E. Dynamic instabilities of frictional sliding at a bimaterial interface. J. Mech. Phys. Solids
**2016**, 89, 149–173. [Google Scholar] [CrossRef][Green Version] - Bar-Sinai, Y.; Aldam, M.; Spatschek, R.; Brener, E.A.; Bouchbinder, E. Spatiotemporal Dynamics of Frictional Systems: The Interplay of Interfacial Friction and Bulk Elasticity. Lubricants
**2019**, 7, 91. [Google Scholar] [CrossRef][Green Version] - Asaro, R.J.; Tiller, W.A. Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion. Metall. Trans.
**1972**, 3, 1789–1796. [Google Scholar] [CrossRef] - Grinfeld, M.A. Instability of the interface between a nonhydrostatically stressed elastic body and a melt. Sov. Phys. Dokl.
**1986**, 31, 831. [Google Scholar] - Spatschek, R.; Hartmann, M.; Brener, E.; Müller-Krumbhaar, H.; Kassner, K. Phase field modeling of fast crack propagation. Phys. Rev. Lett.
**2006**, 96, 015502. [Google Scholar] [CrossRef][Green Version] - Brener, E.A.; Marchenko, V.I.; Spatschek, R. Influence of strain on the kinetics of phase transitions in solids. Phys. Rev. E
**2007**, 75, 041604. [Google Scholar] [CrossRef] [PubMed][Green Version] - Spatschek, R.; Müller-Gugenberger, C.; Brener, E.; Nestler, B. Phase field modeling of fracture and stress-induced phase transitions. Phys. Rev. E
**2007**, 75, 066111. [Google Scholar] [CrossRef][Green Version] - Hecht, F. New development in FreeFem++. J. Numer. Math.
**2013**, 20, 251–266. [Google Scholar] [CrossRef] - Bhogireddy, V.S.P.K.; Hüter, C.; Neugebauer, J.; Steinbach, I.; Karma, A.; Spatschek, R. Phase-field modeling of grain-boundary premelting using obstacle potentials. Phys. Rev. E
**2014**, 90, 012401. [Google Scholar] [CrossRef][Green Version] - Pundt, A.; Kirchheim, R. Hydrogen in Metals: Microstructural Aspects. Annu. Rev. Mater. Res.
**2006**, 36, 555. [Google Scholar] [CrossRef] - Jülich Supercomputing Centre. JURECA: Modular supercomputer at Jülich Supercomputing Centre. Journal of large-scale research facilities. J. -Large-Scale Res. Facil. JLSRF
**2018**, 4, A132. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Cementite in a lower bainite of 22MnB5. Left: SEM image with the backscattered electron detector. Right: Orientation map on the same region, IPFcolor coded. The bainitic microstructure was formed after isothermally holding at 420 °C for 12 s after austenization at 900 °C

**Figure 2.**Precipitation in the bulk and near free or confined surfaces. The left panel illustrates the case of precipitate formation in the bulk, sufficiently far away from interfaces and surfaces. A lattice mismatch between the precipitate and matrix leads to elastic distortions in both of them, as illustrated here for a precipitate with a larger volume (positive eigenstrain). In the central panel, the precipitate is close to a free surface, where the material can relax in the outward normal direction, such that elastic stresses are reduced. Consequently, the elastic energy is here lower than in the bulk case. In the right panel, the precipitate is close to an interface to an elastically stiffer phase or a rigid substrate. If it is coherently connected to the matrix phase, elastic relaxation is hampered or suppressed near the matrix-substrate interface. In this case, the elastic energy of the system is even higher than in the bulk case, and precipitation therefore unfavorable.

**Figure 3.**Sketch of the investigated phase diagram with the coexistence of a solid solution $\alpha $ phase with a “cementite” phase. The (artificial) neglect of elastic effects leads to a bulk solubility limit, which is marked by the solid red curve. For low temperatures, it is well described by the (dashed, red) Arrhenius approximation, Equation (2). Additional inclusion of elastic effects shifts the curves to the right (corresponding to $\gamma =0$ in Equation (3) for the Arrhenius asymptotics); hence, the solubility limit is increased (black curve pair). From there, the proximity to a free surface decreases $1-\gamma $ and therefore leads to the blue curves for the solubility limit, which is in between the bulk cases with and without elasticity (blue curves). For an explicit example, we mention shear-coupled grain boundary motion in Fe-C with phase coexistence between ferrite ($\alpha $) and cementite Fe

_{3}C, for which ${c}_{\mathrm{cem,0}}=6.67$ wt%. The carbon solubility in ferrite is very low; hence, the description as a dilute alloy is appropriate. According to Equations (3) and (7), the reduction of the solubility limit for carbon near a shear-coupled grain boundary in ferrite is approximately 50% (blue curves) relative to the bulk elastic situation (black curves) for an array of precipitates at room temperature, as discussed in detail in [14].

**Figure 4.**Illustration of the concept of shear-coupled motion. The left panel (

**a**) shows the reference state with a horizontal grain boundary. The dashed line is a marker across the grain boundary to track the motion of lattice planes. Panel (

**b**) shows the case of pure sliding motion. The upper grain is subjected to a lateral motion with velocity ${v}_{\Vert}$ by application of an external shear stress $\tau $. As a result, the upper grain “slides” along the lower one, similar to a frictional process [21,22]. As a consequence, the dashed marker line becomes discontinuous at the grain boundary. The right panel (

**c**) shows the consequences of shear-coupled motion. While the grains have a relative lateral velocity ${v}_{\Vert}$, simultaneously, a normal motion of the grain boundary takes place with velocity ${v}_{n}$. This happens because atoms from the upper grain connect to the lower grain during the motion at the moment when a layer of upper grain atoms aligns with the lattice of the lower grain. Notice that an inversion of the tangential velocity also changes the orientation of the normal interface motion. As a result of the normal grain boundary motion, the marker line develops a kink [14].

**Figure 5.**In the sequence red to green to blue, an array of precipitates approaches an originally planar grain boundary and leads to an increasing morphological change $p\left(x\right)\ne 0$. The parameters are ${d}_{blue}=0.55\mathsf{\mu}\mathrm{m}$, ${d}_{red}=0.85\mathsf{\mu}\mathrm{m}$, ${d}_{green}=1.15\mathsf{\mu}\mathrm{m}$, $E=175$ GPa, ν = 0.25, $R=0.5\mathsf{\mu}\mathrm{m}$, $W=20.55\mathsf{\mu}\mathrm{m}$, $\beta =0.07$, and eigenstrain ${\epsilon}_{0}=0.02$ between the precipitate and matrix [14].

**Figure 6.**Total elastic energy for a precipitate as function of scaled distance $d/W$ from an originally straight grain boundary (W is the lateral distance in an array of precipitates) for $\nu =1/4$. For large distances, the grain boundary does not affect the energy, and it approaches the value ${F}_{\mathrm{min}}={F}_{\mathrm{prec}}$. For lower distances, the energy is reduced and leads to an effective attractive interaction towards the grain boundary and a lower solubility limit. The curves end at the point when the precipitates touch the grain boundary ($d=R$), and a penetration raises the energy again (not shown; see [14] for a discussion). The limit $d=R$ is shown as the dashed curve for arbitrary ratio $d/W$. Altogether, an energetic reduction by up to 16% can be achieved [14].

**Figure 7.**Illustration of the concept for the calculation of the stored elastic energy. The (coherent) precipitate of radius R is located at a distance d away from a grain boundary (center-to-center distance), which has a thickness ${D}_{\mathrm{GB}}$. The entire FEM simulation domain has a cylindrical geometry with a radius ${R}_{c}\gg R$, such that the mantle surfaces do not influence the precipitate. All surfaces are traction free.

**Figure 8.**Calculated elastic energy (scaled to the energy of an isolated precipitate in a defect-free and infinite matrix) as a function of the precipitate’s distance to the spatially extended grain boundary, using ${D}_{\mathrm{GB}}/R=0.8$. If the precipitate is far away from the grain boundary, the energy reaches a plateau at $1-\gamma =1$. If it approaches the top (free) surface of the cylinder (here, for $d/R>3.5$; see also Figure 7), the energy drops according to the analysis in Section 2. If the precipitate approaches the grain boundary, the energy change depends on the ratio of Young’s moduli in the grain boundary and bulk, ${E}_{\mathrm{GB}}/E$. For ${E}_{\mathrm{GB}}/E<1$ (${E}_{\mathrm{GB}}/E>1$), we obtain an energy reduction (increase), corresponding to an attractive (repulsive) interaction between the precipitate and the grain boundary. If the precipitate intersects the grain boundary (blue shaded region), this trend continues.

**Figure 9.**Matching between the microscopic description of the interaction between a coherent precipitate and a shear-coupled grain boundary (see Section 3) to a mesoscopic description with a finite width grain boundary layer. By the proper choice of the grain boundary thickness ${D}_{\mathrm{GB}}$ and the elastic softening ${E}_{\mathrm{GB}}/E$, a reasonable matching can be achieved outside the grain boundary layer.

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**MDPI and ACS Style**

Wang, K.; Weikamp, M.; Lin, M.; Zimmermann, C.; Schwaiger, R.; Prahl, U.; Hunkel, M.; Spatschek, R.
Influence of Interface Proximity on Precipitation Thermodynamics. *Metals* **2020**, *10*, 1292.
https://doi.org/10.3390/met10101292

**AMA Style**

Wang K, Weikamp M, Lin M, Zimmermann C, Schwaiger R, Prahl U, Hunkel M, Spatschek R.
Influence of Interface Proximity on Precipitation Thermodynamics. *Metals*. 2020; 10(10):1292.
https://doi.org/10.3390/met10101292

**Chicago/Turabian Style**

Wang, Kai, Marc Weikamp, Mingxuan Lin, Carina Zimmermann, Ruth Schwaiger, Ulrich Prahl, Martin Hunkel, and Robert Spatschek.
2020. "Influence of Interface Proximity on Precipitation Thermodynamics" *Metals* 10, no. 10: 1292.
https://doi.org/10.3390/met10101292