A novel method for accurate and efficient evaluation of the change in energy barriers for carbon diffusion in ferrite under heterogeneous stress is introduced. This method, called Linear Combination of Stress States, is based on the knowledge of the effects of simple stresses (uniaxial or shear) on these diffusion barriers. Then, it is assumed that the change in energy barriers under a complex stress can be expressed as a linear combination of these already known simple stress effects. The modifications of energy barriers by either uniaxial traction/compression and shear stress are determined by means of atomistic simulations with the Climbing Image-Nudge Elastic Band method and are stored as a set of functions. The results of this method are compared to the predictions of anisotropic elasticity theory. It is shown that, linear anisotropic elasticity fails to predict the correct energy barrier variation with stress (especially with shear stress) whereas the proposed method provides correct energy barrier variation for stresses up to ∼3 GPa. This study provides a basis for the development of multiscale models of diffusion under non-uniform stress.
REFERENCES
1.
D. A.
Porter
and K. E.
Easterling
, Phase Transformation in Metals and Alloys
(Chapman and Hall
, London
, 1992
), p. 514
.2.
K. A.
Taylor
and M.
Cohen
, “Ageing of ferrous martensites
,” Prog. Mater. Sci.
36
, 225
–272
(1992
).3.
M.
Hillert
, “Diffusion and interface control of reactions in alloys
,” Metall. Mater. Trans. A
6
, 5
–19
(1975
).4.
5.
M.
Weller
, “Point defect relaxations
,” Mechanical Spectroscopy Q−1
(Trans Tech Pub.
, 2001
), p. 683
.6.
J. L.
Snoek
, “Effect of small quantities of carbon and nitrogen on the elastic and plastic properties of iron
,” Physica
8
(7
), 711
–733
(1941
).7.
M.
Weller
, “Anelastic relaxation of point defects in cubic crystals
,” J. Phys. IV
6
(C8
), 63
(1996
).8.
M.
Hillert
, “The kinetics of the first stage of tempering
,” Acta Metall.
7
(10
), 653
–658
(1959
).9.
M.
Cohen
, “Self-diffusion during plastic deformation
,” Trans. JIM
11
, 145
–151
(1970
).10.
A.
Cottrell
and B.
Bilby
, “Dislocation theory of yielding and strain ageing of iron
,” Proc. Phys. Soc. A
62
, 49
–62
(1949
).11.
K.
Kamber
, D.
Keefer
, and C.
Wert
, “Interactions of interstitials with dislocations in iron
,” Acta Metall.
9
, 403
–414
(1961
).12.
V.
Massardier
, N.
Lavaire
, M.
Soler
, and J.
Merlin
, “Comparison of the evaluation of the carbon content in solid solution in extra-mild steels by thermoelectric power and by internal friction
,” Scr. Mater.
50
, 1435
–1439
(2004
).13.
M.
Miller
, E.
Kenik
, K.
Russell
, L.
Heatherly
, D.
Hoelzer
, and P.
Maziasz
, “Atom probe tomography of nanoscale particles in ODS ferritic alloys
,” Mater. Sci. Eng. A
353
(1–2
), 140
–145
(2003
).14.
D.
Frenkel
and B.
Smit
, Understanding Molecular Simulations: From Algorithms to Applications
(Academic Press
, 2002
).15.
A. F.
Voter
, “Introduction to kinetic Monte-Carlo
,” in Radiation Effects in Solids
, NATO Science Series
Vol. 235
, edited by K. E.
Sickafus
, E. A.
Kotomin
, and B. P.
Uberuaga
(Springer
, Netherlands
, 2007
), pp. 1
–23
.16.
D. P.
Landau
and K.
Binder
, A Guide to Monte-Carlo Simulations in Statistical Physics
(Cambridge University Press
, Cambridge, England
, 2005
).17.
G.
Subramanian
, D.
Perez
, B. P.
Uberuaga
, C. N.
Tomé
, and A. F.
Voter
, “Method to account for arbitrary strains in kinetic Monte-Carlo simulations
,” Phys. Rev. B
87
, 144107
(2013
).18.
S.
Garruchet
and M.
Perez
, “Modeling the Snoek peak by coupling molecular dynamics and kinetic Monte-Carlo methods
,” Comput. Mater. Sci.
43
, 286
–292
(2008
).19.
R. G. A.
Veiga
, M.
Perez
, C. S.
Becquart
, C.
Domain
, and S.
Garruchet
, “Effect of the stress field of an edge dislocation on carbon diffusion in α-iron: Coupling molecular statics and atomistic kinetic Monte Carlo
,” Phys. Rev. B
82
, 054103
(2010
).20.
N.
Castin
and L.
Malerba
, “Calculation of proper energy barriers for atomistic kinetic Monte Carlo simulations on rigid lattice with chemical and strain field long-range effects using artificial neural networks
,” J. Chem. Phys.
132
, 074507
(2010
).21.
L. K.
Béland
, P.
Brommer
, F.
El-Mellouhi
, J. F.
Joly
, and N.
Mousseau
, “Kinetic activation-relaxation technique
,” Phys. Rev. E
84
, 046704
(2011
).22.
G.
Henkelman
and H.
Jònsson
, “Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points
,” J. Chem. Phys.
113
, 9978
(2000
).23.
G.
Henkelman
, B.
Uberuaga
, and H.
Jònsson
, “A climbing image nudged elastic band method for finding saddle points and minimum energy paths
,” J. Chem. Phys.
113
, 9901
(2000
).24.
S. A.
Trygubenkoa
and D. J.
Wales
, “A doubly nudged elastic band method for finding transition states
,” J. Chem. Phys.
120
, 2082
(2004
).25.
G. T.
Barkema
and N.
Mousseau
, “Event-based relaxation of continuous disordered systems
,” Phys. Rev. Lett.
77
, 4358
(1996
).26.
N.
Mousseau
and G.
Barkema
, “Traveling through potential energy landscapes of disordered materials: The activation-relaxation technique
,” Phys. Rev. E
57
, 2419
(1998
).27.
N.
Mousseau
and G.
Barkema
, “Activated mechanisms in amorphous silicon: An activation-relaxation-technique study
,” Phys. Rev. B
61
, 1898
–1906
(2000
).28.
G.
Henkelman
and H.
Jónsson
, “A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives
,” J. Chem. Phys.
111
, 7010
–7022
(1999
).29.
A.
Voter
, F.
Montalenti
, and T. C.
Germann
, “Extending the time scale in atomistic simulation of materials
,” Annu. Rev. Mater. Res.
32
, 321
–346
(2002
).30.
C. S.
Becquart
, J. M.
Raulot
, G.
Bencteux
, C.
Domain
, M.
Perez
, and S.
Garruchet
, “Atomistic modeling of an Fe system with a small concentration of C
,” Comput. Mater. Sci.
40
, 119
(2007
).31.
M. I.
Mendelev
, D. J.
Srolovitz
, S.
Han
, A. V.
Barashev
, and G. J.
Ackland
, “Development of an interatomic potential for phosphorus impurities in α-iron
,” J. Phys.: Condens. Matter
16
, S2629
–S2642
(2004
).32.
R. G. A.
Veiga
, C. S.
Becquart
, and M.
Perez
, “Comments on “Atomistic modeling of an Fe system with a small concentration of C
,” Comput. Mater. Sci.
82
, 118
–121
(2014
).33.
S.
Plimpton
, “Fast parallel algorithms for short-range molecular dynamics
,” J. Comput. Phys.
117
, 1
–19
(1995
).34.
A.
Stukowski
, “Visualization and analysis of atomistic simulation data with OVITO - The Open Visualization Tool
,” Model. Simul. Mater. Sci. Eng.
18
, 015012
(2010
).35.
D.
Sheppard
, R.
Terrell
, and G.
Henkelman
, “Optimization methods for finding minimum energy paths
,” J. Chem. Phys.
128
, 134106
(2008
).36.
L. J.
Munro
and D. J.
Wales
, “Defect migration in crystalline silicon
,” Phys. Rev. B
59
, 3969
(1999
).37.
E.
Clouet
, S.
Garruchet
, H.
Nguyen
, M.
Perez
, and C.
Becquart
, “Dislocation interaction with C in α-Fe: A comparison between atomic simulations and elasticity theory
,” Acta Mater.
56
, 3450
(2008
).38.
D. J.
Bacon
, D. M.
Barnett
, and R. O.
Scattergood
, “Anisotropic continuum theory of lattice defects
,” Prog. Mater. Sci.
23
(2–4
), 51
–262
(1980
).39.
R. G. A.
Veiga
, M.
Perez
, C. S.
Becquart
, E.
Clouet
, and C.
Domain
, “Comparison of atomistic and elasticity approaches for carbon diffusion near line defects in α-iron
,” Acta Mater.
59
, 6963
(2011
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
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