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Atomistic simulations of solute-interface interactions in iron Jin, Hao 2014

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Atomistic Simulations of Solute-InterfaceInteractions in IronbyHao JinB.Sc., Shandong University, 2006M.Sc., Shandong University, 2009a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Materials Engineering)The University of British Columbia(Vancouver)June 2014© Hao Jin, 2014AbstractThe kinetics of the recrystallization and austenite-ferrite (fcc-bcc) phase trans-formation in steels are markedly affected by substitutional alloying elements.Nevertheless, the detailed mechanisms of their interaction with the grain boundariesand interfaces are not fully understood. Using density functional theory, wedetermine the segregation energies of commonly used alloying elements (e.g. Nb,Mo, Mn, Si, Cr, Ni) in the Σ5 (013) tilt grain boundary in bcc and fcc Fe, andthe bcc-fcc interfaces. We find a strong interaction between large solutes (e.g. Nb,Mo and Ti) and grain boundaries or interfaces that is consistent with experimentalobservations of the effects of these alloying elements on delaying recrystallizationand the austenite-to-ferrite transformation in low-carbon steels. In addition, wecompute the solute-solute interactions as a function of solute pair distance in thegrain boundaries and interfaces, which suggest co-segregation for these large solutesat intermediate distances in striking contrast to the bulk.Besides the prediction of solute segregation, the self- and solute-diffusion in Fe-based system are also investigated within a framework combining density functionaltheory calculations and kinetic Monte Carlo simulations. Good agreement betweenour calculations and the measurements for self- and solute diffusion in bulk Feis achieved. For the first time, the effective activation energies and diffusioncoefficients for various solutes in the α-Fe Σ5 (013) grain boundary are determined.The results demonstrate that grain boundary diffusion is significantly faster than forlattice diffusion, confirming grain boundaries are fast diffusion paths. By contrast,the effective activation energy of self-diffusion in a bcc-fcc Fe interface is close tothe value of fcc bulk self-diffusion, and the investigated bcc-fcc interface provides amoderate “fast diffusion” path.iiPrefaceThis dissertation is written based on original research conducted by theauthor, Hao Jin. All of the work presented henceforth was conducted in theMaterials Engineering Department of the University of British Columbia,at the Point Grey Campus. My supervisor Dr. Matthias Militzer wasinvolved in all stages of the project, provided guidance and assisted withthe manuscript composition.Figures 1.1, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10 2.11 2.12 2.132.14 2.15 2.16 2.17 2.18 2.19 and 4.1 in the introduction, literature reviewand methodology chapters have been taken with permission from the citedsources.The contents of Chapter 5 were published in: Hao Jin, Ilya Elfimov andMatthias Militzer, “Study of the interaction of solutes with Σ5 (013) tiltgrain boundaries in iron using density-functional theory”, J. Appl. Phys,vol. 115, p. 093506, 2014. Part of the simulation results was also presentedin a conference: Hao Jin, Ilya Elfimov and Matthias Militzer, “Study of theinteraction of solutes with interfaces in iron using density-functional theory”.2011 TMS Annual Meeting and Exhibition, San Diego, CA, 2011.Chapters 6 and 7 are based on the simulation work I conducted, and itwill be submitted for publication. Part of the simulation results was alsopresented in a conference: Hao Jin, Ilya Elfimov and Matthias Militzer,“First-principles Simulations of the Interaction of Alloying Elements withthe Austenite-ferrite (fcc-bcc) Interface in Iron”. 2014 TMS Annual Meetingand Exhibition, San Diego, CA, 2014.Chapter 8 is based on the simulation work I conducted. Section 8.3iiiwas presented in a conference: Hao Jin, Ilya Elfimov and Matthias Militzer,“Atomistic simulations of solute interface interactions in iron”. 2012 TMSAnnual Meeting and Exhibition, Orlando, FL, 2012.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . xxvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Solute Segregation on Grain Boundaries . . . . . . . . . . . . 62.2.1 Simulations of Grain Boundary Segregation . . . . . . 62.2.2 Binding Energy . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Effects of Solutes on Grain Boundary Properties . . . 132.3 Solute Diffusion in Grain Boundaries . . . . . . . . . . . . . . 162.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Solute Diffusion in α-Fe . . . . . . . . . . . . . . . . . 17v2.3.3 Point Defects in Grain Boundaries . . . . . . . . . . . 202.3.4 Diffusion Mechanisms in Grain Boundaries . . . . . . 222.4 Austenite (fcc) and Ferrite (bcc) Interface . . . . . . . . . . . 262.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.2 Approaches to Study Magnetism in γ-Fe . . . . . . . . 262.4.3 Effects of Solutes on Magnetism of γ-Fe . . . . . . . . 292.4.4 DFT Studies for Heterophase Interfaces . . . . . . . . 312.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . 364 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . 384.1.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 384.1.2 Exchange-correlation Energy . . . . . . . . . . . . . . 394.1.3 Structure Optimization . . . . . . . . . . . . . . . . . 404.1.4 Simulation Software Package . . . . . . . . . . . . . . 414.1.5 Domain Size . . . . . . . . . . . . . . . . . . . . . . . 434.1.6 Parameter Setup and Formulas for DFT Calculations 484.1.7 Simulation Procedure for Solute Segregation . . . . . . 484.1.8 Simulation Procedure for Diffusion . . . . . . . . . . . 494.1.9 Hessian Matrix and Vibrational Frequencies . . . . . . 514.2 Kinetic Monte Carlo Simulations . . . . . . . . . . . . . . . . 524.3 Molecular Statics Simulations . . . . . . . . . . . . . . . . . . 545 Interaction of Solutes with the Σ5 (013) Tilt GrainBoundary in Iron . . . . . . . . . . . . . . . . . . . . . . . . . 555.1 Grain Boundary Structure . . . . . . . . . . . . . . . . . . . . 555.2 Grain Boundary Energy . . . . . . . . . . . . . . . . . . . . . 565.3 Single Solute Segregation Energies . . . . . . . . . . . . . . . 575.4 Solute-solute Interactions at the Grain Boundary . . . . . . . 635.5 Molecular Statics Simulations . . . . . . . . . . . . . . . . . . 716 First-Principles Study of Face-Centered Cubic γ-Iron . . . 746.1 Bulk and Grain Boundary Structures in γ-Fe . . . . . . . . . 74vi6.2 Magnetic Configurations for γ-Fe . . . . . . . . . . . . . . . . 776.3 Single Solute in γ-Fe . . . . . . . . . . . . . . . . . . . . . . . 806.4 Solute-solute Interactions in γ-Fe . . . . . . . . . . . . . . . . 836.5 Solute Segregation in Σ5 Grain Boundaries in fcc γ-Fe . . . . 867 Interaction of Solutes with α-γ Iron Interface . . . . . . . . 907.1 Structure of Ferrite-Austenite (bcc-fcc) Interface . . . . . . . 907.2 Interface Energy . . . . . . . . . . . . . . . . . . . . . . . . . 937.3 Magnetic and Electronic Properties for bcc-fcc Interface . . . 957.4 Binding Energies of Solutes with bcc-fcc Interface . . . . . . . 977.5 Solute-solute Interactions in bcc-fcc Interface . . . . . . . . . 1078 Simulation of Self- and Solute-diffusion in Fe . . . . . . . . 1118.1 Diffusion in Fe bcc Lattice . . . . . . . . . . . . . . . . . . . . 1118.2 Diffusion in the Fe fcc Lattice . . . . . . . . . . . . . . . . . . 1178.3 Grain Boundary Diffusion in bcc Fe . . . . . . . . . . . . . . 1228.4 Diffusion in the bcc-fcc Fe Interface . . . . . . . . . . . . . . . 1289 Conclusions and Future Work . . . . . . . . . . . . . . . . . 1329.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136viiList of TablesTable 2.1 Binding energies (in eV) of various non-metallic elementswith special grain boundaries. . . . . . . . . . . . . . . . . 12Table 2.2 Calculated activation energies of fully ordered ferromag-netic state (Q0) and pre-factor (D0) in α-Fe vs. experi-mental data. . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 5.1 Magnetic moments (µB) for single solute in grain boundary(GB) and bulk sites. . . . . . . . . . . . . . . . . . . . . . 68Table 6.1 Predictions of lattice parameters (al), and magnetic mo-ments (MM) for magnetic phases of α and γ-Fe at 0 K.Earlier DFT and experimental results are also shown forcomparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 78Table 6.2 Effective binding energies (Eeffseg ) for selected solutes in Σ5bcc and fcc grain boundary. Unit is eV. . . . . . . . . . . . 88Table 7.1 Surface and interface energies for the equilibrium bcc andfcc surfaces, and bcc-fcc interface, in units of J/m2.* . . . 94Table 8.1 Calculated vacancy formation energy (Ev), migration en-ergy (Em), vacancy-solute binding energy (Eb), and acti-vation energy (Q0) for self- and solute-diffusion in bcc Fe. 113viiiTable 8.2 Vacancy formation energies (Ev), vacancy migration bar-rier (Em), and the activation energy for self-diffusion (Q0)along each path. The migration paths are labeled inFigure 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Table 8.3 Lattice constants (al), atomic volume (V ), vacancy for-mation energy (Ev), effective migration energy (Eeffm ),and effective activation energy (Qeffa ) as a function oftemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . 119Table 8.4 Calculated migration energy for various solutes along grainboundary (Em, in units of eV). Sites a0, b0, c0, and c referto boundary sites as shown in Figure 5.1. . . . . . . . . . 125Table 8.5 The effective activation energy (Qgb) in Σ5 (013) grainboundaries and the ratio of Qgb/Qbulk. . . . . . . . . . . . 127Table 8.6 Migration energies of vacancy for different types of jumppaths as defined in the text in the bcc-fcc interface. Thearrows indicate the spin states of the jump Fe atoms ininitial and final states. . . . . . . . . . . . . . . . . . . . . 130Table 8.7 Activation energy for Fe self-diffusion in the bcc-fcc inter-face, fcc bulk, bcc bulk, and Σ5 grain boundaries (GBs) at0K. Unit is eV. . . . . . . . . . . . . . . . . . . . . . . . . 130ixList of FiguresFigure 1.1 Variation of the interface velocity (ν) versus the applieddriving pressure (Pd) for systems with different soluteconcentrations (Ci, i = 1, 2, 3) [5]. . . . . . . . . . . . . . 2Figure 2.1 Side view of the Fe Σ3 (111) grain boundary [15]. (a)Cluster of 53 atoms and (b) cluster of 91 atoms. Theopen circle represents solute atom. . . . . . . . . . . . . 7Figure 2.2 Side view of the Fe Σ3 (111) and Σ5 (210) grain bound-aries [25]. The lighter and darker circles mark theatoms belonging to two different planes. The interstitialpositions at the grain boundary are indicated by theyellow circles. The substitutional positions in differentgrain layers are labeled by the numbers. Lower panelsshow top view of the supercells taken in the cross-sectionplane passing through the grain boundary (broken line). 8Figure 2.3 < 100 > symmetric tilt grain boundary structures withstructural units outlined for the Σ5(210), Σ29(730),Σ5(310) and Σ17(530) boundaries [19]. Black and whitedenote atoms on different 100 planes. The differentstructural units are labeled as A, B, and C. . . . . . . . . 10xFigure 2.4 MD simulation of the Cu Σ5(210) grain boundary at800K allowing variations in grain boundary density. Theboundary undergoes a first-order phase transformationfrom filled kites (labeled in blue color) to split kites(labeled in red color) nucleating and growing from thesurface [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.5 DFT simulations for segregation energies as a function ofsegregation site for the Fe Σ3 (111) grain boundary withsulfur at the substitutional site [34]. . . . . . . . . . . . . 13Figure 2.6 Calculated binding energies of phosphorous with varyingthe segregation concentration in the Σ3 (111) grainboundary in Fe [35]. . . . . . . . . . . . . . . . . . . . . . 14Figure 2.7 Results of energy difference relative to the equilibriumvs. uniaxial tensile strain for the pristine (solid line withclosed circles (black)), 1 hydrogen atom- (broken linewith open circles (blue)) and 4 hydrogen atoms-trapped(broken line with squares (red)) Σ3(112) grain boundaries[39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.8 Minimum-energy path for carbon diffusion in α-Fe andthe local structures of initial, intermediate, final, andtransition states [9]. . . . . . . . . . . . . . . . . . . . . . 18Figure 2.9 Molecular statics results for vacancy formation energy asa function of the distance from the grain boundary in Cu[42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.10 Examples of vacancy instability in Cu Σ9(12¯2) grainboundary. (a) Perfect grain boundary structure; (b)Unstable vacancy at boundary site 6 [42]. . . . . . . . . . 22Figure 2.11 Possible migration path of He interstitial atom in theFe Σ11 grain boundary. Small sphere represents the Heatom. Arrows indicate possible paths [65]. . . . . . . . . . 23xiFigure 2.12 Diffusion coefficients of hydrogen atoms in bulk, on freesurface and in a Σ5 grain boundary. The activationenergies for hydrogen diffusion in bulk, free surface, andgrain boundary are 0.044 eV, 0.066 eV, and 0.31 eV,respectively [67]. . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.13 Calculated and measured self-diffusion coefficients in a Σ9Cu grain boundary [42]. . . . . . . . . . . . . . . . . . . . 25Figure 2.14 Phase diagram for bcc Fe and fcc Fe with differentmagnetic configurations [9]. . . . . . . . . . . . . . . . . . 27Figure 2.15 Variation of average atomic volume and magnetic moment(MM) per Fe atom of the fcc structured Fe-Cu alloysagainst the Cu concentration [85]. . . . . . . . . . . . . . 30Figure 2.16 Fragment of crystal and magnetic structure of γ-Fe forthe AFMD magnetic ordering [73]. Carbon interstitialimpurity in octahedral position is shown by dark (red)circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.17 Schematic of the (a) Fe(001)-Ag(001) interface and the(b) Fe(110)-Ag(111) interface [88]. The cross symbolsmark the high-symmetry sites with respect to the Agplane for the successive Fe atom at interface. The layersaround the interface are indexed by numbers. . . . . . . . 32Figure 2.18 Evolution of the atomic magnetic moment of the Fe atomsas a function of the position of the layer in the Fe(001)slab (black circles) and in the Au(001)-Fe(001) interface(red squares) [91]. . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.19 Projected densities of the d states of individual Fe atomsin the Au(001)/Fe(001) interface. Fesurf , Fecenter, andFeint denote the Fe atoms at the surface, in the center,and at the interface, respectively [91]. . . . . . . . . . . . 34xiiFigure 4.1 Sketch of the search paths for the minimum of a func-tion (i.e. energy) using the steepest descent (left) andconjugate-gradient approach (right). In both cases, thesearch begins at point 1. The conjugate-gradient methodconverges faster [97]. . . . . . . . . . . . . . . . . . . . . 41Figure 4.2 Convergence test of the total energy with respect toenergy cutoff. . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 4.3 Convergence test of the total energy with respect to k-point mesh for the conventional bcc Fe unit cell. . . . . . 45Figure 4.4 Convergence test of Nb cohesive energy with respect tosupercell size. . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 5.1 (a) Side view of the Σ5 (013) symmetrical tilt grainboundary. The lighter and darker circles represent Featoms in different (100) atomic planes. (b) Atomicstructure of the boundary sites. The letters a, b, andc refer to different boundary positions. The superscript(′) indicates the symmetric position with respect to the(013) boundary plane. The subscript identifies differentboundary unit cells of the Σ5 (013) grain boundary. . . . 56Figure 5.2 Grain boundary energy as a function of distance betweengrain boundaries. . . . . . . . . . . . . . . . . . . . . . . . 57Figure 5.3 Binding energies for substitutional solutes at the Σ5 (013)tilt grain boundary as a function of distance from theboundary plane; a, b, c, b′, and c′ refer to the boundarysites as labeled in Figure 5.1. . . . . . . . . . . . . . . . . 58Figure 5.4 Total grain boundary segregation of Nb as a function oftemperature in Σ5 grain boundary in Fe. The symbolsare calculated according to Equation 5.3, while the line isthe Langmuir-McLean fit. . . . . . . . . . . . . . . . . . . 59xiiiFigure 5.5 (a) The average binding energies, and (b) the effectivebinding energies of solutes as a function of solute atomicvolume. The dotted line indicates the atomic volume ofFe in the bcc lattice. . . . . . . . . . . . . . . . . . . . . . 60Figure 5.6 Valence electron density difference contour maps in the(100) plane for (a) Co-, and (b) Nb- segregated atboundary position a with respect to the non-segregatedgrain boundary (in units of e/A˚3 with a contour spacing of0.01 e/A˚3). The solid red and dashed blue lines indicategains and losses in electron density, respectively. . . . . . 62Figure 5.7 Binding energies for a second solute atom segregatingat (a) boundary position a, and (b) boundary positionb when the same solute is already present at anotherboundary site. The square symbols indicate the bindingenergies of a single solute atom. . . . . . . . . . . . . . . . 64Figure 5.8 Pair interactions for (a) Nb, (b) Cr, (c) Cu, and (d) Si inthe Fe bulk and at the Σ5 (013) grain boundary. . . . . . 66Figure 5.9 Binging energies for the grain boundary with Nb segregat-ing at (a) boundary position a, and (b) boundary positionb when another solute (i.e. Mo, Ti, Mn, Cr, V, Cu, orSi) is already present at a neighboring site indicated bythe second letter. The dotted line indicates the bindingenergies for a single Nb atom. . . . . . . . . . . . . . . . 70Figure 5.10 Site volume distribution in Σ5 (36.9°), Σ29 (43.6°) andΣ61 (10.4°) grain boundaries. . . . . . . . . . . . . . . . . 72Figure 5.11 Averaged atomic volume as a function of grain boundarymisorientation angle. The bars indicate the maximumand the minimum atomic volume of the boundary sites. . 73Figure 6.1 Fe fcc lattice structure with AFMD magnetic configura-tions. Red and blue circles indicate spin-up and spin-down magnetic states. The letters o, p, q, w, e, and e′ areused to label the different positions in the fcc bulk. . . . . 74xivFigure 6.2 SQS-32 structure for fcc Fe lattice. The red and blue ballindicate spin-up and spin-down states. . . . . . . . . . . . 75Figure 6.3 (a) Side view of the Fe Σ5 fcc grain boundaries withAFMD magnetic configurations. (b) Top view of thesupercells. Red and blue circles indicate spin-up and spin-down magnetic states. . . . . . . . . . . . . . . . . . . . . 76Figure 6.4 Various magnetic configurations vs. lattice constant infcc Fe. The bcc ground state is set to be the reference(zero) state. . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 6.5 Magnetic moments for solute atoms in bcc and fcc Fematrix. The x-axis is arranged in ascending order ofatomic number. . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 6.6 Calculated partial density of states (DOS) of 3d-orbitalfor solutes in the bcc (left) and fcc (right) phases. TheFermi levels (EF ) are set to zero . . . . . . . . . . . . . . 82Figure 6.7 Pair interactions for (a) Nb, (b) Cr, (c) Mn, (d) Ni, (e)Cu, and (f) Si in γ- and α-phase Fe matrix. . . . . . . . . 84Figure 6.8 Binding energies for substitutional solutes at the Σ5fcc grain boundary as a function of distance from theboundary plane; a, b, c, b′, and c′ refer to the boundarysites as labeled in Figure 6.3. . . . . . . . . . . . . . . . . 87Figure 6.9 Volume distribution per periodic unit cell for Σ5 grainboundaries in bcc and fcc Fe. . . . . . . . . . . . . . . . . 88Figure 7.1 Structure of bcc-fcc interface with K-S orientation rela-tionship. (a) AFMD and (b) SQS configuration for fcc.The white circles represent atoms in the bcc Fe grain, andthe red (blue) circles indicate atoms in the fcc Fe grainwith spin up (down) states. . . . . . . . . . . . . . . . . . 92Figure 7.2 bcc-fcc interface energy as a function of (a) the bcc grainsize while the length of the fcc grain is fixed to 25.4 A˚;and (b) the fcc grain size while the length of the bcc grainis fixed to 21.3 A˚. . . . . . . . . . . . . . . . . . . . . . . 93xvFigure 7.3 Magnetic moments of Fe atoms across the bcc-fcc interface. 95Figure 7.4 Averaged site volume distribution across the bcc-fccinterface. The bars indicate the range of values of sitevolumes for Fe atoms with spin-up or spin-down states. . 96Figure 7.5 Relative energies for (a) Nb, and (b) Mn across theα-γ interface as a function of distance from the habitplane. The bars indicate the range of energies for theSQS simulations, where the replaced Fe atom has spin-upor spin-down states. . . . . . . . . . . . . . . . . . . . . . 98Figure 7.6 Relative energies for substitutional solutes across the bcc-fcc interface as a function of distance from the habit plane.The bars indicate the range of energy values, where thereplaced Fe atom has spin-up or spin-down states. Thefcc grain is described by the AFMD configuration. . . . . 99Figure 7.7 Definition of the binding energy using linear interpolationfrom the bulk energies. . . . . . . . . . . . . . . . . . . . . 100Figure 7.8 Binding energies for substitutional solutes at the bcc-fccinterface as a function of distance from the habit plane.The bars indicate the actual values of binding energies.The fcc grain is described by the AFMD configuration.The interface width is indicated by two dashed lines. . . . 101Figure 7.9 Relative binding energies of Nb at the bcc-fcc interface asa function of the site volume. . . . . . . . . . . . . . . . . 102Figure 7.10 Calculated effective binding energies for selected alloyingelements as a function of solute volume in bcc-fcc inter-face. The dotted line indicates the volume of Fe calculatedfrom the first nearest neighbor distance in the bcc bulk Fe.105Figure 7.11 Calculated effective binding energies for selected alloyingelements as a function of solute volume in bcc-fcc inter-face. The dotted line indicates the volume of Fe calculatedfrom the first nearest neighbor distance in the bcc bulk Fe.106Figure 7.12 Illustrations of interaction profiles between solute atomsand bcc-fcc interface. . . . . . . . . . . . . . . . . . . . . . 107xviFigure 7.13 Pair interactions for (a) Nb, (b) Cr, and (c) Mn in bcc-fcc Fe interface. The first solute occupies its favoriteposition. The subscripts α and γ indicate solute atomoccupies bcc α- or fcc γ-phase grain, respectively. Thesuperscripts ‘1st’ and ‘2nd’ denote the first and the secondsubstitutional solute atom, respectively. The arrow ↑ (↓)represents solute atom has spin-up (spin-down) state. . . 109Figure 8.1 First-neighbor diffusion of the vacancy in bcc Fe. Onevacancy locates in the center, and diffuses along the redline to the corner of the cubic cell (the corner atom diffusesinto the centered vacancy). The calculated migrationbarrier for vacancy is also shown. . . . . . . . . . . . . . . 112Figure 8.2 Relationship between α¯ and ∆M12. The triangles denotethe parameter α¯ is derived from experiments [169–172]and the line gives the model according to Equation 8.4. . 114Figure 8.3 An illustration of the different vacancy hops involved inthe Le Claire nine-frequency model for the correlationfactor for solute diffusion in Fe. X and V denote thesolute atom and the vacancy, respectively. . . . . . . . . 115Figure 8.4 Predicted temperature dependence of the calculated α-Fe self-, Mn, Mo, and Nb solute-diffusion coefficients vs.published experimental data [11, 48, 165, 170, 171, 174?–178]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 8.5 The modified activation energies for Fe self-diffusion inbcc and fcc phase at different temperature. . . . . . . . . 121Figure 8.6 Predicted temperature dependence of the Fe-self diffusioncoefficients vs. published experimental data in bcc and fcc[174, 175]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 8.7 Vacancy formation energies at different sites of Σ5 (013)grain boundaries in α-Fe. The dotted red line indicatesthe value in the bulk lattice. . . . . . . . . . . . . . . . . 123xviiFigure 8.8 Vacancy-solute binding energies at Σ5 (013) grain bound-aries and bulk in α-Fe. Vacancy segregates at (a)boundary a0, and (b) boundary b position. The capitalletters X refer to the solute and V refer to the vacancy . . 124Figure 8.9 Predicted temperature dependence of the calculated Nb,Mo, Mn, and Fe-self diffusion coefficients at grain bound-aries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 8.10 Vacancy formation energies as a function of distance fromthe habit plane. The bars indicate the actual vacancyformation energies, where Fe atom has spin-up or spin-down states. . . . . . . . . . . . . . . . . . . . . . . . . . . 129Figure 8.11 Illustration of various jump paths in the bcc-fcc interface.Blue and red circles indicate fcc Fe atoms with spin-downand spin-up magnetic state, while white circles representbcc Fe atoms. . . . . . . . . . . . . . . . . . . . . . . . . . 129xviiiList of SymbolsAS Area of the boundary or interfaceal Lattice constant~a Acceleration in the Newton’s second lawa, b, c, b′, c′ Different positions in boundariesC11, C12, C44 Elastic constants in the bulkcbulk Solute concentration in the bulkcigb Solute concentration at the boundary site icgb Total solute concentration at the grain boundaryc0gb Fraction of boundary sites with favourable bindingciint Solute concentration at the bcc-fcc interface site icint Total solute concentration at the bcc-fcc interfacec0int Fraction of bcc-fcc interface sites with favourable bindingceqv Equilibrium vacancy concentrationd1, d2, d3 First nearest neighbor distance in the fcc bulk with AFMDstatedoe, doe′ Second nearest neighbor distance in the fcc bulk with AFMDstatexixdgs Grain sizedα bcc grain sizedγ fcc grain sizeD Diffusion coefficientD0 Pre-factor in the Arrhenius expression for diffusion coefficientDapp Apparent diffusion coefficientDb Grain boundary diffusivityDI bcc-fcc interface diffusivityDl Bulk diffusivityDαl bcc bulk diffusivityDγl fcc bulk diffusivityEb Solute-vacancy binding energyEcut Cut-off energyEm Migration energyEf2 Correlation energyEv Vacancy formation energyEX Total energy of the supercell with one soluteEX+Y Total energy of the supercell with two solute atomsE Total energies of the supercell with one vacancyE+X Total energy of the supercell with one vacancy and one soluteEirel Relative energy of the solute at the interface site iEiseg Segregation energy of the solute at the boundary site ixxEgb Total energy of the supercell with grain boundariesEiX+gb Total energy of the supercell with grain boundaries and onesolute atom segregated at the boundary site iEeff Effective solute-solute interactionEF Fermi energy~F Forcef2 Correlation factorFi Fraction of the grain boundary or interface sites that havebinding Eiseg~G Vector of the reciprocal latticeHi,j Hessian matrixHmi,j Mass-weighted Hessian matrixHˆ Hamiltonian operatorHˆH [ρ(~r)] Electrostatic interactions in Kohn-Sham equationsJ1 Exchange parameter in the first coordinate shell~k MomentumK Bulk modulusli Supercell dimension in i directionM Interface or grain boundary mobilitymia Atomic mass∆M12 Sum of the change in the local magnetic moments induced onthe Fe atoms in the first and second neighbor neighbors of ansolute atomxxiMM Local magnetic momentsN Number of atoms in the supercellNitermax Total number of vacancy jumps in the KMC simulationso, p, q, w, e, e′ Different positions in the fcc bulkQ0 Acitvation energy at 0KQa Acitvation energyQP Acitvation energy in paramagnetic stateRN Summation of the jump rate〈r2(t)〉 Mean square displacement of the diffusing solute atoms duringtime t~ri Electron coordinates~Ri Nuclei coordinatess(T ) Reduced magnetization at given temperature TSv Vacancy formation entropyTc Curie temperatureTm Equilibrium melting temperatureTα−γ Ferrite-austenite transition temperatureTˆ [ρ(~r)] Non-interacting kinetic energy in Kohn-Sham equationsUˆext[ρ(~r)] Potential energy in Kohn-Sham equationsurand Random number in the KMC simulaitonVˆxc[ρ(~r)] Exchange-correlation energy in Kohn-Sham equationsxrand Random number in the KMC simulaitonxxiiα Ferriteα¯ Species-dependent parameter that quantifies the dependence ofQa magnetizationαel, γel Parameters in Eshelby elasticity modelγ AusteniteΓj Jump frequency to site jΓX Solute-vacancy exchange jump frequencyδ Effective thickness of the grain boundaryδα Effective thickness of the interface in the bcc grainδγ Effective thickness of the interface in the fcc grainεi Eigenvalues in mass-weighted Hessian matrixθ Misorientation angleν0 Attempt frequencyνi Frequencies of the pure Fe systemνeqi Frequencies of the equilibrium configuration with one vacancyνspi Frequencies of the saddle-point configurationρ Electron charge densityσ Interface or grain boundary energyτkmc Residence time associated with the KMC stepΨ(~R,~r) Wave functions in the many-body Schro¨dinger equationΦi(~r) Wave functions in Kohn-Sham equations<∏l,m >SQS Correlation function for SQS, where {l,m} are the “interac-tion parameters” of figuresxxiiiΩ0 Primitive cell volumexxivList of AbbreviationsAFM Anti-ferromagneticAFMD Double-layer anti-ferromagneticAHSS Advanced high strength steelsbcc Body-centered cubicCE Cluster ExpansionsCPA Coherent potential approximationCG Conjugate-gradientCLS Coincident site latticeDFT Density functional theoryDOS Density of statesDP Dual phaseEAM Embedded atom methodECIs Effective cluster interactionsfcc Face-centered cubicFM FerromagneticGBs Grain boundariesxxvGGA Generalized gradient approximationJMAK Johnson-Mehl-Avrami-KolmogorovLDA Local density approximationMS Molecular staticsMD Molecular dynamicsKMC kinetic Monte CarloK-S Kurdjumov and SachsNC Non-collinearNEB Nudged elastic bandNM Non-magneticNN Nearest-neighborN-W Nishiyama and WassermannPAW Projector augmented wavePBE Perdew, Burke, and EnzerhofPM ParamagneticSQS Special quasi-random structureTMs Transition metalsTST Transition state theoryTRIP Transformation induced plasticityxxviAcknowledgementsFirst of all, I would like to express my sincere gratitude to my supervisor Dr.Matthias Militzer for providing continuous support and wisdom throughoutmy studies at UBC. His diligence, self-discipline and enthusiasm alwaysinspire me and set me a good example to follow for the rest of my life.Secondly, I would like to give a special thank to Dr. Ilya Elfimov. Ilyahas consistently supported and actively contributed to all my work. Healso spent many extra hours to provide professional advices for my graduatestudies.I would also like to extend my thanks to my committee members, Dr.Chad Sinclair, Dr. Jo¨rg Rottler and Dr. Warren Poole for their constructivecriticism and valuable advice on my research.I had a pleasure of spending wonderful time with colleagues and friendsat the Microstructural Engineering group at the Department of MaterialsEngineering. In particular, I would like to thank Benqiang, Liam, Tao, Sina,and Tegar for their stimulating discussions.I would like to thank the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada for the financial support of this work, as wellas WestGrid and Compute/Calcul Canada for computational resources.I want to thank Xiao, who appeared in my life in the dusk of the Ph.D.program and made it so much brighterAbove all, I would like to thank my parents for their love, unconditionalsupport, and tireless dedication throughout my life.xxviiChapter 1IntroductionAs the demand to produce new families of low-carbon high-strength steelsincreases, advancing knowledge of the physical mechanisms of microstruc-ture evolution becomes increasingly important in order to develop steelswith desired properties. These steels are so-called advanced high-strengthsteels (AHSS) and have improved properties. AHSS include dual phase(DP) steel, transformation induced plasticity (TRIP) steel, complex phasesteel and martensitic steel. The tensile strengths range from 500 to 1900MPa. These steels are superior in strength and ductility combination, andthus facilitate the energy absorption during impact and ensure safety whenreducing weight. As a result, with appropriate manufacturing techniques,AHSS offer great opportunities for designing inexpensive, safe and fuel-efficient vehicles by reduction of product weight, enhancement of crashperformance, and improvement of car fuel economy.Using existing facilities to produce AHSS one is required to have anincreased alloying content in terms of substitutional elements, such as V, Mn,Nb, Mo, Cr, Si, and Ti. Due to the presence of a high density of structuralimperfections at interphase boundaries, these solute atoms are more likelyto accumulate at these interfaces. It is believed that solute interact with themigrating interface and this is a basic phenomenon effectively used duringindustrial processing to tailor the phase transformation kinetics.In order to aid the alloy design, great effort has been made to develop1knowledge-based process models for the steel industry in the last decades[1, 2]. However, these process models require numerous empirical parametersthat are typically determined from time-consuming laboratory experiments.In addition, important physical properties in terms of the solute-interfaceinteraction are still essentially unexplored and have become an importantresearch area.By analyzing the physical mechanisms of microstructure evolution, it isknown that the presence of alloying elements can drastically reduce themigration rate of interface in pure metals. For example, substitutionalalloying elements (e.g., Nb, Mo) can significantly retard recrystallizationand the austenite-ferrite phase transformation in steels. This retardation isknown as the solute drag effect, which plays a crucial role in tailoring thematerial properties [3, 4]. In Figure 1.1, the interface velocity as a functionof the driving pressure is shown. For a given driving pressure, the velocitydecreases with the increase in solute concentration [5]. At sufficiently highdriving pressures, the interface velocity approaches the intrinsic mobility ofthe system without solute elements.Figure 1.1: Variation of the interface velocity (ν) versus theapplied driving pressure (Pd) for systems with different soluteconcentrations (Ci, i = 1, 2, 3) [5].2The transition between high and low interface velocity regimes occursfor higher solute concentrations. When the driving pressure increases inthe low velocity regime, the interface breaks up from the solute atoms at acritical velocity. While at high velocity regime, when the driving pressuredecreases, at a critical velocity, the solute atoms can significantly reduce thevelocity of the interface. The transition velocity is not the same for the twoconditions, and thus there can be a jump from high to low velocity regimeas shown in Figure 1.1.The solute drag effect has been studied for several decades. Lu¨cke andDetert developed the first quantitative theory to explain the influence ofsmall amounts of impurities upon recrystallization [6]. Then a continuummodel proposed by Cahn [7] and followed by Hillert [8], formed the basisof the current understanding of the phenomenon. In these approaches,the two solute drag parameters, i.e. binding energy (Eseg) and trans-interface diffusivity (D), and the intrinsic mobility (M) are introduced todescribe the solute-interface interactions. It should be emphasized that theseapproaches are phenomenological models used on the macroscale. Whendescribing experimental observations, the intrinsic mobility and two solutedrag parameters are essentially employed as fitting parameters. Currently,intrinsic mobility and solute drag parameters cannot be determined exactlyby independent experimental studies. To improve the predictive capabilitiesof these models, new theoretical models and simulation tools with aminimum of empirical parameters are required.Recent progress in Computational Materials Science has provided tremen-dous opportunities to formulate models containing fundamental informationon the basic atomic mechanisms of microstructure evolution that can beimplemented across different length and time scales [4]. This approach,also called multi-scale modeling, can connect parameters from the atomisticscale to the macroscopic scale of an industrial product. The multi-scalestudies start from atomistic simulations, e.g. density functional theory(DFT), which provide fundamental properties based on the knowledge atthe atomistic level. The solute drag parameters, i.e. the binding energyand trans-interface diffusivity can be quantitatively determined by the DFT3simulations.DFT is based on the Hohenberg-Kohn theorem, in which the electrondensity is treated as the fundamental variable. Hohenberg and Kohn provedthat the total energy of a many-body system is a unique functional of theelectron density and has a minimum corresponding to the ground statedensity. Within the framework of Kohn and Sham, the many-body problemof interacting electrons in a static external potential is reduced to a problemof non-interacting electrons moving in an effective potential, which can beapplied to a system containing in principle any number of particles. Theconnection between true many body and auxiliary single particle systemsis made through the exchange-correlation potential for which there are noknown analytical expressions. Fortunately, the widely used approximation,i.e. the Generalized Gradient approximation (GGA), gives remarkablyaccurate results. For example, the calculated lattice constant for bcc Fediffers from the experimental result by less than 1% [9]. Note that the DFTcalculations are limited to T = 0K. Nevertheless, it has been shown to bea very powerful theoretical tool to obtain reliable quantitative informationsuch as binding energies of solute at interfaces and activation energies forsolute diffusion [9–11].In grain boundaries and interfaces, the activation energies of the solutesdepend on the boundary positions. Consequently, there are a multiple ofjump rates. To investigate solute diffusivities, the kinetic Monte Carlo(KMC) approach is employed. KMC methods rely on probabilities of events,which can effectively overcome the time and length scale limitation, anddetermine the solute diffusion in a large system. The probabilities areobtained based on the detailed input from DFT calculations.The solute drag parameters influence the overall phase transformationas observed on the industrial line. The motivation of the present work is toprovide more insight into the atomistic mechanisms of the solute-interfaceinteraction and determine the important solute drag parameters by usingatomistic simulations, i.e. DFT and KMC. These values can then be usedin the process model with a greatly reduced number of fitting parameters.The combination of atomistic simulations with process model will provide a4more reliable and efficient way to predict trends on how alloying additionsaffect the phase transformation for advanced steels [12].5Chapter 2Literature Review2.1 IntroductionIn this review, the focus is on the atomistic simulations of solute dragparameters, i.e. the binding energy (Eseg) and the trans-interface diffusivity(D). We first review the current results of solute segregation on grainboundaries in Fe with emphasis on atomistic simulations approaches, e.g.density functional theory and molecular dynamics simulations. The self-and solute-diffusion in the bulk and grain boundaries are then discussed.In the latter part of this chapter, atomistic studies based on the densityfunctional theory for heterophase interfaces and the magnetic ground statesof fcc Fe are reviewed.2.2 Solute Segregation on Grain Boundaries2.2.1 Simulations of Grain Boundary SegregationA grain boundary is the interface between two grains (crystals), which arecritical for many properties of materials. For examples, grain boundaries arewell known to act as sinks for point defects, preferential sites for nucleationof secondary phases and offer the possibility for fast diffusion [13]. Modelingand simulation of segregation to grain boundaries at the atomic scale can6Figure 2.1: Side view of the Fe Σ3 (111) grain boundary [15]. (a)Cluster of 53 atoms and (b) cluster of 91 atoms. The opencircle represents solute atom.provide valuable insight into segregation processes. In the past decades,a large amount of simulation work has been devoted to the structure ofhigh-angle grain boundaries in Fe, especially for special boundaries, suchas symmetrical tilt boundaries and pure twist boundaries [10, 14–20], inwhich the grain boundary is treated as a perfect planar interface betweentwo crystals that have different orientations. Among all these studies,symmetrical low Σ boundaries as defined in the coincident site lattice (CSL)theory [21, 22], were primarily investigated.First-principles calculations are often used to study the solute segrega-tion behavior at grain boundaries and their influence on grain boundarycohesion. These kinds of simulations for grain boundaries in bcc Fe werepioneered by Freeman and Olson [10, 14–17]. In their early studies, a clustermodel (i.e. non-periodic structure) containing 6∼10 atoms was employedto simulate the Σ3 (111) grain boundary in Fe [14]. Soon after it wasrealized that when impurities (e.g. phosphorus) are introduced, the elasticlattice distortions exceed the dimensions of the cluster. In the subsequentcalculations, they expanded the cluster size up to 91 atoms for more realisticsimulations [16, 17]. Figure 2.1 shows the cluster model for the Fe Σ3(111) grain boundary containing 53 and 91 atoms, respectively. In addition,other grain boundaries, for example Σ5 (013) grain boundaries, were also7Figure 2.2: Side view of the Fe Σ3 (111) and Σ5 (210) grainboundaries [25]. The lighter and darker circles mark the atomsbelonging to two different planes. The interstitial positions atthe grain boundary are indicated by the yellow circles. Thesubstitutional positions in different grain layers are labeled bythe numbers. Lower panels show top view of the supercells takenin the cross-section plane passing through the grain boundary(broken line).investigated [23, 24]. In these studies, a cluster model containing up to 196atoms was used.The cluster approach often suffers from finite size effects even when thecell consists of several hundred atoms. As a result, the calculated results maycontain significant errors [15]. This casts some doubts on the applicability ofthese models. To overcome this problem, a supercell approach with periodic8boundary condition is utilized in modern studies of the grain boundariesand interfaces.In the supercell model, the grain boundary structure is created byrotating two grains around the common axis by misorientation angle θ.In order to maintain the periodic boundary conditions, each supercell hasto contain two identical, reversely oriented grain boundaries. The typicalstructures of Σ3 (111) and Σ5 (210) grain boundaries are shown in Figure 2.2[25]. By now, there have been a considerable number of studies dedicatedto these boundaries, and encouraging results have been obtained [25–27].However, it should be emphasized that all these studies are limited to thesespecial low Σ grain boundaries, i.e. Σ3, Σ5, and very recently, Σ11 grainboundaries [25, 27, 28]. The higher Σ and low angle grain boundaries havenever been considered based on the framework of first-principle calculationsdue to the expensive computational costs.To access those general high Σ grain boundaries, molecular dynamics(MD) simulations are a viable approach. These simulations are much lesscomputationally expensive than first-principles calculations, but are limitedby the accuracy or availability of interatomic potentials. Nonetheless, MDsimulations are increasingly being used to study grain boundary segregation.Recently, a series of MD calculations have been performed to systemat-ically study <100>, <110>, and <111> symmetric grain boundaries [18–20]. These boundaries not only contain several low-order coincident sitelattice (CSL) grain boundaries (e.g., Σ3, Σ5, Σ9, and Σ11 boundaries),but also include more general high Σ boundaries. In these studies, thegrain boundaries are analyzed by characterizing the local atomic structureas structural units. One important finding is that the high Σ grainboundaries can be characterized by structural units from two neighboringlow Σ boundaries.Figure 2.3 shows an example of the<100> symmetric tilt grain boundarysystem, where the Σ29(730) boundary is a combination of structural unitsfrom the two Σ5 boundaries [19]. The structural units for the Σ5(310) andΣ5(210) grain boundaries are labeled as B and C, respectively. In a similarmanner, the Σ17 boundary is a combination of the Σ5 structural units C9Figure 2.3: < 100 > symmetric tilt grain boundary structures withstructural units outlined for the Σ5(210), Σ29(730), Σ5(310)and Σ17(530) boundaries [19]. Black and white denote atoms ondifferent 100 planes. The different structural units are labeledas A, B, and C.and structural units of the bulk lattice, i.e. A. This conclusion is general andholds for pure tilt and twist boundary types with low index rotation axes,which implies that the results obtained from special low Σ grain boundariescan be extended to more general high Σ grain boundaries.The effect of temperature on the grain boundary structures was recentlystudied by Asta’s group [29, 30]. With MD simulations, they characterizedmultiple grain boundary phases by structure and atomic densities in thegrain boundary core region. As shown in Figure 2.4, phase transformationsare found in the Σ5(210) grain boundary. During isothermal annealsat temperatures below 1050 K, the grain boundary transforms into itsthermodynamically stable phase at low temperatures, i.e. the split kites(labeled in red color as shown in Figure 2.4). On the contrary, when theboundary with split kites is heated up to 1100 K, its structure transformsto filled kites (labeled in blue color as shown in Figure 2.4). Frolov et al.suggested that this first-order phase transformation was induced by pointdefects, where the absorption of point defects strongly modified the grain10boundary structure, and finally resulted in structural transformations.Figure 2.4: MD simulation of the Cu Σ5(210) grain boundary at 800Kallowing variations in grain boundary density. The boundaryundergoes a first-order phase transformation from filled kites(labeled in blue color) to split kites (labeled in red color)nucleating and growing from the surface [29].2.2.2 Binding EnergyThe binding energy of a solute with a grain boundary (also called segregationenergy) determines thermodynamically the solute segregation behavior.Within the framework of atomistic calculations, the binding energy is definedas the energy difference of the system when solute atoms moved from thebulk to the grain boundaries, which can be expressed as:Eseg = EX,gb − EX,bulk (2.1)where EX,gb and EX,bulk are the total energies of the supercell with oneX solute at the grain boundary or in the bulk. Negative Eseg means that theimpurity segregation at the grain boundary position is favorable. In a seriesof first-principles calculations, the binding energies of various non-metallicelements (i.e. hydrogen, boron, carbon, nitrogen, oxygen, phosphorous,and sulfur) with special grain boundaries have been studied. Table 2.1summarizes these results.Table 2.1 demonstrates that: (1) In general these elements prefer tosegregate at the interstitial boundary positions independent of the boundarytypes. (2) For a given grain boundary, the binding energy varies significantly11Table 2.1: Binding energies (in eV) of various non-metallic elementswith special grain boundaries.Interstitial Position Substitutional PositionΣ3(111) Σ3(112) Σ5(012) Σ5(013) Σ3(111) Σ5(012)H -0.45 [31] -0.34 [32] – -0.3 [33] – –B -2.0 [27] – -3.1 [25] – 0.4 [27] -1.9 [26]C -1.0 [27] – -1.0 [27] -1.8 [27] 0.5 [27] -2.4 [27]N -0.1 [25] – -1.1 [25] – 0.4 [27] -2.0 [27]O – – -1.9 [27] – – -1.8 [27]S -3.1 [27] – -5.0 [27] – -0.2∼-1.4 [27, 34] -1.6 [27]P -3.2 [35] – -4.5 [35] – -0.1 [35] -1.0 [35]for different impurities. For example, the binding energy of interstitialphosphorous with the Σ3 grain boundary is -3.2 eV, while it is an orderof magnitude smaller for nitrogen. (3) The binding energies at the Σ5 (012)grain boundary are larger (absolute value) than Σ3 (111) grain boundaries.At the Σ5 (012) grain boundary, the binding energies are distinctly negativefor all impurities in interstitial and substitutional sites, while they arepositive, i.e. repulsive for boron, carbon and nitrogen at substitutionalpositions in the Σ3 (111) grain boundaries. (4) The binding energy dependson the boundary sites. Figure 2.5 shows the variation of binding energiesas a function of the distance from the center of the boundary plane in aΣ3 (111) grain boundary [34]. It is clear that at different boundary sites,the binding energy of sulfur varies from -0.2 eV to -1.4 eV. These variationsare suggested to be attributed to the local-arrangement of the atoms at thegrain boundary [34, 35].In addition, Figure 2.5 also shows that the binding energy decreasesquickly when sulfur moves into the interior of the grain. Similar resultswere also obtained by Solanki et al., who carried out molecular dynamics12calculations for a more general high-angle grain boundary [36]. In theirstudies, 75 sites within 15 A˚ of the Σ13 (320) grain boundary were selectedto calculate the binding energy of phosphorus. Their results revealed thatthe distance associated with the binding energy approaching the bulk valuewas about 5 A˚. In other words, the influence of the grain boundary on thesolutes was short ranged, and limited to about 4∼5 atomic layers.Figure 2.5: DFT simulations for segregation energies as a function ofsegregation site for the Fe Σ3 (111) grain boundary with sulfurat the substitutional site [34].The binding energies as a function of solute concentration were investi-gated recently. Using DFT, Yamaguchi systematically increased the numberof solute atoms (NP ) in the supercell, where NP ≤ 8 [35], and determinedthe most stable configurations of solute atoms for each concentration.Figure 2.6 shows the example of binding energies of phosphorous. Thetotal binding energy (Esegtot =∑NPi (Esegi )) of NP phosphorous atoms isdenoted by lines and points (left tick mark). The columns indicate bindingenergy (Eseg) of the additional phosphorous atoms (right tick mark). Onecan see that in the range of NP = 1 to 4, Eseg is roughly constant and13Figure 2.6: Calculated binding energies of phosphorous with varyingthe segregation concentration in the Σ3 (111) grain boundaryin Fe [35].the total binding energy increases continuously. This indicates that theinteraction between solute atoms is weak. Afterwards the repulsion betweenthe segregated solute atoms becomes more pronounced and at NP = 7, theincremental segregation energy (Eseg) is positive, i.e. the boundary sites areno longer favorable for P segregation. As a result, the segregation approachessaturation.2.2.3 Effects of Solutes on Grain Boundary PropertiesSolute segregation at grain boundaries can significantly influence themechanical properties of a material through the embrittlement effect. Someimpurities, for example hydrogen, can strongly reduce the grain boundarycohesion, resulting in grain boundary embrittlement. Whereas otherimpurities, e.g. carbon, can increase grain boundary cohesion [4]. Despite ofthe use of relatively small supercells with simple Σ3 or Σ5 grain boundaries,14first-principles calculations have been proved to be very useful in predictingthe embrittlement effect of various impurities.Within the atomistic simulation framework, Rice and Wang postulatedan approach to determine the strengthening or embrittling effect of animpurity on grain boundaries [37]. In this method, the key quantity thatdetermines the solute-induced embrittlement is the strengthening energy,which is defined as the difference between the binding energies of the soluteelements to the grain boundary (Eseg) and to the surface (Esur). If theelements have a stronger binding to the surface than to the grain boundaries,it is expected that segregation of such elements causes grain boundaryembrittlement. Otherwise, they are grain boundary strengthening elements[37]. Based on this criterion, a number of first-principles calculations havebeen conducted to study the effects of various solutes on grain boundariesin bcc Fe [27, 38, 39]. The results can be summarized as follows: hydrogen,oxygen, phosphorous, and sulfur act as embrittlers at grain boundaryinterstitial sites. Carbon and nitrogen increase the grain boundary cohesion,and are expected to be grain boundary strengthening elements. At differentoccupation sites, boron can either increase or decrease the grain boundarycohesion in Fe.Using DFT, Momida et al. calculated the energy-strain curves of Σ3(112)grain boundaries with and without interstitial hydrogen [39]. In Figure 2.7,the energy difference is defined as the work of fracture, which can becalculated as the difference between the grain boundary energy and twofractured-surface energies. It is evident that the work of fracture decreaseswith increasing number of hydrogen, indicating that hydrogen is a grain-boundary embrittler.To analyze these grain boundary weakening/strengthening phenomenonsand elucidate the physical mechanisms of the impurity effects, electronicstructure, charge density, excess volume, etc. have been extensively studied,and two major findings have been identified, i.e. contributions from chemicaleffects, and contributions from size effects.In early studies, based on the charge density analysis, Freeman et al.reported that the chemical bonding of boron and carbon with iron atoms15Figure 2.7: Results of energy difference relative to the equilibriumvs. uniaxial tensile strain for the pristine (solid line with closedcircles (black)), 1 hydrogen atom- (broken line with open circles(blue)) and 4 hydrogen atoms-trapped (broken line with squares(red)) Σ3(112) grain boundaries [39].at the grain boundary was strong, whereas it was dramatically weak forphosphorous and sulfur [14–17]. Similar results were also obtained by Juanet al. [33]. They found that when hydrogen or sulfur was present at theboundary site, all neighboring Fe-Fe bonds were weakened, which givesrise to grain boundary embrittlement. In these studies chemical bondingcharacteristic is concluded to be the key factor determining the embrittlingor cohesion behavior of a segregated impurity at Fe grain boundaries.In subsequent studies, Braithwaite et al. investigated the role ofboron, nitrogen, and oxygen impurities at a Σ5 grain boundary [38]. Indisagreement with the chemical bonding proposal, they found no significantcovalent bonding between the impurity and neighboring Fe atoms. Thissuggests that the grain boundary cohesion is simply related to the size ofthe impurity atom rather than the chemical bonding.Recently, by separating the contributions from solute size and chemical16bonding, Wachowicz et al. pointed out that both chemical bondingand solute size can influence the grain boundary weakening/strengtheningproperties [25–27]. In their studies, the contributions from solute size tend toweaken the grain boundary cohesion, while the contribution from chemicalbonding can either enhance or decrease the grain boundary cohesiondepending on the species. Consequently, cohesion at the grain boundaryis increased when large strengthening chemical contribution dominates.Such conclusions were later supported by Kiejna et al., who systemicallyinvestigated the effects of several impurities (B, C, P, N, O, and S) onstructure, energetic and mechanical properties of Σ5 grain boundaries [27].2.3 Solute Diffusion in Grain Boundaries2.3.1 OverviewInterface diffusion controls the rates of many processes in materials atelevated temperatures, such as phase transformations, grain growth, andsolid-state reactions that makes it a technologically important topic [3, 4, 40].Quantification of solute grain boundary diffusion is therefore importantto estimate and control the microstructure evolution. Unfortunately, dueto the narrow width of grain boundary regions and the small amount ofatoms involved in the diffusion process, its experimental measurement anddetermination is rather challenging. Thus, the understanding of basics ofsolute grain boundary diffusion is still limited. Recently, significant progresshas been achieved through atomistic computer simulations [9, 11, 41–43], which has enabled a more detailed understanding of those diffusionprocesses.2.3.2 Solute Diffusion in α-FeBefore considering grain boundary diffusion, we first briefly review recentfirst-principle results for bulk diffusion. This is because: first, in recent yearsfirst-principle methods have been applied extensively in calculations of self-and solute-diffusivities in bulk α-Fe. Second, this provides a verification of17earlier intuitive atomistic models of diffusion and solute-interaction models,and it also provides an opportunity for a deeper understanding of theatomistic processes.Figure 2.8: Minimum-energy path for carbon diffusion in α-Fe andthe local structures of initial, intermediate, final, and transitionstates [9].In previous studies, defect formation energies, defect-solute interactionenergies and diffusion activation energies in Fe were systematic studied fora variety of impurities. Jiang et al. [9] used the nudged elastic band (NEB)approach in DFT simulations to investigate the activation energy of bulkdiffusion of carbon in α-Fe. Figure 2.8 shows the minimum-energy path forcarbon diffusion in α-Fe and the local structures of initial, intermediate,final, and transition states. The calculation results identify octahedralsites as the lowest energy position for carbon. The diffusion occurs viatetrahedral sites with the barrier of 0.86 eV, which agrees well with theexperimental value of 0.87 eV [44]. In addition, the pre-exponential factorD0 is also calculated by constructing and diagonalizing the Hessian matrix18Table 2.2: Calculated activation energies of fully orderedferromagnetic state (Q0) and pre-factor (D0) in α-Fe vs.experimental data.1Q0 (eV) D0 (cm2/s)DFT Exp. DFT Exp.W 2.80 [11] 3.12±0.28 [47] 1.40 [11] 1.60 [48]Mo 2.60 [11] 2.48 [49] 0.63 [11] 0.57 [50]Cu 2.55 [51] 2.56 [52]Ni 2.71 [53] 2.68 [54]Cr 2.77 [55] 3.06±0.07 [47]Al 2.47 [56] 2.55 [57]C2 0.86 [9] 0.87 [44] 1.44×10−3 [9] 1.67×10−3 [58]H2 0.10 [46] 0.14 [59] 1.37×10−3 [46] 1.40×10−3 [59]1 The model systems employed by above DFT calculations contain54 or 128 atoms in the supercell, which corresponds to impurityconcentration of 1.85% or 0.78%.2 H and C diffuse via interstitial sitesbased on the frozen phonon approach [45]. Ramunni et al. [46] used the sameapproach to calculate the coefficient of interstitial hydrogen diffusion. Thecalculated diffusivity was found to be in good agreement with experimentaldata (see Table 2.2).For the case of substitutional impurities in α-Fe, calculations of diffusiv-ity have been performed for W, Mo, Cu, Ni, Cr, Al, etc. [11, 51, 53, 55, 56].In addition, motivated by the question of whether any of the 5d solutesare slow diffusers in α-Fe, Asta et al. systematically investigated 5dtransition metal elements, where experimental data was lacking [11, 43].They found that Re had the lowest calculated impurity diffusion coefficient,approximately an order of magnitude lower than that for self-diffusion attemperatures near 1000 K. Recently, the effect of stress on self-diffusion in19bcc Fe was also investigated [60].A summary of calculated diffusion parameters is given in Table 2.2. Goodagreement is found for those cases where a comparison with experimentaldata is possible. These encouraging benchmark results provide an incentiveto study solute diffusion in grain boundaries or interfaces, which is a muchmore complicated task than bulk diffusion.2.3.3 Point Defects in Grain BoundariesRecently, the structures of interstitials and vacancies in a variety ofgrain boundaries have been studied based on DFT and molecular staticscalculations [41, 42, 61]. In these studies, the perfect grain boundarystructure was first constructed based on the coincident site lattice theory,and then an atom was either added or removed at a selected boundary siteto form an interstitial or a vacancy. The system was then relaxed to getthe equilibrium structure. Note that there are only a few DFT calculationsfor such systems because the addition of point defects to a grain boundaryrequires much larger supercells and generally reduces the symmetry of thesupercell. Both increase significantly the load on computing resources to thepoint that some calculations can not be conducted using modern softwareand hardware. Consequently, so far the results obtained by DFT are verylimited. Therefore, the following section reviews mainly the results obtainedby molecular statics and dynamics simulations.It was found that formation energies of a point defect can be differentat different boundary sites and depend on the grain boundary structure.Figure 2.9 shows an example of vacancy formation energies (Ev) in Σ5 grainboundaries in fcc Cu obtained from molecular statics calculations [42]. Itcan be seen that at some sites the vacancy formation energy can be up to10% lower as compared to the bulk, while at other sites, it may lie above thebulk value. Suzuki et al. pointed out that the observed spread of the point-defect energies could be attributed to the existence of alternating tension andcompression regions in the grain boundary [41, 42]. Nevertheless, in generalthe vacancy formation energy in grain boundaries is on average lower as20Figure 2.9: Molecular statics results for vacancy formation energy asa function of the distance from the grain boundary in Cu [42].compared to that in the bulk [41, 42, 61, 62]. Similar to solute segregationat the grain boundary, the vacancy formation energy quickly approachesthe bulk value as the distance from the boundary increases. Zhou et al.have studied local structural relaxations and formation energies of vacanciesin various sites in the Σ5 grain boundary in bcc Fe [62]. Using the DFTmethod, they found that the vacancy formation energy in the boundaryregion was 1.43∼2.20 eV.It should be emphasized that in the bulk, the relaxations around thevacancy are typically small, and the vacancy remains at the site where it iscreated. However, this may not be the case for grain boundaries. Brokmanet al. and Kwok et al. reported that vacancies could produce considerablylarge relaxations of neighboring atoms. Moreover, some boundary sites werefound to be unstable for vacancies [63, 64]. Similar findings were reported bySuzuki and Mishin [41, 42]. They systemically investigated a wide variety ofconfigurations in Cu grain boundaries with one vacancy, and reported thatthe vacancy could delocalize in the grain boundary structure.An example of this process is shown in Figure 2.10. In this case, a21Figure 2.10: Examples of vacancy instability in Cu Σ9(12¯2) grainboundary. (a) Perfect grain boundary structure; (b) Unstablevacancy at boundary site 6 [42].vacancy is created at site 6 in the Σ9 (12¯2) grain boundary, however, thissite is unstable for the vacancy and gets filled by atom 2. The vacancy atsite 2 induces large relaxations, and as a result, the atom residing initiallyat site 1′ moves to the midpoint between sites 1′ and 2 after relaxation,which leads to a vacancy delocalization between these two positions. Notethat if the vacancy is generated at site 1′, one can get exactly the sameconfiguration. Such vacancy delocalization phenomenon were also observedin Σ5, Σ7, Σ13 grain boundaries in fcc Cu.2.3.4 Diffusion Mechanisms in Grain BoundariesBecause of the complexity, the understanding of grain boundary diffusionis limited. Nevertheless, some progress has been achieved through DFTand molecular dynamics simulations, which reveal a number of genericproperties of diffusion processes in grain boundaries [41, 42, 65, 66]. Inthese simulations, a single point defect (vacancy or interstitial) is createdand its formation energy is evaluated at various positions in the grainboundary. The defect is then allowed to walk along the grain boundary, andthe atomic trajectories are analyzed to determine the most typical diffusiveevents induced by the defect. The diffusion coefficient is then obtained for22various directions in the grain boundary.Figure 2.11: Possible migration path of He interstitial atom in the FeΣ11 grain boundary. Small sphere represents the He atom.Arrows indicate possible paths [65].Figure 2.11 shows an example of a possible migration path of interstitialhelium in the Fe Σ11 grain boundary [65]. One can see that the interstitialhelium migrates from the hexahedral position below the grain boundaryplane (dotted line) to a similar position above the grain boundary plane,leading to helium diffusion along the [11¯3] direction, as indicated by thearrows. Gao et al. reported strong binding between helium and the grainboundary with diffusion occurring within 3 atomic layers from the boundaryplane. Liu et al. studied interstitial hydrogen diffusion in the Σ5 grainboundary in Fe [67]. They found that in contrast to bulk diffusion, thediffusion coefficient perpendicular to the grain boundary plane is four ordersof magnitude lower than that along the tilt axis. They also found a largebinding energy of hydrogen to the grain boundary (see Table 2.1). Thecalculated diffusion coefficients of hydrogen atoms in the bulk, on the freesurface and in a Σ5 grain boundary are shown in Figure 2.12. The activationenergies for hydrogen diffusion in bulk, surface, and grain boundary are0.044 eV, 0.066 eV, and 0.31 eV, respectively. One can see that at lowtemperature, the diffusivity of interstitial hydrogen in the grain boundaryis significantly lower than in the bulk.In the case of vacancy migration, Ingle and Crocker studied self-diffusionin a Σ3 (110) grain boundary in Fe [68]. It was shown that activation energiesfor a variety of vacancy jumps are substantially lower than in the bulk,23Figure 2.12: Diffusion coefficients of hydrogen atoms in bulk, on freesurface and in a Σ5 grain boundary. The activation energiesfor hydrogen diffusion in bulk, free surface, and grain boundaryare 0.044 eV, 0.066 eV, and 0.31 eV, respectively [67].suggesting faster boundary self-diffusion. Similar results were also reportedby Kwok et al. [64]. They found that the vacancy propagation is almostentirely confined to the grain boundary region. Further, Lei et al. havecalculated the activation energies for vacancy diffusion in Σ3 (0001) grainboundaries of α-Al2O3 using DFT [66]. They found that the activationenergies were up to 50% lower than in the bulk lattice, indicating fasterdiffusion in the boundary than in the bulk.Generally, self-interstitials were found to be relatively immobile and notcontributing to the fast grain boundary diffusion. Balluffi et al. reportedthat the activation energy for self-interstitials in Fe Σ5 grain boundary waseven larger than for self-diffusion in the bulk, suggesting that boundaryself-diffusion was dominated by the vacancy exchange mechanism, and theinterstitial mechanism could be ruled out [63]. Pontikis et al. investigatedthe self-diffusion mechanism in a Σ5 grain boundary in fcc Cu [69]. Theirresults confirmed the immobility of the interstitial in the grain boundary,24which provided further support for the vacancy dominated mechanism ofgrain boundary diffusion.Figure 2.13: Calculated and measured self-diffusion coefficients in aΣ9 Cu grain boundary [42].In spite of the obvious complexity of diffusion processes in grainboundaries, both theoretical and experimental results show that grainboundary diffusion coefficients follow the Arrhenius law. Figure 2.13 showsthe Arrhenius plot of calculated grain boundary self-diffusion coefficients infcc Cu [42]. The experimental results are also shown for comparison. It canbe seen that the calculations are in reasonable quantitative agreement withexperimental measurements, which demonstrates that atomistic simulationsare capable of predicting quantitative information on grain boundarydiffusion.252.4 Austenite (fcc) and Ferrite (bcc) Interface2.4.1 OverviewTo the best of our knowledge, no DFT simulations are available for the bcc-fcc interface in iron. The primary reason is the complexity of the magneticstate of fcc Fe since fcc Fe shows paramagnetic states, which creates achallenge for modeling bcc-fcc interface. In this section, we first review themagnetic properties of Fe within the framework of DFT. In the latter partof this section, DFT studies for heterophase interfaces are also discussed.2.4.2 Approaches to Study Magnetism in γ-FeMagnetism plays an important role in understanding the physical propertiesof iron and iron alloys, including the relative stability of different phases[70]. There are many papers related to this research area. For example,in 1981 Kubler [71] investigated the total energies of nonmagnetic (NM),ferromagnetic (FM) and antiferromagnetic (AFM) states for bcc phases ofiron by using the augmented-spherical-wave (ASW) method. He found thatthe ground state for the bcc structure is the ferromagnetic (FM) state. Suchconclusion was confirmed by recent first-principle DFT calculations [9, 72,73]. The magnetic moment per Fe atom in the bulk was predicted to be 2.20µB [9], which was in good agreement with the experimental value of 2.22µB [74].The situation with γ-Fe (fcc) is much more uncertain. It is known that γ-Fe is stable in the temperature interval from T=1173 K and T=1660 K, andexists naturally in the paramagnetic (PM) state. Recent experiments alsoreported that thin films of γ-Fe could be anti-ferromagnetic or ferromagnetic[75, 76]. The complexity of the magnetic properties of γ-Fe presentssignificant challenge for DFT calculations. To identify the stable magneticconfiguration for γ-Fe, an extensive search was performed based on first-principle DFT calculations [9, 72, 73].The most straightforward approach to simulate the paramagnetic statewithin DFT is to use ordered magnetic arrangements, such as single-26layer antiferromagnetic (AFM) and double-layer antiferromagnetic (AFMD)states. The AFM and AFMD phases refer to the simple layered anti-ferromagnetic structure (↑↓↑↓...) and the bilayer AFM structure (↑↑↓↓...),respectively.Figure 2.14: Phase diagram for bcc Fe and fcc Fe with differentmagnetic configurations [9].In Figure 2.14, the calculated total energies of various magnetic config-urations, i.e. non-magnetic (NM), ferromagnetic (FM), single-layer anti-ferromagnetic (AFM), and double-layer anti-ferromagnetic (AFMD), areshown as a function of lattice parameter. For fcc NM state, the minimumlocates at 3.45 A˚. Once the magnetism is included, the results changesignificantly. For the fcc FM phase, the curve displayed in Figure 2.14presents two distinct and separate minima, a low-spin (LS) FM statewith relatively small lattice constant and a high-spin (HS) FM phase withsomewhat larger volume. In early work for carbon in γ-Fe, Jiang and Carter[9] took the HS FM state as a reference. A difficulty with this is that there isa discontinuous transition between the HS FM phase and the LS FM phase,27which could be triggered by defects. Here, the AFMD state has the lowesttotal energy for fcc Fe. Jiang et al. reported that the predicted latticeparameter for the AFMD state agrees well with those extrapolated frommeasured lattice parameters in γ-Fe alloys [9], which implied that AFMDis a good approximation for the fcc Fe. In addition, the calculated vacancyformation energy in the AFMD structure is 1.84 eV, which is relatively closeto that concluded from experiment (1.71 eV) [77].In recent years, based on the framework of DFT calculations, severalnew methodologies have been developed to treat disordered alloys. Severalgroups have applied these ideas in the studies of paramagnetic materials.The most widely used approaches can be summarized as follows:Coherent Potential Approximation The coherent potential approximation(CPA) treats random A1−xBx alloys by considering the average occupationsof lattice sites by A and B atoms [78]. Since it is a mean field approach, localrelaxations are not considered explicitly and the effects of alloying elementson the distribution of local environments cannot be taken into account,which is a major drawback to the application of this approach.Cluster Expansions Approach In the case of Cluster Expansions (CE)Approach [79], the Ising model is used and the occupations of atoms Aand B in the parent lattice are labelled as +1, and -1. The configurationalenergetics of the system is characterized by using the effective clusterinteractions (ECIs). In order to estimate the ECIs, the energies of multiplepre-selected ordered configurations have to be obtained in their fully relaxedgeometries based on the DFT calculations, which is rather laborious work.Special Quasi-random Structures The concept of Special QuasirandomStructure (SQS) was proposed first by Zunger et al. [80]. The basic ideaof SQS is to mimic random solutions by using a small unit cell with onlya few (4∼32) atoms, which best satisfies near-neighbor pair and multisitecorrelation functions of random substitutional alloys. Compared with the28Coherent Potential Approximation and Cluster Expansions approaches,which treat the random alloys by considering average occupations andrequires dozens of calculations, a single calculation is sufficient for the SQSapproach to obtain the required properties of a random alloy.In recent years, the SQS technique has been successfully applied to studythe paramagnetic state of metals. For example, Shim et al. studied themagnetic transition for Fe2O3 using a 40-atom SQS supercell , and predictedthat in the paramagnetic state, Fe2O3 is metallic, which is consistent withthe experimental observation [81].Recent DFT calculations also showed that the ground state of fcc Fe canbe a spin-spiral state [82, 83]. However, the results indicate that the energydifference to the AFMD order is only a few meV/atom, and the latticeconstant between these two structures differs by only 0.01 A˚. Moreover,Ackland etal. found that most of the non-collinear calculations for γ-Fe withdefects converged to collinear magnetic states [84]. Only few configurationsretained non-collinear structures. However, the energy of the non-collinearconfiguration was only marginally lower than the collinear results. On theother hand, the non-collinear calculations are extremely time-consuming,such that only small periodic supercells (typically consisting of 4∼32 atoms)can be considered.2.4.3 Effects of Solutes on Magnetism of γ-FeThe impurity effects on the magnetic properties of fcc Fe were alsoinvestigated by means of DFT. For substitutional elements, Kong and Liureported that the ferromagnetic state of fcc Fe phase could be stabilizedby Cu atoms, which was attributed to chemical pressure effects due to thelarger atomic volume of Cu [85]. The magnetic moment per Fe atom of thefcc structured Fe-Cu alloys is shown as a function of Cu concentration inFigure 2.15. In agreement with experimental observations, the authors findthat the magnetic moment of Fe increases with Cu concentration. Recently,Medvedeva et al. performed DFT calculations to study the effect of Mn29Figure 2.15: Variation of average atomic volume and magneticmoment (MM) per Fe atom of the fcc structured Fe-Cu alloysagainst the Cu concentration [85].substitution on the structural and magnetic properties of fcc Fe [86]. Theirresults suggested that Mn favored the ferromagnetic coupling which, inprinciple, could lead to long range ferromagnetism.In the case of interstitials, Boukhvalov et al. investigated the localperturbations of the crystal and magnetic structure of fcc Fe near a carboninterstitial impurity [73]. The fragment of crystal and magnetic structure ofγ-Fe for the AFMD ordering is shown in Figure 2.16. By comparing the totalenergy and magnetic moments for FM, AFM, and AFMD structures, theyfound that the ferromagnetic state could be stabilized locally by the carboninterstitial. However, the global magnetic structure is still the AFMD state.Hydrogen is especially important for steels since it can lead to hydrogenembrittlement. Nazarov and co-workers investigated the thermodynamicsof hydrogen-vacancy interaction in fcc Fe. They studied several orderedcollinear magnetic configurations. The results demonstrated that theground state is a locally modified AFMD configuration [87]. It wasfound that hydrogen in interstitial positions significantly changed the localmagnetization and increases the total magnetic moment of the system by0.52 µB.30Figure 2.16: Fragment of crystal and magnetic structure of γ-Fefor the AFMD magnetic ordering [73]. Carbon interstitialimpurity in octahedral position is shown by dark (red) circle.2.4.4 DFT Studies for Heterophase InterfacesRecently, DFT has been used to study solid-solid heterophase interfaces.Figure 2.17 shows the supercell used in a DFT study of an interface betweenbcc-Fe and fcc-Ag [88]. The interfaces were constructed according to theNishiyama and Wassermann (N-W) orientation relationship. An importantoutput of such calculations is the interfacial energy, which is a key parameterdetermining the nucleation barrier and also the shapes of precipitates. Inaddition, the interfacial energy obtained by DFT calculations can be used asthe input parameter for mesoscopic simulations, such as phase-field models.Vaithyanathan et al. demonstrated how DFT and phase-field models can be31Figure 2.17: Schematic of the (a) Fe(001)-Ag(001) interface and the(b) Fe(110)-Ag(111) interface [88]. The cross symbols markthe high-symmetry sites with respect to the Ag plane forthe successive Fe atom at interface. The layers around theinterface are indexed by numbers.combined to build a bridge between micro- and macro-scales [89, 90]. Theyconstructed coherent (100) and semi-coherent (001) interfaces of θ′-Al2Cuin fcc Al solid solution and then calculated interfacial energies using DFT.Using these energies in the phase field model, the authors demonstrated thatthe theoretical precipitate microstructure evolution is in good agreementwith the experimental data.Magnetic properties of the interfaces is another subject of recent DFTstudies. Benoit et al. reported that the magnetic moments of the Fe atomsin the Au(001)/Fe(001) interface are enhanced by about 26% [91]. Thelayer projected magnetic moments are summarized in Figure 2.18, where theinterface position corresponds to the index -1. It is observed that magneticproperties of the interfaces between magnetic and non magnetic materials32Figure 2.18: Evolution of the atomic magnetic moment of the Featoms as a function of the position of the layer in the Fe(001)slab (black circles) and in the Au(001)-Fe(001) interface (redsquares) [91].resemble very closely that of the surface of the magnetic material.The electronic properties of the Au(001)/Fe(001) interface system werealso investigated [91]. Figure 2.19 shows the projected densities of the dstates (PDOS) of individual Fe atoms at the surface, in the center, and inthe Au(001)/Fe(001) interface. Comparing the PDOS of Fe atoms at thesurface or at the interface with the one in the center, one observes that thepresence of the surface or interface shifts the spin-down states by around+2 eV down to 0-0.5 eV. In addition, due to the presence of the surfaceor interface, the spin-up states below the Fermi level shift down to lowerenergies.33Figure 2.19: Projected densities of the d states of individual Fe atomsin the Au(001)/Fe(001) interface. Fesurf , Fecenter, and Feintdenote the Fe atoms at the surface, in the center, and at theinterface, respectively [91].2.5 ConclusionsBased on the first-principle studies, the segregation behavior of non-metallicelements, e.g. hydrogen, boron, carbon, phosphorous, etc., and theireffects on special low Σ grain boundaries have been extensively studied.In addition, some general, e.g. high Σ and low angle, grain boundarieswere also investigated through molecular dynamics simulations. However,the role of transition metals (TMs) such as V, Mn, Ti, Nb, etc. hasreceived somewhat less attention. Experimentally, it is known that theseTMs actually play rather important roles in microstructure evolution andmechanical properties of materials through the solute-drag effect. So far, thequantitative knowledge of the effect of these TMs on the Fe grain boundariesis still limited. Hence, it is necessary to advance our knowledge on theinteractions between TMs and grain boundaries and interfaces using first-principle DFT simulations.Previous studies indicated that fast grain boundary diffusion is governedby a vacancy mechanism in the vast majority of boundaries. The activationenergy of a vacancy jump is found to be substantially lower than in the bulk.34Despite these encouraging results, the understanding of grain boundarydiffusion is still limited. To the best of our knowledge, no work has beendone for substitutional solute atom diffusion in grain boundaries due tothe expensive computational costs for first-principle calculations and thelack of suitable interatomic potentials for molecular dynamics simulations.Nevertheless, previous theoretical simulations have already demonstratedthat atomistic simulations are capable of predicting quantitative informationabout diffusion in metals. So there is a strong motivation to understand moreabout the diffusion mechanisms in grain boundaries in Fe based on cuttingedge modeling tools.So far, the AFMD configuration is considered as a good approximationfor the magnetic ground state of γ-Fe. Previous studies have also shownthat the substitutional and interstitial elements could influence the localmagnetic state, however, the global magnetic structure is still AFMD state.In addition, the new developed special quasi-random magnetic structuresprovide an opportunity to deal with the paramagnetic state of γ-Fe in a morerealistic manner. In the current state, AFMD and quasi-random structuresare promising approximations to simulate the magnetic states of γ-Fe.35Chapter 3Scope and ObjectivesThe aim of this work is to quantify how solute atoms interact with interfacesin Fe. For this purpose, we use first-principle DFT combined with kineticMonte Carlo simulations. The critical solute drag parameters such asbinding and activation energies will be determined and the coefficient forinterface diffusion will be calculated. Special grain boundaries, e.g. Σ5(013) symmetrical tilt grain boundaries in bcc Fe, will be investigated indetail. This kind of special boundary has a small periodic unit cell, whichmakes our proposed DFT calculations feasible. The studies will then beextended to more complex interfaces, i.e. selected bcc-fcc Fe interfaces tostudy the solute-interface interaction and the diffusion mechanisms. Morespecifically, the objectives of this work can be divided into five main goals:1. Investigate special grain boundaries, e.g. the Σ5 (013) symmetricaltilt grain boundary, containing commonly used alloying elements, e.g. Mn,Mo, Nb and V, etc. to determine the binding energies of these solutesto the grain boundary and their activation energies for diffusion along theboundary.2. Select the proper magnetic configurations for fcc Fe and determinethe basic properties of a selected bcc/fcc interface, including the geometric,electronic and magnetic structures.3. Investigate the interactions between alloying elements and the fccgrain boundaries and determine the binding energies of these solutes.364. Quantify the interactions of solutes with a selected bcc-fcc interfaceand determine the binding energies of these solutes.5. Develop a KMC model to determine the diffusion coefficients in grainboundaries and bcc-fcc interfaces.37Chapter 4Methodology4.1 Density Functional Theory4.1.1 FundamentalsTheoretically, an exact treatment of solids can be obtained by solving themany-body Schro¨dinger equation involving both the nuclei and the electrons:HˆΨ( ~R1 ~R2... ~RN , ~r1 ~r2... ~rn) = EΨ( ~R1 ~R2... ~RN , ~r1 ~r2... ~rn) (4.1)where ~Ri are the nuclei coordinates, ~ri are the electron coordinates, Hˆis the Hamiltonian operator, Ψ is the wave function, E is the total energyof the system, N is the total number of nuclei, and n is the total numberof electrons in the system. The Hamiltonian operator Hˆ is the sum ofthe kinetic energy operator Tˆ and the potential energy operator Vˆ . Thewave function Ψ is a measure for the probability of finding the particle ata certain position. The probability density of a particle is the square of theamplitude of the wave function, |Ψ|2. However, it is extremely difficult tosolve the many-body Schro¨dinger equation. In fact, the only system thatcan be solved analytically is the single-electron hydrogen atom. In general,the Schro¨dinger equation has to be solved numerically.Hohenberg and Kohn have shown that the total energy of a system can38be uniquely defined by the electron charge density and has a minimumcorresponding to the ground state density [92], i.e.E0 = min(E[ρ(~r)]) (4.2)Within the framework of Kohn and Sham, the many-body problem ofinteracting electrons in a static external potential is reduced to a problem ofnon-interacting electrons moving in an effective potential, and the originalmany-electron Schro¨dinger equation is then converted into a set of Kohn-Sham equations [93]:(Tˆ [ρ(~r)] + HˆH [ρ(~r)] + Uˆext[ρ(~r)] + Vˆxc[ρ(~r)])Φi(~r) = ξiΦi(~r) (4.3)where Tˆ [ρ(~r)] is the non-interacting kinetic energy, HˆH [ρ(~r)] is theelectrostatic interactions between electrons, Uˆext[ρ(~r)] is the potential energyfrom the external field due to positively charged nuclei, and Vˆxc[ρ(~r)] is theso-called exchange-correlation energy. So far, the exact form of exchange-correlation energy is still unknown. For this reason, suitable approximationshave been introduced.4.1.2 Exchange-correlation EnergyThe most widely used approximation is the so-called Local Density Ap-proximation (LDA), which assumes that the exchange correlation energy isa function of the local charge density [94]. LDA has been demonstratedsuccessful when employed to solids. However, one significant limitation ofLDA is its overbinding of solids: lattice parameters are usually underpre-dicted while cohesive energies are usually overpredicted. In addition, LDAcalculations predicted the hexagonal para-magnetic structure are the moststable state in Fe, whereas it is well known that the ground state of Feshould be the bcc ferromagnetic state.In an effort to rectify the inaccuracies of LDA, the Generalized GradientApproximation (GGA) was introduced [95]. GGA is a natural improvementon LDA by considering not only the local charge density, but also its39gradient. The results calculated by GGA generally agree noticeably betterwith experiments than LDA. It also gives a better description of 3d transitionmetals [96]. For example, the ground state of Fe is correctly predicted withinthe GGA calculations. For these reason, in this project, the GGA proposedby Perdew, Burke, and Enzerhof (PBE) will be adopted rather than LDAfor the exchange-correlation energies.4.1.3 Structure OptimizationThe crucial point when performing DFT calculations is to find the elec-tronic ground state of the system, which is usually referred as structureoptimization (or “relaxation”). In principle, when searching for the optimalgeometry, the DFT code performs several ionic relaxation steps. It movesall nuclei in appropriate directions (according to energy gradients), in whicheach nucleus should have lower potential energy. After each ionic step,the ground state energy of this particular atomic arrangement has to beevaluated by a self-consistent cycle of electronic relaxation steps. Once thedifference of any 2 consecutive energy values is lower than a given thresholdvalue, the self-consistent cycle is finished. Then the energy gradients (orforces on atoms) are evaluated. If they are lower than pre-defined thresholdvalues, the optimized structure is achieved. Otherwise, new positions ofatoms are proposed and repeat from the start. For finding the ground stateaccording to the energy and its gradients, the following numerical algorithmsare widely used in DFT codes:(1) Steepest descent method(2) Conjugate-gradient (CG) methodThe scheme of search paths for locating the minimum of the energy by aniterative process using the steepest descent and conjugate gradient methodsis shown in Figure 4.1. In the steepest descent procedure, the minimizationprocedure starts at an arbitrary point (e.g. point 1 in Figure 4.1) andevaluates the gradient of the energy [97]. The negative of the gradient vectorgives the direction of the steepest descent which is followed until a minimumis reached along this line. At that point, the procedure is repeated until the40Figure 4.1: Sketch of the search paths for the minimum of a function(i.e. energy) using the steepest descent (left) and conjugate-gradient approach (right). In both cases, the search begins atpoint 1. The conjugate-gradient method converges faster [97].value can no longer be lowered within a given convergence threshold. It isevident that this procedure is reasonable, but not efficient.A more efficient approach is the conjugate gradient method, which firstalso starts a steepest descent step with line minimization. However, inthe following step, the search direction contains information not only fromthe gradient at the current point, but also “conjugated” to the previoussearch directions. Sheppard et al. reported that the convergence based onthe conjugate-gradient algorithm is four times faster than that of steepestdescent method [98]. Therefore, in this project, we will use conjugate-gradient algorithm to find the optimized structure.4.1.4 Simulation Software PackageThere are many DFT codes currently available, among which the widelyused software packages are Wien2k, Siesta, FIREBALL, VASP, CASTEP,PWSCF, and ABINIT. Wien2k employs the so-called all-electron DFTmethod, which means all electrons are treated in the same framework. Thistreatment can provide highly accurate band structure, total energy, etc.However, it is also computationally demanding. In contrast, Siesta andFIREBALL use atomic orbitals as basis set, which are much more efficient.The main disadvantage is incompleteness of the smallest basis set whichmeans the calculated properties often show extreme sensitivity to smallchanges in the basis set. To balance the computational load and accuracy,41VASP, CASTEP, PWSCF, and ABINIT codes were developed based on theplane wave basis set and pseudopotential approximations.In these DFT codes, the wave functions are expanded in terms of planewave basis set, i.e.:Φi(~r) =1√Ω0∑~Gci(~G)ei(~k+ ~G)·~r (4.4)where ~G is the vector of the reciprocal lattice, ~k denotes the momentum,and Ω0 represents the primitive cell volume. In practice, only a finite set ofplane waves is used for a given system. The completeness of the basis set iscontrolled by the cut-off energy (Ecut), i.e.(k +G)22< Ecut (4.5)where (k+G)22 denotes the kinetic energy of a plane wave. Only thoseplane-waves with a kinetic energy smaller than Ecut are included in thebasis set. The main disadvantage of a plane wave basis set is that a largenumber of basis functions is required to describe atomic wave functions nearthe nucleus. To overcome this problem, the pseudopotential approximationis employed. This approximation is based on the assumption that onlythe valence electrons have significant effect on the physical and chemicalproperties of the system. Therefore, the core electrons do not need tobe treated explicitly any more. This approximation provides usually asufficiently good result, while it is much more efficient when compared withthe all-electron DFT calculations.It is noticeable that the pseudopotentials for PWSCF and ABINIT codesare still not available for some elements, which limit the application of thesetwo codes. Fortunately, the pseudopotentials for VASP and CASTEP codesare complete and reliable. However, the CASTEP code does not supportsearching transition states, and therefore is not suitable for diffusion studies.Consequently, VASP is used as the DFT algorithm in this work.VASP, i.e. Vienna Ab initio Simulation Package, is a computer program42for atomic scale materials modelling, e.g. electronic structure calculationsfrom first principles. It computes an approximate solution to the many-body Schro¨dinger equation within density functional theory, solving theKohn-Sham equation. The optimized structures, ground state energy,transition states, etc. can be obtained based on the VASP calculations.In previous studies, VASP has been successfully applied to investigate grainboundary segregation and diffusion of solutes [11, 25–27]. Note that thecomputational load of DFT calculations is expensive. A simple structurerelaxation calculation for Σ5 Fe grain boundary containing 120 atoms takes3 days using 32 CPU cores. As a result, the calculation domain is usuallysmall for VASP simulations, which is limited to about 100∼200 atoms. In thepresent work, the calculations are performed using Westgrid supercomputingresources.4.1.5 Domain SizeThe Influence of the Energy Cut-OffAs has been discussed in subsection 4.1.4, the cut-off energy is an importantand sensitive variable, which determines the plane wave basis set. Theinfluence of the cut-off energy on the convergence of the total energy of theconventional bcc Fe unit cell has been evaluated with a fixed k-point meshof 20×20×20. The testing results are shown in Figure 4.2. It is evident thatthe total energy is well converged with kinetic energy cut-offs over 350 eV.Therefore, in this work, the cut-off energy is set to be 350 eV.The Influence of the k-Point SamplingFor a periodic system, integrals in real space over the infinite system arereplaced by summing the values at a finite number of points in the Brillouinzone, i.e. the k-point mesh. Choosing a sufficiently dense mesh of k pointsis crucial for the convergence of the results. Based on the scheme proposedby Monkhorst and Pack, k-points are distributed homogeneously in theBrillouin zone [99]. The distance between two k points in reciprocal space43200 300 400 500 600 700 800-8.32-8.31-8.30-8.29-8.28Total Energy (eV/atom)Cutoff Energy (eV)Figure 4.2: Convergence test of the total energy with respect toenergy cutoff.is given as:dk−points = 1/kili (4.6)where li is the supercell dimension in i direction. ki determines numberof k-points in i direction. For example, k1=k2=k3=3 indicates a 3×3×3 k-point mesh. In order to check the influence of the k-point mesh in reciprocalspace on the convergence of energy values, grids of 1×1×1 up to 20×20×20k-point mesh are evaluated. This gives the k-points spacing from 0.352 A˚−1to 0.018 A˚−1 for the conventional bcc Fe unit cell with the lattice constantsof 2.84 A˚. In Figure 4.3, the total energy of the conventional bcc Fe unit cellis well converged to 0.1% with a 8×8×8 k-point mesh (i.e. the separationis 0.05 A˚−1). In the following calculations, the k-points spacing is set to beequal or smaller than 0.05 A˚−1 within the first Brillouin zone.Size of the SupercellIn this work, the convergence with respect to number of atoms is alsoevaluated. We have adopted four sizes of Fe bcc supercell from 16 atoms2×2×2 supercell, 54 atoms 3×3×3 supercell, 128 atoms 4×4×4 supercell,440 2 4 6 8 10 12 14 16 18 20-11.5-11.0-10.5-10.0-9.5-9.0-8.5-8.0Total Energy (eV/atom)k-Points per AxisFigure 4.3: Convergence test of the total energy with respect to k-point mesh for the conventional bcc Fe unit cell.and 250 atoms 5×5×5 supercell. The cohesive energy for Nb, which is thelargest and least elastically favorable solute studied in this work, is shownin Figure 4.4. The Nb cohesive energies are converged for system sizes of 54atoms and larger to within 0.01 eV. Therefore, in the following calculations,a 3×3×3 or an equivalent supercell is adopted.Special Quasirandom Structure CalculationsAs discussed in the previous sections, the special quasirandom structure(SQS) is proposed to determine properties of random solid solutions througha periodic structure, which is employed here to simulate the paramagneticstate of γ-phase Fe. In order to characterize the statistics of a given atomicarrangement, the correlation functions are introduced.Using the language of Ising models, each site i of the configuration isassigned a spin variable Sˆ, which takes the value -1 if it is occupied byFe↑ (i.e. Fe atom with spin-up state), or +1 if occupied by Fe↓ (i.e. Featom with spin-down state). Furthermore, all the sites can be grouped infigures f(l,m) of l vertices, where l = 1, 2, 3... corresponds to a shape: point,450 50 100 150 200 250-2.13-2.10-2.07-2.04-2.01-1.98Nb Cohesive Energy (eV)Number of Atoms in the SupercellFigure 4.4: Convergence test of Nb cohesive energy with respect tosupercell size.pair, and triplet... respectively, spanning a maximum distance of m, wherem = 1, 2, 3... is the first, second, and third-nearest neighbors, and so forth.For example, pairs of atoms (a figure with l=2 vertices separated by an mthneighbor distance), triangles (l=3 vertices), etc. The correlation functions,∏l,m, are the averages of the products of site occupations of figure k at adistance m. The optimum SQS for a given composition is the one that bestsatisfies the condition:(∏l,m)SQS ∼=<∏l,m>R (4.7)where <∏l,m >R is the correlation function of a real paramagnetic γ-phase Fe. Since there is no correlation for the real paramagnetic γ-phaseFe, the correlation function is simply zero.In the present work, the Alloy Theoretic Automation Toolkit (ATAT)[100] has been used to generate SQS for the fcc structure. In general, thesmaller the unit cell, the worse the correlation functions that match thoseof the real γ-phase Fe with paramagnetic state. Note that the algorithmused in this work is to enumerate every possible supercell and every possible46atomic configuration, which becomes prohibitively expensive as the size ofthe SQS increases. Hence, we use a SQS supercell consisting of 32 atoms tosimulate the paramagnetic state of γ-phase Fe.Grain Boundary EnergyIn order to obtain reliable results, it is necessary to consider a supercellwith reasonable size. In particular, the distance between two adjacent grainboundaries should be sufficiently large to decouple them from each other.For this purpose, the supercell size for grain boundary is determined fromcalculation of the grain boundary energy as a function of inter-boundarydistance. In this work, the grain boundary energy (σgb) is calculated as thedifference between the total energy of the supercell with (Egb) and without(Ebulk) grain boundaries, which can be expressed as:σgb =12AS(Egb − Ebulk) (4.8)where AS is the area of the boundary, which is chosen to be comparablewith a 3×3×3 bulk supercell, i.e. 8.52×8.52 A˚2. Note that there are twograin boundaries in the supercell that is imposed by the periodic boundaryconditions. Therefore, the grain boundary energy is corrected by the factorof 2.α-γ Interface EnergyIn a similar manner, the α-γ interface energy is calculated as follows:σα−γ =12AS(Eα−γ − Eα − Eγ) (4.9)where Eα and Eγ are the total energies of the supercell for bcc and fccgrains, respectively. Eα−γ is the total energy of the supercell consisting ofbcc grain, fcc grain, and the α-γ interfaces.474.1.6 Parameter Setup and Formulas for DFT CalculationsAs has been discussed above, all of the DFT calculations are carriedout using VASP [101, 102], with the projector augmented wave (PAW)method [103], and the Perdew-Burke-Ernzerhof (PBE) generalized gradientapproximation (GGA) to the exchange correlation functional [104]. SinceFe is a magnetic element, its spin-up density (ρ↑) and spin-down density(ρ↓) are not the same. Therefore, the spin-polarized DFT calculations wereperformed. Note that compared with the non-spin polarized calculations,the accuracy is the same on both levels. Which solution is more reasonableonly depends on whether the system is magnetic or not. If it is (either FMof AFM), the calculation has to be done spin-polarized, and the total energyof the cell will be more negative than for the non-polarized calculation. Ifit is not, the results should be the same for both calculations.In this work, a cutoff energy of 350 eV is used to truncate the plane-waveexpansion of the wave functions. Full relaxation of the atomic positions isallowed to an energy convergence of 10−4 eV and force convergence of 10−2eV/A˚ in each case, using a conjugate gradients routine to find the localminimum.4.1.7 Simulation Procedure for Solute SegregationTo study the solute-interface interactions, the binding energy (also referredas segregation energy) of the solutes with the grain boundaries is calculated.A straightforward way to calculate segregation energies would be to comparethe energies of a system with a grain boundary and a solute atom wherein one case the solute is located at the boundary and in another casein the bulk sufficiently far away from the boundary to avoid any elasticcoupling with the boundary. The required large supercell would leadto prohibitive computational cost such that an alternative approach isemployed by executing separate grain boundary and bulk simulations usingsmaller supercells. Hence, the binding energy is expressed as:Eiseg = (EiX+gb + Eref )− (Egb + EX) (4.10)48where Egb and Eref are the total energies of the pure Fe system calculatedwith and without grain boundaries, whereas EiX+gb and EX are the totalenergies of the supercell with and without grain boundaries and one soluteatom. The superscript i denotes the substitutional sites in the grainboundary. A negative value for Eiseg indicates an attractive segregationenergy of the solute atom to the considered grain boundary site.The segregation energy for the second solute is calculated as:Ei,jseg = (Ei,jX+Y+gb + Eref )− (EjY+gb + EX) (4.11)where Ei,jX+Y+gb is the total energy of the supercell containing solutes Xand Y occupying boundary sites i and j, respectably. EjY+gb is the totalenergy of the supercell with solute Y placed at boundary site j.To analyze the solute-solute interaction, the effective interaction (Eeff )is calculated, which is defined as:Eeff = (EX+Y + Eref )− (EX + EY ) (4.12)where EX (EY ) is the total energy of a configuration containing onesingle solute atom and EX+Y is the total energy of a supercell with twosolute atoms. Eref is the total energy of the solute free supercell. A positivevalue of Eeff corresponds to repulsive interaction.4.1.8 Simulation Procedure for DiffusionThe simulation procedure for diffusion employed in this work is outlined asfollows: first, a single point defect is introduced at various positions in thesystem and its formation free energy is calculated using DFT. The vacancyformation energy can be calculated as:Ev = E − Eref + Eref/N (4.13)where E and Eref are the total energies of the pure system calculatedwith and without a vacancy, respectively.In case of solute substitution, one should also calculate the binding49energy for the vacancy-solute pair which can be expressed as:Eb = (E+X + Eref )− (E + EX) (4.14)where E+X (EX) is the total energy of the supercell with (without)vacancy and one solute. We define Eb such that favorable binding is negative.At the next step, migration energies (Em) of the atomic jumps, which arethe energy difference between the saddle-point barrier and the equilibriumposition, are computed using nudged elastic band (NEB) method with theclimbing image algorithm [105, 106]. In implementing this method, N + 2configurations are considered, where N configurations are from intermediatestates along the transition path and the other two are the initial and finalstates of migration, both of which are local minima on the potential energysurface. It is started by minimizing a transition path as a chain of imagesthat connect the initial and a final state. A minimization algorithm isapplied, and the energy is minimized in all directions except for the directionof the reaction path. The images are moved in the direction to minimizethe forces, which gives the minimum energy path, with the highest point onthe path being a best guess of the saddle-point (SP). The atom migrationenergy (Em) is then determined based on the NEB calculations.The activation energy at 0K (Q0) is then given as the sum of thevacancy formation, migration and solute-vacancy binding energies in thefully ordered ferromagnetic state:Q0 = Ev + Eb + Em (4.15)In order to model rate phenomena, harmonic transition-state theory(TST) is used in this work. Harmonic TST offers a straightforwardapproximation to a rate constant, at which atoms move from one site toa neighboring one. In Harmonic TST, the jump frequencies are given by[107]:Γ = ν0 exp(−EmkBT) (4.16)where ν0 denotes the “attempt frequency” telling how often an attempt50is made to exceed the barrier. Typically it is of the order of the Debyefrequency, i.e. 1012 ∼ 1013s−1. The attempt frequency (ν0) is calculated as:ν0 =3N−3∏i=1νeqi /3N−4∏i=1νspi (4.17)where νeqi and νspi are the frequencies of the equilibrium and saddle-pointconfigurations. In this work, the vibrational frequencies were computed fromthe Hessian matrix (see next section for details). In previous studies, Astaet al. explicitly calculated vibrational spectrums for Mo, W, and Fe-selfdiffusion, and found that the attempt frequencies were very similar [11],which can be ascribed to the same diffusion mechanism (vacancy mediated)and the same host. And consequently, in their later studies, they assumedthe attempt frequencies (ν0) were constant for all of the jump rates (Γ) [43].Since Σ5 grain boundary is a simple and ordered structure, we assume thevalues of attempt frequencies in Σ5 grain boundary are comparable to thebulk values. Therefore, in the present work, we only calculate the attemptfrequencies for Fe self-diffusion in the bcc and fcc bulk, and apply thesevalues to the grain boundary diffusion.The rate constants (Γ) are then tabulated in a rate catalogue and usedas input data for the kinetic Monte Carlo (KMC) simulations.4.1.9 Hessian Matrix and Vibrational FrequenciesThe Hessian matrix is the matrix of second derivatives of the energy withrespect to geometry. To calculate the Hessian matrix, finite differences areused, i.e. each ion is displaced in the direction of each Cartesian coordinate,and from the forces the Hessian matrix is determined. In the harmonicapproximation, the elements of the Hessian matrix is given by:Hi,j =∂2E∂xi∂xj(4.18)In order to calculate the vibrational frequencies, the Hessian matrix is51first mass-weighted:Hmi,j =Hi,j√mla ∗mma(4.19)where mia are the atomic mass. Diagonalization of this matrix yieldseigenvalues, ε (force constants), from which the vibrational frequencies canbe calculated:νi =12pi√εi (4.20)In this work, we calculate the vibrational frequencies of the pure Fesystem (νi), bulk Fe with one vacancy at the equilibrium position (νeqi ), andbulk Fe with one vacancy at the saddle-point (νspi ).4.2 Kinetic Monte Carlo SimulationsUnlike the bulk situation, the activation energies of the solutes in theinterfaces depend on the positions. Consequently, there are a multiple ofjump rates. To investigate solute diffusivities, a kinetic Monte Carlo (KMC)model is developed based on the detailed input from DFT calculations.KMC is an efficient method for carrying out dynamical simulations for awide variety of stochastic and/or thermally activated processes. In contrastto the MD method which can generally only be carried out on a time scaleof nanoseconds or less, KMC simulations can effectively overcome the timescale limitation, and the diffusion coefficients can be calculated at interfaces.In the KMC model atoms are located on a rigid lattice. Diffusion occursby vacancy exchange with a first nearest neighbor atom. At each KMC step,there are z possible vacancy jumps, where z is the coordination number.These jumps form a Markov process, which is simulated by the residence-time algorithm. In the present study, the residence time associated with theKMC step is given by [108]:τkmc = − ln(urand)/RN (4.21)where urand is a random number uniformly distributed between 0 and 1,and RN is the summation of the jump rate, i.e. RN =∑zj=1 Γj . Here Γj is52the jump rate, which can be obtained according to Equation 4.16 based onthe DFT calculations. The algorithm for KMC is outlined as the followingprocedure:0. Set the time t = 01. Initialize the lattice structure by assigning the type of atoms to eachlattice site.2. Insert the vacancy into the lattice sites.3. Start KMC loop over Nitermax steps.4. Define the local atomic configuration in each direction for the vacancyand extract the value of the energy barrier from the predefined table. Forma list of the rates Γj of all possible transitions in the system.5. Calculate the cumulative function RN =∑Γj for j = 1...z.6. Get a random number xrand ∈ (0, 1)7. Find the event to carry out j by finding the j for which Rj−1 <xrandRN ≤ Rj .8. Carry out event j9. Get a new random number urand ∈ (0, 1)10. Update the time with tkmc = tkmc + τkmc, where τkmc is determinedby Equation 4.21.11. Set Niter = Niter + 1. If Niter < Nitermax , return to step 4.For grain boundary diffusion, a thin boundary model is constructed.The vacancy moves randomly. At each step, the residence time of thevacancy at the current site is calculated from its jump rates in all possibledirections. The clock is advanced after the vacancy leaves the site. As thevacancy walks, it also moves atoms around and induces their diffusion. Oncethe simulation is complete, the apparent diffusion coefficients (Dapp) of thesolute atoms are recorded according to the Einstein equation:[109]Dapp =〈r2(t)〉6 · t(4.22)where 〈r2(t)〉 is the mean square displacement (MSD) of the diffusingsolute atoms during time t.According to the analysis of Porter and Easterling, if the grain boundary53has an effective thickness δ and the grain size is dgs, the apparent diffusioncoefficient (Dapp) is expressed as [110]:Dapp = (1−δdgs)Dl +δdgsDb (4.23)where Dl and Db are the bulk and grain boundary diffusivities. Since thebulk diffusivities Dl are calculated directly from DFT, the grain boundarydiffusivitiesDb can thus be determined. In this work, the results are obtainedbased on the 105 independent calculations at each temperature.4.3 Molecular Statics SimulationsIn this work, to extend our simulation of special grain boundaries usingDFT calculations to more general grain boundaries, the molecular statics(MS) simulations are employed, which are performed using the LAMMPScode [111]. To obtain a reliable and accurate results, it is critical to choosesuitable potentials U of the system.The most widely used potential theories is embedded atom method(EAM) [112]. So far, there are several EAM based potential available forpure Fe, e.g. Johnson 1989 [113], Ackland et al. 1997 [114], and Mendelev etal. [115]. Recently, Malerba et al. systematically compared the performanceof these potentials for bcc and fcc iron [116]. They found that the results (e.g.lattice constants, bulk modulus, etc.) obtained by Mendelev-type potentialswere closer to DFT results than others, which make them currently thebest choice in order to “extend density functional theory” to larger scales.Therefore, the EAM potential derived by Mendelev et al. for pure iron hasbeen employed in this work [115]. Although MS simulation are much moreefficient than DFT calculations, they are still limited in their applicationsto only a few pure metals, and very few binary and ternary systems due tothe lack of suitable interatomic potentials [36, 41, 42].Similar to the DFT calculations, the models are relaxed using theconjugate gradient method to obtain the equilibrium structures.54Chapter 5Interaction of Solutes withthe Σ5 (013) Tilt GrainBoundary in Iron5.1 Grain Boundary StructureThe Σ5 (013) tilt grain boundary is constructed according to the coincidencesite lattice (CSL) theory by rotating two bcc grains by 36.9° about the〈100〉 axis [63]. This grain boundary has mirror symmetry with respectto the (013) boundary plane. The grain boundary structure is shown inFigure 5.1. As in the bulk calculations, periodic boundary conditions areemployed such that the supercell contains two grain boundaries. Solutesare placed at the substitutional sites within one grain boundary only. InFigure 5.1, the letters a, b, and c refer to different boundary positions.55Figure 5.1: (a) Side view of the Σ5 (013) symmetrical tilt grainboundary. The lighter and darker circles represent Fe atomsin different (100) atomic planes. (b) Atomic structure of theboundary sites. The letters a, b, and c refer to differentboundary positions. The superscript (′) indicates the symmetricposition with respect to the (013) boundary plane. Thesubscript identifies different boundary unit cells of the Σ5 (013)grain boundary.5.2 Grain Boundary EnergyAs discussed in Chapter 4, the supercell size was determined from calculationof the grain boundary energy as a function of inter-boundary distance. InFigure 5.2, it is clear that the grain boundary energy is converged whenthe distances are larger than 9.0 A˚. In order to balance accuracy of resultsand computational load, we adopt a supercell of 120 lattice sites with thedimensions of 8.52×8.98×18.0 A˚3. The calculations were performed using a4×4×2 Monkhorst-Pack k-point mesh.564 6 8 10 12 14 16 180.620.640.660.680.700.720.74  Grain Boundary Energy(J/m2)Distance between two identical GBs (Å)Figure 5.2: Grain boundary energy as a function of distance betweengrain boundaries.5.3 Single Solute Segregation EnergiesWe first investigate the solute segregation behavior in the Σ5 tilt grainboundary in bcc Fe. The binding energy of solutes with the grain bound-aries (i.e. segregation energy) is calculated according to Equation 4.10.The calculated binding energies for various grain boundary positions aresummarized in Figure 5.3. We find that segregation to the grain boundaryis energetically favorable for all solutes and boundary sites considered in thiswork. Furthermore, interactions between solute atoms and grain boundaryare short-range and limited to about 2∼3 atomic layers. On the otherhand, different solutes prefer different positions at the grain boundary. Forexample, the binding energy for Nb, Ti, and Mo atoms are the largest inposition a, whereas other elements such as Si, V, Cr, Mn, Co, Ni, and Cuprefer position b.In order to compare with the experimental results, it would be useful toaverage the binding energy. In this work, two different ways of finding theaverage binding energies are employed, i.e. the simple numerical averaging57-5 -4 -3 -2 -1 0 1 2 3 4 5-0.55-0.50-0.45-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.050.00 Distance from Boundary Plane (Å)cbc'aBinding Energy (eV) Co Si V Cr Mn Ni Cu Ti Mo Nbb'Figure 5.3: Binding energies for substitutional solutes at the Σ5 (013)tilt grain boundary as a function of distance from the boundaryplane; a, b, c, b′, and c′ refer to the boundary sites as labeled inFigure 5.1.method and the Langmuir-McLean approach. The numerical method simplyaverages the binding energies over the five boundary sites, i.e. a, b, c, b′, andc′, whereas the Langmuir-McLean approach is a more sophisticated method,which would be more relevant for solute segregation.Based on the Coghlan-White approach, if the grain boundary hasmultiple types of site, the segregation to each type of site is calculatedindividually, then a weighted average is performed to get the total grainboundary segregation. So for a number of different types of site (i), eachwith binding energy, Eiseg, the segregation to that type of site is foundthrough simple Langmuir-McLean equation [117, 118]:cigb1− cigb=cbulk(1− cbulk)exp (−EisegkBT) (5.1)580 200 400 600 800 1000 12000.30.40.50.60.70.80.91.01.1Nb Segregation, cgbTemperature (K)  Data FittingFigure 5.4: Total grain boundary segregation of Nb as a functionof temperature in Σ5 grain boundary in Fe. The symbolsare calculated according to Equation 5.3, while the line is theLangmuir-McLean fit.where cigb is the solute concentration at the boundary site i, and cbulkis the solute concentration in the bulk. Coghlan and White rewrittenEquation 5.1 as:cigb =cbulk exp (−EisegkBT)1− cbulk + cbulk exp (−EisegkBT)(5.2)Then the total grain boundary segregation is a weighted average summedover all grain boundary sites, which is expressed as:cgb =∑i(Ficigb) (5.3)where Fi is the fraction of the grain boundary sites that have binding597.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0-0.45-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.050.00 Average Binding Energy (eV)Solute Volume (Å3)CoSiVCrMnNiCuTiMoNb(a)7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0-0.45-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.050.00 Effective Binding Energy (eV)Solute Volume (Å3)CoSiVCrMnNiCuTiMoNb(b)Figure 5.5: (a) The average binding energies, and (b) the effectivebinding energies of solutes as a function of solute atomic volume.The dotted line indicates the atomic volume of Fe in the bcclattice.Eiseg. In Figure 5.4, the total segregation (cgb) of Nb as a function oftemperature is plotted. It is clear that cgb decreases with the increaseof temperature. We then fit these data according to Langmuir-McLeanequation and obtain the effective binding energies, i.e. Eeffseg .Figure 5.5a shows the simple numerical averaged binding energies. Theresults are plotted as a function of solute volume (Vs). In Figure 5.5b,the effective segregation energies of solutes, which are obtained based onthe Langmuir-McLean approach, are shown as a function of solute volume.60The two averaging approaches lead to very similar characteristics of solutesegregation in the Σ5 grain boundary.In this work, the solute volume was computed from relaxed atomicpositions in the bulk lattice according to the touching hard-sphere model:the radius of Fe is defined as half of the Fe-Fe nearest neighbor (NN)distance; solute radii are found by subtracting the Fe radius from thesolute-Fe nearest neighbor distance [119]. In Figure 5.5, there is a distinctcorrelation between the average segregation energy and solute volume, i.e.the magnitude of Eseg and thus the tendency for segregation increase withsolute size. Evidently, for large solute, such as Nb, Mo, etc. the elastic straininduced by the substitutional solutes in the bulk lattice can be partiallyrelieved at the grain boundary where the packing density is reduced. Incontrast, solutes whose size is similar than that of Fe do not introducelarge lattice distortions and hence exhibit much smaller segregation energies.Similar size dependent trends were reported for vacancy-solute binding anddiffusion in Mg [119, 120].To further illustrate the influence of the solute volume on segregation,we show in Figure 5.6 the change in electron density in the (100) planeas calculated from the difference between the electron density of the puresystem and the system containing one solute atom in position a. The plotsare shown for Co and Nb because they represent the two extremes of latticedistortion that the investigated substitutional solutes introduce in the bulk.The bulk lattice distortion due to Co is very small (less than 0.003A˚ decreasein Co-Fe distance with respect to Fe-Fe distances in the bulk). As comparedto Fe, Co has one extra nuclear charge which is screened by the electrondensity within about 1.0 A˚ radius from the nucleus (see Figure 5.6(a)).In contrast, Nb 4d orbitals are much more spatially extended leading tosignificant and rather asymmetric leakage of charge towards neighboringFe atoms (see Figure 5.6(b)). The complexity of this charge distributiondemonstrates the difficulty in defining an effective solute volume at thegrain boundary where the symmetry constraints are lifted. This could beexceptionally problematic when the valence states of the solute and the hostare of very different origin.610.0600.100.98-3 -2 -1 0 1 2 3-3-2-10123(a)Feb'Feb0.0300.030-0.070-0.630.0300.0800.0800.030-3.00.080-3 -2 -1 0 1 2 3-3-2-10123FebFeb'(b)Figure 5.6: Valence electron density difference contour maps in the(100) plane for (a) Co-, and (b) Nb- segregated at boundaryposition a with respect to the non-segregated grain boundary(in units of e/A˚3 with a contour spacing of 0.01 e/A˚3). The solidred and dashed blue lines indicate gains and losses in electrondensity, respectively.62A comparison between theory and experiments is limited by the lack ofexperimental data (especially for the Σ5 (013) tilt boundary). Nonetheless,the reported segregation enthalpy for Si (-9±3.5kJ/mol) [121] agrees wellwith our calculation of -8.7kJ/mol for the effective binding energy basedon the Langmuir-McLean approach. The experimental segregation energiesof Mn and Nb to random grain boundaries are -11kJ/mol and -29kJ/mol[122, 123], respectively. These values are comparable with our predictions of-14kJ/mol and -39kJ/mol, respectively, for the Σ5 boundary. In addition,our calculation indicates that the binding energies of Nb, Mo, and Ti withthe grain boundary are comparatively large, and the segregation of Nb isthe most pronounced among all investigated cases. Overall larger bindingenergies indicate stronger interactions with the grain boundary. Such stronginteractions are expected to delay grain growth and recrystallization. Allthese effects are actually observed in the case of Nb and Mo in agreementwith our prediction of rather large binding energies for these elements [123,124].5.4 Solute-solute Interactions at the GrainBoundaryAfter having considered segregation of a single solute atom, let us nowturn to solute-solute interaction and its contribution to the grain boundarysegregation energies. As has been discussed above, the size differencebetween smaller and larger solutes is reflected in their preference to segregateat particular sites. Smaller solutes prefer to occupy b positions whereaslarger solutes segregate at position a (Figure 5.3). Therefore, we selectthese two positions for detailed analysis of solute-solute interactions at thegrain boundary.Figure 5.7 shows the calculated segregation energies according to Equa-tion 4.11 for various solutes segregating at boundary site a (Figure 5.7(a)) orb (Figure 5.7(b)) while the same kind of solute is already present at anotherboundary position. The segregation energies of a single solute are also shownfor comparison. The predicted trends are similar in both scenarios, i.e.63Co Si V Cr Mn Ni Cu Ti Mo Nb-0.5-0.4-0.3-0.2-0.10.00.1 a ab,  ds-s=2.52Å aa0, ds-s=4.71ÅBinding Energy (eV)(a)Co Si V Cr Mn Ni Cu Ti Mo Nb-0.6-0.4-0.20.00.20.40.6 b bb', ds-s=2.18Å ba, ds-s=2.52Å bb'1, ds-s=3.59ÅBinding Energy (eV)(b)Figure 5.7: Binding energies for a second solute atom segregating at(a) boundary position a, and (b) boundary position b when thesame solute is already present at another boundary site. Thesquare symbols indicate the binding energies of a single soluteatom.Figure 5.7(a) and 5.7(b). For Co, Ni and Cu, co-segregation energies arevery close to the segregation energies of individual solutes. For all the otherinvestigated cases, the presence of one solute atom decreases the solute-grainboundary segregation energies for the second solute atom when two soluteatoms occupy nearest neighbour positions.For short solute-solute distances, i.e. 2.18 A˚, which is about 90% of thenearest neighbor distance in the pure bulk lattice, the repulsive interaction64may be strong enough to make segregation of the second solute atomunfavorable (Figure 5.7(b)). Nonetheless, for spacing of 2.52 A˚ and larger,the segregation energies for the second solute atom are negative except forSi at 2.52A˚, suggesting that these positions remain favorable for solutesegregation albeit with a reduced tendency of segregation. An exceptionto this tendency are the larger solutes (Nb, Ti, Mo) which show an increaseof up to 30% in the magnitude of the segregation energy when they occupytwo neighboring b-sites (Figure 5.7(b)). As the solute pair distance becomeslarger, the influence of the first solute on the segregation energies of thesecond solutes gradually vanishes. As shown in Figure 5.7(a), when the twosolutes occupy neighboring sites a separated by a distance of 4.71 A˚, thesegregation energies of the second solute atoms are, within the accuracy ofthe calculations, the same as the segregation energies of a single solute atomat the same site. These results suggest that even a high symmetry grainboundary, such as Σ5, can support a rather significant solute enrichment.To analyze the solute-solute interaction in more detail, we calculate theeffective interaction (Eeff ) as a function of solute-solute distance for boththe bulk and the grain boundary according to Equation 4.12. Evidently,full mapping of such interactions for the grain boundaries involves a ratherlarge number of permutations. To minimize the number of calculations, wefix one solute atom in position b while the other solute atom occupies otherboundary positions. The results for solute-solute interactions as a functionof solute pair distance in the grain boundary are shown in Figure 5.8. Forcomparison, the effective interactions between these solute pairs in the bulklattice from first to fifth nearest-neighbor positions are also plotted.In Figure 5.8, Cu shows attractive interactions. While for all the otherinvestigated cases, repulsive interactions are observed. Overall, we find fourmajor solute-specific contributions to the effective interactions: contribu-tions from the elastic energy induced by atomic size misfit, contributionsfrom competing magnetic interactions, contributions from magneticallydriven clustering, and contributions from chemical bonding.The elastic contribution to solute-solute interaction can be estimatedusing the classical elasticity theory [125, 126]. According to the Eshelby652.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2-0.10.00.10.20.30.40.5(a)5nn Eeff in GBs Eeff in Bulk Eelastic in BulkEffective Interaction Energy, Eeff (eV)Nb pair distance (Å)1nn2nn3nn4nnbcbabb1bb'1bc'ba2bb'02.4 2.8 3.2 3.6 4.0 4.4 4.8 5.20.000.050.100.150.20(b) Eeff in GBs Eeff in Bulk Eelastic in BulkEffective Interaction Energy, Eeff (eV)Cr pair distance (Å)2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2-0.08-0.040.000.04(c) Eeff in GBs Eeff in Bulk Eelastic in BulkEffective Interaction Energy, Eeff (eV)Cu pair distance (Å)2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2-0.050.000.050.100.150.200.25(d) Eeff in GBs Eeff in Bulk Eelastic in BulkEffective Interaction Energy, Eeff (eV)Si pair distance (Å)Figure 5.8: Pair interactions for (a) Nb, (b) Cr, (c) Cu, and (d) Si inthe Fe bulk and at the Σ5 (013) grain boundary.model, the elastic interaction energy between two solute atoms as a functionof their distance d is given by:εelast =0.3γ2el∆V 2(αeld3) (5.4)where ∆V is the volume difference between solute and Fe atom. Theparameters αel and γel are related to the elastic constants Cij of pure Fe:αel = C11 −C12 − 2C44, and γel = (C11 + 2C12)/3C11. The following elasticconstants were obtained from DFT calculations for bcc bulk Fe: C11 = 243.6GPa, C12 = 137.4 GPa, and C44 = 120.9 GPa. Then, the bulk modulusK = (C11 + 2C12)/3 = 172.8 GPa is obtained which agrees well with theexperimental value of 171.6 GPa [127] and the previous DFT result of 175.0GPa [128]. The calculated elastic interactions for solute pairs in the bulkare depicted in Figure 5.8 by the solid lines.66The elastic contribution to the solute-solute interaction is determined bythe relative size of solutes with respect to Fe. As the size of solutes increases,the elastic contribution to solute-solute interactions becomes increasinglymore important. As shown in Figure 5.8(a), Nb-Nb interactions in the bulkare comparable to those predicted by the classical elasticity theory. Theyare, however, different at the grain boundary. When two Nb atoms occupygrain boundary positions at intermediate pair distances (bb′1 and bc′ pairs),the effective interaction between Nb becomes attractive. This is because thepacking density is low at these boundary positions, which gives rise to largeexcess volume. As a result, the substantial elastic strain can be efficientlyrelieved thereby favoring the Nb-Nb binding. Similar trends were also foundfor other larger solute atoms, i.e. Mo-Mo and Ti-Ti (see Figure 5.7(b)).From elastic considerations alone, one would expect to find rather smalleffective interactions between Mn, Cr, V, Co and Si in Fe. However, a strongrepulsion between two Cr atoms is clearly evident in our DFT calculations(see Figure 5.8(b)). In fact, it is only a factor of two smaller than the Nb-Nbnearest neighbor repulsion, which is predominantly elastic. Such a strongrepulsion can be attributed to magnetic frustration. To better show theinfluence of the competing magnetic interactions, the magnetic moments foreach solute atom in Fe matrix are calculated. Since solute-matrix magneticinteraction varies from one solute to another, the single solute bulk and grainboundary calculations were performed with ferro- and antiferro- initial spinarrangements between solute and matrix.In agreement with previous work, we find anti-ferromagnetic (AFM)ground state for Ti, V, Cr, Mn, Nb and Mo [129, 130]. For Co, Ni andCu, the ferromagnetic (FM) state has the lowest total energy. In systemswith two solutes and for large solute-solute distances, the initial magneticconfigurations are chosen according to the solute-matrix coupling foundin the bulk calculations. However, we check if other possible magneticconfigurations can be converged for the short solute-solute distances. Incases where several magnetic solutions are found, we use the one with thelowest total energy.The calculated magnetic moments for solute atoms in grain boundaries67Table 5.1: Magnetic moments (µB) for single solute ingrain boundary (GB) and bulk sites.1a b c average bulkSi -0.12 -0.14 -0.11 -0.12 -0.10Nb -0.70 -0.60 -0.69 -0.66 -0.75Mo -0.81 -0.66 -0.69 -0.70 -0.77Ti -0.79 -0.64 -0.69 -0.70 -0.77Cu 0.09 0.14 0.11 0.12 0.17Ni 0.78 0.77 0.81 0.79 0.84Co 1.65 1.60 1.69 1.65 1.71Mn -2.40 -1.81 -1.85 -1.94 -1.77Cr -2.33 -1.68 -1.71 -1.82 -1.79V -1.41 -1.09 -1.21 -1.20 -1.251 The sign of the solute magnetic moment is positive(negative) if the magnetic moment is parallel (anti-parallel) to that of Feand the bulk structure are listed in Table 5.1. Note that the predictedmagnetic moments are essentially the same in the bulk and in the grainboundary. Further, they change by less than 1% if two solute atoms arebrought close to each other. Thus, magnetic moments of single solute atomsas shown in Table 5.1 can be used to infer magnetic interactions.The magnetic moments for Cr (Mn and V) are large compared withthose of other alloying elements, suggesting that the effects due to magneticfrustration are of significance for Cr (Mn and V). In the dilute limit, thisAFM coupling increases the energy of the system when two solutes occupyadjacent sites in an otherwise ferromagnetic (FM) host material. Olssonet al. reached similar conclusions in recent bulk calculations [130]. Moreover,68Levesque et al. showed that by constraining the Cr local moments to zero,one can reduce Cr-Cr interactions to negligible values [131]. Our findingsdemonstrate that the interactions between 3d TMs in the grain boundaryare similar to those in the bulk.Compared to Nb and Cr, the Cu case appears at first surprisingas one would expect relatively weak repulsive Cu-Cu interactions dueto its moderate size and a lack of local magnetic moment. As shownin Figure 5.8(c), however, Cu shows an attractive interaction indicatingpossible clustering, which is consistent with the observations that Cuimpurities have a pronounced tendency to segregate from the matrix[132, 133]. Since all Cu d-orbitals are occupied, it is natural to think of Cuas non-magnetic solute. By employing the simple Heisenberg model, one canestimate that a single Cu impurity would cost the magnetic energy by 8J1in bcc iron, where J1 is the exchange parameter in the first coordinate shell.Evidently, it is doubled when two Cu impurities are far apart. By combiningthem in a pair, one can reduce the penalty in magnetic energy by 2J1 thataccounts for about 70% of the interaction energy using the value calculatedby Ruban et al [134]. This is, of course, a rather crude estimate since itonly accounts for the nearest neighbor exchange interactions. Nonetheless,we find that Cu-Cu interaction energy is reduced by an order of magnitudewhen the magnetism is switched off in the calculations. Thus, the attractiveCu-Cu interaction can be viewed as a magnetically driven clustering.From Figure 5.8(d), it is evident that a short-range repulsive interactionoccurs when Si atoms come sufficiently close to each other, within 3 or 4A˚.Unlike the TMs, the origin of this repulsive force is based on the fact thatSi atoms prefer to bond with Fe atoms instead of with other Si atoms. Infact, the binding energy per one Si atom in bulk Si is -1.14 eV, which isappreciably smaller (absolute value) than those in Fe-Si alloys, i.e. -1.87eV. This indicates that the electronic levels of Fe-Si bonding are formed inan energetically deeper range than those of Si-Si bonding. In this case, mostelectrons of Si are expected to contribute to Fe-Si bonds rather than Si-Sibonds. As a result, a short-range repulsive interaction occurs for Si-Si pairin the Fe matrix.69Nb-Mo Nb-Ti Nb-Mn Nb-Cr Nb-V Nb-Cu Nb-Si-0.6-0.5-0.4-0.3-0.2-0.10.0  Segregation Energy (eV) ab, ds-s=2.52Å aa0, ds-s=4.71Å(a)Nb-Mo Nb-Ti Nb-Mn Nb-Cr Nb-V Nb-Cu Nb-Si-0.5-0.4-0.3-0.2-0.10.0  Segregation Energy (eV) ba,  ds-s=2.52Å bb'1, ds-s=3.59Å bb'0,  ds-s=5.20Å(b)Figure 5.9: Binging energies for the grain boundary with Nbsegregating at (a) boundary position a, and (b) boundaryposition b when another solute (i.e. Mo, Ti, Mn, Cr, V, Cu,or Si) is already present at a neighboring site indicated by thesecond letter. The dotted line indicates the binding energies fora single Nb atom.Steels usually contain several alloying elements each intended to affectmicrostructure and properties in specific ways. Here, we study thesegregation energy of Nb, the largest solute in our set, to sites a and b in thegrain boundary when another solute of a different type is already present at aneighboring site. As shown in Figure 5.9, the segregation energy is stronglydependent on the relative solute positions. The variation is particularlypronounced when the other solute is of large size, i.e. for the Nb-Mo and70Nb-Ti combinations. For a solute-solute distance of 2.52 A˚, the magnitudeof the Nb segregation energy is reduced in these two cases by about 60%(Figure 5.9(a)) and 75% (Figure 5.9(b)), respectively, as compared to thecase of a single Nb atom. The reduction is about 50% for Mn, Cr, or V. Forthe Si-Nb and Cu-Nb cases the segregation energy of Nb is only marginallyaffected (by less than 30 meV). When the separation distance between thetwo sites is increased, the Nb segregation energy quickly approaches thatfor a single Nb atom. For intermediate distances, however, an increase inthe magnitude of the segregation energy is predicted for selected cases, inparticular for Ti-Nb and Mo-Nb (see Figure 5.9b). These findings are similarto those for the Nb-Nb co-segregation (see Figure 5.7).5.5 Molecular Statics SimulationsThe above DFT calculations indicate that the excess volume at boundarysites is an important parameter to predict the segregation tendency of solutesat grain boundaries. This is in particular of significance for large soluteatoms (i.e. Nb, Mo, and Ti) that are crucial alloying or microalloyingelements in advanced high-strength steels. Since most steels and otherpolycrystalline materials are primarily composed of random boundariesrather than special Σ boundaries, it is useful to evaluate to what extent DFTcalculations for these special boundaries remain of merit for more generalgrain boundaries. Here, we consider the excess volume of boundary sitesas an indicator for this evaluation. For this purpose, a series of molecularstatics (MS) simulations were performed.As in DFT calculations, the symmetrical [100] tilt grain boundaries wereconstructed by rotating two bcc grains about the [100] axis with the rotationangles ranging from 10° to 44° to obtain higher order Σ boundaries. Inthis work, nine different tilt grain boundaries, i.e. Σ5, Σ13, Σ17, Σ25,Σ29, Σ37, Σ41, Σ53, and Σ61, are employed to study the excess volumeat the boundary region. For all these investigated cases, two identicalgrain boundaries were included in the MS simulation block to maintain theperiodic boundary conditions. The length between two grain boundaries710.95 1.00 1.05 1.10 1.15 1.20 1.25 1.300.000.050.100.150.200.250.300.350.400.45 FrequencySite volume/bulk atomic volumeMisorientation: 61 (10.4 ) 5   (36.9 ) 29 (43.6 )5bulkFigure 5.10: Site volume distribution in Σ5 (36.9°), Σ29 (43.6°) andΣ61 (10.4°) grain boundaries.is separated by at least 20 nm, which is sufficient large to eliminate theinteraction effects from each other.The atomic volumes are calculated by constructing Voronoi polyhedra(VP) for each site in the grain boundary. For a particular center atom, VP isdefined as the volume of space containing all points closer to this center atomthan to any other atoms. The faces of this polyhedron are perpendicularbisectors of the vectors connecting the center atom to its nearest neighbors.Figure 5.10 shows the site volume distributions for Σ5 (36.9°), Σ29 (43.6°)and Σ61(10.4°) grain boundaries. The Σ29 boundary is selected because ithas the largest misorientation angle in our set. On the other hand, the Σ61boundary has the least number of coincident sites in our set and hence isthe most random boundary considered. The site volumes are given in unitsof the atomic volume in the bulk. One can see that these distributions areasymmetric and extend to up to 25% increase in volume with respect to thebulk value.7210 15 20 25 30 35 40 450.91.01.11.21.36141253713175329Site Volume/Bulk Atomic VolumeMisorientation ( )5bulkFigure 5.11: Averaged atomic volume as a function of grain boundarymisorientation angle. The bars indicate the maximum and theminimum atomic volume of the boundary sites.The average volumes for all nine boundaries are shown in Figure 5.11as a function of misorientation angle with the vertical lines representing therange of values for each boundary. The average excess volume for the Σ5grain boundary is similar to that of more general boundaries but the rangeis narrower. Nevertheless, based on the data shown in Figure 5.10 and 5.11,it is reasonable to assume that, in a first approximation, calculations for theΣ5 grain boundary provide trends which remain applicable to random grainboundaries. This is also consistent with the predicted trend of segregationenergies that matches general observations of the role of the investigatedsolutes on overall recrystallization and phase transformation behavior inlow-carbon steels.73Chapter 6First-Principles Study ofFace-Centered Cubic γ-Iron6.1 Bulk and Grain Boundary Structures in γ-FeFigure 6.1: Fe fcc lattice structure with AFMD magnetic configu-rations. Red and blue circles indicate spin-up and spin-downmagnetic states. The letters o, p, q, w, e, and e′ are used tolabel the different positions in the fcc bulk.For fcc bulk simulations, the calculations are performed using a supercellof 256 lattice sites for fcc Fe with double-layer anti-ferromagnetic (AFMD)74Figure 6.2: SQS-32 structure for fcc Fe lattice. The red and blue ballindicate spin-up and spin-down states.arrangement. For this supercell, the Monkhorst-Pack 2×2×2 k-point meshis used [99] to sample the Brillouin zone. Figure 6.1 shows the fcc bulkstructure with AFMD magnetic configurations. It is noticeable that thereexists local structure distortions. As illustrated in Figure 6.1b, each site(o) has three types of nearest neighbors, namely p, q, and w. The distancebetween Fe↑-Fe↓, i.e. d3 is 0.1 A˚ shorter than the distance between Fe↑-Fe↑(Fe↓-Fe↓), i.e. d2, while d1 keeps the same value before and after relaxation.Such distortion only occurs within the first nearest neighbor separation. Forthe second nearest neighbors, we get doe=doe′ .In this work, the non-magnetic (NM), ferromagnetic (FM), and anti-ferromagnetic (AFM) states are also studied to check the relative stabilityof the collinear magnetic configurations. For this purpose, a conventionalfcc Fe unit cell containing 4 lattice sites with a k-point mesh of 8×8×8 isemployed.To investigate the special quasi-random structure (SQS) and non-collinear magnetic arrangement, a supercell of 32 lattice sites is used in thiswork. The optimum SQS-32 configuration is shown in Figure 6.2, whichis identical with Pezold’s work [135]. The non-collinear configuration is75employed from the work of Antropov et al. [136]. The calculations areperformed using a 4×4×4 Monkhorst-Pack k-point mesh.Figure 6.3: (a) Side view of the Fe Σ5 fcc grain boundaries withAFMD magnetic configurations. (b) Top view of the supercells.Red and blue circles indicate spin-up and spin-down magneticstates.In a similar manner to the bcc grain boundary, the Σ5 fcc Fe grainboundary is constructed according to the coincidence site lattice (CSL)theory by rotating two bcc grains about the 〈100〉 axis [63]. The grain76boundary structure is shown in Figure 6.3. To describe the magnetic stateof fcc phase, AFMD magnetic configuration is employed. As illustrated inFigure 6.3b, along a-axis, i.e. 〈100〉 direction, the AFMD phase has twolayers spin up and the next two layers spin down. In this work, we adopt asupercell of 80 lattice sites with the dimensions of 7.1×7.94×16.0 A˚3. Thecalculations are performed using a 4×4×2 Monkhorst-Pack k-point mesh.6.2 Magnetic Configurations for γ-FePrevious DFT calculations indicate that the double-layer anti-ferromagnetic(AFMD) state has the lowest total energy for fcc Fe (see Chapter 2 fordetails). Therefore, in the following calculations, we mainly focus on theAFMD magnetic configuration. In Figure 6.4, the AFMD, special quasi-random structure (SQS), and non-collinear (NC) configurations vs. theatomic volume for Fe are displayed. The calculated lattice parameters (al),and local magnetic moments (MM) for various magnetic phases of bulk α-and γ-Fe at 0 K are shown in Table 6.1. For comparison, the previous DFTcalculations and experimental results are also listed.In this work, we have not only concentrated on (001) magnetic layeredstructures, i.e. AFMD[001], but also considered other planar structures, suchas AFMD[111] and AFMD[110] (i.e. stacked along [111] and [110] direction).In agreement with previous work, the AFMD[001] state was found to possessthe lowest total energy among all investigated collinear states. In this study,the SQS magnetic configuration is considered to represent the paramagneticarrangements in the same way as one deals with random alloys. In SQSstructure, the spin-up and spin-down states are quasi-randomly distributed,which can give more realistic results for paramagnetic γ-Fe.According to the Heisenberg model, it is possible to use non-collinearmagnetism and start by setting up randomized initial spins, which wasconsidered in earlier studies of the non-collinear magnetism in γ-Fe [82, 83].In agreement with the previous work [82], our results indicate that theground state of γ-Fe is a non-collinear state. However, in Figure 6.4, theenergy difference between non-collinear state and the AFMD[001] order is77Table 6.1: Predictions of lattice parameters (al), and magnetic moments(MM) for magnetic phases of α and γ-Fe at 0 K. Earlier DFT andexperimental results are also shown for comparison.Systemal (A˚) MM (µB)Our resultsPreviousExp.1Our resultsPreviouswork workα-FeFM 2.84 2.84 [137] 2.85 [138] 2.20 2.20 [74]γ-FeNM 3.45 3.45 [137]LS FM 3.48 3.48 [137] 1.04 1.00 [137]HS FM 3.64 3.64 [137] 3.65 [138] 2.56 2.60 [137]AFM2[001] 3.49 3.49 [137] 1.34 1.33 [137]AFMD2[001] 3.55 3.54 [137] 1.98 1.94 [137]AFMD3[111] 3.47 0.91AFMD4[110] 3.51 1.70SQS 3.52 1.615NC (PM) 3.55 3.55 [82] 3.56 [138]1 The lattice constants are extrapolated to 0K according to the linear thermalexpansion2 Stacked along [001] orientation3 Stacked along [111] orientation4 Stacked along [110] orientation5 Averaged results for Fe with spin-up (or spin-down) states789 10 11 12 13 14050100150200250300350Energy (meV/atom)Volume (Å3/atom) bcc FM fcc AFMD[001] fcc AFMD[110] fcc AFMD[111] fcc SQS fcc NCFigure 6.4: Various magnetic configurations vs. lattice constant in fccFe. The bcc ground state is set to be the reference (zero) state.only a few meV/atom, and the lattice constants for these two structuresare found to be the same. Moreover, Ackland et al. found that mostof the non-collinear calculations for the γ-Fe with defects converged tocollinear magnetic states [84]. Only few configurations remain non-collinearstructures. However, the energy of the non-collinear configuration was onlymarginally lower than the collinear results. On the other hand, the non-collinear calculations are extremely time-consuming. For these reasons, wehave not used it for the further calculations. In the following calculations, weuse the AFMD[001] and SQS configurations to describe the magnetic statesof the fcc Fe.79Si Ti V Cr Mn Fe Co Ni Cu Nb Mo-2-10123Magnetic Moments (B) fcc AFMD fcc SQS bccFigure 6.5: Magnetic moments for solute atoms in bcc and fcc Fematrix. The x-axis is arranged in ascending order of atomicnumber.6.3 Single Solute in γ-FeIn this section, we examined single solute atom substituted in bulk γ-Fe. In the calculations, one Fe atom with spin-up state (i.e. positivemagnetic moment) is replaced by the solute atom. Note that the resultsare independent of the spin state of the replaced Fe atom. The magneticmoments for solute atoms in fcc Fe with AFMD and SQS arrangementsare shown in Figure 6.5. The values in the bcc phase are also presentedfor comparison. Except for Si and Cu, whose magnetic moments are foundto be close to zero in both phases, for all the other investigated cases, themagnetic moments in the fcc phase differ from the values in the bcc phase.In general, these elements can be categorized into three groups: the non-magnetic elements, i.e. Ti, V, Nb, and Mo; elements that have the oppositespin state with the Fe atoms in the bcc phase, i.e. Cr, and Mn; and elementsthat have the same spin state with the Fe atoms in the bcc phase, i.e. Co,and Ni.80It is noticeable that Ti, V, Nb, and Mo atoms have remarkable magneticmoments in the bcc phase, even though these alloying elements are non-magnetic elements. Previous studies have shown that these large magneticmoments are induced by the neighboring iron atoms. By contrast, themagnetic moments for Ti, V, Nb, and Mo in the fcc phase become negligible.For Cr and Mn, the magnetic moments show opposite behavior in bccand fcc phase. It is known that these two solutes prefer anti-ferromagneticcoupling. As a result, in ferromagnetic host material, i.e. bcc Fe phase, theyare antiferromagnetically coupled to the iron with large intrinsic magneticmoments. While in the fcc phase, since the nearest-neighbor Fe atomshave either spin-up or spin-down states, they maintain the same magneticarrangement with the replaced Fe atom.In the case of Co and Ni, the local magnetic moments are positive andare coupled ferromagnetically to the host Fe in the bcc phase. Whereas inthe fcc phase, the magnetic moments for Co and Ni are heavily suppressedrelative to their pure reference state value (µrefCo = 1.63 and µrefNi = 0.59 µB),which is consistent with previous DFT results, i.e. 0.08 µB for Ni [84].In Figure 6.6, the partial density of states (DOS) of 3d-orbital for solutesin the fcc phase with AFMD and SQS configuration are presented. Forcomparison, the partial DOS for solutes in the bcc phase are also shown. Weselect Ti, Cr, and Ni because these elements represent three different groupsas has been discussed above. For Fe in the bcc phase, Figure 6.6a showsthat the spin-up states (also called majority states) are mostly occupied.The spin-down states (also called minority states), on the other hand, areonly partially filled. Unequal occupancy of the spin states gives rise to largemagnetic moments, which are consistent with previous results [139, 140].Comparing the projected local DOS of Fe atoms in the fcc phase with theone in the bcc phase, one observes that the peaks of the spin-down states ataround +2 eV are shifted to -0.5∼1.0 eV, and more 3d electrons filled thespin-down states in the fcc phase.With the substitution of Ti, the valance 3d electrons are mainlyaccommodated in the spin-down states in bcc, which contribute to theinduced magnetic moments of Ti. By contrast, both spin-up and spin-81-1.50.01.53.0bcc phaseFe     FM-1.50.01.53.0 AFMD SQS    Fefcc phase-1.50.01.53.0 FMbcc phasebcc phasebcc phase   DOS (Arb. units)-1.50.01.53.0 AFMD SQSfcc phase    TiCrNi-1.50.01.53.0 FM   -1.50.01.53.0 AFMD SQSfcc phase   -8 -6 -4 -2 0 2 4 6 8-3.0-1.50.01.53.0 FMTiCr  Energy (eV)Ni-8 -6 -4 -2 0 2 4 6 8-3.0-1.50.01.53.0 AFMD SQSfcc phase  Figure 6.6: Calculated partial density of states (DOS) of 3d-orbitalfor solutes in the bcc (left) and fcc (right) phases. The Fermilevels (EF ) are set to zero82down states are almost equally occupied by the 3d electrons, which givesrise to zero magnetic moment in fcc (see Figure 6.5). Compared with Ti,the spin down states of the Cr 3d-orbital are shifted toward lower energyin the bcc phase. Thus, by adding electrons, the unoccupied states movecloser to EF . In addition, the 3d electrons mainly occupy the spin-downstates, and align their magnetic moments in an opposite sense to the hostmagnetic moments. This AFM coupling increases the energy of the systemwhen two Cr atoms occupy adjacent sites in the ferromagnetic host material,which leads to the magnetic frustration. In fcc, both spin-up and spin-downstates are occupied, and a positive magnetic moment for Cr is obtained (seeFigure 6.5).The results for Ni indicate that the spin-up states are almost completelyfilled and couple ferromagnetically to the host Fe atoms in bcc. By contrast,in fcc, the occupation of electrons in spin-up and spin-down states are almostequal, resulting in a small magnetic moment.In Figure 6.5 and Figure 6.6, it is evident that the results obtainedby AFMD and SQS magnetic configurations are remarkably close to eachother. The largest deviation of the magnetic moments obtained by twodifferent magnetic configurations is less than 16%, suggesting that theAFMD structure is a good representative and can be used to predict solutesegregation and diffusion properties in fcc Fe as an alternative to the SQSmagnetic structure.6.4 Solute-solute Interactions in γ-FeIn this section, we investigated the solute pair interactions in γ-Fe withthe AFMD configuration. The solute-solute interactions are calculatedaccording to Equation 4.12. Due to the asymmetric structure of fcc Fewith AFMD magnetic configuration, each site (o) has three types of nearestneighbors, namely p, q, and w, as shown in Figure 6.1. Therefore, threetypes of solute-solute interactions are calculated at the first nearest neighborseparation. The calculated effective interactions as a function of solute-solute distance are shown in Figure 6.7.832 3 4 5 6 7 8 90.00.10.20.30.40.5p Eeff in fcc Eeff in bccEffective Interaction Energy, Eeff (eV)Nb pair distance (Å)wq(a)2 3 4 5 6 7 8 90.000.050.100.150.20(b) Eeff in fcc Eeff in bccEffective Interaction Energy, Eeff (eV)Cr pair distance (Å)wqp2 3 4 5 6 7 8 90.000.070.140.21 Eeff in fcc Eeff in bccEffective Interaction Energy, Eeff (eV)Mn pair distance (Å)qpw(c)2 3 4 5 6 7 8 9-0.030.000.030.06 Eeff in fcc Eeff in bccEffective Interaction Energy, Eeff (eV)Ni pair distance (Å)wpq(d)2 3 4 5 6 7 8 9-0.10-0.050.000.05(e) Eeff in fcc Eeff in bccEffective Interaction Energy, Eeff (eV)Cu pair distance (Å)wpq2 3 4 5 6 7 8 9-0.050.000.050.100.150.200.25 Eeff in fcc Eeff in bccEffective Interaction Energy, Eeff (eV)Si pair distance (Å)(f)wpqFigure 6.7: Pair interactions for (a) Nb, (b) Cr, (c) Mn, (d) Ni, (e)Cu, and (f) Si in γ- and α-phase Fe matrix.84A strong repulsive interaction between Nb atoms is observed in γ-Fe.The repulsive energy is up to 0.4 eV at the first nearest-neighbor distance,which is comparable with the case in the bcc α-Fe lattice. Such strongrepulsive interaction can be ascribed to the elastic strain induced by thelarge size mismatch. Similar trends are also found for other large soluteatoms, i.e. Mo-Mo, and Ti-Ti.The solute-solute interactions between Cr (Mn) atoms are repulsive.However, compared with the case of Nb, the values are much smaller. InFigure 6.7b, the repulsive interaction between Cr atoms is less than 0.1 eVat first nearest neighbor separation. For Mn atoms, the repulsive energyfor the Mn-Mn pair is found to be negligible in fcc γ-Fe, in contrast tothe modest repulsive interaction in bcc α-Fe. This is because the strongrepulsive interaction is caused by the magnetic frustration in ferromagneticbcc Fe. When the magnetic arrangements have been changed, i.e. AFMDin fcc Fe, the influence of the magnetism on the solute pair interactionsbecomes smaller. Consequently, weak repulsive interactions are observedfor Cr (Mn). In addition, we also analyse the structure distortion inducedby the substitutional Cr (Mn) atom in fcc γ-Fe. The results indicate thatbond lengths from single Cr (Mn) solute to its nearest-neighbor shell, i.e.dCr−Fe (dMn−Fe) differs from pure Fe, i.e. dFe−Fe by at most 0.015 (0.01)A˚. This less than 1% effect exists only for the first nearest neighbor shelland decays rapidly with distance. So one would expect rather small effectiveinteractions from elastic aspects.Figure 6.7d shows that the Ni-Ni interaction depends on the occupationsites. Specifically, when two Fe atoms with different spin states are replacedby the Ni atoms, the Ni pair interaction behaves repulsive, otherwise, itshows attraction. This is because Ni prefers ferromagnetic coupling. Thisferromagnetic coupling increases the energy of the system when two Nisolutes occupy adjacent sites with opposite spin states. Recall that themagnetic moments for Ni are small (see Figure 6.5). As a result, the effectsof magnetism to the repulsive interactions are much smaller as comparedwith the effects for Cr (Mn, V) in the bcc lattice.Cu shows an attractive interaction in the fcc lattice (see Figure 6.7e),85indicating possible clustering similar to that predicted in the bcc lattice. Asdiscussed in Chapter 5, such attractive Cu-Cu interaction can be viewed asa magnetically driven clustering. The Si-Si interactions as a function of Sipair distance is shown in Figure 6.7f. It is clear that a short-range repulsiveinteraction occurs between Si atoms. The origin of this repulsion is basedon the fact that Si atoms prefer to bond with Fe atoms instead with otherSi atoms.6.5 Solute Segregation in Σ5 Grain Boundaries infcc γ-FeIn this section, we describe the simulation results for the solute segregationbehavior in the Σ5 tilt grain boundary in fcc γ-Fe. The binding energies ofsolutes with the grain boundaries are calculated according to Equation 4.10.In the calculations, one Fe atom with spin-up state is replaced by the solute.Note that the results are independent of the spin states of the replaced Featom. In Figure 6.8, the calculated binding energies for different solutes atvarious grain boundary positions are presented. The results indicate thatsimilar to the cases in the bcc phase, the influence of the grain boundary onthe binding energies of solutes are short ranged and limited to about 2∼3atomic layers in the fcc phase. In addition, different solutes prefer differentpositions at the grain boundary. For example, the binding energy for Nband Mo atoms have the largest segregation energies in position a, whereasother elements such as Si, Cr, and Mn prefer position b.Despite of these similarities, we found positive binding energies for somerelatively small solute atoms, such as Cr and Si in fcc phase grain boundaries.In addition, no segregation tendency is found for Ni. By contrast, for bigsolute atoms, such as Nb and Mo, large segregation energies are obtained.According to Langmuir-McLean equation, we calculate the effectivebinding energies, Eeffseg (see Chapter 5 for details). Note that in some cases,e.g. Cr and Si, only a few boundary sites may be available for grain boundary86-5 -4 -3 -2 -1 0 1 2 3 4 5-0.6-0.5-0.4-0.3-0.2-0.10.00.1b' Distance from Boundary Plane (Å)cbc'aBinding Energy (eV) Si Cr Mn Ni Mo NbFigure 6.8: Binding energies for substitutional solutes at the Σ5 fccgrain boundary as a function of distance from the boundaryplane; a, b, c, b′, and c′ refer to the boundary sites as labeled inFigure 6.3.segregation. As a results, Equation 5.1 should be modified as:cigbc0gb − cigb=cbulk(1− cbulk)exp (−EisegkBT) (6.1)where c0gb is the fraction of boundary sites with favourable binding. Forexample, c0gb=0.8 for Cr, whereas c0gb=0.4 for Si (see Figure 6.8). Theeffective segregation energies of selected solutes in Σ5 grain boundary inbcc and fcc Fe are listed in Table 6.2.It can be seen that for large solutes, i.e. Nb and Mo, the interactionsbetween these solute atoms and the grain boundary in fcc are even strongerthan that in the bcc phase, which demonstrated that segregation of thesesolute atoms to the grain boundary in fcc are more favorable. For example,the binding energies of Nb at boundary site a in fcc increase by 15% (see87Table 6.2: Effective binding energies (Eeffseg ) for selected solutes in Σ5bcc and fcc grain boundary. Unit is eV.Si Cr Mn Ni Mo Nbbcc -0.09 -0.12 -0.14 -0.19 -0.31 -0.40fcc -0.10 -0.10 -0.08 – -0.35 -0.471.00 1.05 1.10 1.150.000.050.100.150.200.250.300.350.400.45 Volume distributionAtomic volume (unit: bulk value)5 GBs fcc phase bcc phasebulkFigure 6.9: Volume distribution per periodic unit cell for Σ5 grainboundaries in bcc and fcc Fe.Figure 5.1 and Figure 6.8). This can be understood based on the fact thatfcc Fe is the closest-packed structure. Evidently, the elastic strain inducedby these big solutes in fcc bulk are larger than that in bcc bulk. Therefore,when the grain boundary is present where the packing density is significantlyreduced, substantial strains can be partially relieved, which give rise to largersegregation energies.We also plot the site volumes for bcc and fcc grain boundaries. Theresults are shown in Figure 6.9. The site volumes are given in units of theatomic volume in the bulk, which is an important parameter to predict thesegregation tendency of large solutes at grain boundaries as discussed in88Chapter 5. It is clear that the site volumes in the fcc grain boundary havelarger values when compared with the bcc grain boundary. Especially atsite c, it shows the biggest increase in size when comparing fcc and bccand the binding energies increase accordingly by 20% (see Figure 5.1 andFigure 6.8). As a result, remarkable attractive interactions between largesolutes and the fcc Σ5 grain boundary are achievedFor Ni, it prefers to stay at the bulk position rather than at the boundarysites. A possible explanation is that Ni prefers to bind with the host atoms,and consequently, it is unfavorable for Ni to be placed at the boundary site,since this will result in longer distance between nearest neighbors due to theexcess volume. This finding is consistent with Enomoto et al.’s report thatno appreciable segregation of Ni to fcc boundaries is observed [141]. Theexperimental binding energy of Mn to austenite grain boundaries is -9±3kJ/mol [141], which is also comparable with our predictions of -7.7 kJ/molfor the fcc Σ5 grain boundary.89Chapter 7Interaction of Solutes withα-γ Iron Interface7.1 Structure of Ferrite-Austenite (bcc-fcc)InterfaceThe structure and orientation of austenite-ferrite interfaces has been studiedextensively by experiments because of its significance in controlling themicrostructure of steels [142]. The most frequently detected orientationrelationships for fcc and bcc lattices are those derived by Kurdjumov andSachs (K-S) in 1930 and by Nishiyama and Wassermann (N-W) in 1935[143, 144]. The orientations can be described by a set of planes that areparallel in the lattices, i.e.:N-W orientations: (111)γ‖(110)α and [11¯0]γ‖[001]αK-S orientations: (111)γ‖(101)α and [101¯]γ‖[111¯]αBoth models assume unanimously that a close packed {110} plane of thebcc lattice is arranged parallel to a closed packed {111} of the fcc lattice.However, the alignment of the planes to each other is different. It seems thatthese two relationships are quite different from each other, however, in fact,they are only 5.26° apart. In this work, we describe the bcc-fcc interfaceusing a selected K-S orientation relationship.90Note that the domain of DFT calculations is relatively small. Tominimize the lattice mismatch and construct a coherent interface, wecarefully studied the interface structure and select the (12¯1) plane as thehabit plane. This habit plane is also observed in a number of experimentalstudies [145, 146]. Accordingly, the three orthogonal directions for thefcc grain, x, y, and z, are chosen along the [101¯], [111], and [12¯1] crystaldirections, respectively. The z-direction is perpendicular to the interphaseboundary, i.e. (12¯1) is the habit plane. At the atomic level, the (12¯1) habitplane is not flat, but consists of structural ledges composed of (11¯1) and(02¯0) facets [147].The periodicity in the x- and y-directions for the bcc crystal is chosenclose to those of the fcc crystals to ensure coherency. Thereby, x and y arechosen along [111¯] and [15¯4] crystal directions, respectively. The orientationof the bcc grain perpendicular to the interface, i.e. along z-direction, is[31¯2]. In order to maintain periodic boundary conditions, two interphaseboundaries were included in the simulation block. The structure of the fcc-bcc Fe interface with AFMD and SQS magnetic configurations are shown inFigure 7.1.91Figure 7.1: Structure of bcc-fcc interface with K-S orientationrelationship. (a) AFMD and (b) SQS configuration for fcc. Thewhite circles represent atoms in the bcc Fe grain, and the red(blue) circles indicate atoms in the fcc Fe grain with spin up(down) states.927.2 Interface Energy6 8 10 12 14 16 18 20 220.340.360.380.400.42 bcc grain(a)Interface Energy (J/m2)(b)   8 10 12 14 16 18 20 22 24 260.400.420.440.460.480.50 fcc grainInterface distance (Å)   Figure 7.2: bcc-fcc interface energy as a function of (a) the bcc grainsize while the length of the fcc grain is fixed to 25.4 A˚; and (b)the fcc grain size while the length of the bcc grain is fixed to21.3 A˚.The supercell size was determined from calculation of the interfaceenergy as a function of interface distance (see Chapter 4). The results areshown in Figure 7.2. Note the fcc grain is described by the AFMD magneticconfiguration.Since the interface structure is asymmetric, we check the fcc and bccgrain size separately. Figure 7.2a shows the interface energy as a functionof the thickness of the bcc grain. In order to minimize the influence of the93fcc grain, we fix the length of the fcc grain to 25.4 A˚. The results show thatthe interface energy is converged when the bcc grain thickness is 10.6 A˚ andlarger. We then fix the length of the bcc grain to 21.3 A˚ and systematicallyincrease the thickness of the fcc grain. Figure 7.2b shows that the interfaceenergy is converged when the fcc grain is 17.4 A˚ and larger. The α − γinterface energy (σα−γ) is determined to be 0.41 J/m2 (see Table 7.1).In this work, we also calculated the surface energies for differentequilibrium bcc and fcc surfaces. This is because the surface energies canbe more accurately measured than interface energies, which provide a goodbenchmark for our simulations. In Table 7.1, the surface energies for bcc(31¯2) and fcc (12¯1) are only slightly (around 5%) larger when compared totheir closest-packed surface, i.e. bcc (110) and fcc (111) surface. In addition,our calculations agree well with the experimental measurements [148, 148].These results suggest that our calculations are reasonable and reliable.Table 7.1: Surface and interface energies for the equilibrium bcc andfcc surfaces, and bcc-fcc interface, in units of J/m2.*PresentPreviousExp.theoretical workFe(110)bcc 2.44 2.43 [149] 2.41 [148]Fe(31¯2)bcc 2.59Fe(111)fcc 2.21 2.18 [150] 2.11 [151]Fe(12¯1)fcc 2.30Interface 0.41 ∼0.5 [152]* The magnetic structure of the fcc Fe is described using AFMDconfiguration.It is known that fcc Fe exists in paramagnetic state. In order to check theinfluence of magnetic structure on the interface energy, the SQS magneticconfigurations are employed to describe the fcc phase. The calculated α− γinterface energy is 0.45 J/m2. The results indicate that the interface energies94obtained by these two different magnetic structures (i.e. AFMD and SQS)are close to each other, and they are comparable with the experimentalestimates [152].7.3 Magnetic and Electronic Properties forbcc-fcc Interface-10 -8 -6 -4 -2 0 2 4 6-2.3-2.2-2.1-2.0-1.91.92.02.12.22.32.4fcc grainMagnetic Moments (B)Distance from Interface (Å)bcc grainfcc bulk Fe with spin-down statebcc bulk Fefcc bulk Fe with spin-up stateFigure 7.3: Magnetic moments of Fe atoms across the bcc-fccinterface.We now turn to the analysis of the magnetic profile around the fcc-bccinterface. Figure 7.3 shows the local magnetic moments of the Fe atoms asa function of their positions in the bcc-fcc interface. The results indicatethat the oscillations of magnetic moments occur in the vicinity of the fcc-bccinterface. In addition, the magnetic moments of the Fe atoms in both phasesare enhanced at the interface, and the highest magnetic moment is 2.35 µBin the bcc phase. At a distance of 3.0 A˚ away from the interface, the localmagnetic moment converges to the bulk value in both phases, suggesting95the influence of the interface on the magnetic moments is short-ranged.Similar fluctuations in the local magnetic moments are observed notonly in the bcc-fcc Fe interface, but also in Fe grain boundaries [28, 153],Fe surfaces [154], and Fe nanowires [155]. The origin of the high magneticmoments of the Fe interface atoms has been discussed earlier [25, 156, 157].Cˇa´k et al. attributed such high magnetic moment at the boundary or surfaceto the magnetovolume effect, i.e. magnetic transition leads to volume change[153].-8 -6 -4 -2 0 2 4 60.960.981.001.021.041.06   Distance from Interface (Å)Site Volume/ bcc Bulk Atomic VolumeFigure 7.4: Averaged site volume distribution across the bcc-fccinterface. The bars indicate the range of values of site volumesfor Fe atoms with spin-up or spin-down states.In Figure 7.4, the local atomic volume for each Fe atom is calculated byconstructing a Voronoi polyhedron around each site. The calculated atomicvolume is 11.4 A˚3 and 11.1 A˚3 for bcc and fcc phases, respectively. While atthe interface, the atomic volume is increased up to 12.0 A˚3. In general, thehigh local magnetic moments are associated with increased Voronoi volumes,which is similar to what has been reported for Fe bcc tilt grain boundaries[28, 153]. In addition, from Figure 7.4, it is evident that the thickness of thepure bcc-fcc interface is about 7.0 A˚.967.4 Binding Energies of Solutes with bcc-fccInterfaceIn this section, the segregation behavior of different solutes at variousinterface positions is investigated. The relative energy of solutes with thebcc-fcc interfaces is calculated as:Eirel = (EiI+X + EI)− (Eα+X + Eα) (7.1)where EiI+X and EI are the total energies of bcc-fcc interface with andwithout solute. Eα+X and Eα are the total energies of bcc bulk withand without solute. The superscript i denotes the substitutional sites inthe interface. A negative value for Eirel indicates an attractive interactionbetween solute and the considered interface site.Since Nb has the largest segregation energy among our investigatedcases, while Mn is one of the most common alloying elements in steels, wefirst select these two elements for detailed analysis. The calculated relativeenergies for Nb and Mn across the bcc-fcc interface are shown in Figure 7.5.Both AFMD and SQS magnetic configurations are employed to describedthe magnetic states of the fcc grain.Despite some deviations, the segregation behavior obtained by the SQSmagnetic configuration is comparable with the AFMD results. For Nb,strong interactions between Nb and the bcc-fcc interface are obtained. Basedon the SQS magnetic configuration, the energy difference between austeniteand ferrite for Nb is 0.18 eV, which is close to the AFMD results, i.e. 0.21 eV.In addition, the energy difference between Nb in the interface and in ferriteis 0.18 eV, which differs from the AFMD result by 0.01 eV. For Mn, bothSQS and AFMD results indicate that there exist large chemical potentialdifference between ferrite and austenite. The energy difference between Mnin the interface and in austenite is 0.10 eV, which is the same with the valueobtained by using the AFMD arrangement. Therefore, one can concludethat AFMD and SQS magnetic configurations for fcc-Fe lead to very similarcharacteristics of solute segregation in the bcc-fcc interface. Thus, in thefollowing calculations, we focus on the computationally less costly AFMD97-6 -4 -2 0 2 4 6-0.3-0.2-0.10.00.10.20.3AFMD fcc bulk  Nb SQS Nb AFMD Distance from Interface (Å) Relative Energy (eV)fcc grainSQS fcc bulkbcc grainbcc bulk(a)-6 -4 -2 0 2 4 6-0.6-0.5-0.4-0.3-0.2-0.10.00.1AFMD fcc bulk Mn SQS Mn AFMD(b)Relative Energy (eV)Distance from Interface (Å)SQS fcc bulkfcc grainbcc grainbcc bulkFigure 7.5: Relative energies for (a) Nb, and (b) Mn across the α-γinterface as a function of distance from the habit plane. Thebars indicate the range of energies for the SQS simulations,where the replaced Fe atom has spin-up or spin-down states.98-6 -4 -2 0 2 4 6-0.2-0.10.00.10.20.3 Nb Distance from Interface (Å) Relative Energy (eV)fcc grainfcc bulkbcc grain(a)bcc bulk-6 -4 -2 0 2 4 6-0.2-0.10.00.1 Mo Distance from Interface (Å) Relative Energy (eV)fcc grainfcc bulkbcc grain(b)bcc bulk-6 -4 -2 0 2 4 6-0.35-0.30-0.25-0.20-0.15-0.10-0.050.000.05 Ni Distance from Interface (Å) Relative Energy (eV)fcc grainfcc bulkbcc grainbcc bulk(c)-6 -4 -2 0 2 4 6-0.5-0.4-0.3-0.2-0.10.00.1 Mn Distance from Interface (Å) Relative Energy (eV)fcc grainfcc bulkbcc bulkbcc grain(d)-6 -4 -2 0 2 4 6-0.060.000.060.120.18 Cr Distance from Interface (Å) Relative Energy (eV)fcc grainfcc bulkbcc grainbcc bulk(e)-6 -4 -2 0 2 4 6-0.12-0.060.000.060.12 Si Distance from Interface (Å) Relative Energy (eV)fcc grainfcc bulkbcc grainbcc bulk(f)Figure 7.6: Relative energies for substitutional solutes across the bcc-fcc interface as a function of distance from the habit plane. Thebars indicate the range of energy values, where the replaced Featom has spin-up or spin-down states. The fcc grain is describedby the AFMD configuration.magnetic configurations.In Figure 7.6, the relative energies for Nb, Mo, Ni, Mn, Cr, and Si acrossthe bcc-fcc interface are shown. These alloying elements are crucial forsteels by controlling the austenite to ferrite transformation. We find that99different solutes prefer different phases. Mn and Ni are austenite stabilizerand prefer the γ-phase, whereas Nb, Mo, Cr, and Si are ferrite stabilizer andprefer the α-phase. As a result, a chemical potential difference exists for thesubstitutional elements between ferrite and austenite. Figure 7.6 shows thatthe energy difference between ferrite and ausenite for Mo and Si are small,i.e. less than 0.06 eV. This difference becomes much larger for Mn and Ni,i.e. up to 0.35 eV. Consequently there is a considerable driving force for Mnand Ni transfer from ferrite to austenite, which is expected to result in a‘spike’ in austenite [158]. These predictions are consistent with the role ofthese alloying elements in stabilizing austenite and ferrite, respectively. Forexample, Ni is a critical alloying element for austenitic stainless steels andCr for ferritic stainless steels.As shown in Figure 7.6, a deep valley is found for Nb, Mo, and Si,indicating strong binding of these elements with the bcc-fcc interface. Onthe other hand, there is a gradual change between ferrite and austenite forNi, Mn, and Cr, and no obvious binding is observed. To give a more realisticrepresentation of the binding energy, we use the average values between thebulk values as the reference point [158, 159].-6 -4 -2 0 2 4 6-0.2-0.10.00.10.20.3 Nb Distance from Interface (Å) Relative Energy (eV)fcc grainbcc grainBinding EnergyFigure 7.7: Definition of the binding energy using linear interpolationfrom the bulk energies.100-6 -4 -2 0 2 4 6-0.4-0.3-0.2-0.10.00.1 Nb Distance from Interface (Å)Binding Energy (eV)fcc grain bcc grain(a)-6 -4 -2 0 2 4 6-0.3-0.2-0.10.00.1 Mo Distance from Interface (Å)Binding Energy (eV)fcc grainbcc grain(b)-6 -4 -2 0 2 4 6-0.3-0.2-0.10.00.1 Ni Distance from Interface (Å)Binding Energy (eV)fcc grain bcc grain(c)-6 -4 -2 0 2 4 6-0.3-0.2-0.10.00.1 Mn Distance from Interface (Å)Binding Energy (eV)fcc grain bcc grain(d)-6 -4 -2 0 2 4 6-0.2-0.10.00.1 Cr Distance from Interface (Å)Binding Energy (eV)fcc grainbcc grain(e)-6 -4 -2 0 2 4 6-0.2-0.10.00.1 Si Distance from Interface (Å)Binding Energy (eV)fcc grainbcc grain(f)Figure 7.8: Binding energies for substitutional solutes at the bcc-fccinterface as a function of distance from the habit plane. Thebars indicate the actual values of binding energies. The fcc grainis described by the AFMD configuration. The interface widthis indicated by two dashed lines.101For each interface position, we calculate the average energy using linearinterpolation between the two bulk values. Then the binding energy for eachposition is defined as the difference between the DFT results (as shown inFigure 7.6) and the interpolation from the bulk energies. Figure 7.7 shows anexample of this analysis for Nb. Note that the widths of the bcc-fcc interfacedepend on the substituted solute atoms, which vary from 7.0 A˚ to about 9.0A˚, and show certain asymmetric behaviour. In this work, the boundaries ofthe interface are defined as the positions where the variation of the relativeenergies due to magnetism (i.e. spin-up and spin-down) is less than 0.01eV (see the bars in Figure 7.6). Using this procedure, we calculate thebinding energies of selected solutes across the bcc-fcc interface. The resultsare shown in Figure 7.8.0.96 0.98 1.00 1.02 1.04 1.06-0.3-0.2-0.10.00.10.20.3 Nb Relative Energy (eV)Site VolumeFigure 7.9: Relative binding energies of Nb at the bcc-fcc interface asa function of the site volume.Significant binding energies are found for large, elastically dominatedsolutes (Nb, Mo). The maximum energy difference between the interfaceand the reference point for Nb (Mo) is 0.39 eV (0.24 eV). Such strong102binding energies indicate that bcc-fcc interface positions are favorable forlarge solute elements segregation. In particular, we find the larger bindingenergy of Nb (Mo) corresponds to the larger excess volume of the interfacesites. Figure 7.9 plots the relative energies of Nb as a function of the sitevolume. It is evident that there is a distinct correlation between Eirel andthe site volume, i.e. Eirel increasing with the site volume. This trend can beunderstood in terms of a strain-relief argument, whereby the elastic straininduced by the large substitutional solutes (e.g. Nb, Mo, etc.) in the bulklattice can be partially relieved at the interface due to the excess volume.In the case of Ni, however, the interaction between Ni and bcc-fccinterface is found to be rather weak, which is consistent with previousexperimental observations [141, 158]. Recall that there is no segregationtrend for Ni in the fcc Σ5 grain boundary (see Chapter 6). A possibleexplanation is that Ni prefers to bind with Fe atoms. From ferrite toaustenite, the coordination number is increased from 8 to 12, resulting inlower values for Ni in austenite than in ferrite and at the interface positions.For Mn, the energy difference between the interface and the reference pointis -0.20 eV, which is comparable with the values at the Σ5 bcc and fcc grainboundaries. For Cr, relatively small binding energies are observed, whereasmoderate binding energies are found for Si.In Figure 7.8, the bars indicate the actual values of binding energies,where the replaced Fe atom has spin-up or spin-down states. It is evidentthat the spin states play an important role in binding energies. This isbecause the distance between two Fe atoms with the same spin states islarger than those with opposite spin states (see Chapter 6, Figure 6.1),which give rise to larger excess volume. For large solutes, e.g. Nb and Mo,the larger excess volume favors the solute-interface binding. For magneticelements, i.e. Ni, Mn, and Cr, the binding energies are influenced by themagnetic interactions between solute and the host atoms. As discussedin Chapters 5 and 6, the Cr-Fe (Mn-Fe) and Ni-Fe interactions are anti-ferromagnetic and ferromagnetic, respectively. At the interface, there existdifferent magnetic arrangements, which contribute to the variation of thebinding energies. For Si, however, the influence of the spin states on the103binding energies is small.In a similar manner with the grain boundary, we calculate the effectivebinding energies of solutes in the bcc-fcc interface. Based on the Coghlan-White approach, the segregation to different sites i in the bcc-fcc interface,each with the binding energy Eiseg, is given by [117, 118]:ciint1− ciint=cbulk(1− cbulk)exp (−EisegkBT) (7.2)where cbulk is the solute concentration in the bulk, and ciint is the soluteconcentration at the interface site i. Equation 7.2 can be rewritten as:ciint =cbulk exp (−EisegkBT)1− cbulk + cbulk exp (−EisegkBT)(7.3)ciint is calculated for each interface site. Then the total segregation cintis a weighted average summed over all interface sites, which is given by:cint =∑i(Ficiint) (7.4)where Fi is the fraction of the interface sites that have binding Eiseg. InFigure 7.10, the calculated total segregation for Nb in Fe bcc-fcc interfaceare plotted. We fit these data according to the Langmuir-McLean equationand obtain the effective binding energies (Eeffseg ). Here three fitting equationsare employed in this work, i.e.:Fitting 1:cintc0int − cint=cbulk(1− cbulk)exp (−EeffsegkBT) (7.5a)Fitting 2:cint1− cint=cbulk(1− cbulk)exp (−EeffsegkBT) (7.5b)Fitting 3: cint = cbulk exp (−EeffsegkBT) (7.5c)In Equation 7.5a, c0int is the fraction of interface sites with favourable1040 200 400 600 800 1000 12000.00.20.40.60.81.0    Nb Fitting 1 (E=-0.26 eV) Fitting 2 (E=-0.24 eV) Fitting 3 (E=-0.23 eV)Solute Segregation, cintTemperature (K)Figure 7.10: Calculated effective binding energies for selected alloyingelements as a function of solute volume in bcc-fcc interface.The dotted line indicates the volume of Fe calculated from thefirst nearest neighbor distance in the bcc bulk Fe.binding. For example, 20 of 24 interface sites have the favourite binding forNb, i.e. c0int=0.83. Though Equation 7.5a is the most rigid expression, itcan not be used in the experiments. This is because it is extremely difficultto obtain c0int based on the experimental measurements. Note that the datacannot be fitted perfectly across the whole temperature region using onebinding energy since there are unlike for the Σ5 boundaries rather largefluctuations in the individual binding energies for different interface sites.Since we are interested in α−γ transformation temperature, i.e. 773∼1184K,we focus on this high temperature range. The results based on the differentfitting equations are essential the same at high temperature. To make ourresults, i.e. Eeffseg , more comparable with those measurements, Equation 7.5bis employed in the following study.The effective binding energies by fitting Equation 7.5b of selected solutesare summarized in Figure 7.11. The results are plotted as a function of1057.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0-0.30-0.25-0.20-0.15-0.10-0.050.00 Binding Energy (eV)Solute Volume (Å3)SiCrMnNiMoNbFigure 7.11: Calculated effective binding energies for selected alloyingelements as a function of solute volume in bcc-fcc interface.The dotted line indicates the volume of Fe calculated from thefirst nearest neighbor distance in the bcc bulk Fe.solute volume (Vs), which was computed from relaxed atomic positions inthe bcc bulk according to the hard sphere model. Similar with the case ofbcc Σ5 grain boundary, there is a distinct correlation between the effectivebinding energy and solute volume, i.e. the magnitude of Eeffseg and thusthe tendency for segregation increase with solute size. Compared with thevalues in bcc and fcc grain boundaries, the binding energies of large solute,i.e. Nb and Mo, are smaller in the bcc-fcc interface. This can be ascribedto the smaller excess volume in the bcc-fcc interface than in the Σ5 bcc orfcc grain boundaries (see Figure 6.9 and Figure 7.4). On the other hand,the binding energies of relatively small alloying elements, e.g. Si and Mn,are comparable with the values in bcc and fcc grain boundaries.The experimental results for interaction parameters between Fe andsolutes in the interface are limited. Nevertheless, Zurob et al. estimated thebinding energies for several alloying elements with the phase boundary using106controlled decarburization experiments. The experimental binding energiesfor Cr and Mo are -3∼-6 kJ/mol (-0.03∼-0.06 eV) and -15∼-24 kJ/mol (-0.16∼-0.25 eV) [152, 158, 160], which are comparable with our predictions,i.e. -5.8 kJ/mol (-0.06 eV) and -15.4 kJ/mol (-0.16 eV), respectively. Inaddition, our calculations indicate that the binding energies of Nb and Mowith the interface are relatively large. Overall, such large binding energiesindicate strong interactions with the interface, and are expected to delayausenite-to-ferrite transformation, which is consistent with experimentalobservations [158, 161, 162].In general, the solute segregation behavior is illustrated in Figure 7.12.For large solute elements, e.g. Fe-Nb and Fe-Mo systems, the interactions arerather strong, which can be rationalized with the elastic strains, where theselarge solute elements bind to the interface as a means of relieving strain onthe Fe matrix. For Fe-Mn and Fe-Ni systems, the energy difference betweenferrite and austenite is comparatively large, which provides considerabledriving force for solute transfer from ferrite to austenite. The interactionbetween Ni and the interface is weak, while it becomes much larger for Mn.For ferrite stabilizers with similar atom size than Fe, relative small (Cr) tomoderate (Si) binding energies are predicted.Figure 7.12: Illustrations of interaction profiles between solute atomsand bcc-fcc interface.7.5 Solute-solute Interactions in bcc-fcc InterfaceIn order to analyze the segregation characteristics of solutes in moredetail, we calculate the effective interaction (Eeff ) as a function of solute-107solute distance according to Equation 4.12. To minimize the number ofcalculations, we fix one solute atom in its favorite position based on thesingle solute segregation profile while the other solute atom occupies otherinterface positions.The results for Nb-Nb, Cr-Cr, and Mn-Mn are shown in Figure 7.13.We select these solutes because they are some of the most common alloyingelements in steels. In addition, these examples represent two majorsolute-specific contributions to the effective interactions: contributions fromthe elastic energy induced by atomic size misfit, and contributions fromcompeting magnetic interactions.The elastic contribution to the solute-solute interaction is determined bythe relative size of solutes with respect to Fe. As the size of solutes increases,the elastic contribution to solute-solute interactions becomes increasinglymore important. In Figure 7.13a, the repulsive energy for Nb-Nb pair is upto 0.4 eV within the first nearest neighbor distance, which is comparablewith the values of bcc and fcc Fe bulk. In addition, we find apparentnegative binding energies (about -0.1 eV) at intermediate pair distance. Suchattractive interactions occur when two Nb atoms belong to different grains.Similar with the case of Σ5 tilt grain boundaries, the attractive interactioncan be ascribed to the excess volume at the interface (see Figure 7.4),whereby the elastic strain at these positions can be partially relieved.In chapter 5, we have discussed that strong repulsive interactionsbetween Cr atoms can be attributed to magnetic frustration. Accordingto the model of Moriya [129], the Cr-Fe and Cr-Cr interactions areantiferromagnetic (AFM). In the dilute limit, this AFM coupling increasesthe energy of the system when two Cr atoms occupy adjacent sites in anotherwise ferromagnetic (FM) host material. Figure 7.13b shows that in thebcc-fcc interface, when two Cr atoms occupy adjacent sites in the bcc phase,strong repulsive energy is obtained. By contrast, when the second Cr atomoccupies the fcc grain position and has the opposite spin states than thefirst Cr atom, the repulsive interactions for Cr solute pair becomes weaker,i.e. less than 0.1 eV within the first nearest neighbor distance.Similar results are also found for Mn-Mn interactions. In Figure 7.13c,1082 3 4 5 6-0.10.00.10.20.30.40.5 1stNb2ndNb 1stNb2ndNbInteraction Energy, Eeff (eV)Nb pair distance (Å)(a)2 3 4 5 6-0.050.000.050.100.150.200.25 1stCr2ndCr 1stCr2ndCr 1stCr2ndCrInteraction Energy, Eeff (eV)Cr pair distance (Å)(b)2 3 4 5 6-0.050.000.050.100.150.200.25 1stMn2ndMn 1stMn2ndMn 1stMn2ndMn(c)Interaction Energy, Eeff (eV)Mn pair distance (Å)Figure 7.13: Pair interactions for (a) Nb, (b) Cr, and (c) Mn in bcc-fccFe interface. The first solute occupies its favorite position. Thesubscripts α and γ indicate solute atom occupies bcc α- or fccγ-phase grain, respectively. The superscripts ‘1st’ and ‘2nd’denote the first and the second substitutional solute atom,respectively. The arrow ↑ (↓) represents solute atom has spin-up (spin-down) state.109remarkable repulsive interactions are obtained when two Mn atoms occupyadjacent sites with the same spin states. On the other hand, when twoMn atoms occupy positions in the fcc phase with opposite spin states, theinteraction between the Mn pair becomes much smaller, i.e. less than 0.06eV. This can be ascribed to the AFMD magnetic arrangements that reducesthe influence of the magnetic frustration.110Chapter 8Simulation of Self- andSolute-diffusion in Fe8.1 Diffusion in Fe bcc LatticeIn this work, we consider diffusion occurring through vacancy mediated,thermally activated lattice jumps. The mechanism of vacancy diffusion inbcc Fe is illustrated in Figure 8.1. For pure Fe, the calculated vacancyformation energy (Ev) and migration energy (Em) are 2.21 eV and 0.63 eV,respectively, which are consistent with the range of values of Ev=2.18∼2.23eV and Em=0.63∼0.68 eV from previous DFT calculations [11, 163, 164].The calculated vacancy formation, migration and binding energies, as wellas the associated activation energies for self- and solute-diffusion in theferromagnetic and paramagnetic state are summarized in Table 8.1.Table 8.1 shows that for all investigated cases, the binding energies ofsolute-vacancy pairs are negative (attractive). Compared with other soluteelements, Nb is found to display the largest binding energy to a neighboringvacancy, which can be understood in terms of a strain-relief argument. Thedistortion of the Fe lattice due to the oversized atomic radius of Nb resultsin large attractive interactions between Nb atoms and vacancies.The migration energies for the solutes in Table 8.1 correspond to anearest-neighbor exchange with a vacancy. All of the solutes considered here1110.0 0.5 1.0 1.5 2.0 2.5 3.0-0.10.00.10.20.30.40.50.60.7 Migration Energy (eV)Diffusion Coordinate (Å)Figure 8.1: First-neighbor diffusion of the vacancy in bcc Fe. Onevacancy locates in the center, and diffuses along the red line tothe corner of the cubic cell (the corner atom diffuses into thecentered vacancy). The calculated migration barrier for vacancyis also shown.show smaller values of Em than that for Fe self-diffusion, and Mn exhibits thelowest value. When the vacancy formation, binding and migration energiesare combined to compute the ferromagnetic activation energies (Q0) fromEquation 4.15, it is seen that all solutes have a smaller calculated value thanthat for Fe self-diffusion in the ordered ferromagnetic state.We focus for the remainder of this section on Mn, Mo, and Nb solute-diffusion. The diffusion coefficient D is expressed in the form of theArrhenius expression:D = D0 exp(−Qa/kBT ) (8.1)where D0 is the pre-exponential factor and Qa is the activation energyabove 0K. The influence of the bulk magnetization on the diffusion activationenergy is well known [165]. In the whole temperature range across the Curietemperature Tc, Qa is expressed in the following form:Qa = Q0(1 + α¯s(T )21 + α¯) (8.2)where s(T ) = M(T )/M(T = 0K) is the reduced magnetization at112Table 8.1: Calculated vacancy formation energy (Ev), migrationenergy (Em), vacancy-solute binding energy (Eb), and activationenergy (Q0) for self- and solute-diffusion in bcc Fe.Ev (eV) Em (eV) Eb (eV) Q0 (eV)Fe 2.21 0.63 2.84Si 0.43 -0.33 2.31Ti 0.42 -0.30 2.33V 0.47 -0.08 2.60Mn 0.39 -0.23 2.37Nb 0.40 -0.40 2.21Mo 0.51 -0.19 2.53the temperature T , which varies continuously in the ferromagnetic phasebetween one at 0 K, and zero at the Curie temperature, i.e. Tc=1043 K inFe (s(T) = 0 for all T in the paramagnetic state). s(T ) can be calculatedbased on the Onsager and Yang model, which is expressed as [166, 167]:s(T ) = (1− T/Tc)1/8 (8.3)In Equation 8.2, the parameter α¯ is a species-dependent parameter thatquantifies the dependence of Qa on magnetization. Nitta et al. indicatedthat for solute species where the parameter α¯ has not been measured, itsvalue can be estimated from a linear relationship between the magnitudesof α¯ and ∆M12 [168], which is given by:α¯ = 0.053∆M12 + 0.163 (8.4)where ∆M12 is the sum of the change in the local magnetic momentsinduced on the Fe atoms in the first and second nearest neighbors of an113-2.8 -2.1 -1.4 -0.7 0.0 0.7 1.40.000.040.080.120.160.200.24   MnMoTiSiVNbFeCoValue of M12( B) This work Exp.Figure 8.2: Relationship between α¯ and ∆M12. The triangles denotethe parameter α¯ is derived from experiments [169–172] and theline gives the model according to Equation 8.4.solute atom, i.e.:∆M12 =8∑i=1∆M1sti +6∑i=1∆M2ndi (8.5)∆M12 can be obtained based on our DFT calculations. In Figure 8.2,the results for measured values of α¯ are plotted with triangle symbols. Thesolid line represents the linear relationship as indicated by Equation 8.5.In order to compute solute diffusion coefficients, we employ the approachproposed by Asta et al., which is based on the framework of Le Claire’s nine-frequency model [11, 43]. The pre-exponential factor (D0) can be expressedin terms of the lattice constant (al), the correlation factor (f2), the attemptfrequency for the hop of a solute atom to a nearest-neighbor vacancy (ν0),and the entropy of vacancy formation in Fe (Sv), i.e.:D0 = a2l f2ν0 exp(Sv/kB) (8.6)where kB is the Boltzmann constant. In Equation 8.6, the correlationfactor (f2) can be quantitatively evaluated using the nine-frequency model114Figure 8.3: An illustration of the different vacancy hops involved inthe Le Claire nine-frequency model for the correlation factor forsolute diffusion in Fe. X and V denote the solute atom and thevacancy, respectively.developed by Le Claire [173]. This approach is depended on the jumpfrequencies as illustrated in Figure 8.3 and are given by:f2 =1 + t11− t1(8.7)where t1 can be expressed as:t1 =ΓXΓX + 3Γ12 + 3Γ13 + Γ15 −Γ12Γ21Γ21+0.512Γ24− 2Γ13Γ31Γ31+1.536Γ0 −Γ15Γ51Γ51+3.584Γ0(8.8)Here ΓX , Γij , and Γ0 are the solute-vacancy exchange jump frequency,Fe-vacancy jump frequency from site i to j as illustrated in Figure 8.3,and Fe-vacancy jump frequency in pure Fe. These jump frequencies arecalculated according to Equation 4.16It has been demonstrated that the correlation factors (f2) display goodArrhenius behavior [119], which can be modeled as:f2 = f0 exp(−Ef2/kBT ) (8.9)where Ef2 is defined as the “correlation energy”. For Mn, we findEf2 is less than 0.01 eV. While for large solutes, i.e. Mo and Nb, the115calculated correlation energies have small contributions (3% ∼5%) to theoverall activation energies, which are in agreement with the previous resultsobtained by Asta et al. [11].In Equation 8.6, the vacancy-formation entropy Sv can be calculated asthe difference of the vibrational entropy for a supercell with and withoutone vacancy, i.e [11]:Sv = kB[3(N−1)∑i=1ln(kBThνeqi)−N − 1N3N∑i=1ln(kBThνi)] (8.10)where νeqi denotes the vibrational frequencies in a supercell with onevacancy at equilibrium configuration, while νi corresponds to the frequenciesin a pure Fe system, which are computed from the hessian matrix (seeChapter 4 for detail).Figure 8.4 provides a direct comparison between calculated and measuredtemperature-dependent diffusion coefficients for Fe self-diffusion, Mn, Moand Nb solute-diffusion. For Fe self-diffusion the agreement betweencalculations and measurements in the ferromagnetic phase is excellent,while the calculated values underestimate some of the measured values inthe paramagnetic phase at the highest temperatures [174–176]. For Mndiffusion, our results are slightly (by a factor of 2) higher than the valuesreported by Lu¨bbehusen et al. in paramagnetic phase [170], while agreewell with Kirkaldy et al.’s work [177]. For Mo diffusion, the calculateddiffusion coefficients show good overall agreement with the experiments,and are roughly in the middle range of the measured values obtained byNitta et al. [172] and Alberry et al. [48]. In addition, Figure 8.4c showsprevious DFT results for the Mo diffusion coefficient obtained by Asta etal. [11], which indicate that our results are consistent with previous DFTcalculations. For Nb diffusion, the calculated diffusion coefficients showgood overall agreement with the measured values in the ferromagnetic phaseobtained by Herzig et al. [171]. The calculations are in good agreement withthe measurements of Geise et al. [178] in the paramagnetic phase as well.Overall, our calculations predict reasonable results for self- and solute-1168 9 10 11 12 131E-231E-211E-191E-171E-15Fe self-diffusion   T-1/10-4 (K-1) D (m2s-1) Present work Iijima et al. Lübbehusen et al. Hettich et al. Buffington et al. Walter et al.T=1184KTC=1043K8 9 10 11 12 131E-201E-181E-161E-14   T-1/10-4 (K-1) D (m2s-1) Present work Lübbehusen et al. Kirkaldy et al.T=1184KTC=1043KMn diffusion in -Fe8 9 10 11 12 131E-211E-191E-171E-15   T-1/10-4 (K-1) D (m2s-1) Present work Asta et al. (DFT) Nitta et al. Alberry et al.T=1184KTC=1043KMo diffusion in -Fe8 9 10 11 12 131E-201E-181E-161E-14   T-1/10-4 (K-1) D (m2s-1) Present work Geise et al. Herzig et al.T=1184KTC=1043KNb diffusion in -FeFigure 8.4: Predicted temperature dependence of the calculated α-Feself-, Mn, Mo, and Nb solute-diffusion coefficients vs. publishedexperimental data [11, 48, 165, 170, 171, 174? –178].diffusion in the bcc Fe lattice in both para- and ferromagnetic states, whichare comparable with the experiments and also previous DFT calculations.The agreement between the computed and measured results demonstratedthat the computational methodology presented in this project is capableof predicting quantitative information of the atomistic diffusion processes,which gives us confidence to investigate solute diffusion in grain boundariesand interfaces.8.2 Diffusion in the Fe fcc LatticeIn this section, AFMD magnetic arrangement is first employed to describethe fcc phase. The calculated vacancy formation energy in pure fcc Fe is1.72 eV. This value is significantly lower than that in bcc Fe, i.e. 2.21117eV, suggesting that vacancies are much easier to generate in the fcc phase.Such lower vacancy formation energy can be ascribed to the higher packingdensity in the fcc phase. The migration barriers are then determined usingnudged elastic band (NEB) method [105, 106]. In the AFMD configuration,each site (o) has three types of nearest neighbors, namely p, q, and w, whichare labeled in Figure 6.1. As a result, three migration paths have to beconsidered. The results are listed in Table 8.2.Table 8.2: Vacancy formation energies (Ev), vacancy migrationbarrier (Em), and the activation energy for self-diffusion (Q0)along each path. The migration paths are labeled in Figure 6.1.AFMDSQSPath o− p o− q o− wEv (eV) 1.72 1.72 1.72 1.92∼1.99Em (eV) 1.09 0.74 0.93 0.51∼0.68Q0 (eV) 2.81 2.46 2.65 2.46∼2.67It should be noted that along path o− w, the initial and final magneticmoments for the migrating atom have the opposite sign and magneticmoment flips are required along the migration path. Nevertheless, ourcalculations indicate that the energy cost of magnetic moment flipping isless than 0.05 eV, i.e. less than 5% of the migration energy, and thereforehave a relatively small effect on the migration barrier. Overall, the estimatedmigration energies for vacancies in fcc γ-Fe are in the range of 0.74∼1.09eV, which is about 16% ∼ 70% higher than the values in the bcc phase.Combined with the vacancy formation energy, the activation energy for self-diffusion in γ-Fe lies between 2.46 eV and 2.81 eV.For comparison, the vacancy formation energy, migration energy, andactivation energy are also investigated by using the SQS magnetic arrange-ment. Note that in SQS magnetic structure, the vacancy formation energyand migration energy vary from site to site. In this work, six different118configurations are considered. The results are shown in Table 8.2, fromwhich one can see that though there is some deviations between SQS andAFMD for vacancy formation energies and migration barriers, the activationenergies calculated by the two independent magnetic arrangements fall intothe same range.The experimental results of activation energies for fcc self-diffusion are2.80∼2.90 eV [175, 179], which are about 10% higher than our calculatedvalues. This is because our calculations are performed at 0K, whereasthe experiments are carried out at high temperature (i.e. T > 1173 K).To correlate such deviations, the corresponding lattice constants at hightemperature should be employed.Table 8.3: Lattice constants (al), atomic volume (V ), vacancyformation energy (Ev), effective migration energy (Eeffm ), andeffective activation energy (Qeffa ) as a function of temperature.al (A˚) V (A˚3) Ev (eV) Eeffm (eV) Qeffa (eV)bcc (0K) 2.84 11.45 2.21 0.63 2.84bcc (800K) 2.866 11.77 2.24 0.63 2.87fcc (0K) 3.55 11.18 1.72 0.87 2.59fcc (293K) 3.59 11.57 1.87 0.83 2.70fcc (1184K) 3.65 12.16 1.97 0.81 2.78To address the influence of the lattice constants due to the thermalexpansion, we calculated the vacancy formation energy, effective migrationenergy, and effective activation energy using different lattice constants.These lattice constants are employed from the experimental measurementsat a given temperature T [138, 180]. The results are listed in Table 8.3. Inthis work, the effective migration energy is calculated via KMC simulations.With the increase of temperature, the effect of lattice constant on theactivation energy in the fcc phase is remarkable, while it is rather small inthe bcc phase. This is because fcc is the closest-packed structure, which has11912 nearest neighbors. The change of lattice constants induced by thermalexpansion results in a larger influence of nearest neighbors Fe atoms onactivation energy than that in the bcc phase. Consequently, the activationenergy of the fcc phase obtained at T=1184K is increased to 2.78 eV, whichis consistent with the experimental measurements, i.e. 2.80 eV [179]. Itshould be noted that for the bcc phase, the magnetic transformation hasa significant influence on the activation energy, and the activation energyhas to be modified according to Equation 8.2. Recall that α=0.163 for Fe(see Figure 8.2). Thus the activation energy at a given temperature T isexpressed as:Qa = Q0(1 + 0.163s(T )21.163) (8.11)where s(T ) is calculated according to Equation 8.4. The modified acti-vation energies for Fe self-diffusion in bcc and fcc at different temperaturesare plotted in Figure 8.5. At low temperature, the activation energy is lowerin fcc than in bcc, while it becomes opposite at high temperature.The diffusion coefficients for Fe self-diffusion in the fcc phase arecalculated in the Arrhenius form, i.e.:D = D0 exp(−Qeffa /kBT ) (8.12)where D0 is the pre-factor, which is given by:D0 = a2l f2ν0 exp(Sv/kB) (8.13)where the correlation factor f2=0.7815 for fcc. Based on our calculations,the attempt frequencies ν0 for different jump paths in fcc Fe are very similar,i.e. 9.0 THz, 6.6 THz, and 7.4 THz for jump paths o− p, o− q, and o− w,respectively. For simplicity, we take the average value 7.7 THz, which is alsoclose to the value in the bcc phase, i.e. 8.5 THz. The vacancy formationentropy (Sv) in the fcc phase is 2.8kB, which is about 60% of the valuein bcc. Consequently, the pre-factor (D0) in fcc is 9.6×10−6 m2/s at zerotemperature. The influence of the lattice constants on the pre-factor arealso tested. The results indicate that the difference of D0 is less than 3%.1200 200 400 600 800 1000 12002.42.52.62.72.82.9 fcc Fe bcc FeActivation Energy (eV)Temperature (K)TcFigure 8.5: The modified activation energies for Fe self-diffusion inbcc and fcc phase at different temperature.Thus, the pre-factor can indeed be considered as a constant over the relevanttemperature range, i.e. 0∼1200K.Figure 8.6 presents the calculated Fe-self diffusion coefficients in bcc andfcc. At lower temperature, the predicted diffusion coefficient is larger in fccthan in bcc, while it becomes opposite at higher temperature. In addition,the calculated self-diffusion coefficients in bcc and fcc at higher temperatureagree well with experimental observations [174, 175].1216 12 18 24 30 361E-541E-491E-441E-391E-341E-291E-241E-191E-14T (K) T-1/10-4 (K-1)     Self-Diffusion Coefficient (m2/s) This work (bcc) This work (fcc) Exp. (bcc) Exp. (fcc)TC=1043KT=1184K16001200 800 400 Figure 8.6: Predicted temperature dependence of the Fe-self diffusioncoefficients vs. published experimental data in bcc and fcc [174,175].8.3 Grain Boundary Diffusion in bcc FeThe vacancy formation energies (Ev) calculated for several atomic positionswithin the grain boundary are shown in Figure 8.7. The results indicate thatthe influence of the grain boundary on Ev is short-ranged and is limited toabout 3∼4 atomic layers. In addition, it was found that Ev displays verystrong site-to-site variations, i.e. at boundary position b, Ev is 0.77 eV (35%)below the bulk value of 2.21 eV, while at boundary site a, the difference is lessthan 10%. Such large variations in Ev can be attributed to the existence ofalternating tension and compression regions in the boundary areas. We findthat the nearest Fe-Fe distance at the boundary site b (dbb′ , see Figure 5.1) isonly 2.18A˚. It is about 12% shorter than that in the bulk (2.46A˚), which givesrise to high strain at this position. While at boundary site a, the nearest Fe-Fe distance (dab) increases to 2.51A˚, resulting in relatively low strain at this122position. Similar results were also reported for Cu grain boundaries [41, 42].Evidently, lower formation energies favor the accumulations of vacancies inthe grain boundary.-4 -3 -2 -1 0 1 2 3 41.41.61.82.02.22.4 Vacancy Formation Energy (eV)  Distance from Boundary Plane (Å) Boundary bulkab b'Figure 8.7: Vacancy formation energies at different sites of Σ5 (013)grain boundaries in α-Fe. The dotted red line indicates thevalue in the bulk lattice.Figure 8.8 shows the calculated binding energies for vacancy-solute pairs.The capital letter X refers to the solute atom while V stands for the vacancy.The subscripts a, b, and c indicate grain boundary positions. A negativesign denotes favorable binding. Overall, the formation of the solute-vacancypair is energetically favorable. However, similar to Ev, the vacancy-solutebinding energies (Eb) vary from site to site. In most of the cases, however,the vacancy-solute binding energies in grain boundaries are comparable withthe values in the bulk. While in a couple of cases, the interactions areremarkably weaker than in the bulk, or even show positive (repulsive) values.The migration energy barriers (Em) for various solutes along differentpaths in the Σ5 (013) grain boundary are listed in Table 8.4. The jump b0−cbridges two neighboring structural units. The jump vector is normal to thetilt [001] axis, so this jump does not contribute to diffusion along the tilt axis.For other jumps, they have components along the tilt axis and can contribute123Si V Mn Ti Mo Nb0.10.0-0.1-0.2-0.3-0.4  Vacancy-Solute Binding Energy (eV) Xb0Va0 XcVa0 bulk(a)Si V Mn Ti Mo Nb0.10.0-0.1-0.2-0.3-0.4  Vacancy-Solute Binding Energy (eV) XaVb XcVb bulk(b)Figure 8.8: Vacancy-solute binding energies at Σ5 (013) grainboundaries and bulk in α-Fe. Vacancy segregates at (a)boundary a0, and (b) boundary b position. The capital lettersX refer to the solute and V refer to the vacancyto diffusion in both directions in the boundary plane. It is seen that themigration energies can vary for the different paths along the boundary. Ingeneral, solutes have a smaller calculated migration energy than that forFe self-diffusion in the grain boundary, suggesting faster diffusion of soluteatoms than Fe self-diffusion.Unlike the bcc bulk situation, the activation energies of the solutes in thegrain boundary depend on the positions, and the jump rates vary at different124Table 8.4: Calculated migration energy for various solutes along grainboundary (Em, in units of eV). Sites a0, b0, c0, and c refer toboundary sites as shown in Figure 5.1.Path a0-b0 b0-a0 a0-c c-a0 b0-c0 c0-b0 b0-c c-b0Fe 0.91 0.32 0.68 0.40 0.39 0.71 0.45 0.77Si 0.57 0.42 0.63 0.32 0.28 0.53 0.37 0.62Ti 0.93 0.18 0.62 0.24 0.18 0.48 0.30 0.60V 0.88 0.55 0.64 0.28 0.55 0.69 0.41 0.55Mn 0.84 0.32 0.59 0.21 0.32 0.44 0.31 0.43Nb 0.98 0.19 0.66 0.22 0.16 0.41 0.24 0.49Mo 0.85 0.23 0.60 0.31 0.27 0.46 0.34 0.54sites. To investigate solute diffusion and obtain the effective activationenergies, a kinetic Monte Carlo (KMC) model is developed based on thedetailed input from DFT calculations. In this work, a thin boundary modelwith periodic boundary conditions is constructed. As discussed in Chapter5, the effective thickness of the grain boundary (δ) is about 6 atomic layers(∼4.5 A˚). The ratio of the effective thickness of the grain boundary (δ) to thegrain size (dgs) is set to 0.25, i.e. dgs = 4δ. In the KMC model, the supercellconsists of N=8× 104 lattice sites with the dimension of 142× 180× 36 A˚3.The vacancy concentration is given by 1/N . To correlate our KMC vacancyconcentration with the real case, the physical time is rescaled using therelation [181]:τr = τkmccv/ceqv (8.14)where ceqv is the equilibrium vacancy concentration, which is given by aBoltzmann distribution, i.e.ceqv = exp(SvkB) exp[−Ev + EbkBT] (8.15)12512 13 14 15 16 17 18 19 201E-271E-251E-231E-211E-191E-17T-1/10-4 (K-1)Diffusion Coefficients (m2s-1) Fe      Mo Mn NbFigure 8.9: Predicted temperature dependence of the calculated Nb,Mo, Mn, and Fe-self diffusion coefficients at grain boundaries.The apparent diffusion coefficients are determined by Equation 4.22based on the KMC simulations. According to Equation 4.23, the grainboundary diffusivity is given by:Db =dgsδ(Dapp − (1−δdgs)Dl) (8.16)When Dapp  Dl, diffusion through the bulk lattice can be ignoredin comparison to grain boundary diffusion. This is indeed the case in oursimulations. At 800K (the highest temperature used in the simulations), theobtained apparent diffusion coefficients for Nb is 1.54 × 10−17 m2/s, whichis two orders of magnitude larger than the bulk diffusion, i.e. 1.26 × 10−19m2/s (see Figure 8.4d). The difference is even larger for self-diffusion, whichis up to 3 orders of magnitude. Thus, Equation 8.16 can be rewritten as:Db =dgsδDapp (8.17)The calculated Nb, Mo, Mn, and Fe-self diffusion coefficients in Σ5 (013)126tilt grain boundary are presented in Figure 8.9. Though the activationenergy varies for different diffusion paths, the diffusion coefficients followthe Arrhenius law quite well, which is consistent with the experimentalobservations for Fe self-diffusion and previous theoretical results for Cu self-diffusion in grain boundaries [41, 42, 182]. In addition, Nb is found to havea much larger diffusion coefficient, which is approximately two orders ofmagnitude higher than that for Fe self-diffusion. Such an enhanced diffusioncoefficient can be ascribed to both the attractive vacancy-solute bindingenergies and the reduced value of the migration energy (see Figure 8.8 andTable 8.4). The effective activation energy in grain boundaries (Qgb) canbe extracted from the slope in the Arrhenius plot shown in Figure 8.9. Theresults are listed in Table 8.5. Compared with the activation energy in bccFe, the effective activation energy in grain boundaries is reduced by 15-20%,which indicates that grain boundaries offer a fast diffusion path.Table 8.5: The effective activation energy (Qgb) in Σ5 (013) grainboundaries and the ratio of Qgb/Qbulk.Fe Si Ti V Mn Nb MoQgb (eV) 2.27 1.96 1.92 2.14 1.98 1.83 2.10Qgb/Qbulk 0.80 0.85 0.82 0.82 0.84 0.83 0.83Experimental values for diffusion in grain boundaries are limited.Nevertheless, the calculated activation energy for self-diffusion in pure Fegrain boundary are comparable with experimental data, which lies between1.46∼2.34 eV when extrapolated to the ferromagnetic phase [182? , 183]. Itis interesting to note that Turnbull and Hoffman who studied self-diffusionin [001] grain boundaries in silver as a function of crystal misorientation (θ)found that within the experimental uncertainty, the self-diffusion coefficientsin different grain boundaries are independent of θ. Similar results were alsoobtained by Suzuki and Mishin [42]. They studied Cu self-diffusion in severaltilt grain boundaries by molecular dynamics simulations and found that the127conclusions reached for the Σ5 grain boundary are general and hold forother grain boundaries as well. Thus, one would expect that the tendencyfor solute diffusivities at the considered Σ5 grain boundary can be extendedto more general boundaries. Further calculations are needed to confirm thisassumption.8.4 Diffusion in the bcc-fcc Fe InterfaceFigure 8.10 presents the vacancy formation energy at the bcc-fcc interfaceas a function of distance from the habit plane. The AFMD configurationis employed to describe the magnetic states of fcc. The results indicatethat there exists a chemical potential gradient for vacancies between ferriteand austenite. Vacancies form much easier in austenite than in ferrite. Inaddition, lower vacancy formation energies are observed at the interfacepositions on the austenite side. Similar to the case of grain boundaries, theinfluence of the interface on the vacancy formation energy is short-ranged,i.e. the bulk value of the vacancy formation energy is achieved approximately4 A˚ away from the interface.We then calculated the migration energy of vacancies in the bcc-fccinterface. Due to the asymmetric structure, the jump rates vary from site tosite. The jump paths are illustrated in Figure 8.11, which can be categorizedinto five groups: (I) migration from the second layer to the first layer in bcc;(II) migration within the first layer in bcc; (III) migration from the bccside to the fcc side of the interface; (IV ) migration within the first layer infcc; and (V ) migration from the second layer to the first layer in fcc. Thecalculated migration energies (Em) for different types of jumps are listedin Table 8.6. In addition, the migration energy for opposite jump direction(E′m) are also shown.The results indicate that in general, Em is smaller compared with thebulk value. For example, the migration energies from the second layer tothe first layer in bcc (group I) are 0.55 eV, which are about 13% lower thanthe bcc bulk value, i.e. 0.63 eV. In addition, the migration energies havelower values when the vacancy moves from the inner grain (second layer) to128-8 -6 -4 -2 0 2 4 61.31.41.51.61.71.81.92.02.12.22.32.42.5fcc grainbcc bulk Vacancy Formation Energy (eV)Distance from Interfacefcc bulkbcc grainFigure 8.10: Vacancy formation energies as a function of distancefrom the habit plane. The bars indicate the actual vacancyformation energies, where Fe atom has spin-up or spin-downstates.Figure 8.11: Illustration of various jump paths in the bcc-fccinterface. Blue and red circles indicate fcc Fe atoms with spin-down and spin-up magnetic state, while white circles representbcc Fe atoms.129Table 8.6: Migration energies of vacancy for different types of jumppaths as defined in the text in the bcc-fcc interface. The arrowsindicate the spin states of the jump Fe atoms in initial and finalstates.Em E′mI (↑↑) 0.55 0.69II (↑↑) 0.60 0.56III (↑↑) 0.52 0.71III (↑↓) 0.58 0.68IV (↑↑) 0.65 0.66IV (↓↓) 0.58 0.76IV (↑↓) 0.58 0.49V (↑↑) 0.61 0.66V (↓↓) 0.70 0.76V (↑↓) 0.70 0.83the interface (first layer) than when it moves in the opposite directions. Asa result, vacancies are expected to be confined to the interface, which areconsistent with the experimental observations that interfaces act as a sinkfor vacancies.Table 8.7: Activation energy for Fe self-diffusion in the bcc-fccinterface, fcc bulk, bcc bulk, and Σ5 grain boundaries (GBs)at 0K. Unit is eV.Interface fcc bulk bcc bulk Σ5 GBsQ0 2.46 2.59 2.84 2.27The apparent diffusion coefficients in the bcc-fcc interface is determinedby Equation 4.22 based on the KMC simulations. In a similar manner with130the grain boundary, if the interface has an effective thickness δα in bcc andδγ in fcc, and the bcc (fcc) grain size is dα (dγ), the apparent diffusioncoefficient (Dapp) is expressed as:Dapp =dα − δαdα + dγDαl +dγ − δγdα + dγDγl +δα + δγdα + dγDI (8.18)where Dαl , Dγl and DI are the bcc bulk, fcc bulk, and bcc-fcc interfacediffusivities. As discussed in Chapter 7, the effective thickness of the bcc-fccinterface is about 7.0 A˚. We set dα/δα=dγ/δγ=5. The supercell consistsof N = 2.16 × 105 lattice sites with the dimension of 197 × 184 × 67 A˚3.Dαl and Dγl are obtained directly from the DFT calculations (see sections8.1 and 8.2). Since the influence of the thermal expansion and magnetictransformation on the interface diffusion is unclear, we did not include theseeffects in our calculations. So in the simulations, we use the activationenergies Q0 at T=0K to calculate the Fe self-diffusions without any furthercorrections for lattice constants and magnetism at higher temperature. Theinterface diffusivity DI is thus determined by Equation 8.18.The effective activation energy in the bcc-fcc interface can be extractedfrom the slope in the Arrhenius plot for the interface diffusivity. The resultis shown in Table 8.7. For comparison, the activation energies for Fe self-diffusion in the fcc bulk, bcc bulk, and in the Σ5 bcc grain boundary are alsolisted. Compared with the values in bulk diffusion, bcc-fcc interface providea moderate “fast diffusion” path. However, it should be noted that atelevated temperature, the thermal expansion and magnetic transformationhave significant influence on the activation energies for diffusion in fcc andbcc, respectively. As a result, it can be expected that the activation energyin the bcc-fcc interface is modified at higher temperature as well. Therefore,new theoretical models and simulation tools are required.131Chapter 9Conclusions and FutureWork9.1 ConclusionsThis work demonstrates an atomistic modeling scheme for the simulationof solute-interface interactions in Fe, which provides a basis for furtherinvestigations. The binding energies of solutes with grain boundariesand bcc-fcc interfaces are determined based on the framework of DFTcalculations. The calculated jump rates are set as the input data forKMC simulations to obtain the diffusion coefficients and effective activationenergies. The main simulation results are summarized below:• Strong interactions between large solutes (i.e. Nb, Mo, and Ti) andΣ5 grain boundaries in bcc and fcc Fe as well as the bcc-fcc interface areobtained. Such strong interactions are expected to delay grain growth,recrystallization, and ausenite-to-ferrite transformation. This can beunderstood in terms of the strain-relief argument, where large solutes bindto the grain boundary or interface as a means of relieving strain on the Fematrix.• It is found that segregation to the Σ5 bcc grain boundaries arefavorable for all solutes and boundary sites considered in this work, and132the magnitude of the segregation energy of the solute atoms increases withthe solute atom volume.• Our DFT calculations indicate that the excess volume at boundariesand interfaces is an important parameter to predict the segregation tendencyof solutes. This is in particular of significance for large solute atoms (i.e.Nb, Mo, and Ti). Further, molecular statics simulations reveal that thoughthe site volume distribution is much narrower for the Σ5 grain boundary, theaverage excess volume is comparable to that for high Σ grain boundaries,suggesting that the tendency to accommodate solute atoms at the consideredΣ5 grain boundary can, to a large extent, be applied to more generalboundaries.• A large set of possible reference states for γ-Fe at 0 K have beenevaluated and the AFMD and SQS configurations are found to be themost suitable magnetic states. Though SQS may give more realistic resultsfor paramagnetic γ-Fe, the properties such as vacancy formation energy,solute pair interactions, etc. vary from site to site, which requires a largecomputational load. Further calculations indicate that the results obtainedby AFMD and SQS magnetic structures are comparable to each other,suggesting AFMD structure is a good and more convenient representativefor paramagnetic γ-Fe.• It is found that some of the boundary sites are not favorable forsolute segregation at the Σ5 grain boundary in fcc Fe. Positive segregationenergies are observed for Cr and Si. In addition, no segregation tendency ofNi to fcc grain boundary is found, which implies that Ni prefers to stay atthe bulk position rather than at the boundary sites. On the contrary, Largebinding energies for Nb and Mo are obtained at the Σ5 grain boundary infcc Fe. The interactions between these solutes and the fcc grain boundaryare even stronger than those at the bcc grain boundary.• In the bcc-fcc interface, a chemical potential difference exists for thesubstitutional elements between ferrite and austenite. This difference israther large for Mn and Ni, which provided considerable driving force forsolute transfer from ferrite to austenite.133• Using the average energies between the two bulk values as thereference point, we calculate the effective binding energies of selected solutesin the bcc-fcc interface based on the Langmuir-McLean equation. Goodagreement with the experimental observations for solute segregation areshown. The reported interaction parameters for Nb, Mo Mn, and Siagree well with our predictions. The tendency predicted for segregation,in particular the strong segregation of Nb, is consistent with observationsof the role of Nb in delaying austenite-ferrite transformation in low-carbonsteels, which at least in part have been attributed to solute drag.• The solute-solute interactions and their predicted contributions tothe segregation energies indicate that there is a massive penalty to thebinding energy when the solute-solute distance is short. While at someintermediate solute-solute distance, co-segregation effects are observed forlarge solute elements (i.e. Nb, Mo, and Ti) and the binding energies areincreased by about 30%. A detailed analysis reveals that for TMs, thereare mainly three aspects that contribute to the solute-solute interactions,i.e. elastic strains, competing magnetic interactions, and contributions frommagnetically driven clustering. In addition, for non-metal element suchas Si, the chemical bonding effects may also contribute to the solute pairinteractions.• The self- and solute-diffusivities in bcc and fcc bulk lattice havebeen computed within a framework combining DFT calculations andKMC simulations. Good agreement between our calculations and themeasurements for self- and solute diffusion in bulk Fe is achieved, whichdemonstrates the computational methodology presented in this project iscapable of predicting quantitative information of the atomistic processes.• Our simulations indicate that the effective activation energies for theΣ5 bcc grain boundary diffusion are 80%∼85% of those for lattice diffusionfor different solutes, suggesting grain boundaries can provide fast diffusionpaths. By contrast, the effective activation energy of self-diffusion in bcc-fcc Fe interface is close to the value of fcc self-diffusion at 0K, and bcc-fccinterface provides a moderate “fast diffusion” path.1349.2 Future WorkTo further enhance the present understanding, several suggestions can bemade for further investigations as follow:• The present study focuses on the special grain boundaries, i.e.Σ5 grain boundary. Further work is required to understand the solutesegregation behavior at more general, e.g. high Σ or low angle, grainboundaries. This could be achieved by employing the hybrid method,i.e. quantum mechanics/molecular mechanics (QM/MM) approach, whichcombines the strengths of the accuracy of quantum mechanics and theefficiency of molecular mechanics [184, 185].• It is demonstrated that the lattice constants due to the thermalexpansion and the magnetic transformation at elevated temperature have asignificant influence on the diffusion coefficients in fcc and bcc, respectively.Thus, new theoretical models and simulation tools are required to modifythe activation energies for diffusion across the bcc-fcc interface at highertemperature.• Carbon is one of the most important elements in steels. Further workwill have to be done to incorporate carbon into the system. 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