# Atomistic Study of the Role of Defects on α → ϵ Phase Transformations in Iron under Hydrostatic Compression

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Interatomic Potentials

#### 2.2. MD Simulation Setup

^{−1}in an NVE ensemble, and the maximum strain was 10%. Periodic boundary conditions were applied in all three normal directions for the crystal containing twins. The Verlet scheme with a time step of 0.5 fs was used in the simulations.

#### 2.3. Monte Carlo Simulation Setup

#### 2.4. Atomic Structure Identification

## 3. Results and Discussion

#### 3.1. Frank–Read Source in bcc Iron

^{−1}.

#### 3.2. Pressure-Induced Phase Transformation

#### 3.3. Edge Dislocation

#### 3.4. Cottrell Atmosphere

#### 3.5. Comparison

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bancroft, D.; Peterson, E.L.; Minshall, S. Polymorphism of Iron at High Pressure. J. Appl. Phys.
**1956**, 27, 291–298. [Google Scholar] [CrossRef] - Jamieson, J.C.; Lawson, A.W. X-ray Diffraction Studies in the 100 Kilobar Pressure Range. J. Appl. Phys.
**1962**, 33, 776–780. [Google Scholar] [CrossRef] - Bassett, W.A.; Huang, E. Mechanism of the Body-Centered Cubic–Hexagonal Close-Packed Phase Transition in Iron. Science
**1987**, 238, 780–783. [Google Scholar] [CrossRef] [PubMed] - Dewaele, A.; Denoual, C.; Anzellini, S.; Occelli, F.; Mezouar, M.; Cordier, P.; Merkel, S.; Véron, M.; Rausch, E. Mechanism of the α-ϵ phase transformation in iron. Phys. Rev. B
**2015**, 91, 174105. [Google Scholar] [CrossRef] - Kadau, K. Microscopic View of Structural Phase Transitions Induced by Shock Waves. Science
**2002**, 296, 1681–1684. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kalantar, D.H.; Belak, J.F.; Collins, G.W.; Colvin, J.D.; Davies, H.M.; Eggert, J.H.; Germann, T.C.; Hawreliak, J.; Holian, B.L.; Kadau, K.; et al. Direct Observation of the α-ϵ Transition in Shock-Compressed Iron via Nanosecond X-ray Diffraction. Phys. Rev. Lett.
**2005**, 95, 075502. [Google Scholar] [CrossRef] [PubMed] - Boehler, R.; Santamaría-Pérez, D.; Errandonea, D.; Mezouar, M. Melting, density, and anisotropy of iron at core conditions: New x-ray measurements to 150 GPa. J. Phys. Conf. Ser.
**2008**, 121, 022018. [Google Scholar] [CrossRef] - Sinmyo, R.; Hirose, K.; Ohishi, Y. Melting curve of iron to 290 GPa determined in a resistance-heated diamond-anvil cell. Earth Planet. Sci. Lett.
**2019**, 510, 45–52. [Google Scholar] [CrossRef] - Gunkelmann, N.; Tramontina, D.R.; Bringa, E.M.; Urbassek, H.M. Morphological changes in polycrystalline Fe after compression and release. J. Appl. Phys.
**2015**, 117, 085901. [Google Scholar] [CrossRef] - Ekman, M.; Sadigh, B.; Einarsdotter, K.; Blaha, P. Ab initio study of the martensitic bcc-hcp transformation in iron. Phys. Rev. B
**1998**, 58, 5296–5304. [Google Scholar] [CrossRef] - Herper, H.C.; Hoffmann, E.; Entel, P. Ab initio full-potential study of the structural and magnetic phase stability of iron. Phys. Rev. B
**1999**, 60, 3839–3848. [Google Scholar] [CrossRef] - Friák, M.; Šob, M. Ab initio study of the bcc-hcp transformation in iron. Phys. Rev. B
**2008**, 77, 174117. [Google Scholar] [CrossRef] - Jafari, M.; Nobakhti, M.; Jamnezhad, H.; Bayati, K. Density functional theory study of the α → ω martensitic transformation in titanium induced by hydrostatic pressure. Condens. Matter Phys.
**2013**, 16, 33703. [Google Scholar] [CrossRef] - Errandonea, D.; Meng, Y.; Somayazulu, M.; Häusermann, D. Pressure-induced α → ω transition in titanium metal: A systematic study of the effects of uniaxial stress. Phys. B Condens. Matter
**2005**, 355, 116–125. [Google Scholar] [CrossRef] - Trinkle, D.R.; Hennig, R.G.; Srinivasan, S.G.; Hatch, D.M.; Jones, M.D.; Stokes, H.T.; Albers, R.C.; Wilkins, J.W. New Mechanism for the α to ω Martensitic Transformation in Pure Titanium. Phys. Rev. Lett.
**2003**, 91, 025701. [Google Scholar] [CrossRef] - Friák, M.; Hickel, T.; Körmann, F.; Udyansky, A.; Dick, A.; von Pezold, J.; Ma, D.; Kim, O.; Counts, W.; Šob, M.; et al. Determining the Elasticity of Materials Employing Quantum-mechanical Approaches: From the Electronic Ground State to the Limits of Materials Stability. Steel Res. Int.
**2011**, 82, 86–100. [Google Scholar] [CrossRef] - Fisher, J. Application of nucleation theory to isothermal martensite. Acta Metall.
**1953**, 1, 32–35. [Google Scholar] [CrossRef] - Luu, H.T.; Gunkelmann, N. Pressure-induced phase transformations in Fe-C: Molecular dynamics approach. Comput. Mater. Sci.
**2019**, 162, 295–303. [Google Scholar] [CrossRef] - Harrison, R.J.; Voter, A.F.; Chen, S.P. Atomistic Simulation of Materials: Beyond Pair Potentials; Number 1; A Division of Plenum Publishing Corporation: New York, NY, USA, 1994; pp. 281–286. [Google Scholar] [CrossRef]
- Gunkelmann, N.; Bringa, E.M.; Kang, K.; Ackland, G.J.; Ruestes, C.J.; Urbassek, H.M. Polycrystalline iron under compression: Plasticity and phase transitions. Phys. Rev. B
**2012**, 86, 144111. [Google Scholar] [CrossRef] - Kadau, K.; Germann, T.C.; Lomdahl, P.S.; Holian, B.L. Atomistic simulations of shock-induced transformations and their orientation dependence in bcc Fe single crystals. Phys. Rev. B
**2005**, 72, 064120. [Google Scholar] [CrossRef] - Kadau, K.; Germann, T.C.; Lomdahl, P.S.; Albers, R.C.; Wark, J.S.; Higginbotham, A.; Holian, B.L. Shock Waves in Polycrystalline Iron. Phys. Rev. Lett.
**2007**, 98, 135701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Venables, J.A. The martensite transformation in stainless steel. Philos. Mag. J. Sci.
**1962**, 7, 35–44. [Google Scholar] [CrossRef] - Yang, X.S.; Sun, S.; Wu, X.L.; Ma, E.; Zhang, T.Y. Dissecting the Mechanism of Martensitic Transformation via Atomic-Scale Observations. Sci. Rep.
**2014**, 4, 6141. [Google Scholar] [CrossRef] [PubMed] - Veiga, R.G.A.; Perez, M.; Becquart, C.S.; Domain, C. Atomistic modeling of carbon Cottrell atmospheres in bcc iron. J. Phys. Condens. Matter
**2012**, 25, 025401. [Google Scholar] [CrossRef] [Green Version] - Veiga, R.; Goldenstein, H.; Perez, M.; Becquart, C. Monte Carlo and molecular dynamics simulations of screw dislocation locking by Cottrell atmospheres in low carbon Fe–C alloys. Scr. Mater.
**2015**, 108, 19–22. [Google Scholar] [CrossRef] - Queyreau, S.; Marian, J.; Gilbert, M.R.; Wirth, B.D. Edge dislocation mobilities in bcc Fe obtained by molecular dynamics. Phys. Rev. B
**2011**, 84, 064106. [Google Scholar] [CrossRef] [Green Version] - Daw, M.S.; Foiles, S.M.; Baskes, M.I. The embedded-atom method: A review of theory and applications. Mater. Sci. Rep.
**1993**, 9, 251–310. [Google Scholar] [CrossRef] - Gunkelmann, N.; Bringa, E.M.; Tramontina, D.R.; Ruestes, C.J.; Suggit, M.J.; Higginbotham, A.; Wark, J.S.; Urbassek, H.M. Shock waves in polycrystalline iron: Plasticity and phase transitions. Phys. Rev. B
**2014**, 89, 140102. [Google Scholar] [CrossRef] - Gunkelmann, N.; Bringa, E.M.; Urbassek, H.M. Influence of phase transition on shock-induced spallation in nanocrystalline iron. J. Appl. Phys.
**2015**, 118, 185902. [Google Scholar] [CrossRef] - Amadou, N.; de Resseguier, T.; Dragon, A.; Brambrink, E. Coupling between plasticity and phase transition in shock- and ramp-compressed single-crystal iron. Phys. Rev. B
**2018**, 98, 024104. [Google Scholar] [CrossRef] - Hepburn, D.J.; Ackland, G.J. Metallic-covalent interatomic potential for carbon in iron. Phys. Rev. B
**2008**, 78, 165115. [Google Scholar] [CrossRef] - Tschopp, M.A.; Solanki, K.N.; Gao, F.; Sun, X.; Khaleel, M.A.; Horstemeyer, M.F. Probing grain boundary sink strength at the nanoscale: Energetics and length scales of vacancy and interstitial absorption by grain boundaries in α-Fe. Phys. Rev. B
**2012**, 85, 064108. [Google Scholar] [CrossRef] - Hirel, P. Atomsk: A tool for manipulating and converting atomic data files. Comput. Phys. Commun.
**2015**, 197, 212–219. [Google Scholar] [CrossRef] - Osetsky, Y.N.; Bacon, D.J. An atomic-level model for studying the dynamics of edge dislocations in metals. Model. Simul. Mater. Sci. Eng.
**2003**, 11, 427–446. [Google Scholar] [CrossRef] - Bacon, D.; Osetsky, Y.; Rodney, D. Dislocation–Obstacle Interactions at the Atomic Level. In Dislocations in Solids; Elsevier: Amsterdam, The Netherlands, 2009; Chapter 88; pp. 1–90. [Google Scholar] [CrossRef]
- Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] - Vo, N.Q.; Averback, R.S.; Bellon, P.; Caro, A. Limits of hardness at the nanoscale: Molecular dynamics simulations. Phys. Rev. B
**2008**, 78, 241402. [Google Scholar] [CrossRef] - Waseda, O.; Veiga, R.G.A.; Morthomas, J.; Chantrenne, P.; Becquart, C.S.; Ribeiro, F.; Jelea, A.; Goldenstein, H.; Perez, M. Formation of carbon Cottrell atmospheres and their effect on the stress field around an edge dislocation. Scr. Mat.
**2016**, 129, 16–19. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Nelson, D.R.; Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B
**1983**, 28, 784–805. [Google Scholar] [CrossRef] - Honeycutt, J.D.; Andersen, H.C. Molecular dynamics study of melting and freezing of small Lennard-Jones clusters. J. Phys. Chem.
**1987**, 91, 4950–4963. [Google Scholar] [CrossRef] - Kelchner, C.L.; Plimpton, S.J.; Hamilton, J.C. Dislocation nucleation and defect structure during surface indentation. Phys. Rev. B
**1998**, 58, 11085–11088. [Google Scholar] [CrossRef] - Ackland, G.J.; Jones, A.P. Applications of local crystal structure measures in experiment and simulation. Phys. Rev. B
**2006**, 73, 054104. [Google Scholar] [CrossRef] - Stukowski, A. Structure identification methods for atomistic simulations of crystalline materials. Model. Simul. Mater. Sci. Eng.
**2012**, 20, 045021. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool. Model. Simul. Mater. Sci. Eng.
**2009**, 18, 015012. [Google Scholar] [CrossRef] - de Koning, M.; Cai, W.; Bulatov, V.V. Anomalous Dislocation Multiplication in FCC Metals. Phys. Rev. Lett.
**2003**, 91, 025503. [Google Scholar] [CrossRef] [Green Version] - Shimokawa, T.; Kitada, S. Dislocation Multiplication from the Frank-Read Source in Atomic Models. Mater. Trans.
**2014**, 55, 58–63. [Google Scholar] [CrossRef] - Li, X.Y.; Yang, W. Atomistic simulations for the evolution of a U-shaped dislocation in fcc Al. Phys. Rev. B
**2006**, 74, 144108. [Google Scholar] [CrossRef] [Green Version] - Tsuru, T.; Aoyagi, Y.; Kaji, Y.; Shimokawa, T. Influence of Competition between Intragranular Dislocation Nucleation and Intergranular Slip Transfer on Mechanical Properties of Ultrafine-Grained Metals. Mater. Trans.
**2013**, 54, 1580–1586. [Google Scholar] [CrossRef] [Green Version] - Fitzgerald, S.P.; Aubry, S.; Dudarev, S.L.; Cai, W. Dislocation dynamics simulation of Frank-Read sources in anisotropic α-Fe. Model. Simul. Mater. Sci. Eng.
**2012**, 20, 045022. [Google Scholar] [CrossRef] - Po, G.; Cui, Y.; Rivera, D.; Cereceda, D.; Swinburne, T.D.; Marian, J.; Ghoniem, N. A phenomenological dislocation mobility law for bcc metals. Acta Mater.
**2016**, 119, 123–135. [Google Scholar] [CrossRef] [Green Version] - Po, G.; Ghoniem, N. A variational formulation of constrained dislocation dynamics coupled with heat and vacancy diffusion. J. Mech. Phys. Solids
**2014**, 66, 103–116. [Google Scholar] [CrossRef] - Domain, C.; Monnet, G. Simulation of Screw Dislocation Motion in Iron by Molecular Dynamics Simulations. Phys. Rev. Lett.
**2005**, 95, 215506. [Google Scholar] [CrossRef] [Green Version] - Vitek, V.; Smith, D.A.; Pond, R.C. Structure of tilt grain boundaries in b.c.c. metals. Philos. Mag. A
**1980**, 41, 649–663. [Google Scholar] [CrossRef] - Hahn, E.N.; Fensin, S.J.; Germann, T.C.; Meyers, M.A. Symmetric tilt boundaries in body-centered cubic tantalum. Scr. Mater.
**2016**, 116, 108–111. [Google Scholar] [CrossRef] [Green Version] - Forbes, J.W. Shock Wave Compression of Condensed Matter; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar] [CrossRef]
- Burgers, W. On the process of transition of the cubic-body-centered modification into the hexagonal-close-packed modification of zirconium. Physica
**1934**, 1, 561–586. [Google Scholar] [CrossRef] - Wang, F.M.; Ingalls, R. Iron bcc-hcp transition: Local structure from x-ray-absorption fine structure. Phys. Rev. B
**1998**, 57, 5647–5654. [Google Scholar] [CrossRef] - Meyers, M.; Jarmakani, H.; Bringa, E.; Remington, B. Chapter 89 Dislocations in Shock Compression and Release. In Dislocations in Solids; Elsevier: Amsterdam, The Netherlands, 2009; pp. 91–197. [Google Scholar] [CrossRef]
- Turneaure, S.J.; Renganathan, P.; Winey, J.; Gupta, Y. Twinning and Dislocation Evolution during Shock Compression and Release of Single Crystals: Real-Time X-ray Diffraction. Phys. Rev. Lett.
**2018**, 120, 265503. [Google Scholar] [CrossRef] - Meyers, M.A.; Murr, L.E. (Eds.) Shock Waves and High-Strain-Rate Phenomena in Metals; Springer US: New York, NY, USA, 1981. [Google Scholar] [CrossRef]
- Vítek, V. Dissociation of Dislocations on {110} Planes in Anisotropie B.C.C. Metals. Phys. Status Solidi B
**1966**, 15, 557–566. [Google Scholar] [CrossRef] - Kuznetsov, A.; Gornostyrev, Y.; Katsnelson, M.; Trefilov, A. Effect of the dislocations on the kinetics of a martensitic transition. Mater. Sci. Eng. A
**2001**, 309–310, 168–172. [Google Scholar] [CrossRef] - Cohen, J.; Hinton, R.; Lay, K.; Sass, S. Partial dislocations on the {110} planes in the b.c.c. lattice. Acta Metall.
**1962**, 10, 894–895. [Google Scholar] [CrossRef] - Clouet, E.; Garruchet, S.; Nguyen, H.; Perez, M.; Becquart, C.S. Dislocation interaction with C in α-Fe: A comparison between atomic simulations and elasticity theory. Acta Mater.
**2008**, 56, 3450–3460. [Google Scholar] [CrossRef] - Wilde, J.; Cerezo, A.; Smith, G.D.W. Three-dimensional atomic-scale mapping of a Cottrell atmosphere around a dislocation in iron. Scr. Mater.
**2000**, 43, 39–48. [Google Scholar] [CrossRef] - Grujicic, M.; Olson, G.B. Dynamics of Martensitic Interfaces. Interface Sci.
**1998**, 6, 155–164. [Google Scholar] [CrossRef] - Smith, R.F.; Eggert, J.H.; Swift, D.C.; Wang, J.; Duffy, T.S.; Braun, D.G.; Rudd, R.E.; Reisman, D.B.; Davis, J.P.; Knudson, M.D.; et al. Time-dependence of the alpha to epsilon phase transformation in iron. J. Appl. Phys.
**2013**, 114, 223507. [Google Scholar] [CrossRef] [Green Version] - Shao, J.L.; Wang, P.; Zhang, F.G.; He, A.M. Hcp/fcc nucleation in bcc iron under different anisotropic compressions at high strain rate: Molecular dynamics study. Sci. Rep.
**2018**, 8, 7650. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The simulation setup used to predict the behavior of crystals containing two coherent twin boundaries. Periodic boundary conditions were used for all three normal directions. The blue surface marks the (121) plane.

**Figure 2.**Simulation box containing a single $1/2\left[111\right]\left(\overline{1}01\right)$ edge dislocation (green line). Periodic boundary conditions were used along $\left[1\overline{2}1\right]$ (x) and $\left[111\right]$ (z) directions, whereas rigid boundary conditions were applied along the $\left[\overline{1}01\right]$ (y) direction.

**Figure 3.**Model of the Frank–Read source in iron showing a dislocation segment of length 9 nm terminated by cylindrical holes of radius 0.5 nm.

**Figure 4.**Dislocation mobility of an edge dislocation under shear deformation from the Frank–Read source in iron. Top: DDD simulation; Middle and bottom: MD simulations under shear deformation with a strain rate of $5\times {10}^{8}$ s

^{−1}for two different slip systems. The dislocation is detected by the Dislocation analysis algorithm implemented in OVITO [45]. It is surrounded by disordered atoms in grey.

**Figure 5.**Visualization of the system configuration before and after relaxation on $\left(\overline{1}01\right)$ plane of a $\Sigma 3$ coherent twin boundary crystal. Atoms are colored according to crystal structure types (blue, bcc; grey, disordered atoms).

**Figure 6.**Snapshot of one layer of atoms on the $\left(\overline{1}01\right)$ plane of a single-crystal containing four coherent twin boundaries at 0 ps and 30 ps (3.0% strain). Atoms are colored according to crystal structure types (blue, bcc; yellow, hcp; grey, disordered atoms) and red arrows are displacement vectors. Connections between atoms are shown by black lines.

**Figure 7.**Snapshot of the $\left(\overline{1}01\right)$ plane of a single-crystal containing 4 coherent twin boundaries at a strain of 4% and 10% (40 ps and 100 ps) on the left and the right, respectively. Atoms are colored according to crystal structure types (blue, bcc; green, fcc; yellow, hcp; grey, disordered atoms).

**Figure 8.**Phase fraction versus strain of a single-crystal containing four coherent twin boundaries under hydrostatic compression with a strain rate of $1\times {10}^{9}$ s

^{−1}.

**Figure 9.**Dislocation shapes of the 1/2$\left[111\right]\left(\right)open="("\; close=")">\overline{1}01$ dislocation at 0 ps and 27.5 ps (2.5% strain). The original dislocation is on the left and the dislocation after compression at a strain of 2.75% (27.5 ps) is on the right. Atoms are colored according to their shear stress in GPa.

**Figure 10.**Snapshots showing side view, front view and a cross section of the dislocation (in green) and surrounding atoms at strain of 3% (30 ps). The side view shows the buckling of surrounding atoms and the nucleation of the hcp phase with a closer look of the nucleation of the hcp phase on the $\left(\right)$ plane. Atoms are colored according to crystal structure types (blue, bcc; green, fcc; yellow, hcp; grey, disordered atoms).

**Figure 11.**Snapshots showing the Carbon Cottrell atmosphere for the system equilibrated by MD simulations. (

**Left**) Top view of the distribution of carbon atoms (red) in a radius of 4.5 nm from the edge dislocation, bcc iron is shown in blue. (

**Right**) Side view where only carbon and disordered iron atoms (white) along the edge dislocation are shown.

**Figure 12.**Snapshots of the dense phase at 6% strain of the sample containing an edge dislocation and a Cottrell atmosphere are shown on top and bottom, respectively. Atoms are colored by their local structure or atom type (red, carbon; yellow, hcp; green, fcc).

**Figure 13.**Hexagonal close-packed phase fraction versus strain for simulations of the sample with two twin surfaces (red), four twin surfaces (green), edge dislocations (blue) and Cottrell atmosphere (cyan) and pure iron single-crystals (black). Fcc fractions for all five simulations are embedded in the top left figure.

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**MDPI and ACS Style**

Luu, H.-T.; Veiga, R.G.A.; Gunkelmann, N.
Atomistic Study of the Role of Defects on *α* → *ϵ* Phase Transformations in Iron under Hydrostatic Compression. *Metals* **2019**, *9*, 1040.
https://doi.org/10.3390/met9101040

**AMA Style**

Luu H-T, Veiga RGA, Gunkelmann N.
Atomistic Study of the Role of Defects on *α* → *ϵ* Phase Transformations in Iron under Hydrostatic Compression. *Metals*. 2019; 9(10):1040.
https://doi.org/10.3390/met9101040

**Chicago/Turabian Style**

Luu, Hoang-Thien, Roberto G. A. Veiga, and Nina Gunkelmann.
2019. "Atomistic Study of the Role of Defects on *α* → *ϵ* Phase Transformations in Iron under Hydrostatic Compression" *Metals* 9, no. 10: 1040.
https://doi.org/10.3390/met9101040