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Atomistic simulations of dynamic interaction between grain boundaries and solute clusters Wicaksono, Aulia Tegar 2015

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Atomistic Simulations of DynamicInteraction between Grain Boundariesand Solute ClustersbyAulia Tegar WicaksonoM.Sc., National University of Singapore, 2010M.Eng., Massachusetts Institute of Technology, 2009B.Eng., Nanyang Technological University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Materials Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2015c© Aulia Tegar Wicaksono 2015AbstractMicrostructure evolution during material processing is determined by a number of factors,such as the kinetics of grain boundary migration in the presence of impurities, which cantake form of solid solution, second-phase precipitates or clusters. The dynamic interactionbetween grain boundaries and clusters has not been explored. In this work, a varietyof simulation tools are utilized to approach this problem from an atomistic perspective.Atomistic simulations are first implemented to explore the parameter space of the solutedrag problem, i.e. grain boundary migration in a binary ideal solid solution system, via akinetic Monte Carlo framework. Depending on their diffusivity, solute atoms are capableof modifying the structure of a migrating boundary, leading to a diffusion-dependent dragpressure. A phenomenological model adapted from the Cahn model is proposed to explainthe simulation results.The interaction between clusters and a migrating grain boundary is studied next usingmolecular dynamics simulations. The iron helium (Fe-He) system is chosen as the objectof the study. A preliminary step towards such a study is to investigate the grain bound-ary migration in pure bcc Fe. An emphasis is placed upon demonstrating the correlationbetween the migration of curved and planar boundaries. Evidence that verifies such a cor-relation is established, based on the analyses on the shapes, the kinetics and the migrationmechanism of both types of boundaries. Next, the formation of He clusters in the bulkand grain boundaries of Fe is examined. The cluster formation at the boundary occurs ata lower rate relative to that in the bulk. This is attributed to the boundary being a slowdiffusion channel for interstitial He atoms. The overall effect of clusters on the boundarymigration is twofold. Clusters reduce the boundary mobility via segregation; the magni-tude of their effect can be rationalized using the Cahn model in the zero velocity limit.Clusters also act as pinning sources, delaying or even completely halting the boundarymigration. A phenomenological model adapted from the Zener pinning model is used todiscuss the role of clusters on grain boundary migration.iiPrefaceThis dissertation is written based on original research conducted by the author,Aulia Tegar Wicaksono. All of the work presented henceforth was conducted inthe Department of Materials Engineering of The University of British Columbia,at the Point Grey campus. My advisors, Prof. Matthias Militzer and Prof. ChadW. Sinclair, were involved in all stages of the project; they provided guidance andassisted with the manuscript composition.Figures 2.2, 2.3, 2.6, 2.7, and 2.12 in Chapter 2 Literature Review have beentaken with permission from the cited sources.The contents of Chapter 5 were published in: A. T. Wicaksono, C. W. Sinclair,M. Militzer, “A three-dimensional atomistic Kinetic Monte Carlo study of dynamicsolute-Interface interaction”, Modelling and Simul. Mater. Sci and Eng, 21, 085010(2013). Part of the simulations results was also presented in a conference: A. T.Wicaksono, C. W. Sinclair, and M. Militzer, “The interaction between a migratinginterface and mobile non-interacting solutes: a kinetic Monte Carlo simulation”,6th International Conference on Multiscale Materials Modelling, Singapore, 2012.Chapter 6 is based on the simulation work I conducted, and it has been submit-ted for publication. Part of the simulation results was also presented in a confer-ence: A. T. Wicaksono, C. W. Sinclair, M. Militzer, H. Song and J. J. Hoyt “A molec-ular dynamics study of symmetric grain boundaries migration in α-Fe”, 143rd TMSAnnual Meeting, San Diego, CA, United States, 2014.The contents of Chapter 7 were published in: A. T. Wicaksono, M. Militzer, C.W. Sinclair, “Atomistic simulations of the effect of helium clusters on grain bound-ary mobility in iron”, IOP Conference Series: Materials Science and Engineering,89, 012048, (2015). Part of the simulation results was also presented in a confer-ence: A. T. Wicaksono, M. Militzer, and C. W. Sinclair, “Atomistic Determinationof Grain Boundary Mobility in Fe-He alloys”, 7th International Conference on Mul-iiitiscale Materials Modelling, Berkeley, CA, United States, 2014.Chapter 8 is based on the simulation work I conducted, and it will be submittedfor publication. Part of the simulation results was also presented in a conference:A. T. Wicaksono, M. Militzer, and C. W. Sinclair, “Atomistic simulations of grainboundary mobilities in the iron-helium system”, 7th International Conference onSolid-Solid Phase Transformations in Inorganic Materials, Whistler, BC, Canada,2015.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Grain boundary structure and energy . . . . . . . . . . . . . . . . . 52.1.1 Crystallographic construction . . . . . . . . . . . . . . . . . 52.1.2 Grain boundary energy . . . . . . . . . . . . . . . . . . . . . 72.1.3 Grain boundary roughening transition . . . . . . . . . . . . 92.2 Grain boundary segregation . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Experimental observation . . . . . . . . . . . . . . . . . . . . 102.2.2 Models for solute segregation . . . . . . . . . . . . . . . . . 112.3 Grain boundary migration in a pure system . . . . . . . . . . . . . 132.3.1 Motion of grain boundaries . . . . . . . . . . . . . . . . . . . 132.3.2 Grain boundary mobilities . . . . . . . . . . . . . . . . . . . 142.3.3 Experimental measurements . . . . . . . . . . . . . . . . . . 17v2.4 Grain boundary migration in alloys . . . . . . . . . . . . . . . . . . 192.4.1 Experimental observations . . . . . . . . . . . . . . . . . . . 192.4.2 Solute drag models . . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Particle pinning models . . . . . . . . . . . . . . . . . . . . . 252.4.4 Cluster drag models . . . . . . . . . . . . . . . . . . . . . . . 292.5 Perspective from atomistic simulations . . . . . . . . . . . . . . . . 302.5.1 Overview of atomistic simulation techniques . . . . . . . . 302.5.2 Progress from recent atomistic studies . . . . . . . . . . . . 322.6 Outstanding questions . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Kinetic Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.2 Model material . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.3 Rates of atomistic events . . . . . . . . . . . . . . . . . . . . 434.1.4 Running a KMC simulation . . . . . . . . . . . . . . . . . . . 454.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Interatomic potentials . . . . . . . . . . . . . . . . . . . . . . 474.2.3 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.4 Running an MD simulation . . . . . . . . . . . . . . . . . . . 514.2.5 Relevant algorithms . . . . . . . . . . . . . . . . . . . . . . . 535 Kinetic Interplay between Ideal Solute Atoms and a Migrating GrainBoundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3.1 Grain boundary migration in a pure bicrystal . . . . . . . . 655.3.2 Grain boundary migration in the presence of diffusing so-lutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68vi5.4 Effect of solute diffusivity on the structure of grain boundary . . . 735.5 Modified solute drag model . . . . . . . . . . . . . . . . . . . . . . . 755.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 Grain Boundary Migration in Pure BCC Iron Bicrystals . . . . . . . . 786.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3.1 Lattice parameter of bulk bcc Fe at finite temperatures . . . 806.3.2 Reduced mobilities of curved boundaries . . . . . . . . . . . 816.3.3 Absolute mobilities of planar symmetric twin boundaries . 836.3.4 Absolute mobilities and energies of planar inclined bound-aries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.4 Correlation between the migration of curved and planar bound-aries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.4.1 Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.4.2 Mobilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.4.3 Atomistic mechanisms . . . . . . . . . . . . . . . . . . . . . 936.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Interaction of Helium Clusters with Non-driven Grain Boundaries inBCC Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3.1 Properties of He in the bulk Fe crystal . . . . . . . . . . . . . 1007.3.2 Properties of He in grain boundaries of Fe . . . . . . . . . . 1047.3.3 Mobilities of cluster-enriched grain boundaries . . . . . . . 1097.4 Cluster drag coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 1117.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148 Effect of Helium Clusters on Grain Boundary Migration in BCC Iron 1158.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115vii8.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.3.1 Trapping of segregated He atoms . . . . . . . . . . . . . . . 1188.3.2 Effect of He clustering and segregation on grain boundarymigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.3.3 Effect of pre-segregated He atoms on grain boundary mi-gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.3.4 Effect of monosized clusters on grain boundary migration . 1308.4 Cluster pinning pressure . . . . . . . . . . . . . . . . . . . . . . . . 1338.5 The effective cluster pinning model . . . . . . . . . . . . . . . . . . 1368.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 1419.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146AppendicesA Running MD simulations in LAMMPS . . . . . . . . . . . . . . . . . . 173A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173A.2 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174A.2.1 File management . . . . . . . . . . . . . . . . . . . . . . . . . 174A.2.2 Molecular statics (MS) and molecular dynamics (MD) . . . 175A.2.3 Input file for MS calculation . . . . . . . . . . . . . . . . . . 175A.2.4 Input file for MD simulation . . . . . . . . . . . . . . . . . . 180A.2.5 Supplementary files . . . . . . . . . . . . . . . . . . . . . . . 184A.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.3.1 Thermodynamic output (log file) . . . . . . . . . . . . . . . 188A.3.2 Per-atom output (dump files) . . . . . . . . . . . . . . . . . . 189A.4 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189viiiList of Tables2.1 Arrhenius parameters of grain boundary mobilities, see Eq. (2.6), ob-tained from studies employing molecular dynamics simulations. . . . . 344.1 Types of octahedral sites and their ω−values, see Eq. (4.2). . . . . . . . . 434.2 Crystallographic axes of bicrystals containing a U-shaped grain boundary(the type-1 cell) investigated in this study, see Figure 4.4(b) for symboldefinition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Geometrical setup of the type-2 bicrystals used for determining the ab-solute mobility of planar boundaries, their common XA/XB axis being〈110〉, see Figure 4.4(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Formation energy of a single He atom in the bulk crystal of bcc Fe. . . . . 605.1 Material properties, which act as the input for KMC simulations. . . . . . 655.2 The number of solute atoms introduced to KMC simulations. . . . . . . 685.3 Solute diffusivities and the parameters for drawing fitting curves in Fig-ure 5.4(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.1 Dimensions of the type-2 bicrystals used for determining the energy andabsolute mobility of planar boundaries. Refer to Figure 4.4(d) and Table4.3 for the illustration and the crystallographic axes of type-2 bicrystals,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Arrhenius parameters of the reduced mobilities of curved twin bound-aries, shown in Figure 6.2(b). . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Arrhenius parameters of the absolute mobilities of planar symmetric twinboundaries, shown in Figure 6.8. . . . . . . . . . . . . . . . . . . . . . . 876.4 Fitting parameters in Eqs. (6.1) and (6.3) to draw the continuous approx-imation of γ (ϕ) and M (ϕ) in Figure 6.9. . . . . . . . . . . . . . . . . . 90ix6.5 Reduced mobilities extracted from simulations [Table 6.2] and a contin-uum model [Eq. (6.5)] where ∆ = (M∗mod - M∗) / M∗. . . . . . . . . . . . 937.1 Geometrical setup of simulation boxes and their solute contents. Referto Figure 4.4(d) and Table 4.3 for the illustration and the crystallographicaxes of type-2 bicrsytals, respectively. . . . . . . . . . . . . . . . . . . . 997.2 Arrhenius parameters of the single-atom diffusivity in Figure 7.6. . . . . 1057.3 Properties describing the interaction between He and Fe grain bound-aries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.4 Cluster drag coefficients α obtained from the simulations and the Cahnmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.1 Types of initial configuration and quantities of He atoms being intro-duced to a type-1 bicrystal that contains the incoherent curved twin bound-ary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.1 Line-by-line description of a setfl file. . . . . . . . . . . . . . . . . . . . 184A.2 The position vectors of the 8 nearest neighbours of atoms that belong tothe bulk grain of either side of the planar coherent twin boundary. . . . . 187xList of Figures2.1 (a) A construction of a symmetric grain boundary of misorientation θ, (b)an asymmetric boundary lying at an inclination angle ϕ from the bound-ary in (a), (c) and (d) the construction of the same symmetric and inclinedboundaries via the CSL framework. CSL unit cells and its coincidencesites are indicated by the dashed squares and solid points, respectively. . 62.2 Computed grain boundary energies in Al (a) as a function of misorien-tation for [001] grain boundaries, showing a trend consistent with theRead-Shockley model, Eq. (2.1), for small θ [61], and (b) as a functionof inclination for a given misorientation θ of [110] grain boundaries, thetrends deviating from the Read-Shockley model, Eq. (2.2) [63]. . . . . . . 82.3 Variation of high-angle grain boundary energy in Pb with temperature,as measured from the dihedral angle experiments [65] . . . . . . . . . . 92.4 Modes of grain boundary motion according to Cahn’s framework. Upperrow: original bicrystals; lower row: bicrystal evolution due to bound-ary motion. (a) pure normal motion, (b) pure sliding motion, (c) coupledtranslational-normal motion, (d) grain rotation. Thick and thin solid ar-rows indicate the direction of the driving force and the resulting motion,respectively. Dashed thin arrows in (c) and (d) indicate the direction ofsecondary coupled motions. . . . . . . . . . . . . . . . . . . . . . . . . 142.5 The migration of a U-shaped boundary, resulting from the collective mi-gration of segments that make up the curved portion of the boundary, e.g.segments A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16xi2.6 Temperature dependence of (a) absolute mobilities and (b) reduced mo-bilities of 85◦-misorientation boundaries in Zn, being obtained using mag-netic field and capillarity pressure, respectively. The bicrystal shape usedon each measurement technique is illustrated by the insets [9]. . . . . . . 182.7 (a) Grain boundary velocities measured from recrystallization experimentsin zone-refined Pb doped with Sn [10], (b) the effect of Sn on the activa-tion energy for the migration of special and random boundaries in Pb [11].Reprinted with permission of The Minerals, Metals & Materials Society. . 202.8 Examples of (a) diffusivity profiles D(z) and (b) binding energy profileEb(z), δ and Eb0 being the grain boundary width and the maximum bind-ing energy, respectively, (c) solute drag pressure dependence on bound-ary velocity; data points are the solutions to Eqs. (2.11) and (2.12) whilethe dashed lines are the model from Eq. (2.13). . . . . . . . . . . . . . . 222.9 The velocity-driving pressure relationship predicted by the Cahn solutedrag model under different binding energies. The curve labelled ’2Eb0’ il-lustrates a hysteresis denoting different paths for abrupt break-away tran-sitions, the dotted part representing the so-called unstable regime. . . . . 232.10 (a) The pinning force of a particle, (b) the total pinning pressure due tomultiple particles that are in contact with the boundary, i.e. particles lo-cated within the interaction zone, i.e. the volume that extends to a dis-tance rp from both sides of the boundary. . . . . . . . . . . . . . . . . . 262.11 The velocity-driving pressure relationship under different particle pin-ning models (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.12 Snapshots that show correlated atomic motion governing the migrationof grain boundaries, from (a) MD simulations [183] (b) high-resolutionTEM [184], and (c) an illustrative sketch. Column (a) shows the view nor-mal to the boundary plane of the [001] θ = 28◦ boundary in Au at 500K. Column (b) shows the [112] θ = 14◦ grain boundary in Au at 893 K,the migration direction highlighted with an arrow. Column (c) shows asketch of correlated motion of grain boundary atoms in a tilt CSL bicrys-tal, dashed lines indicating the position of grain boundary. . . . . . . . . 35xii4.1 (a) A KMC simulation box where two crystals of the same orientationare separated by a flat {001} interface, the interface being considered as aKMC representation of the grain boundary, (b) the initial position of thegrain boundary, (c) a flipping event occurred at a spin i whose neighboursare j, (d) the boundary average position advances upon several flippingevents of the spins belonging to one of the grains. . . . . . . . . . . . . . 404.2 (a) A portion of symmetric tilt boundary in a CSL bicrystal. CSL unitcells of grain A and grain B are indicated and are rotationally related toeach other through correlated atomic motion, shown in Figure 2.12(c);(b) a KMC representation of the bicrystal in (a), where CSL unit cells arerepresented by black (S = 12 ) and white (S = − 12 ) spins, see also Figure4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 (a) An octahedral site in a bcc lattice surrounded by its six neighbour-ing solvent atoms, (b) Different types of octahedral site as defined by Eq.(4.2), see also Table 4.1, (c) Two types of terrace site (ω = 1.0) found in aflat interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 (a) The type-0 simulation cell consisting of a single bcc crystal with di-mensions of L30, (b) the type-1 simulation cell containing a U-shaped half-loop grain boundary, and (c) its three-dimensional view, (d) the type-2simulation cell containing two planar grain boundaries. . . . . . . . . . . 494.5 An illustration of convex-hull technique to determine the shape of shrink-ing grain in a type-1 bicrystal. . . . . . . . . . . . . . . . . . . . . . . . 584.6 A technique implemented in this work to calculate the average curvatureκ(t) of a migrating curved boundary at a given time t, see text. . . . . . 594.7 An illustration of dendrogram, showing the progress in ∆σ2 (see text) asclusters grow with the number of iteration. λc is the clustering cut-offparameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.8 (a) The position of atoms that constitute a cluster, (b) the convex-hull con-struction to determine the volume and radius of a cluster. . . . . . . . . 635.1 The boundary roughness as a function of temperature for different bound-ary dimensions, NXNY. The roughening temperature Tc is indicated. . . . 66xiii5.2 The velocity of the grain boundary as a function of driving pressure for apure bicrystal at different temperatures. . . . . . . . . . . . . . . . . . . 675.3 (a) Grain boundary velocity as a function of driving pressure for bicrys-tals containing different solute contents, the diffusivity of which has anactivation barrier Qd of 26 kJ mol−1, (b) The calculated drag pressure cal-culated and plotted versus velocity based on data in (a). The solid curveswere drawn using Eq. (5.2). . . . . . . . . . . . . . . . . . . . . . . . . 695.4 Drag pressure plotted as a function of (a) velocity, and (b) normalizedvelocity for bicrystals with a constant C0 of 3at%, but with different solutediffusivities. The normalized velocity is the velocity multiplied by theratio of lattice parameter and diffusivity. Dashed lines indicate the fitbetween simulation results and the modified solute drag model, Eq. (5.2),the fit parameters being given in Table 5.3. . . . . . . . . . . . . . . . . 715.5 Snapshots of grain boundary cross-sectional view during its steady-statemigration at the velocity corresponding to maximum drag pressure andinteracting with (a) slow-diffusing solute and (b) fast-diffusing solute.Yellow pixels indicate interfacial solvent atoms i whose height is equal tothe average position of the boundary (hi = h). White, orange and salmonpixels indicate hi = h − a0, hi = h + a0 and hi > h + a0, respectively.Regions corresponding to bulges in (b) are highlighted. . . . . . . . . . 725.6 (a) The time-average boundary roughness (Eq. (5.1)) plotted against thenormalized boundary velocity, i.e. velocity × lattice parameter/bulk dif-fusivity, for varying solute diffusivity; (b) A time-resolved trace of bound-ary roughness corresponding to the peak drag pressure for interface in-teracting with fast-diffusing and slow-diffusing solutes, i.e. points P1 andP2 in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.7 (a) The drag pressure data from Figure 2.8(c), now being fit to the mod-ified solute drag model. (b) The PFC simulation results from the litera-ture [161] on the drag pressure variation with velocity for different solutediffusivities, being fit to both the original and the modified solute dragmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76xiv6.1 The lattice parameter of bcc Fe crystal, aT, as a function of temperature be-tween 800 and 1200 K, showing good agreement with experiments [237,238] and molecular dynamics simulations that employed the same poten-tial used in this work (the Ackland04 potential) [206] and a different typeof potential (the Dudarev07 potential) [207]. . . . . . . . . . . . . . . . . 806.2 (a) An example of the grain shrinkage due to curvature-driven boundarymigration, data set being taken from a 25-nm coherent loop at 900 K (b)Arrhenius plot of reduced mobilities, dashed lines indicating the Arrhe-nius fit (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3 The shape of (a) 25-nm diameter coherent loop and (b) 35-nm diameterincoherent loop during their steady-state migration (> 1 ns) at 1000 K,atoms coloured based on their potential energy. . . . . . . . . . . . . . . 826.4 The velocity of planar incoherent twin boundaries driven via ADF tech-nique by 53.7 MPa at 900 K. (a) Average order parameter profile η(z) at1.5 ns and (b) the corresponding profile of order parameter gradient ∆η(z)∆zat 1.5 ns, (c) the evolution of grain boundary positions, i.e. position of thetwo peaks in panel (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5 (a) The velocity of planar incoherent twin as a function of driving pres-sure at several temperatures, linear regression being applied for P < 80MPa, (b) Arrhenius plot of the absolute mobilities of the planar incoherenttwin, obtained from the ADF technique . . . . . . . . . . . . . . . . . . . 846.6 (a) The average position of the incoherent and coherent twin as a func-tion of time, (b) the mean-squared displacement (MSD) obtained fromthe evolution shown in (a), error bars and lines representing the standarddeviation and linear regression, respectively. . . . . . . . . . . . . . . . . 856.7 The distribution of boundary displacement at 1000 K for several time in-tervals τ, evaluated from the h(t) of (a) the planar coherent twin bound-ary, and (b) the planar incoherent twin boundary, shown in Figure 6.6(a). 866.8 Arrhenius plot of the absolute mobility of the planar incoherent twin,computed from the RW technique and the ADF technique. Fitting pa-rameters are given in Table 6.3. . . . . . . . . . . . . . . . . . . . . . . 87xv6.9 (a) Grain boundary energy γ and (b) absolute mobility M as a functionof inclination ϕ computed from the simulations at several temperatures;dashed lines in (a) and (b) are fitting curves for each temperature. . . . . 886.10 Superimposition of the predicted shapes of curved boundaries [solid redlines, see Eq. (6.4)] into the shapes observed from simulations (i.e. Figure6.3) for (a) the coherent loop and (b) the incoherent loop at 1000 K. TheΓ (φ) and M (φ) used in the continuum model are shown on the right sideof the figure. Note that the Γ (φ) and M (φ) data set are the same for bothboundaries, but shifted along the φ-axis by 90◦. . . . . . . . . . . . . . . 916.11 Geometrical features of a curved boundary as a visual for Eq. (6.5). . . . 926.12 Snapshots taken from the migration of (a) the curved incoherent twin, (b)the curved coherent twin, (c) the planar boundary of 27◦ inclination, and(d) the planar incoherent twin at 1000 K. Arrows indicate displacementvectors of each atom from 0.5 to 0.9 ns after the steady-state migrationstarted. The planar boundaries in (c) and (d) were driven using the ADFtechnique. Dashed boxes indicate the random shuffling of atoms, whiledotted ovals indicate the cooperative atomic motion (see text). . . . . . . 947.1 (a) The evolution of mean-squared displacement of a single He atom attetrahedral sites in bulk Fe at several temperatures, (b) the bulk diffusivityof He atom as a function of temperature, calculated from (a) via Eq. (4.27). 1007.2 Snapshots of a portion of the type-0 cell with 1.0at% He at 1000 K cap-tured at 3 ns showing (a) both Fe (black) and He (magenta) atoms, (b)only He atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.3 The cluster size distribution in the bulk crystal of Fe captured after 3 nssimulations for bulk concentration C0 of (a) 0.1at%, (b) 0.5at%, and (c)1.0at%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.4 (a) Average cluster radius rc and (b) r2c as a function of cluster size c ob-tained from cluster characterization at 1000 K. . . . . . . . . . . . . . . . 1037.5 The profile of the binding energy of a single He atom to several planarboundaries calculated by molecular statics at 0 K, the regime of non-zerobinding energy being defined as the boundary width δ. . . . . . . . . . 104xvi7.6 Grain boundary diffusivities of a single He atom as a function of tempera-ture for grain boundaries of different inclinations. Bulk diffusivities fromFigure 7.1(b) have been included for comparison. Dashed lines are theArrhenius relationship, D = D0 exp (−Qd/RT), with parameters givenin Table 7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.7 (a) A snapshot of a 44.7◦-bicrystal containing 0.5 at% He taken at 3 ns,showing only He atoms coloured according to cluster size; (b) evolutionof cluster size distribution in the bulk, and (c) at the boundary portion ofthe bicrystal shown in (a). . . . . . . . . . . . . . . . . . . . . . . . . . 1067.8 (a) Average segregation for different boundaries as a function of C0, errorbars are the standard deviation (b) the effective binding energy accordingto the Langmuir-McLean model, Eq. (7.2). . . . . . . . . . . . . . . . . . 1077.9 (a) The evolution of average boundary position, h(t), for the cluster-enriched (C0 = 0.5at%) 44.7◦-boundary at 1000 K, t? indicating the timeat which cluster size distribution at the boundary reached steady state(see text), (b) h(t) for the pure boundary of the same inclination. . . . . 1097.10 (a) Distribution of boundary displacement at an interval τ of 0.2 ns, forthe pure and cluster-enriched (C0 = 0.5at%) 44.7◦-boundary at 1000 K; themean-square displacement as a function of τ of (b) the pure and (c) thecluster enriched boundaries. . . . . . . . . . . . . . . . . . . . . . . . . 1107.11 (a) Effective mobility of pure and enriched grain boundaries, (b) Clusterdrag contribution, M−1clust in Eq. (7.3), as a function of bulk solute content. 1117.12 Assumed profile of (a) binding energy and (b) diffusivity to estimate αfrom Eq. (2.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.1 Different cases of solute configuration investigated in this chapter. . . . . 1168.2 A series of snapshots of the boundary migration in the type-i bicrystalcontaining 100 ppm He. The oval regimes indicate the trapping of segre-gated He atoms, see text. . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.3 Atomistic mechanism of the trapping of segregated He atoms. . . . . . . 1198.4 Evolution of the shrinking grain volume in the type-i bicrystals for differ-ent concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120xvii8.5 Evolution of the shrinking grain volume in the type-i bicrystals for C0 of400 to 700 ppm from two simulation runs, differing in the initial place-ment of He atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.6 (a) The steady state portion of the shrinkage evolution from Figures 8.4and 8.5, (b) the steady state velocity as a function of C0 for the type-ibicrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.7 Snapshots of the type-i boundary for different C0 at a time during theirsteady-state migration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238.8 (a) The local boundary curvature profile κ(z) for the shapes presented inFigure 8.7, (b) the evolution of average curvature κ(t) for the case of 0(pure) and 500 ppm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.9 The effective average curvature of the boundary from the type-i simula-tions, 〈κ〉eff, normalized by the average curvature of the pure boundary,〈κ〉pure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.10 An illustration of the procedure to determine the number of clusters inthe curved boundary: (a) a snapshot of type-i bicrystal with 500 ppm He(run #1) at 4 ns, (b) Fe atoms that make up the boundary, (c) He clustersclassified based on their location in the bicrystal, (d) He clusters in thecurved boundary, characterized based on their size, see Section fordetails. The size of atoms is exaggerated for clarity. . . . . . . . . . . . . 1258.11 (a) Total number of He atoms during the period of steady-state migrationfor the type-i bicrystal with 500 ppm He, (b)-(g) the cluster size distribu-tion taken from different times during such a period. . . . . . . . . . . . 1268.12 The average number of boundary clusters Nc vs. cluster size c, taken byaveraging the size distribution captured every 0.1 ns during the period ofsteady-state migration, the error bars representing the standard deviationover the same time period. . . . . . . . . . . . . . . . . . . . . . . . . . 1278.13 (a) The average number of boundary He atoms as a function of C0 in type-i bicrystals, (b) the boundary velocity as a function of the average numberof He atoms, i.e. the ordinate from Figure 8.6(b) vs. the ordinate from part(a) of this figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128xviii8.14 (a) The shrinking grain volume evolution in the type-s bicrystals for dif-ferent concentrations, (b) the velocity of unpinned boundaries at steadystate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.15 Snapshots of the type-s boundary that initially contained (a) 30 and (b) 50He atoms, taken at a time during steady-state migration after unpinningoccurs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.16 The simulation results of the type-c boundary. (a) The steady-state effec-tive boundary velocity, Veff, and (b) the average number of monosizedclusters that are present at the boundary, Nc, as a function of the bulkcluster concentration C0, dotted lines drawn as a visual guide, (c) the or-dinate from (a) plotted as a function of the ordinate from (b), dashed linesindicating the linear fit following Eq. (8.1), see text. . . . . . . . . . . . . 1318.17 Pinning pressure of a single cluster ρc as a function of cluster size. . . . . 1328.18 Average number of boundary clusters, Nc, as compared to the expectednumber of boundary clusters for a given number of bulk clusters (dashedlines), i.e. C0vs/vbulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358.19 Comparison between the boundary velocities from the type-i simulations(Figure 8.6(b)) and those predicted by the effective cluster pinning model(Eq. (8.5)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137A.1 Block 1 of an input file for MS calculation. . . . . . . . . . . . . . . . . . 175A.2 Block 2 of an input file for MS calculation. . . . . . . . . . . . . . . . . . 176A.3 Block 3 of an input file for MS calculation. . . . . . . . . . . . . . . . . . 178A.4 Block 4 of an input file for MS calculation. . . . . . . . . . . . . . . . . . 179A.5 Block 5 of an input file for MS calculation. . . . . . . . . . . . . . . . . . 180A.6 Block 1 of an input file for MD simulation. . . . . . . . . . . . . . . . . . 180A.7 Block 2 of an input file for MD simulation. . . . . . . . . . . . . . . . . . 181A.8 Block 3 of an input file for MD simulation. . . . . . . . . . . . . . . . . . 181A.9 Block 4 of an input file for MD simulation. . . . . . . . . . . . . . . . . . 182A.10 Block 5 of an input file for MD simulation. . . . . . . . . . . . . . . . . . 183xixA.11 The content of the neighbour list files for computing the order parameterin LAMMPS, based on Table A.2, at 1000 K (aT at 1000 K is 2.882Å, seeFigure 6.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.12 An example of the log file from an MD simulation, based on the input filediscussed in Section A.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . 189xxList of Symbols and AbbreviationsRoman SymbolsA Grain boundary area [m2]A1, ... , A4 Model parameters in the Read-Shockley model: A1, A3 [MPa], A2,A4 [dimensionless],a0 Lattice parameter of a bcc Fe crystal at 0 K [Å]aT Lattice parameter of a bcc Fe crystal at a finite temperature T [Å]B1, ... , B3 Parameters of the fitting curves to describe the variation of absolutemobility and energy of inclined boundary on the inclination angle.C(z) Solute concentration profile along the boundary normal z [at%],[ppm]C0 Bulk solute concentration [at%], [ppm]Cgb Solute concentration at the grain boundary [at%], [ppm]C0gb Fraction of grain boundary sites available for segregated atoms atsaturation [at%], [ppm]c Cluster size, i.e. the number of atoms in a cluster [atoms]D(z) Diffusivity profile along the boundary normal z [m2 s−1]Dbulk Bulk diffusivity of a single solute atom [m2 s−1]Dgb Grain boundary diffusivity of a single solute atom [m2 s−1]DU Diameter of a half-loop curved grain boundary [nm]xxids Length of a segment in a curved grain boundary [nm]Eia Activation energy of an atomistic event i [eV/atom]Eb(z) Binding energy profile along the boundary normal z [kJ mol−1],[eV]Eb0 Maximum solute binding energy to the boundary [kJ mol−1], [eV]E∗b0 Effective binding energy obtained from the segregation level Cgb viathe Langmuir-McLean model [kJ mol−1], [eV]Etot Total potential energy of an MD box upon relaxation using molecu-lar statics [eV]Et→s Energy change that accompanies the transition of a He atom froman interstitial site to a substitutional site [eV]∆E Energy difference between the initial and final state of a KMC event[eV]eT Potential energy of an Fe atom in the bulk of b.c.c. Fe crystal at afinite temperature T [eV]Fext Forces imposed to an atom due to an externally applied force [N]FZ Pinning force imposed by a particle on a grain boundary [N]Fint Forces imposed to an atom by other atoms [N]FEbulk,t/o/s Formation energy of a single He atom occupying a tetrahedral (t),octahedral (o) or substitutional (s) site in bcc Fe [eV]fs Ratio of the total volume of particles in contact with the grain bound-ary and the swept volume [dimensionless]fv Volume fraction of pinning particles [dimensionless]h(t) Average position of grain boundary at an instantaneous time t [Å]xxii∆h2Mean-squared displacement of grain boundary position [Å2]K Cluster pinning coefficient, independent of cluster size [MPa]kcr Geometric pinning coefficient [Pa m−2]kB Boltzmann constant, 1.38 × 10−23 J K−1kr Proportionality constant relating the squared radius of a cluster r2cand the cluster size c via r2c = krc [Å2]kZ Zener pinning coefficient [dimensionless]LX, LY, LZ Dimensions of an MD simulation box [nm]M Absolute mobility of a planar grain boundary [m4 J−1 s−1]M∗ Reduced mobility of a curved grain boundary [m2 s−1]M0 Pre-exponential factor of the absolute mobility [m4 J−1 s−1]M∗0 Pre-exponential factor of the reduced mobility [m2 s−1]Mclust Cluster contribution to the effective absolute mobility of an enrichedplanar boundary [m4 J−1 s−1]Meff Effective absolute mobility of an enriched boundary [m4 J−1 s−1]Meff Average effective absolute mobility of a curved boundary in a bi-nary system [m4 J−1 s−1]Mpure Absolute mobility of a clean planar boundary [m4 J−1 s−1]Mpure Average absolute mobility of a curved boundary in a pure system[m4 J−1 s−1]Nact Areal density of active sites for shuffling in the boundary [m−2]Nc Average number of clusters with a given size cxxiiiNevent Number of events in a KMC simulationNFe Number of Fe atoms in an MD boxNHe Number of He atoms in an MD boxNp Number of pinning particlesNrx Number density of recrystallization nuclei [m−3]NX, NY, NZ Number of unit cells in X, Y and Z directions contained in a box forKMC simulationsP Driving pressure for grain boundary migration [MPa]P Average driving pressure of a curved boundary migration [MPa]Ppure Average driving pressure of a curved boundary in a pure system[MPa]Peff Average driving pressure of a curved boundary in a binary system[MPa]Pd Solute drag pressure [MPa]Pint The driving pressure to drive the boundary at its intrinsic mobility[MPa]PZ Zener pinning pressure [MPa]p, q Parameters of the fitting curves to describe the variation of absolutemobility of inclined grain boundary on the inclination angle.Qd Activation energy of diffusion [kJ mol−1], [eV]Qm Activation energy of grain boundary migration [kJ mol−1], [eV]RA, B A reference list of vectors indicating the position of nearest neigh-bours of an atom in grain A, BxxivR Ideal gas contant, 8.314 J mol−1 K−1ri Vector position of atom irc Radius of a cluster [Å]rgb Grain boundary roughness [nm]rp Radius of a pinning particle [Å]S Spin parameter of main lattice atoms in the KMC model, ±0.5 [di-mensionless]Tφ Unit tangent vector of a boundary segment at a given inclination φT Absolute temperature [K]Tc Roughening temperature [K]t Time [s]U Interatomic potential energy [eV]Umax Maximum added energy per atom in the ADF technique [eV]UADFi Extra energy added to an atom i due to the ADF method [eV]V Grain boundary velocity [m s−1]Vcrit Critical boundary velocity at which the particle pinning behaviourtransition from reducing the boundary mobility to reducing the driv-ing pressure for migration, according to Novikov model [m s−1]Veff Effective velocity of an enriched grain boundary [m s−1]Vpure Velocity of a solute-free grain boundary [m s−1]vbulk Volume of the bicrystal portion that contains monosized clusters ofsize c [nm3]xxvvc Volume of a cluster [Å3]vs Swept volume, or volume within which Np particles are in contactwith the grain boundary [nm3]vT Volume of a simulation cell at temperature T [nm3]vtot Volume of a sample containing Np number of particles [nm3]vU Volume of the shrinking grain of a bicrystal that contains a curvedgrain boundary [nm3]X, Y, Z Vectors representing mutually orthogonal axes of a crystalXrx Fraction of recrystallized grains [dimensionless]z Compressibility factor [dimensionless]Greek Symbolsα, β Cahn’s solute drag parameters: α [m−4 J s], β [m−1 s]Γ Grain boundary stiffness [J m−2]γ Grain boundary energy [J m−2]γpm Interface energy between a particle and the matrix [J m−2]γmin, γmax Parameters of the fitting curves to describe the variation of energyof inclined grain boundary on the inclination angle [J m−2]δ Grain boundary width [nm]ηAi Normalized order parameter of atom i with grain A as a referenceΘ Contact angle between a particle and a grain boundary [◦]θ Misorientation angle [◦]ϑ A randomly generated number between 0 and 1xxviκi Local curvature of a grain boundary segment i [nm−1]κ(t) Average curvature of the boundary at a given time t [nm−1]〈κ〉eff Effective average curvature of the boundary in a binary system [nm−1]〈κ〉pure Net average curvature of the boundary in a pure system [nm−1]Λ Sum of the events’ rate at a given step of KMC simulation [s−1]λc Cutoff parameter for characterizing cluster size [nm2]ν→ Rate of forward boundary migration in the KMC model [s−1]ν← Rate of reverse boundary migration in the KMC model [s−1]νsolute Rate of interstitial solute jump in the KMC model [s−1]νD Debye frequency [s−1]νi Rate of a KMC event, i [s−1]ξAi Unscaled order parameter of atom i with grain A as a referenceρc Pinning pressure of an individual cluster of a given size c [MPa]σ2k In-cluster variance of a cluster k [nm2]ς The occupancy of an interstitial site in the KMC model, 0 or 1τ Time period during which MSD analysis is performed [ns]ϕ Inclination angle relative to the planar coherent twin boundary [◦]φ Inclination angle [◦]Ω Atomic volume [m3]ω Parameter indicating the binding type of a boundary interstitial sitewith a solute atom in the KMC model such that the effective bindingenergy is equal to ωEb0 [dimensionless]xxviiAbbreviationsADF Artificial Driving ForceCSL Coincidence Site LatticeDFT Density Functional TheoryDGSOS Discrete Gaussian Solid on SolidEAM Embedded Atom ModelsGB Grain BoundaryHSLA High-Strength Low AlloyJMAK Johnson-Mehl-Avrami-KolmogorovKMC Kinetic Monte CarloLAMMPS Large-scale Atomic/Molecular Massively Parallel SimulatorMD Molecular DynamicsMMC Metropolis Monte CarloMS Molecular StaticsMSD Mean-Squared DisplacementNFA Nanostructured Ferritic AlloysPFC Phase Field CrystalsRW Random WalkSIA Self Interstitial AtomsTEM Transmission Electron MicroscopeTMCP Thermo-Mechanical Control ProcessingxxviiiAcknowledgmentsIt has been a great honour of mine to work with many dedicated and brilliant peo-ple throughout the course of this project. I have been privileged to be welcome asa student of Prof. Matthias Militzer and Prof. Chad W. Sinclair. Their guidance,encouragement and persistent challenge to improve upon myself have helped mepainting a much clearer shape of my future. I am particularly grateful for thesupport I have received in presenting my work regularly in scientific conferencesduring which I had the opportunity to meet and learn from many excellent re-searchers.The readership of my departmental committee members, Prof. Daan Maijerand Prof. Warren Poole, is acknowledged with gratitude. I am also grateful to myuniversity examiners, Prof. Rizhi Wang and Prof. Alireza Nojeh, and my externalexaminer, Prof. Charlotte Becquart from Université des Sciences et Technologiesde Lille, for their time and critical feedback.My sincere gratitude is due to Prof. Jeffrey J. Hoyt and Wilson Song from Mc-Master University for their fruitful discussion and warm hospitality during mystay at their computational materials science group. I learned a great deal aboutmolecular dynamics simulation from them, particulary the random walk and theartificial driving force techniques for extracting grain boundary mobilities.It was a pleasure to have had stimulating conversations with Dr. MichaelGreenwood from CanmetMATERIALS and Prof. Michel Perez from Universitéde Lyon, INSA-Lyon, during the early stage of this project. Insightful commentsby Prof. Michael Baskes, presently with Mississippi State University, during TMS2015 Annual Meeting and Prof. Jörg Neugebauer from Max-Planck-Institut fürEisenforschung, during his visit to UBC are much appreciated as well.Dr. Doug S. Phillips from University of Calgary deserves a great deal of thanksfor his time and assistance in ensuring my customized LAMMPS software was upxxixand running on Parallel, a Westgrid computing cluster.The warm and efficient help of Fiona Webster, Michelle Tierney, Norma Donaldand Mary Jansepar at Materials Engineering Department, UBC have enabled menavigating around the department seamlessly.Adding rich succulent icing to this proverbial academic cake is the good fortuneof camaraderie of which I have been lucky to be a part: Jennifer Reichert, a greatfriend and underrated lunchtime psychiatrist with whom I have shared many ex-citing adventures and who has, hopefully, now forgiven me for appropriating herdesk; Michael Mahon, a talented person with a great penchant for cuisine prepa-ration; Beth, Seb, Sina, Guillaume L, Guillaume B, Hao, Liam, and Thomas, withwhom I have shared drinks and memorable moments at UBC. I am also gratefulto have had collaborations with dedicated fellow students on many projects underthe Joint Student Chapter umbrella: Lina, Shima, Victor, Millie, Luis, Sophia, MJ,Vivian, Rafay, Elaine, and Masoumeh.Ultimately, this work has been so much more than the fruit of my interactionwith colleagues and authorities in academia. I have been blessed to have met JaredRudek, whose presence has been nothing less than a wonder. Through him I cameto know Loraine and Jim Rudek, who have graciously welcome me to their warmhome. I will always cherish many pleasant memories at Chez Rudek, particularlyFriday dinners and Sunday brunches. The last few months of this program havebeen manageable thanks to their support.Last and certainly never the least, I am proud to thank my parents, H. AhmadDimyati and Dra. Hj. Endang Purwaningsih, Apt. in Indonesia and my sister,Diah Ayu Istiqomah in Ghana. Their unconditional love, unlimited support andcare has made this journey possible. Though we may not see each other as often asyou deserve, you are always in my thoughts. Terima kasih untuk Papa dan Mamaatas dukungan, doa dan pengertiannya. Mil sayang Papa, Mama dan Kubo.xxxChapter 1IntroductionAlloying elements hold a vital role in controlling microstructures and, therefore,bulk properties of engineering materials. The average grain size, for example, de-pends on the rate at which grain boundaries, the interface between crystals of dif-ferent orientation, migrate as they interact with solutes. In the absence of solute,the velocity at which a grain boundary migrates increases linearly with the magni-tude of driving pressure causing the grain boundary migration [1, 2]. The origin ofthese driving pressures can be attributed to elastic strain [3, 4], curvature [5, 6, 7]or applied magnetic field [8, 9].The rate of migration changes as grain boundaries interact with alloying ele-ments, existing either as stationary precipitate particles or as mobile solute atomsin solid solutions. The presence of solute even in very small concentrations influ-ences the kinetics of migration, drastically reducing grain boundary velocities bya few orders of magnitude [10, 11, 12, 13]. Solute segregation to grain boundariesis energetically preferable, in general, as evident from solute enrichment in thevicinity of grain boundaries in most systems [14, 15].Solute segregation may lead to an improvement or degradation of materialproperties. A practically important example of the former is the role of niobium(Nb) as a microalloying element in high-strength low alloy (HSLA) steels. The Nb-microalloying of steels has enabled modern thermo-mechanical control processing(TMCP) since it delays austenite recrystallization to a much larger degree thanother alloying elements [16]. Such a strategy allows the pancaked austenite mi-crostructure to be retained at the end of controlled rolling, thereby promoting theformation of a fine-grained ferrite structure during the subsequent austenite de-composition. This is industrially desirable because there is an associated increasein strength and toughness due to these microstructural features [17].An example of the deleterious effect of solute segregation is illustrated with the1role of helium in irradiated steels. Helium (He) atoms, a byproduct of transmu-tation reactions, tend to form clusters in steel-based reactor pressure vessels [18]due to their low solubility [19]. Cluster formation is significantly enhanced arounddefects such as grain boundaries [20], and may eventually progress into the nucle-ation of nano-voids, or bubbles, above a certain critical size [21]. At longer times,helium bubbles coarsen into visible pores [22], which are precursors to structuraldeteriorations such as swelling [23] and intergranular cracking [24]. Probing thisphenomenon to develop an accurate view of the bubble-boundary interaction iscritical because the interaction affects the structural integrity of technologicallyimportant materials such as Nanostructured Ferritic Alloys (NFA), material can-didates for future generation of nuclear reactors [25]. NFAs are designed to havean average grain size that is within tens of nanometers to maximize their strength[26]. These alloys will be exposed to a high dose of dissolved helium during theirservice [27, 28, 29]. Studies that investigate the interaction between helium clustersand grain boundaries in iron are paramount to this application, particularly for thedevelopment of models for controlling the microstructure of these alloys.These examples briefly illustrate the multitude of underlying factors that gov-ern the interaction between migrating grain boundaries and solute atoms. Becausesuch an interaction is of vital importance to the microstructural evolution of ma-terials, it has received a number of theoretical treatments proposed by many au-thors over the decades. Theories on solute-boundary interaction can be groupedinto two categories based on the assumption about the size and mobility of soluteatoms. Solute drag theories [30, 31, 32, 33, 34, 35] investigate the effect of non-interacting solute atoms on a moving grain boundary. Segregation of solute atomsgenerates a drag pressure that has a non-linear dependence on the grain boundaryvelocity. On the other hand, theories on particle pinning [36, 37, 38] consider thepinning effect of stationary precipitates on grain boundary motion. The modelspredict a pinning pressure that is independent of the velocity.There is a gap between models for solute drag and those for particle pinningin which a situation where grain boundaries interacting with solute clusters hasnot been fully explored. Solute clustering, though less studied, is of significantmetallurgical importance, e.g. Nb clusters in HSLA steels or He clusters in reactor2vessels. A cluster typically consists of a few interacting solute atoms [39], appearsindistinguishable relative to the matrix in TEM images [40], and has no distinctcrystal structure [41].The continuum approach to modelling, which is the framework of classical so-lute drag and particle pinning models, may not be able to treat clusters properlybecause the properties of clusters, e.g. their size and diffusivity, depend on theirlocal atomic configuration [26, 42]. Due to the atomistic nature of clusters, simula-tion tools at the atomic scale provide a feasible avenue to explore their behaviourduring their interaction with a migrating grain boundary. These tools facilitate awell-controlled environment for probing atomic-scale mechanisms that are other-wise inaccessible through experiments. A variety of atomistic simulation tools isavailable, the application of which depends on the time and length scale of theproblems under consideration.The present work aims to develop an atomistically informed understanding ofthe solute cluster-grain boundary interaction, a problem involving events that spanover a wide range of spatiotemporal dimensions. For the binary Fe-He system, thisproblem has a length scale that extends from the characteristic distance of a singleatomic hop (3 to 5 Å) to measurable displacements of grain boundaries (> 10 nm).The time-scale of the problem demands the capability of resolving atomic vibration(1 fs). It is emphasized that these scales represent a lower limit since helium is afast diffuser in bcc iron [27]. For other binary systems, e.g. the Fe-Nb system [43],the time- and length-scale of this problem can be three to four orders of magnitudehigher than those for the Fe-He system.Two types of atomistic simulation tools will be utilized in this work: kineticMonte Carlo (KMC) and molecular dynamics (MD). KMC simulations permit theexploration of parameter space of atomic-scale processes such as grain boundarymigration in alloys. In order to implement this tool, however, the basic propertiesof the material (e.g. lattice parameter and activation energy for grain boundary mi-gration) and the governing atomistic events must be known in advance. While thematerial properties can be obtained from experiments, it is often the case that theatomistic events that are relevant for KMC simulations may not be readily known.A useful tool to access such atomistic information is MD simulations. A typical3MD simulation involves the evolution of approximately 103 to 106 atoms (equiv-alent to 10 to 104 nm3) within a time period of 10 to 100 ns. To complement thelimited spatiotemporal scale of MD simulations, KMC simulations can be devel-oped based on the information derived from MD simulations. The KMC methodallows an increase of two to three orders of magnitude in terms of the domain sizeand time scale of simulations.This work begins by considering a binary model material that behaves as anideal solid solution. The parameter space of solute-grain boundary interaction insuch a system will be explored using KMC simulations. The focus is then shiftedtowards the iron-helium system as implemented in the framework of MD simu-lations. As a benchmarking case for subsequent analyses, the dynamics of grainboundary migration in pure bcc iron crystals is first examined. This will be fol-lowed by a series of tasks investigating the behaviour of grain boundaries in bcciron as they interact with helium atoms. The segregation behaviour of heliumclusters and their size distribution in grain boundaries of iron will be discussed interms of their atomic-scale mechanisms and the kinetic properties of the cluster-boundary interaction. Finally, the retardation effect of clusters on the migration ofgrain boundaries is quantified.Conclusions derived from these atomistic simulations are discussed and as-sessed against relevant experiments and, most primarily, classical models. Basedon these discussions, an atomistically informed understanding on the cluster-grainboundary interaction is developed. Additionally, suggestion on the direction offurther research is presented.4Chapter 2Literature Review2.1 Grain boundary structure and energy2.1.1 Crystallographic constructionA grain boundary (GB) is the interface between crystals of different orientation.Figure 2.1(a) illustrates a bicrystal constructed from a single crystal, half of its por-tion having been rotated counterclockwise about [001] axis by an angle 12θ and theother half by −12θ. To describe the geometry of such a bicrystal, five crystallo-graphic parameters, or commonly referred to as macroscopic degrees of freedom,must be defined. The bicrystal contains a grain boundary of misorientation θ, i.e.the first macroscopic degree of freedom. The coordinate system of one of the grainsis defined by two independent mutually orthogonal axes. Each axis is specified bytwo independent parameters, which represent the remaining four macroscopic de-grees of freedom.A boundary plane that acts as a mirror plane for the grains on either side, e.g.Figure 2.1(a), is referred to as a symmetric boundary. Grain boundaries in a typ-ical microstructure, however, rarely exhibit such a symmetry element [44]; manyof them are considered asymmetric boundaries [45]. An example of an asymmet-ric boundary is presented in Figure 2.1(b), which shows a clockwise rotation ofthe symmetric boundary plane about the [001] axis by an angle ϕ. The generatedboundary is said to be at an inclination ϕ relative to the symmetric boundary.At the atomic-scale study of grain boundaries, it is desirable to construct theboundary directly from geometric information rather than following the sequen-tial scheme shown in Figure 2.1(a) and (b). A mathematical framework called thecoincident site lattice (CSL) enables such a construction [46, 47, 48, 49]. It createsa grain boundary by stacking a family of crystallographic planes from one crys-5tal adjacent to another crystal such that a fraction of the atomic sites at the grainboundary coincides with ideal sites of the lattice of both adjacent grains. The co-incidence among these lattice sites maintains the crystal periodicity except for thepresence of the grain boundary. Furthermore, crystal orientations of both grainscan be expressed as a set of orthogonal vectors, i.e. (X, Y, Z), each being a resultantof an integer multiple of the principal axes. Figure 2.1(c) and (d) show a symmetricand inclined boundary, respectively. Grain orientations in these cells are denotedby two sets of axes, A and B, each being generated from the CSL framework.Figure 2.1: (a) A construction of a symmetric grain boundary of misorientation θ, (b) anasymmetric boundary lying at an inclination angle ϕ from the boundary in (a), (c) and(d) the construction of the same symmetric and inclined boundaries via the CSL frame-work. CSL unit cells and its coincidence sites are indicated by the dashed squares andsolid points, respectively.6In addition to the macroscopic degrees of freedom, there are three microscopicdegrees of freedom representing the atomic-scale translation of one grain with re-spect to the other: two for the translation along the boundary plane and one for thetranslation normal to the boundary plane. Unlike their macroscopic counterparts,these parameters can not be choosen arbitrarily since they are determined from theminimization of grain boundary energy [50]. The normal component of the trans-lation, for example, is responsible for the interfacial volumetric expansion (or theexcess free volume), which controls to a large extent properties such as the grainboundary diffusivity and the material response under mechanical loading [1].If the basis of the crystal contains multiple types of atom, e.g. in ceramics, anextra degree of freedom is required [2]. This parameter specifies the position ofboundary plane with respect to the crystal basis.2.1.2 Grain boundary energyThe structure of a grain boundary determines its properties, for example, grainboundary energy. The grain boundary energy γ, i.e. the extra energy per unit areabrought about by the presence of boundary [2], is one of the key properties thatdrives many kinetic parameters of thermomechanical processing, e.g. the rate ofgrain growth [51] and the nucleation rate of precipitate particles [52].A number of different models have been proposed to describe grain boundarystructure and grain boundary energy [53, 54, 55, 56]. Among the most commonlycited models is the Read-Shockley model in which low-angle grain boundaries aretreated as a network of dislocations and their elastic energy interpreted using theframework of continuum elasticity theory [53]. The model predicts the depen-dence of boundary energy γ on misorientation θ as [53]γ = A1θ (A2 − ln θ) (2.1)where A1 and A2 are model parameters that depend on material properties, e.g.shear modulus, Poisson’s ratio and Burgers vector [53]. The Read-Shockley modelhas been shown to reasonably fit the grain boundary energy data obtained from ex-periments for small misorientations, i.e. θ ≤ 20◦ [57, 58, 59]. As the misorientation7increases, the model is unable to provide a good estimate for grain boundary en-ergy, as the dislocation cores overlap and, therefore, the framework of continuumelasticity can no longer be applied [60].To estimate the energy of high-angle boundaries, atomistic calculations can beimplemented in place of continuum elasticity. Figure 2.2(a) shows an example ofsuch a calculation [61] where the variation of grain boundary energy with misori-entation in the limit of small θ is consistent with the dislocation model. As a largerspace of misorientation is explored, several cusps in the energy-misorientation plotobtained atomistically are apparent and further rationalized as a result of symme-try elements of special boundaries [2]. The existence of a number of these cusps isconfirmed by comparing boundary energies obtained from experiments and thosefrom calculations [62].Figure 2.2: Computed grain boundary energies in Al (a) as a function of misorientationfor [001] grain boundaries, showing a trend consistent with the Read-Shockley model, Eq.(2.1), for small θ [61], and (b) as a function of inclination for a given misorientation θ of[110] grain boundaries, the trends deviating from the Read-Shockley model, Eq. (2.2) [63].The Read-Shockley model also offers a prediction on the dependence of grainboundary energy on inclination ϕ, i.e. [53]γ = A3 (sin ϕ+ cos ϕ)[A4 − sin 2ϕ2 −sin ϕ ln(sin ϕ) + cos ϕ ln(cos ϕ)sin ϕ+ cos ϕ](2.2)where A3 and A4 are parameters that also depend on material properties. Experi-8ments and simulations have shown that the model in Eq. (2.2) produces a poor pre-diction of grain boundary energy relationship with inclination, see Figure 2.2(b) foran example [63]. This is possibly attributed to the structure of inclined boundariesbeing asymmetric, thus rendering the dislocation network model inadmissible.2.1.3 Grain boundary roughening transitionGrain boundary energy γ generally varies weakly with temperature [64]. Figure2.3 shows a slight dependence of the energy of high-angle grain boundary on tem-perature for Pb at above 170◦C [65]. There is a considerable discontinuous jump ingrain boundary energy at around 170◦C, indicating a first-order transition. Such atransition is considered equivalent to a phase transformation, having been invokedto explain discontinuities in grain boundary properties observed from experiments[66]. By treating grain boundaries as two-dimensional phases, an analytical frame-work called grain boundary thermodynamics was constructed to rationalize thepossible types of structural transition that grain boundaries may undergo [67].Figure 2.3: Variation of high-angle grain boundary energy in Pb with temperature, as mea-sured from the dihedral angle experiments [65]An example of grain boundary phase transitions, which has been observed ex-perimentally, is the faceting/roughening transition [68]. A planar grain boundary9may transform into a faceted structure consisting of two or more facets that coexistalong their lines of intersections as temperature decreases. The faceted structuresbecome increasingly rough when temperature is increased. The temperature atwhich the transition from a faceted to a rough boundary occurs is referred to as theroughening temperature Tc. A similar transition has also been reported from re-cent atomistic simulations [66, 69]. The structural change of the boundary reportedin [66] has been associated with improved point defect absorption at grain bound-aries, a critical mechanism for radiation damage healing in irradiated metals. Theboundary phase transition has also been correlated [70] with the non-Arrheniusbehaviour of Ag diffusion in a Cu grain boundary reported from experiments [71].2.2 Grain boundary segregation2.2.1 Experimental observationThe lattice disorder at the grain boundary provides solute atoms with sites havingdifferent energies than those in the bulk of the crystal. This energy difference actsas the driving force for solute segregation, i.e. an accumulation of solute atomsin the form of a solid solution at grain boundaries [64]. In some cases, solute en-richment level may exceed the solubility limit, thereby promoting the nucleationof precipitate particles [50].Through their interaction with grain boundaries, segregated alloying elementscan introduce advantageous effects, e.g. grain refinement in steels due to Nb addi-tion [72], improvement of sinterability in yttria-doped alumina [73] and enhance-ment of creep resistance in alumina by Nd-doping [74]. In other situations, segre-gation can be detrimental to mechanical properties, e.g. Cu embrittlement due toBi segregation [75], sulphur-induced stress relief cracking in ferritic steels [76] andembrittlement of Ni superalloys from hydrogen segregation [77].An example of segregation that is particularly relevant for this work is the caseof helium (He) segregation in ferrite grain boundaries. The Fe-He alloy, brieflyintroduced in Chapter 1, is one of the important systems for future generationnuclear reactors. He atoms have a solubility of less than a few ppm in a ferritematrix [19] and are only present as byproducts of irradiation [18]. Since they are10smaller than Fe atoms, He atoms reside in tetrahedral interstitial sites of the matrix[78]. From an energetic perspective, however, He atoms have minimum energywhen they occupy substitutional sites of the matrix [79]. There is also a decreasein total energy associated with combining a number of He atoms into a heliumcluster [79]. Cluster formation occurs rapidly due to the fast diffusivity of heliumin iron matrix, the activation energy for He diffusion being 4 kJ mol−1 [80]. Clusterformation may eventually lead to the growth of visible pores that deteriorates thestructural integrity of ferrite matrix [18].Microstructural features such as grain boundaries have been postulated to actas traps for helium [81]. This has motivated the development of nanosized ferriticalloys (NFA) [25], candidates for future generation of nuclear reactors with a highvolume fraction of grain boundaries. The boundary trapping of helium is consis-tent with recent atomistic calculations that have reported the binding energies ofhelium to Fe grain boundaries as approximately 140 kJ mol−1 [82, 83].Segregated clusters demonstrate a different size distribution compared to bulkclusters [84]. The difference in size distribution has been attributed to many fac-tors, including grain boundary diffusivity [85] and the presence of carbide pre-cipitates at the boundary [84, 86]. It has been hypothesized that segregation canlower the amount of He in the bulk, thereby reducing the susceptibility of matrixto irradiation-induced swelling [22] despite potentially promoting grain bound-ary embrittlement [87]. A recent experimental study alluded to supporting thishypothesis [86] although the presence of nano-carbides may have enhanced theeffectiveness of grain boundaries for trapping helium clusters [84].2.2.2 Models for solute segregationSolute segregation holds a prominent role in materials performance [88], thus mo-tivating a large number of studies including the development of theoretical modelsto complement experimental efforts [14, 50, 89]. One of the simplest segregationmodels is the Langmuir-McLean model [14], which provides an estimation on thefraction of solute enrichment at grain boundary Cgb via11Cgb = C0gb[1+(1− C0)C0exp(Eb0kBT)]−1(2.3)where C0gb is the fraction of boundary sites available for solutes at saturation, C0is the bulk solute content (atomic fraction), Eb0 is the binding energy of solute tothe boundary, kB is the Boltzmann constant and T is the absolute temperature. Anegative Eb0 indicates an attractive solute-boundary interaction.The Langmuir-McLean model describes the equilibrium enrichment level in analloy system under the assumption of non-interacting solute atoms, i.e. an idealsolution. These conditions were attained in many systems, including In segrega-tion in Ni [90], Si segregation in α-Fe [91], and O segregation in Nb [92], where Eq.(2.3) has been shown to provide an adequate fit to the experimental observations.The agreement between solute segregation reported from experiments and thatpredicted by the Langmuir-McLean model is observed only for a limited numberof binary systems [50, 89]. This is not surprising as the Langmuir-McLean modelimplemented simplified assumptions. For example, grain boundaries are oftenrough [93], their thickness being about 3 to 4 lattice spacings [66], and they containsites with binding energies that depend on local atomic structure [94].Revisions to the original model have been made to account for these features.White and Coghlan adapted the Langmuir-McLean model by recognizing a spec-trum of binding energies that represent the topological variation of segregationsites [95]. Seah and Hondros extended the boundary into a multilayer structure[96], essentially drawing an analogy from the surface science community where asimilar model has been used, i.e. the Brunauer-Emmett-Teller (BET) model [97].By adapting the concept of site competition among adsorbed species from the reg-ular solution model proposed by Fowler and Guggenheim [98], Guttman devel-oped a model that considers the interactions between two co-segregating speciesin ternary solutions [99]. While these revisions have been found to provide a bet-ter description to a number of experimental results [50], they are insufficient whenapplied to dynamic situations, e.g. during materials processing [100]. In suchdynamic cases, the segregation is far from equilibrium and the role of processingparameters such as thermal and deformation cycle becomes important [101].122.3 Grain boundary migration in a pure system2.3.1 Motion of grain boundariesThe motion of grain boundaries is thermodynamically viewed as a kinetic pathto energy minimization, causing the growth of some grains at the expense of oth-ers. This is true only if the direction at which a boundary segment moves is notperpendicular to the normal vector of the boundary segment [102]. A theoreticalframework proposed by Cahn et al [103] distinguishes four modes of grain bound-ary motion, as illustrated in Figure 2.4.(a) Pure normal motion, or grain boundary migration. The motion of grainboundary is along a direction that is parallel to the boundary normal.(b) Pure grain translation (sliding) along the boundary plane. At low tempera-tures, grain sliding occurs if the applied shear stress exceeds a certain thres-hold [102].(c) Coupled motion between grain sliding and grain boundary migration. Anormal motion of a grain boundary can be triggered as a secondary effectof applied shear stress that is parallel to the boundary plane. The normalmotion has a velocity that depends on the pure sliding component and thecoupling factor that varies with misorientation. This mode of motion hasbeen reported experimentally in a limited number of cases [104, 105]. Thecondition for shear coupled motion is only satisfied in bicrystals with specialmisorientations and the coupling factor diminishes with increasing tempera-ture due to grain boundary premelting [106].(d) Grain rotation that is always accompanied by a relative grain translationalong the boundary plane. This process changes the misorientation betweentwo adjacent grains. It is driven by the local boundary curvature and hasbeen observed experimentally, for example, during plastic deformation ofnanocrystalline thin films [107, 108].Each mode of motion may operate at different stages during microstructuralevolution. While the overall motion of grain boundaries is likely to involve a com-13bination of these modes, the grain boundary migration is often the most dominantmode due to the associated energy decrease [2].Figure 2.4: Modes of grain boundary motion according to Cahn’s framework. Upper row:original bicrystals; lower row: bicrystal evolution due to boundary motion. (a) pure nor-mal motion, (b) pure sliding motion, (c) coupled translational-normal motion, (d) grainrotation. Thick and thin solid arrows indicate the direction of the driving force and theresulting motion, respectively. Dashed thin arrows in (c) and (d) indicate the direction ofsecondary coupled motions.It has been established from experiments that employed pure materials that theboundary velocity V is proportional to driving pressure P [3, 4, 5, 8],V = MP (2.4)where M is the absolute mobility of the boundary. It is also postulated that thelinear relationship between V and P prevails when driving pressures are low [109].2.3.2 Grain boundary mobilitiesMany seminal experiments have reported that grain boundary migration is a ther-mally activated process with a certain activation energy Qm [3, 4, 8]. Several mod-els have been proposed to explain such a behaviour by considering the rates ofrelevant atomistic events [1, 2]. The boundary migration is assumed to be the re-sult of atomic shuffling in the forward and reverse directions. In the absence of14driving pressure, the forward shuffling has a rate ν→ that is equal to that of thereverse shuffling ν← and the net velocity is zero. At a finite driving pressure P, theboundary velocity is proportional to the difference between these rates [109],V = nNactΩ [ν→ − ν←]= nNactΩ[νD exp(− QmkBT)− νD exp(−Qm + PnΩkBT)](2.5)where n is the average number of atoms transferred in a shuffle, Nact is the densityof active sites for shuffling on the boundary, Ω is the atomic volume, and kB is theBoltzmann constant. Since both n and Nact vary with grain boundary structure,they represent the anisotropy of grain boundary migration, i.e. the dependenceof migration on the crystallographic structure of the boundary. The rates ν→ andν← scale with the atomic vibrational frequency, or Debye frequency νD, and differonly by their energy barrier, i.e. Qm and (Qm + PnΩ), respectively, where P is thedriving pressure. In most cases, it is valid to invoke the low-pressure approxima-tion, i.e. PnΩ kBT [1, 2]. Through the Taylor polynomial expansion of Eq. (2.5),one can arrive at Eq. (2.4) where M is a function of temperature, following [109]M = M0 exp(− QmkBT)(2.6)and M0 is equal to n2Ω2NactνD/kBT. The rate theory model provides a simpleexplanation for two phenomenological trends: (1) the linear relationship betweenV and P for small P and (2) the Arrhenius dependence of the absolute mobility.Implicit in the rate model is an assumption that the driving pressure is constanteverywhere in the boundary. Any anisotropic behaviour observed in the bound-ary migration is attributed entirely to the anisotropy of absolute mobility via nand Nact in Eq. (2.5). This assumption may be reasonable for planar boundaries,but not so for curved boundaries. A large number of experiments employed theboundary curvature as the driving pressure for measuring grain boundary effec-tive mobilities, e.g. via the grain growth technique [3, 4, 11]. The driving pressurein this technique varies locally along the curvature as illustrated in Figure 2.5.15Figure 2.5: The migration of a U-shaped boundary, resulting from the collective migrationof segments that make up the curved portion of the boundary, e.g. segments A and B.The migration of a curved boundary is a result of the migration of individualsegments i, each being driven by a capillarity pressure Pi acting along the seg-ment’s normal, i.e. [110]Pi = Γiκi (2.7)where κi is the local curvature of segment i and Γi is the stiffness of segment i, i.e.a grain boundary property that varies with inclination ϕ following [110]Γ = γ+d2γdϕ2(2.8)The variation of stiffness Γ with inclination ϕ is an anisotropic property of ma-terial [110, 111]. Since the absolute mobility M is also a property that depends oninclination, both M and Γ are often reduced into a single parameter M∗, i.e. [112]M∗ = MΓ (2.9)where M∗ is referred to as the reduced mobility. Grain boundary mobilities ob-tained from experiments which employed capillarity as the driving pressure aretherefore reduced mobilities. Many of these studies, however, reported their re-sults in terms of the absolute mobilities, implicitly invoking an assumption that γis independent of ϕ, and therefore Γ = γ [3, 4, 11].162.3.3 Experimental measurementsTo measure grain boundary mobilities experimentally, bicrystal specimens are of-ten preferred over polycrystals [1]. The latter, while convenient in practice, arenon-ideal as the relationship between the structure and property of grain bound-aries can not be established from polycrystalline results.In bicrystal experiments, specimens contain exactly two grains separated by ageometrically well-defined planar or curved grain boundary. The planar shape, seeFigure 2.1(c) and (d), is the most ideal geometry since the migration kinetics canbe analyzed directly via the evolution of grain boundary position. Bicrystals withplanar boundaries, however, require substantial effort to manufacture with preciseorientations [1]. Additionally, this shape is mostly relevant for magnetically drivenmigration, practically limiting the type of materials that can be studied [113].Such a limitation is not present in bicrystals with curved boundaries, for exam-ple, the U-shaped half-loop bicrystal in Figure 2.5. The half-loop shape is prefer-able over other curved bicrystals (e.g. the wedge or quarter-loop shape) since thehalf-loop segment migrates as a whole, thus the driving pressure is maintainedas constant during the migration [114]. The main drawback of this technique isthat the measured mobility is an average quantity over the contribution from allsegments along the loop, each having different inclinations [112].Mobility measurements rely upon the techniques to record grain boundary dis-placement. A detailed survey of these techniques is available elsewhere [1]. Insummary, these techniques can be grouped into two methods: discontinuous andcontinuous methods. In the former method, the boundary position is determinedat discrete time intervals by revealing its intersection with the sample surface via agroove which forms through sample cooling or by chemically etching the samplesurface [115]. In the latter method, the boundary position is tracked at all times. Itis therefore necessary to have the identification of boundary location automated.Various techniques that permit such an automation include the polarized light re-flection [1] and the X-ray diffraction [116].Grain boundary velocities measured from experiments using high purity met-als (>99.99%), e.g. Pb [4], Al [5], Bi [8], and Zn [9], are of the order of 1-100 µm s−1,17when the driving pressures are between 0.01 to 100 MPa. Additionally, both ab-solute and reduced mobilities determined from experiments follow an Arrheniusrelationship; the activation energy Qm is of the order of 50-200 kJ mol−1[4, 5, 8, 9].The correlation between absolute and reduced mobilities is the subject of a re-cent study [9]. In this study, the migration of magnetically driven planar bound-aries in Zn and the capillarity driven migration of quarter loop boundaries ofsimilar misorientations were measured from experiments. The Arrhenius plot ofmobilities measured from these experiments is presented in Figure 2.6(a) and (b),where the activation energies are 77 and 174 kJ mol−1 for the planar and curvedboundaries, respectively [9]. The discrepancy between the migration energies ofboth types of boundaries was suggested as an indication that a curved boundarymay migrate under a different mechanism than a planar boundary [9]. This suppo-sition, also suggested in [5], departs from the commonly accepted view that con-siders the migration mechanism to be independent of the type of driving pressureand the boundary geometry [51, 117].Figure 2.6: Temperature dependence of (a) absolute mobilities and (b) reduced mobilitiesof 85◦-misorientation boundaries in Zn, being obtained using magnetic field and capil-larity pressure, respectively. The bicrystal shape used on each measurement technique isillustrated by the insets [9].18The kinetics of grain boundary migration is also affected by the rougheningtransition discussed in Section 2.1.3. This effect was investigated in a series of graingrowth experiments in BaTiO3 at temperatures for which faceted boundaries wereevident [118]. The migration velocities were found to scale linearly with drivingpressures only if the driving pressure exceeded a critical value, the magnitudeof which varied with misorientation. Such a velocity-driving pressure trend wasattributed to the presence of facets in the grain boundary structure [118].2.4 Grain boundary migration in alloys2.4.1 Experimental observationsThe development of process models play a crucial part in modern material process-ing routes. Integrated in these models are sub-models that monitor the evolution ofmicrostructural features such as the fraction of recrystallized grains [119]. A num-ber of models that have been developed for the recrystallization kinetics employedthe Johnson-Mehl-Avrami-Kolmogorov (JMAK) approach [51, 120, 121], e.g.Xrx = 1− exp[−ˆ t0ksN˙rx(Veff(t))3dt](2.10)where Xrx is the recrystallized fraction, ks is the JMAK shape factor (4pi/3 forsphere), N˙rx is the nucleation rate of recrystallized nuclei, and Veff(t) is the effectivegrain boundary velocity at time t [51].Impurities lower the rate of grain boundary migration, effectively affecting mi-crostructural evolution, i.e. via Veff(t) in Eq. (2.10) [51]. The addition of 10−3 wt%of Sn to Pb, for example, decreases the grain boundary velocity of zone-refined Pbby at least an order of magnitude, see Figure 2.7(a) [10]. Further, an order of mag-nitude increase in the Sn content doubles the migration energy of random grainboundaries in Pb, see Figure 2.7(b) [11]. Similar trends have also been observed inother systems, including Al-Mg [122], αFe-Al [123] and αFe-Nb [124].The solute-grain boundary interaction not only controls the microstructure dur-ing materials processing but also has a key role during the service life of materials.19For example, material issues due to helium clusters can become critical after thereactors have been operational for an extended period of time, or the reactors havereached an age where a scheduled repair is required. At this stage, the repair ofdegraded reactor components will be performed, usually involving conventionalwelding techniques [125]. Thermal stresses generated during welding may causethe grain boundary migration [126]. The accumulation of helium bubbles as grainboundaries migrate across the grain interior is thus anticipated [126]. The dynamicinteraction between helium clusters and grain boundaries during such processescritically affects material properties. A study reported that, depending on the rateof boundary migration, the helium content of migrating boundaries may exceed athreshold value above which intergranular cracking is initiated [20].Figure 2.7: (a) Grain boundary velocities measured from recrystallization experiments inzone-refined Pb doped with Sn [10], (b) the effect of Sn on the activation energy for themigration of special and random boundaries in Pb [11]. Reprinted with permission of TheMinerals, Metals & Materials Society.Models attempting to capture the dynamic interaction between grain bound-aries and impurities have been proposed and can be grouped into two generalcategories: solute drag models and particle pinning models. In solute drag mod-els, impurities are assumed to form an ideal solid solution with the matrix andare relatively mobile [31]. A general prediction of the solute drag models is thenon-linear dependence of velocity on the driving pressure. Depending on the ki-netic parameters, there may be an abrupt transition where the boundary breaks20away from the solute cloud. In particle pinning models, impurities are treatedas stationary precipitate particles that are distributed randomly within the matrix[36]. There is a threshold driving pressure below which the boundary is immobile.Above the threshold, the boundary velocity scales linearly with driving pressureand the proportionality constant equals the mobility of a pure boundary.The interaction between grain boundaries and impurities that form clustersconsitutes an intermediate case between solute drag and particle pinning mod-els because clusters of a few interacting solute atoms migrate at a much lower ratethan individual atoms. A cluster in this context is defined as a group of interactingsolute atoms [39] which often provides a contrast similar to that of the matrix inTEM images [40] and has no distinct crystal structure [41]. The effect of clusters ongrain boundary mobilities has not been extensively explored in spite of its techno-logical importance, e.g. embrittlement in irradiated steel vessels due to He clusters[20] or grain refinement of high-strength low-alloy steels due to Nb clusters [16].2.4.2 Solute drag modelsInitially proposed by Lücke-Detert [30], revised by Cahn [31], Lücke-Stüwe [32]and later extended by Hillert [33, 34, 35], solute drag models predict the drag ef-fect during the steady-state migration of a grain boundary. These models share anumber of similarities; Cahn’s version will be used as an illustration for its sim-plicity.During their interaction with a moving boundary, solute atoms maintain a con-centration profile C(z), where z is the distance from the grain boundary plane, thatdepends on the solute diffusivity profile D(z), the binding energy profile Eb(z) andthe boundary velocity V. At steady state, the concentration profile can be obtainedby first assuming a reasonable profile of D(z) and Eb(z), e.g. Figure 2.8 (a), andthen solving the following second-order linear diffusion equation, [31]C′′ +(E′b +D′D+VD)C′ +(D′DE′b + E′′b)C = 0 (2.11)where [′] is a differential operator with respect to z. The drag-pressure Pd is the21sum of force exerted by each solute atom on the boundary, i.e. [31]Pd = −Ω−1ˆ +∞−∞[C(z)− C0] E′bdz (2.12)where Ω is the bulk atomic volume.Figure 2.8: Examples of (a) diffusivity profiles D(z) and (b) binding energy profile Eb(z),δ and Eb0 being the grain boundary width and the maximum binding energy, respectively,(c) solute drag pressure dependence on boundary velocity; data points are the solutions toEqs. (2.11) and (2.12) while the dashed lines are the model from Eq. (2.13).While the coupled Eqs. (2.11) and (2.12) can be solved analytically in the limit oflow or high velocity, a closed form expression for the drag pressure is not available.To approximately merge responses obtained in the high and low velocity limit,Cahn proposed the following expression [31],Pd =αC0V1+ (βV)2(2.13)where the parameters α and β depend on D(z) and Eb(z) via [31]α =4kBTΩˆ +∞−∞sinh2 [Eb(z)/2kBT]D(z)dz (2.14)22β2 = αkBT[Ωˆ +∞−∞[E′b(z)]2 D(z)dz]−1 (2.15)Comparisons between the model in Eq. (2.13) and a numerical solution to thecoupled Eqs. (2.11) and (2.12) are shown in Figure 2.8(c). The poor fit quality ofEq. (2.13) is noted in the case where there is a change in solute diffusivity at thegrain boundary, i.e. a triangular D(x) profile, see Figure 2.8(a).Figure 2.9: The velocity-driving pressure relationship predicted by the Cahn solute dragmodel under different binding energies. The curve labelled ’2Eb0’ illustrates a hysteresisdenoting different paths for abrupt break-away transitions, the dotted part representingthe so-called unstable regime.With regard to grain boundary migration, solute drag models postulate thatthe driving pressure P is dissipated via the intrinsic migration of the boundary,Pint, and the transport of segregated solute atoms along with the boundary as theboundary migrates, Pd, i.e. [31]P = Pint + Pd (2.16)where Pint is given by the ratio of the velocity V and the mobility M, while Pd isgiven by Eq. (2.13). The solute drag component Pd reduces the energy available todrive the boundary, resulting in a low mobility when the driving pressure is small.23If the driving pressure increases, the boundary may gradually escape from thesolute cloud, causing the migration to transition from a solute-loaded state (lowmobility) to a solute-free state (high mobility), see Figure 2.9. Such a transitioncan also occur in the opposite direction, i.e. the case where the driving pressure isinitially large (the high velocity regime) and progressively decreasing. Dependingon the interaction parameters, the transition may be abrupt and the driving pres-sure where the transition occurs may depend on the direction at which the drivingpressure changes, i.e. from low P to high P, and vice versa, see Figure 2.9.The Cahn model has been revisited on several occasions. Lücke-Stüwe [32]provided a correction to account for site-saturation, leading to a segregation thatis consistent with the Langmuir-McLean model, i.e. Eq. (2.3). Hillert generalizedthe model by substituting the binding energy profile with the chemical potentialgradient profile [35], extending the model’s compatibility to the case of interphaseinterface in a multicomponent system.One of the main differences between the Cahn model and the Hillert model isthat in the high-velocity regime, the former predicts a zero drag pressure while thelatter predicts a finite drag pressure that is independent of the boundary velocity.Furthermore, the Hillert model for the case of the interphase interface migrationassumes the balance between three force components, namely the capillarity forcedue to the interface curvature, the intrinsic friction force of the interface, and thesolute drag force due to segregation. Hillert argued that, when the size of equilib-rium daughter phase is small relative to the parent phase (e.g. after nucleation),the interphase migration may proceed along the direction opposite to the capillar-ity force. Along with the intrinsic friction, the capillarity force may act to retardthe growth of the daughter phase. The force component left to consider is the so-lute drag force, which, by means of a mathematical necessity, must act in the samedirection as the net migration. Hillert [35] concluded that solute atoms at the inter-phase interface, thus, may act as a driving force for the interface migration duringphase transformations.Many studies have implemented solute drag models to rationalize experimen-tal results on solute-boundary migration. The Cahn model, for example, was usedto study the Nb effect on the recrystallization and grain-growth of α-Fe, and en-24abled the estimation of parameters such as the binding energy of Nb to α-Fe grainboundaries and trans-interface diffusivity of Nb [124]. In another study, the Cahnmodel was used to identify the optimal Mn content in a low-carbon high-Nb steelto retard the kinetics of static recrystallization [121], thereby allowing steels toachieve a final grain-size of less than 3 µm [72].While experimental observations seem to align with solute drag models, sev-eral predictions have yet to be experimentally verified. For example, the modelpredicts that the drag due to solute with an attractive binding energy is equal tothat of solute with repulsive binding energy. Designing experiments to verify sucha prediction may be challenging as relevant parameters may not be easily tuned.2.4.3 Particle pinning modelsA number of solute atoms may group to form precipitate particles and restraingrain boundary migration. Models that estimate the pinning effect of such parti-cles on the grain boundary migration, i.e. the particle pinning models, have beenproposed by several authors, e.g. Zener via Smith [36], Gottstein-Shvindlerman[37], and Novikov [38]. These models share a number of similarities, that is, theyconsist of three parts [127, 128]:1. The individual pinning force of a particle acting on a grain boundary.2. The total pinning pressure of multiple particles on a grain boundary.3. The effect of total pinning force on the grain boundary velocity.The above concepts will be discussed using the framework of the Zener model. Astationary particle of radius rp that intercepts a grain boundary is considered, seeFigure 2.10(a). The particle induces a pinning force, FZ, on the migrating boundarydue to the partial removal of the boundary area. The force balance at the particle-boundary contact perimeter, e.g. point J in Figure 2.10(a), yields a net force perlength of γ sinΘ. Assuming that the interface energy between the particle and thematrix, γpm, is isotropic, the force balance also produces a net force that is inde-pendent of γpm, see point K in Figure 2.10(a). Along the line of contact perimeter25(i.e. 2pirp cosΘ), the total force FZ acting on the boundary is 2pirpγ cosΘ sinΘ. Themaximum pinning force occurs when Θ = pi/4, i.e. [36]FZ = pirpγ (2.17)Figure 2.10: (a) The pinning force of a particle, (b) the total pinning pressure due to multi-ple particles that are in contact with the boundary, i.e. particles located within the interac-tion zone, i.e. the volume that extends to a distance rp from both sides of the boundary.At a given time, there is a number of particles, Np, interacting with a migratinggrain boundary, as shown in Figure 2.10(b). If each particle is assumed to imposethe maximum pinning force on the boundary, FZ , the total pinning pressure dueto these particles is [36]PZ = NpFZ/A = 2pir2pγNp/vs = ρZNp (2.18)where A is the grain boundary area, vs is the swept volume and ρZ is the individualparticle pinning pressure. The swept volume is the interaction zone where parti-cles are in contact with the grain boundary, see Figure 2.10(b). Since the bound-ary is treated as a sharp boundary, i.e. its width δ = 0, the swept volume is thusvs = 2rp A. By considering that the swept volume extends to both sides of the26boundary, it is implicitly assumed that the pinning force from the particles in frontof the boundary acts on the same direction as the pinning force from the parti-cles behind the boundary [129]. The volume fraction of the particles in the sweptvolume, fs, is fs = 43pir3pNp/vs. If particles are further assumed to be randomlydistributed throughout the entire matrix, the global volume fraction of particles fvcan be taken as equal to fs. These relationships are then substituted to Eq. (2.18),resulting in [36]PZ = 32 fvγ/rp = kZ fvγ/rp (2.19)where kZ is the pinning coefficient, i.e. 3/2 in the original Zener model.The particle pinning pressure acts to reduce the available driving pressure. Thegrain boundary velocity under the Zener pinning effect is given by [36]V = Mpure (P− PZ) (2.20)where Mpure is the boundary mobility in a pure system, PZ is the pinning pressuregiven by Eq. (2.19). If the driving pressure is below PZ, the boundary is completelypinned. The complete pinning condition can be used to estimate the limiting grainsize for a given set of particles with radius rp and volume fraction fv [128].A number of studies, both experimental [128, 130, 131] and simulations [132,133], found that the Zener model underestimates the pinning pressure by a factorof 2 to 10, motivating the need to revisit the model assumptions. Efforts to addressthis problem have been proposed. Ashby et al [134] relaxed the assumption ofisotropic boundary-particle energy, γpm, to treat coherent particles. Machlin [135],and recently Hersent et al [136, 137], considered the pinning effect of individualstationary atoms. Fullman [138] extended the monosized assumption to enable asize distribution of particles. The influence of non-spherical particle shape on thepinning pressure have been analyzed [139], e.g. cuboids by Ringer et al [140] andellipsoids by Ryum et al [141]. Louat [142, 143] and Hunderi et al [129] investigatedthe assumption of equal pinning force between particles in the front of and behindthe boundary. Non-random spatial distributions of particles have been explored byAnand and Gurland [130] and more rigorously by Liu and Patterson [144] wherethe deviation from randomness is represented by a dimensionless parameter called27the stereological factor [145]. The modifications proposed by these authors pertainto the first two parts of the Zener model, i.e. the pinning force of a particle (Eq.(2.17)) and the total pinning pressure from multiple particles (Eq. (2.18)). In otherwords, while the expression for PZ due to these models is different from Eq. (2.19),the pinning effect on the boundary velocity still follows Eq. (2.20).Figure 2.11: The velocity-driving pressure relationship under different particle pinningmodels (see text).Other models revisited the velocity-driving pressure relationship in the pres-ence of particles. Gottstein and Shvindlerman [37] extended the model to the caseof mobile particles. Their additional assumptions include the size-dependent im-purity mobility and the impurity velocity being a product of its mobility and itsattractive force to the boundary. As a consequence of the finite mobility of parti-cles, the boundary is able to migrate at a non-zero mobility despite the low driv-ing pressure. The boundary detaches from particles when its velocity exceeds acritical value Vcrit, above which the velocity-driving pressure trend follows thatof a pure boundary, see Figure 2.11(b). Gottstein and Shvindlerman [37] arguedthat the overall trend in Figure 2.11(b) is similar to the sharp transition betweenthe solute-loaded (low-mobility) and the solute-free (high-mobility) regime in theCahn solute drag model, e.g. the ’1.5Eb0’ curve in Figure 2.9.Extending the Gottstein-Shvindlerman model, Novikov performed a series ofstatistical simulations on the particle pinning effect on grain growth [38]. He con-cluded that depending on the kinetics of mobile particles relative to that of the28grain boundary, particles can serve a dual function when impeding the boundarymigration, i.e. reducing either the boundary mobility or the driving pressure. Inthe low-velocity regime, the boundary migrates at a reduced mobility; the overalltrend is similar to the solute-loaded regime of the Gottstein-Shvindlerman model.In the high-velocity regime, the migration is driven by a reduced driving pressurewhile the mobility is that of pure boundary, which is similar to the high-velocityregime of the Zener model, see Figure 2.11(c).Particle pinning models have been employed to explain microstructural fea-tures such as the size and morphology of grains resulting from thermo-mechanicaltreatments. The presence of Mn precipitates, for example, has been shown todrastically alter the recrystallization behaviour of Al-Mg alloys leading to a fineequiaxed grain structure [146]. Limiting grain size for various alloy systems havealso been deduced using these models [128].2.4.4 Cluster drag modelsThere has only been one study in the literature that attempted to model the interac-tion between solute clusters and a migrating grain boundary [147]. Grey and Hig-gins developed a model to interpret a series of normal grain growth experimentson dilute lead-tin (Pb-Sn) alloys [148] where an apparent change in the exponentof the classical power-law growth model was observed.The model assumes that the drag effect can be decoupled into two components:a drag component Pd and a pinning component PZ [147]. The former is equal tothe the sum of the intrinsic mobility of the grain boundary, Eq. (2.4), and the low-velocity regime of the Cahn model, i.e. when βV  1 in Eq. (2.13). The latter isconsidered as equivalent to the Zener pinning pressure PZ in Eq. (2.20).The cluster drag model considers a binary system just above its solubility limitwhere solute atoms partition into those that form clusters and those that are insolid solution. Clusters are treated as particles, which impede grain boundarymigration by pinning, i.e. the Zener pressure PZ. The remaining solute atomsparticipate in the velocity-dependent drag pressure Pd. Using such a treatment,these authors explained that the exponent change in the growth model was a resultof the transition from the velocity-dependent regime (solute drag) to the velocity-29independent regime (particle pinning). A similar conclusion was also obtainedwhen the model was applied to different binary alloy systems, e.g. tin-antimony(Sn-Sb) [149].The Grey-Higgins cluster drag model has, however, been criticized. Fiset et al[150] argued that the exponent change in the growth kinetics reported in Ref. [148]should be attributed to the development of a polygonized substructure instead ofa kinetic transition. Galibois et al [151] found several inconsistencies in the analysisof experimental results carried out by Grey and Higgins [147] and concluded thatthe agreement between the two was more of an exception than a general case.These critics acknowledged, however, that the decoupling between solute dragcomponent and cluster pinning component may be reasonable.2.5 Perspective from atomistic simulations2.5.1 Overview of atomistic simulation techniquesIn order to develop continuum models on the cluster-boundary interaction, it isnecessary to characterize the formation of clusters at the boundary and quantifytheir kinetic effect on the rate of grain boundary migration. Atomistic simula-tions can provide access to such information because they are capable of resolvingatomic-scale processes that are otherwise difficult to probe in experiments.A variety of atomistic simulation techniques have been developed and can begrouped into two general categories: deterministic methods and stochastic meth-ods. Both categories differ in their underlying principle; the former implementsdeterministic reversible Newtonian dynamics to identify atomistic events and thelatter uses probabilistic approach to simulate the occurrence of atomistic events,enabling the parameter space exploration of the problems.An example of deterministic methods is molecular dynamics (MD), a tool toexamine the behaviour of materials by simulating the interaction between theirconstitutive atoms. The repulsive and attractive force among these atoms are dic-tated by mathematical functions referred to as the interatomic potentials. Thesepotentials are numerically developed to fit the basic properties of materials suchas elastic constants or diffusivities. The analytical form of these potentials varies30depending on the materials that they model, e.g. Lennard-Jones potentials for gas,liquid and solid such as Ar, embedded atom models (EAM) for metals and alloysand Stillinger-Weber potentials for covalent crystals such as semiconductors [152].At its simplest form, an interatomic potential U is a function of the position ofeach atom i, ri, i.e. U = U ({ri} , i ∈ [1 : N]). It consists of a short-range repulsivecomponent and a long-range attractive component. The dynamics of each atomfollows Newton’s 2nd law, the net force being equal to the gradient of potential.Upon solving the Newtonian equation for each atom at every time step, the evo-lution of atomic positions and velocities can be obtained. Thermodynamic statefunctions of the system, e.g. temperature and pressure, can be determined fromthe position and velocity of each atom [153].MD simulations have been applied to examine atomic-scale interactions in manymaterial systems, including metals and binary alloys. The interaction between so-lute and dislocations [154], the martensite nucleation from the fault-band inter-sections in austenite [155] and the migration of austenite-ferrite interface [156] aresome of the examples of complex microstructural phenomena explored using MDsimulations. The time scale of simulations is in the order of ns. Such a short timescale is inherent of MD simulations because the time-step for integrating Newton’s2nd law is in the order of the atomic frequency (10−15 s) [157].A longer simulation time scale is afforded in a deterministic technique calledPhase Field Crystals (PFC). In PFC simulations, atoms are represented by a den-sity field whose peaks correspond to locations where atoms would be found withhigh probability [158], in contrast with MD simulations where atoms are discretepoint objects. The system progresses by evolving the density field via a dissipativedynamic equation [159]. The PFC technique is considered relatively new and thushas not been as extensively explored as MD. Among examples of PFC applicationsinclude the microstructural evolution during structural transformations in metals[160] and the solute drag in a two-dimensional binary model material [161].Metropolis Monte Carlo (MMC) and kinetic Monte Carlo (KMC) are examplesof stochastic methods. Both techniques evolve a molecular system towards itssteady state but only KMC does so in a manner that physically attempts to capturethe time evolution of the system. KMC simulations are therefore more suitable for31dealing with dynamic processes. MMC simulations, on the other hand, are idealfor obtaining thermodynamic state functions of materials.In a KMC simulation, a list of Nevent atomistic events is tabulated. Each event iis assigned a known rate νi, i.e. νi = νD exp(−Eia/kBT)where νD is the vibrationalfrequency and Eia is the activation energy of the event [162, 163]. At each timestep, the rates are computed and an event is selected with a probability that isproportional to its rate. The chosen event is then executed, followed by an updateof each rate. After the execution of the chosen event, the clock that represents theactual time is advanced using the residence time algorithm [162] where the actualtime lapse between two simulation steps is inversely proportional to the total rate.Typical atomistic KMC simulations have a length scale of 10−1 to 102 µm anda time scale of 10−2 to 103 s [163, 164]. These scales highlight the appeal of KMCfor simulating phenomena that occur in a diffusive time scale, e.g. precipitation inFeCu alloys [165] and thin film growth by surface adsorption [166]. KMC is suit-able for these cases because the governing atomistic events and their correspond-ing rate are established prior to performing simulations, having been identifiedfrom other atomistic tools, e.g. ab-initio calculations [167] or MD simulations [168].2.5.2 Progress from recent atomistic studiesA variety of microstructural processes in many systems has been investigated us-ing atomistic simulations, some of which having been further validated againstexperiments. Only a few, however, are dedicated to investigating the interactionbetween solute atoms and migrating grain boundaries.To examine the kinetics of grain boundary migration in pure metals, severaltechniques appropriate for MD simulations have been developed and benchmarked.Simulation techniques capable of extracting grain boundary mobilities include thecapillarity technique [112], the artificial driving force (ADF) technique [169] andthe random walk (RW) technique [170].In the capillarity technique, the boundary is driven by the capillarity pressuredue to the boundary curvature, see Figure 2.5. The computed mobilities are re-duced mobilities, Eq. (2.9), which represent the average migration of segmentsalong the curvature. This technique provides the most direct analysis among other32driven techniques since it involves no artificial assumptions about the manner bywhich the driving pressures are applied. Consequently, it is compatible for alloys.The ADF technique is designed to examine the migration of planar boundaries.First, atoms whose local neighbours differ from those of bulk atoms, i.e. grainboundary atoms, are identified. These atoms are assigned an extra energy, thegradient of which is the driving force for migration. The technique has been ap-plied to several fcc metals, e.g. Al [169, 171] and Ni [172, 173], where qualitativeagreement between simulations and experiments were observed. Recent studiesnoted that considerable care must be taken when analyzing the absolute mobilitiesdetermined from this technique [171, 174, 175]. It is unable to clearly distinguishbetween two grains that form low misorientation angle boundaries [171], thus ne-cessitating a post-simulation correction to the mobility analysis [174, 175]. Ad-ditionally, the technique fails to discriminate between boundary atoms and bulkatoms that have been displaced by a repulsive force due to its short-ranged inter-action with solute atoms. This limits the use of the ADF technique to pure metals.In contrast to previous techniques, the RW technique, or equivalently the bound-ary fluctuation technique [176], computes the absolute mobility of a planar grainboundary in a non-driven bicrystal. Above the roughening temperature [69] (Sec-tion 2.1.3), boundaries undergo temporally uncorrelated spatial fluctuations in-dicative of a random walk process [170]. The mean-squared displacement of roughboundaries can be recorded over time to extract their absolute mobility [170]. Whilethe absolute mobility computed via this technique may not be computed in a con-ventional manner, i.e. via the migration of a grain boundary, the RW techniqueprovides the benefit of being in the limit of zero driving pressure. Additionally,the technique is also compatible for alloy systems.Alternative techniques for computing the mobilities, other than those listedabove, include the elastic anisotropy technique [177, 178]. Previous work reporteda consistency among the absolute mobilities obtained from techniques that em-ploy planar boundaries, i.e. the ADF technique, the RW technique and the elasticanisotropy technique [171, 179]. It is not clear if such a consistency can be extendedto the capillarity technique since no present literature investigates the correlationbetween the curved and planar boundary migration.33Table 2.1: Arrhenius parameters of grain boundary mobilities, see Eq. (2.6), obtained from studies employing moleculardynamics simulations.Mater- Methods for Grain boundary structure M0 Qm Temperature Ref.ials driving the boundary [rotation axis], θ [m4J−1s−1] [eV] [kJ mol−1] range [K]Cu Elastic anisotropy [001] 53.1◦ 4.23× 10−4 0.640 61.7 [800, 1100] [180]AlCapillarity† [111] 27.8◦ 8.32× 10−7 0.075 7.23 [305, 427] [112]Random walk [111] 38.2◦ 3.06× 10−4 0.460 44.4 [500, 775] [181]ADF [112] 7.8◦ 4.44× 10−5 0.116 11.2 [300, 700] [171]NiRandom walk[001] 36.9◦1.72× 10−5 0.440 42.4 [800, 1200][179]ADF 2.01× 10−5 0.450 43.3 [800, 1200]Elastic anisotropy [001] 53.1◦ 2.18× 10−6 0.260 25.1 [800, 1400] [178]†The unit of M0 for mobilities obtained from the capillarity technique is in [m2s−1].34Table 2.1 lists the Arrhenius parameters of grain boundary mobilities obtainedfrom some MD simulations. The migration energies Qm given in Table 2.1 arein the range of 7 to 70 kJ mol−1, approximately an order of magnitude smallerthan typical values obtained from experiments. Such a discrepancy has often beenattributed to the inevitable presence of impurities in experiments [172, 182].Figure 2.12: Snapshots that show correlated atomic motion governing the migration ofgrain boundaries, from (a) MD simulations [183] (b) high-resolution TEM [184], and (c) anillustrative sketch. Column (a) shows the view normal to the boundary plane of the [001]θ = 28◦ boundary in Au at 500 K. Column (b) shows the [112] θ = 14◦ grain boundaryin Au at 893 K, the migration direction highlighted with an arrow. Column (c) shows asketch of correlated motion of grain boundary atoms in a tilt CSL bicrystal, dashed linesindicating the position of grain boundary.In addition to computing grain boundary mobilities, MD simulations have alsobeen implemented to investigate the atomic-scale mechanisms underlying grainboundary migration. Correlated motion of a group of atoms was identified as themechanism responsible for the elastically driven migration of planar grain bound-aries [183, 185], see Figure 2.12(a). This finding was consistent with the time-lapseobservation from high-resolution TEM [184, 186], see Figure 2.12(b). Figure 2.12(c)35presents a sketch that illustrates the correlated atomic motion using a simple sym-metric tilt grain boundary of a CSL bicrystal.Grain boundary migration in alloy systems has not been fully explored in atom-istic simulations. This may be due to the time-scale for solute diffusion being typ-ically slower than the time-scale for boundary migration. For example, in orderto migrate a distance of 1 nm at 300 K from molecular dynamics simulations, agrain boundary in pure Al takes about 0.01 ns [171], while an interstitial hydrogenatom in Al matrix takes about 1 ns to travel the same distance [187]. The rates ofsolute diffusion and grain boundary migration increase by at least two orders ofmagnitude for substitutional solute [187, 188]. Recent studies have implementedshortcuts to bypass the slow diffusion process in MD simulations when investi-gating the dynamic boundary-solute interaction and extracting the mobilities ofsolute-enriched planar grain boundaries [189, 190]. It is not clear if the results pre-sented in these studies are independent of the implemented techniques.Another study examined the curvature-driven grain boundary migration in abinary Fe-Cr system [191]. The migration of U-shaped half-loop coherent twinboundaries in this alloy was found to take place via a mechanism similar to thecorrelated atomic motion identified from the migration of planar boundaries inpure systems [192]. It is not clear, however, if the mechanism of migration is af-fected by the presence of Cr atoms as this was not discussed [191].Other studies implemented different simulation tools to approach the prob-lem of grain boundary migration in alloys. Metropolis Monte Carlo simulations(MMC) have been applied at the atomic scale to investigate the boundary veloc-ity dependence on the solute content [193, 194]. Using a simplified construction,MMC simulations were designed to capture expected trends such as solute dif-fusion and grain boundary migration [193]. Several discrepancies between MMCresults and predictions from classical solute drag models were reported. For exam-ple, the classical models predicted that the solute drag is independent of the signof the binding energy, i.e. whether there is a solute enrichment or depletion at theboundary. MMC simulations found that such a prediction is only valid in the limitof high driving pressure [193]. At low driving pressures, the sign of interactionwas found to play a significant role on the magnitude of drag effect [194].36Another study employed Phase Field Crystals (PFC) simulations to study thesolute drag problem at the atomic scale [161]. The simulation introduced soluteatoms to a two-dimensional bicrystal, the solute having a finite self-interactionenergy, i.e. a non-ideal solution. The kinetics of migration were examined forvarious levels of solute content and diffusivity. A qualitative agreement with theCahn model was observed, despite the ideal solution assumption employed bythe model. Several discrepancies were reported, e.g. there is a minimum drivingpressure below which the interface is completely stationary [161]. PFC results alsosuggest a decrease of the maximum drag pressure with increasing solute diffusiv-ity, this being in contrast with the prediction from classical models [161].2.6 Outstanding questionsAtomistic simulations have been demonstrated to offer valuable insight into grainboundary migration in metals. Several important questions pertaining to thistopic, however, are still outstanding. It has not been ascertained if the migrationof curved grain boundaries operates under a different mechanism than the migra-tion of planar grain boundaries, as suggested by the experimental study presentedin Figure 2.6. No evidence has been presented either to verify that the migrationof grain boundaries operated by correlated motion of atoms provides a dynamicresponse to solute segregation that is consistent with the prediction from classicalmodels. Finally, the role of solute clusters in retarding the kinetics of grain bound-ary migration remains largely unexplored.37Chapter 3Scope and ObjectivesThe goal of this work is to develop a model that describes the role of clusters ongrain boundary migration using atomistic simulations. For this purpose, kineticMonte Carlo and molecular dynamics simulations will be used independently.The solute-grain boundary interaction in an ideal binary solution will first beexamined using KMC simulations. It provides a benchmarking analysis of atom-istic simulations against classical solute drag models. The focus is then shiftedto the iron-helium (Fe-He) system, a non-ideal solution where cluster formationoccurs. A series of MD simulations will be performed, starting with a study onthe migration of grain boundaries in pure Fe. The formation of He clusters in Febicrystals will be considered next. The size distribution of segregated clusters andits effect on the boundary structure will be rationalized using the properties of theFe-He system, e.g. the binding energies and diffusivities of He. Finally, the inter-action between helium clusters and migrating grain boundaries will be analyzedto identify atomistic interaction mechanisms and to quantify the retardation effectof clusters on grain boundary migration.In summary, the objectives of this work are to:1. Explore the parameter space of an atomistic solute drag model in an idealsolution and compare against predictions from classical models.2. Quantify the migration of twin boundaries in pure Fe bicrystals and deter-mine the correlation between the mobilities of curved and planar boundaries.3. Characterize the size distribution of He clusters in Fe bicrystals and deter-mine their effect on the absolute mobilities of grain boundaries.4. Develop a quantitative model that explains the effect of He clusters on thekinetics of grain boundary migration.38Chapter 4Methodology4.1 Kinetic Monte Carlo4.1.1 FundamentalsKinetic Monte Carlo (KMC) simulations extend the length and time scale of typicalatomistic simulations, e.g. MD simulations, into a mesoscopic range, i.e. 10−1 to102 µm and 10−2 to 103 s. The critical input to KMC simulations is a list of possibleatomistic events that are relevant to the problem under consideration. Associatedwith each event will be a rate denoting its frequency of occurence. During eachsimulation step, one of these events will be chosen based on a probability whichis a function of the ratio between the individual rate and the sum of all rates. Theselected event will be performed and the rate of other events updated. The actualtime interval between two consecutive events follows that of a Poisson process[195], i.e. it has an exponential distribution. Such a distribution enables the paral-lelism between the simulation time and the actual time.In this work, KMC simulations will be used to explore the parameter space ofa migrating grain boundary in a binary ideal solution. The objective is to exam-ine, from an atomistic perspective, the solute drag effect in a system that exhibitsno clustering. An atomistic KMC model is developed, capturing many of the ba-sic features of a migrating grain boundary interacting with mobile solute atomswithout describing in detail the crystallography of the boundary itself. The find-ings from the KMC simulations will then be compared with predictions from theclassical solute drag model.394.1.2 Model materialThe model material used for the KMC simulations is a bcc crystal, having dimen-sions of NX× NY× NZ unit cells. In this construction, rather than being defined asthe region separating two crystals of different orientations, the boundary is deter-mined by first assigning a ‘spin’ property to each atom.The atoms in one half of the bicrystal are assigned a ‘spin’ S of +12 (dark-coloured atoms in Figure 4.1) while those in the other half of the box are assigned a‘spin’ of−12 (light-coloured atoms in Figure 4.1). The grain boundary is defined byatoms that have at least one nearest neighbour that belongs to the other grain. Pe-riodic boundary conditions are applied to the walls perpendicular to the boundaryplane (XZ and YZ planes), while a toroidal boundary condition is applied to thewalls parallel to the interface plane (XY plane), i.e. spin changes from S to −S oncrossing these walls [194]. These boundary conditions ensure that the simulationbox contains only one grain boundary.Figure 4.1: (a) A KMC simulation box where two crystals of the same orientation are sep-arated by a flat {001} interface, the interface being considered as a KMC representationof the grain boundary, (b) the initial position of the grain boundary, (c) a flipping eventoccurred at a spin i whose neighbours are j, (d) the boundary average position advancesupon several flipping events of the spins belonging to one of the grains.40Coinciding with this simplified boundary structure, a simple mechanism forgrain boundary migration is adopted. Only solvent atoms residing at the bound-ary plane, see Figure 4.1(b), can switch their membership from their current grainto the adjacent one by flipping their spin, effectively shifting the average positionof the boundary, see Figure 4.1(d). The rate of these flipping events for a givenatom is assumed to depend on its local environment. Without an energy bias be-tween the two crystals, flipping occurs randomly. This results in a boundary thatroughens but whose average position remains the same. If, however, one of thecrystals is assumed to have a higher energy than the other, flipping will be bi-ased in one direction causing the grain boundary to migrate. The spin-flip eventcan be considered equivalent to the cooperative motion of atoms underlying theboundary migration. Figure 4.2(a) and (b) show a simple illustration that draws aparallelism between a grain boundary in a CSL bicrystal and the grain boundarydeveloped for the KMC model in this study.Figure 4.2: (a) A portion of symmetric tilt boundary in a CSL bicrystal. CSL unit cells ofgrain A and grain B are indicated and are rotationally related to each other through corre-lated atomic motion, shown in Figure 2.12(c); (b) a KMC representation of the bicrystal in(a), where CSL unit cells are represented by black (S = 12 ) and white (S = − 12 ) spins, seealso Figure 4.1.By taking the energy of a flat grain boundary as a reference, the excess energyassociated with a non-flat grain boundary is formulated as [196],41Epure = γNXNY∑i=14∑j=1(hi − hj)2 (4.1)In this description, similar to the Discrete Gaussian Solid on Solid (DGSOS) model[197], the excess energy of a rough grain boundary is characterized by a set ofhalf-integral multiple of lattice parameter, hi. This indicates the position along thez-direction of a grain boundary atom of a given spin from an arbitary XY-plane ofreference, which is taken here as the initial grain boundary position. The excessenergy is given by the sum of the square of the differences between the heightof each atom hi and those of adjacent grain boundary atoms j, hj|X+, hj|X−, hj|Y+and hj|Y−, where i and j belong to the same type of spin, see Figure 4.1(c). Themagnitude of energy increase due to roughening scales by γ, which can be thoughtof as an effective boundary energy with units of energy per unit area.Solute atoms are introduced to reside on octahedral interstitial sites of the bcclattice. Identical behaviour for solute in the two grains is assumed and no solute-solute interactions are included.Figure 4.3: (a) An octahedral site in a bcc lattice surrounded by its six neighbouring solventatoms, (b) Different types of octahedral site as defined by Eq. (4.2), see also Table 4.1, (c)Two types of terrace site (ω = 1.0) found in a flat interface.The interaction between solute atoms and the grain boundary are designedsuch that the solute prefers to sit in positions where it is surrounded by solventatoms from both sides of the boundary. An octahedral site is surrounded by sixsolvent atoms, which are paired into three groups: (S1 : S3), (S2 : S4) and (S5 : S6),see Figure 4.3(a), the midpoint of each being the octahedral site itself. The total42energy of the system is modified when a solute atom occupies a site that separatestwo solvent atoms belonging to different grains. Using this convention a bulk oc-tahedral site is defined as one surrounded entirely by solvent atoms from the samegrain. The energy of non-bulk sites are scaled based on the parameter ω, whereω = 0.5 (|S1 − S3|+ |S2 − S4|+ 2|S5 − S6|)with Si = ±0.5 (for left/right grain) (4.2)Based on Eq. (4.2), five distinct types of octahedral sites can be identified. Thefactor two in the third term of Eq. (4.2) is included to ensure that the two types ofoctahedral site found in a perfectly flat boundary, i.e. T1 on the face center and T2on the edge midpoint (see Figure 4.3(c)), have ω = 1. The values of ω and theirmultiplicity are listed in Table 4.1.Table 4.1: Types of octahedral sites and their ω−values, see Eq. (4.2).Symbol Site Type ω−values Multiplicity at 0 K× Bulk 0.0 2B Ledge 0.5 3 Terrace 1.0 3♦ Kink 1.5 14 Island 2.0 1The net interaction between a solute atom and any octahedral site is taken tobe proportional to the ω-value of the site and is given by Esol-GB = Eb0ω where Eb0is the binding energy to a site in a flat grain boundary, a negative value of whichindicating an attractive interaction between solute atom and grain boundary.4.1.3 Rates of atomistic eventsThe system is simulated using a kinetic Monte Carlo scheme where the kineticsare dictated by changes in the total energy of the system before and after an eventoccurs [198, 199]. In a bicrystal containing solute atoms, the total energy of thesystem relative to that of a solute-free system having a flat grain boundary is43E = Epure + Esol-GB= γ(NXNY∑i=14∑j=1(hi − hj)2)+ Eb0(6NXNYNZ∑l=1ωlζl)(4.3)where the index l is over all octahedral sites and ζ denotes the occupancy of site l,i.e. one if occupied and zero otherwise.The rates of two fundamental events have to be considered in this model: (1)the flipping of spins at the boundary from one grain to the other, representing theforward and reverse migration of the grain boundary and (2) the jump of soluteatom from one interstitial site to another. In the simplest case, the forward migra-tion is taken to occur with a rate,ν→ = νD exp(−Qm + ∆E/2kBT)(4.4)where νD is the attempt frequency, Qm is the activation barrier for grain boundarymigration and ∆E is the difference between the total energy of the system beforeand after the event, kB is the Boltzmann constant and T is the absolute temperature[198, 199]. The energy change ∆E in this case arises from changes of grain bound-ary energy, via(hi − hj)from the first term in Eq. (4.3), as well as changes in thenumber of occupied boundary octahedral sites due to grain boundary migratingaway from the segregated solutes, via ωl from the second term in Eq. (4.3).Under these conditions, the average position of the grain boundary will fluctu-ate around its initial average position. To drive the boundary in one direction, adifference between the energy of solvent atoms belonging to the two grains mustbe imposed. This is done by raising the energy of solvent atoms on one side ofthe grain boundary by an amount of PΩ, P being the driving pressure and Ω theatomic volume. The rate of flipping in one direction is then still governed by Eq.(4.4) while the rate of flipping in the other direction is given byν← = νD exp(−Qm + PΩ+ ∆E/2kBT)(4.5)44Written in this way, the grain boundary migration for a pure system is consis-tent with the classic Burke-Turnbull relationship [109] described in Section 2.3.2where the grain boundary velocity V is proportional to the driving pressure Pwhen the driving pressure is low. In a more general form, the Burke-Turnbull rela-tionship dictates a maximum velocity limit at which the grain boundary migratesdespite being driven by an increasingly high driving pressure. Such a trend is alsosatisfied by the rates described in Eqs. (4.4) and (4.5).In addition to the rate given by Eqs. (4.4) and (4.5), the simulation also definesthe jump rate for a solute atom, i.e.νsolute = νD exp(−Qd + ∆E/2kBT)(4.6)where Qd is the activation barrier for bulk diffusion and the energy change ∆E inthis case comes from the changes in the occupancy parameter ζ from Eq. (4.3). Theattempt frequency, νD, in Eqs. (4.4) to (4.6) is assumed to be identical.4.1.4 Running a KMC simulationThe kinetic competition between two classes of species, i.e. the species that governsgrain boundary migration and the species responsible for solute diffusion, will besimulated. At each time step, the rate for each event i, νi, is tabulated and the ratesum of all Nevent events is computed, i.e. Λ = ∑Neventi=1 νi. A random number ϑ1(where 0 < ϑ1 ≤ Λ) is drawn and the event k that satisfies ∑k−1i=1 < ξ1 ≤ ∑ki=1 isexecuted. The rate table is then updated to reflect the repercussions of the executedevent on future events. The simulation timescale is advanced using the residencetime algorithm [162] where the time-interval between two simulation steps is givenby δt = − ln ϑ2Λ and ϑ2 is a number randomly drawn from 0 < ϑ2 ≤ 1. A fast-searching algorithm [200] is also implemented to speed up the simulations.454.2 Molecular Dynamics4.2.1 FundamentalsMolecular dynamics (MD) simulations are a computational tool to simulate thebehaviour of materials by considering the motion of their constituent atoms ormolecules individually. Classical mechanics is applied to determine the trajectoriesof these atoms in response to the intrinsic forces arising from their interaction withother atoms Fint and the external forces being acted upon the system, Fext, e.g.tensile loading. The interatomic interaction can further be modelled as a potentialenergy that is a function of positions of all N atoms, U (r1, . . ., rN), the gradient ofwhich is equal to the force acting on an atom. By applying Newton’s 2nd law, themotion of an atom i under zero external force followsmid2ridt2= −∇iU (r1, . . ., rN) (4.7)where mi and ri are the mass and position vector of atom i, respectively.A physically meaningful MD simulation relies upon two key factors, i.e. theintegrator algorithms for numerically solving 3N coupled differential equations ofmotion in Eq. (4.7) and the interatomic potentials for modelling the interactionamong constituent atoms. The choice of integrator algorithms is guided by theneed to produce self-consistent calculations that obey statistical mechanics princi-ples [153, 201]. This is satisfied by the use of a timestep that is in the order of atomicvibrational frequency (10−15 s). The chosen algorithm must also maintain the en-ergy and momentum conservation principles at all times. Few algorithms thatmeet such requirements include the Verlet algorithm, the velocity Verlet algorithmand the leap-frog algorithm [153, 202]. The velocity Verlet algorithm will be usedin this work, being implemented in the LAMMPS (Large-scale Atomic/MolecularMassively Parallel Simulator) framework, an open-source MD code [203]. The ver-sion of LAMMPS used in this work is the lammps-26May-13 version as compiledusing Intel C++ 12.1 compilers on Parallel and Jasper systems, provided by West-Grid facilities, a division of Compute Canada. A detailed discussion of the proce-dure to run MD simulations using LAMMPS is presented in Appendix A.464.2.2 Interatomic potentialsAn equally critical ingredient to MD simulations is the interatomic potentials, whichreflect the bonding between atoms. The most common type of potentials usedto describe metals and alloys are the embedded atom method (EAM) potentials[204], which express the potential energy as the sum of two-body (pairwise) in-teraction between a given atom and its neighbours within a cutoff distance, andmany-body attractive interaction that accounts for the contribution from a largergroup of neighbouring atoms to the electron density of a given atom.Interatomic potentials to simulate a binary system such as the iron-helium (Fe-He) system require at least three components: the Fe-Fe, the He-He and the Fe-He interactions. The literature provides few choices for each component; thesechoices, however, are not independent of each other. The interatomic potentialschosen in this work are selected based on the choice of the Fe-He interaction sincethis interaction governs the behaviour of He and Fe grain boundaries. A briefdescription of each potential component is presented as follows, while their math-ematical details can be found in Appendix .(1) Fe-Fe interactionThe potential developed by Ackland et al [205], i.e. the Fe-Fe Ackland04 potential,is used in this work. It has been verified to produce equilibrium and dynamicproperties that are consistent with those of ferrite [206]. The potential is an EAMpotential, and its formulation is given by [205]UFe:Fe =NFe−1∑i=1NFe∑j=i+1Upair,Fe:Fe(rij)+NFe∑i=1GFe (ρi) where ρi =NFe∑j=1ΦFe,i(rij)(4.8)where rij is the interatomic distance between Fe atom i and j, Upair,Fe:Fe is a two-body potential describing the short-range interaction among two Fe atoms, GFe isthe energy required to embed an Fe atom i into a uniform electron density ρi andΦFe,i is the spherically averaged electron density experienced by an Fe atom i dueto all of the other Fe atoms in the system.The equilibrium crystal structure from this potential is bcc with a lattice param-eter at 0 K, a0, of 2.8553 Å. Other Fe-Fe potentials available in the literature include47the Dudarev07 potential [207] and the Chamati06 potential [208].(2) The He-He interactionThis work employs the potential proposed by Aziz et al [209], i.e. the He-He Aziz95potential, which is a two-body potential that considers only the Coulombic repul-sive interaction among He atoms Urep,He:He [209], i.e.UHe:He =NHe−1∑i=1NHe∑j=i+1Upair,He:He(rij)(4.9)The Aziz95 potential has an advantage over the other He-He potential in the lit-erature, i.e. the Beck68 potential [210], because the former has been benchmarkedextensively against various empirical data and ab-initio calculations.(3) The Fe-He interactionThe potential proposed by Gao et al [211], i.e. the Gao11 Fe-He potential, is used inthis study. It was developed from fitting analytical functions to ab-initio calculationresults. The Gao11 Fe-He potential is also an EAM type, given by [211]UFe:He =NFe∑i=1NHe∑j=1Upair,Fe:He(rij)+NFe∑i=1FFe (ρsi ) where ρsi =NHe∑j=1ΦHe,i(rij)(4.10)where rij is the distance between an Fe atom i and a He atom j, Urep,Fe:He is a two-body potential describing the short-range interaction between Fe and He atoms,FFe is the energy required to embed an Fe atom i into an electron density ρsi andΦHe,i is the average electron density of Fe atom i due to all He atoms in the system.Other Fe-He potentials in the literature include the Wilson72 potential [212], theJuslin08 potential [213], the Stoller10 potential [214], the Gao10 potential [215] andthe Chen10 potential [216]. The Gao11 potential was developed to complementthe Ackland04 Fe-Fe potential [205] and the Aziz95 He-He potential [209]. This setof potential was chosen because it has been benchmarked for applications that arerelevant to this work. The potential reproduces, for example, the activation energyfor bulk diffusion of He, which is 0.04 eV atom−1 (4 kJ mol−1) [80].484.2.3 Simulation setupFirst, benchmarking simulations are conducted to determine basic properties ofthe Fe-He system, including the lattice parameter of bcc Fe at finite temperatures,aT, as well as the formation energy and bulk diffusivity of a single He atom in bulkbcc Fe crystal. These properties are computed from simulations that employ thetype-0 cell, i.e. a single bcc crystal cell with a dimension of L30 as shown in Figure4.4(a). Periodic boundary conditions are applied in all directions.Figure 4.4: (a) The type-0 simulation cell consisting of a single bcc crystal with dimensionsof L30, (b) the type-1 simulation cell containing a U-shaped half-loop grain boundary, and(c) its three-dimensional view, (d) the type-2 simulation cell containing two planar grainboundaries.To incorporate grain boundaries in subsequent simulations, the following twosetups are adopted. In the type-1 cells, a bicrystal is created whose grains areseparated by a U-shaped half-loop grain boundary (Figure 4.4(b)). In the type-2cells, the bicrystal cell contains two planar boundaries (Figure 4.4(d)).49The type-1 cells are constructed by first creating a single bcc crystal with or-thogonal axes of (XA, YA, ZA). Atoms in the dark-shaded region in Figure 4.4(b)are removed and replaced by atoms occupying bcc lattice sites of different orien-tation, i.e. (XB, YB, ZB). The resulting grain boundary is a U-shaped half-loopboundary, consisting of a curved end cap side and planar sides, see Figure 4.4(b).In this work, two different type-1 cells will be prepared, their axes being definedin Table 4.2. The planar sides in both cells are symmetric coherent and incoherenttwin boundaries. The twofold symmetry of〈110〉-axis renders the planar sidescomplementary to each other, e.g. if the θ⊥ side is the coherent twin, the θ|| sidewill be the incoherent twin. To simplify the nomenclature, the curved portion ina type-1 cell is referred to based on the segment at the loop tip, e.g. “the coherentloop” is a curved boundary with the symmetric coherent twin at its tip.Table 4.2: Crystallographic axes of bicrystals containing a U-shaped grain boundary (thetype-1 cell) investigated in this study, see Figure 4.4(b) for symbol definition.Orthogonal axesDesignationCoherent loop Incoherent loopXA, XB〈1 1 0〉YA〈1 1 2〉 〈1 1 1〉ZA 〈1 1 1〉 〈1 1 2〉YB〈1 1 2〉 〈1 1 1〉ZB〈1 1 1〉 〈1 1 2〉Periodic boundary conditions are applied to walls normal to the X- and Z-axes.In order to prevent the interaction between the dark-shaded grain in the type-1 celland its mirror images arising from the periodic boundary conditions, their distance(i.e. LZ − DU) is set to be 5 nm, as determined from sensitivity analysis on the cellsize. The position of walls normal to the Y-axis are set as stationary, i.e. a shrink-wrapped boundary condition [203], effectively creating two free surfaces. Atomicplanes within a distance of 1 nm from each free surface are fixed to prevent grainrotation [112].The type-2 cells are constructed by populating the light- and dark-shaded re-gion in Figure 4.4(d) with atoms occupying bcc lattice sites of orientation (XA, YA,ZA) and (XB, YB, ZB), respectively. Table 4.3 provides a list of type-2 bicrystals that50have been investigated in this study, each generating an inclined planar boundary,i.e. the boundary that forms an inclination angle ϕ with the planar coherent twin.This is equivalent to an inclination of (90◦ − ϕ) when measured with respect to theplanar incoherent twin. Periodic boundary conditions are applied in all directions,thus creating two planar boundaries of the same inclination, see Figure 4.4(d).Table 4.3: Geometrical setup of the type-2 bicrystals used for determining the absolutemobility of planar boundaries, their common XA/XB axis being〈110〉, see Figure 4.4(d).Inclination Orthogonal axesϕ [◦] XA, XB YA ZA YB ZB0.0*〈1 1 0〉〈1 1 1〉 〈1 1 2〉 〈1 1 1〉 〈1 1 2〉8.1 〈3 3 4〉 〈2 2 3〉 〈11 11 8〉 〈4 4 11〉15.8 〈4 4 7〉 〈7 7 8〉 〈2 2 1〉 〈1 1 4〉19.5 〈1 1 2〉 〈1 1 1〉 〈5 5 2〉 〈1 1 5〉26.8 〈3 3 8〉 〈4 4 3〉 〈19 19 4〉 〈2 2 19〉35.3 〈1 1 4〉 〈2 2 1〉 〈1 1 0〉 〈0 0 1〉44.7 〈1 1 8〉 〈4 4 1〉 〈17 17 4〉 〈2 2 17〉54.7 〈0 0 1〉 〈1 1 0〉 〈2 2 1〉 〈1 1 4〉63.2 〈2 2 19〉 〈19 19 4〉 〈4 4 3〉 〈3 3 8〉70.5 〈1 1 5〉 〈5 5 2〉 〈1 1 1〉 〈1 1 2〉74.2 〈1 1 4〉 〈2 2 1〉 〈7 7 8〉 〈4 4 7〉81.9〈4 4 11〉 〈11 11 8〉 〈2 2 3〉 〈3 3 4〉90.0** 〈1 1 2〉 〈1 1 1〉 〈1 1 2〉 〈1 1 1〉*coherent twin, **incoherent twinThe effect of solute on the migration of grain boundaries is examined by placinga number of He atoms on the simulation cells. The type of sites, the correspondingnumber of He atoms, as well as the dimensions of the simulation cells, will be setdepending on the purpose of simulations, see Chapters 7 and Running an MD simulationAn MD simulation is executed by first subjecting the simulation box to an opti-mization procedure called molecular statics (MS). The MS calculation iterativelysearches for the optimal position of atoms that satisfy a specific objective, e.g. thelowest total energy, under a certain set of constraints, e.g. a zero-pressure con-51straint in all directions achieved by varying the cell volume. The iteration endswhen either the number of iterations has reached a certain limit or the optimumconfiguration has been found, satisfying the objective function within a certain tol-erance [157]. The configuration that satisfies a zero-pressure constraint is referredto as the relaxed configuration [152].In addition to providing the initial configuration of MD simulations, MS calcu-lations can be used to determine the equilibrium properties of the system at 0 K,e.g. the grain boundary energy and the binding energy of He to grain boundariesin Fe. To extract the energy of planar boundaries, the equilibrium structure of thetype-2 cells at 0 K must be identified. A procedure described in [61, 217] is adoptedto obtain these equilibrium structures, i.e. (1) one of the grains is displaced along agiven translational vector that is parallel to the boundary plane, (2) atoms that arewithin a cut-off distance (0.6-0.8a0) from each other are removed, (3) the displacedbicrystal is relaxed and its energy calculated. The translational vectors are gener-ated from the resultants of the integer multiple of the unit vectors of (X, Y) in Table4.3. The cell with the lowest energy is referred to as the equilibrium structure.To bring a relaxed simulation box to a desired temperature, the cell dimensionsare expanded uniformly based on the appropriate lattice parameter aT [206] andthe atomic velocities initialized. The magnitude of velocity is generated from arandom distribution such that the average kinetic energy of the system is propor-tional to the desired temperature. The direction of velocity vectors is assigned suchthat the center of mass of the system remains at rest. The dynamics of each atomthen follows Newton’s equation of motion, i.e. Eq. (4.7).In addition to numerically solving the equations of motion, several other al-gorithms may be run concurrently to maintain certain constraints throughout thesimulation, e.g. the constant-temperature and constant-pressure conditions. Thesealgorithms, being referred to as thermostat and barostat, respectively, must becarefully chosen such that relevant statistical mechanics principles are not violatedthroughout the simulation, see [153] for details. Several thermostat and barostatalgorithms that satisfy such a requirement have been proposed in the literature,including the Nosé-Hoover thermostat, the Langevin thermostat, the Parrinello-Rahman barostat, and the Berendsen barostat [152]. In this work, the Nosé-Hoover52thermostat and the Parrinello-Rahman barostat will be used to control the temper-ature and pressure of the system, respectively [218, 219, 220, 221].At the end of an MD simulation, a number of post-simulation processing maybe required in order to extract useful information from the simulation results.The following sub-sections elaborate relevant techniques to process results post-simulations. These techniques will be employed in Chapters 6 to 8 to determinethe properties of grain boundaries in iron and their interaction with helium atoms.4.2.5 Relevant algorithms4.2.5.1 Order parameterA critical component in analyzing the cluster-boundary interaction via atomisticsimulations is the ability to identify grain boundary atoms. Each atom will beassigned order parameters η, a unitless property due to their nearest neighbourpositions [169]. In this work, the existing LAMMPS sub-routine for this techniquehas been made compatible with bcc crystals.Consider a bicrystal containing grains A and B, see Figure 4.4(d). An atom iin the bulk of grain A is expected to have 8 first nearest neighbours j located at adistance of 12 aT√3 from i, where aT is the lattice parameter at temperature T. Therelative position of neighbours j from atom i is tabulated in RAij , i.e. the referenceneighbour list for atoms in grain A.An unscaled order parameter ξAi can be assigned to each atom i based on thedifference between the relative position of neighbours j from atom i, rij, and thereference positions RAij , and is defined as [169]ξAi =1NactualNactual∑j=1min(rij − RAij)(4.11)where Nactual ≤ 8. The min function ensures that a given actual neighbour j willbe paired with a reference neighbour from the list RAij such that their difference,rij −RAij , is minimum. Based on this definition, if an atom i is in the bulk of crystalA, each of its neighbour j will be paired with one from the list RAij such that rij 'RAij . As a result, the unscaled order parameter of atom i will be ξAi = 0.53A similar procedure can be applied to atoms belonging to grain B whose neigh-bour list is denoted by RBij. The order parameter of an atom in the bulk of grain Bwhen measured using grain A as the reference is indicated by ξAB [169], i.e.ξAB =1N1nnN1nn∑j=1min(RBij − RAij)(4.12)To standardize the range of order parameter across different bicrystals, i.e. dif-ferent pairs of (RAij ,RBij), the normalized order parameter ηAi is defined as [169]ηAi =0 if ξAi < ξlo = KloξAB(ξAi −ξlo)(Khi−Klo)ξAB if ξAi ∈ [ξlo, ξhi]1 if ξAi > ξhi = KhiξAB(4.13)where Klo, Khi are cutoff values chosen between 0 and 1. A sensitivity analysis ofcutoff values was discussed in [169]; here, they are set as 0.25 and 0.75 respectively.To identify grain boundary atoms, the following selection criteria are adopted.A set of cut-off values (ηlo, ηhi) is chosen. An atom is considered to belong tograin A if ηA < ηlo, to grain B if ηA > ηhi, and to the grain boundary otherwise.Subsequent analyses show that the mobility results are independent of the cut-offvalues when 0 ≤ ηlo ≤ 0.3 and 0.7 ≤ ηhi ≤ 1. The mobility calculation was alsofound to be insensitive of the choice of the reference grain. Grain boundary energyThe energy of a planar boundary at temperature T, γ, will be computed using theexcess energy technique [222], where the boundary energy γ is equal to the extraenergy due to the boundary presence normalized by the boundary area A, i.e. [222]γ = (Etot − NFeeT)/2A (4.14)where Etot is the average potential energy of the system, NFe is the number of Featoms, and eT is the potential energy of an Fe atom in a bcc crystal at temperatureT. The factor 2 is to account for periodic boundary conditions, see Figure 4.4(d).544.2.5.3 Grain boundary mobilityThe capillarity techniqueIn the migration of a U-shaped half-loop boundary with an initial diameter of DU,see Figure 4.4(b), the initial driving pressure P is given by [112]P = Γpi/DU (4.15)where Γ is the stiffness, i.e. Eq. (2.8). The boundary shape during the steady-state migration may be different from its initial half-loop shape. As a result, theeffective driving pressure at steady state may be different from the initial drivingpressure. Both experimental [6, 9, 114] and simulation [112, 223] studies, however,typically employ the initial diameter DU to characterize the driving pressure sincethe shape change from the initial state to the steady state migration is a secondorder effect. This work will implement the same approach, i.e. the driving pressureis considered to be given by Eq. (4.15). Assuming the velocity V as a product ofabsolute mobility M and driving pressure P, V can be expressed as [112]V = M∗pi/DU (4.16)where M∗ has been defined in Eq. (2.9).In MD simulations, V is extracted by monitoring the rate at which the shrink-ing grain volume, vU, decreases during the steady-state migration, i.e.dvUdt =DULXdLUdt = DULXV [112], orV =1LXDUdvUdt(4.17)Substituting Eq. (4.16) yields [112]M∗ = 1piLXdvUdt=ΩpiLXdNUdt(4.18)whereΩ is the bulk atomic volume and NU is the number of atoms in the shrinkinggrain, having been identified using the order parameter η, see Section Artifical Driving Force (ADF) techniqueThe ADF technique will be one of the two techniques used in this work to extractthe absolute mobilities of planar grain boundaries [169, 172]. In this technique, thepotential energy of atoms belonging to one grain of a bicrystal is raised. The addedenergy drives the boundary migration, shrinking the volume of high-energy grain.The energy increase of an atom i, UADFi , varies with the atomic local environment,as represented by the order parameter ηi, via [169]UADFi =12Umax (1− cos(piηi)) (4.19)where Umax is the maximum energy added to any atoms. The cosine functionis an arbitrary choice, representing a continuous variation of UADFi with ηi. Theresulting mobilities have been found to be insensitive to the choice of interpolationfunction [171]. Since the interatomic force is proportional to the energy gradient,the extra force resulting from the ADF energy is non-zero only for atoms in or nearthe boundary where ηi varies locally from 0 to 1. The non-zero force affects thedynamics of boundary atoms, causing the boundary to migrate.The link between the added energy Umax and the driving pressure P can beestablished using a reasonable thermodynamic relationship, i.e. [169, 172]P = Umax/(12 a3T) (4.20)where aT is the equilibrium lattice parameter at finite temperature T.To extract the grain boundary velocity at a given driving pressure, the averagegrain boundary position must be determined as a function of time. The bicrystalis first cut into slices of 1 Å thick along the direction of boundary normal. At agiven time, the average position of each slice, zk, is obtained by averaging the z-coordinate of all atoms within that slice. The average order parameters on eachslice, i.e. ηA(zk) and ηB(zk), are calculated in a similar manner. Using these pro-files, the gradient of order parameter difference is calculated, i.e. ∆η(zk)−∆η(zk−1)zk−zk−1where ∆η(zk) = ηA(zk)− ηB(zk). Such a profiling is performed every 0.05-ns in-terval to produce the average position of grain boundaries as a function of time.The slope of linear regression of such data indicates the boundary velocity. In the56limit of low driving pressure, the boundary velocity increases linearly with increas-ing driving pressure. The slope of linear regression between velocity and drivingpressure is equal to the absolute mobility, see Eq. (2.4).The Random Walk (RW) techniqueThe other technique implemented in this work to extract the absolute mobility ofplanar grain boundaries is the RW technique. In this technique, the average posi-tion of the grain boundary, h(t), is obtained by taking the average of Z-coordinateof boundary atoms, which have been identified using the method discussed in Sec-tion The squared-displacement of the boundary for a given time-interval τis computed from ∆h2(τ) = [h (t + τ)− h (t)]2. For a given τ, there are Ni waysto choose the initial time t (t = 0, τ, ... , Niτ), where Ni = ttot/τ and ttot is thetotal simulation time. The mean-squared displacement (MSD) can be calculatedby taking the average of ∆h2, i.e.〈∆h2〉=(∑Nii=0 ∆h2i)/(Ni + 1). If the boundaryis rough at a given temperature T, it performs a random walk that exhibits diffu-sive characteristics, i.e. the evolution of its MSD yields a kinetic coefficient directlyrelated to the absolute mobility M and the boundary area A, via [170]〈∆h2〉= 2MkBTτ/A (4.21) Shape of a shrinking grainThe initial U-shaped half-loop of the boundary in a type-1 bicrystal, see Figure4.4(b), may evolve during the course of its migration. It is useful to track the shapeof the boundary as it migrates since the boundary shape can be used to determinethe position of He atoms relative to the boundary, and consequently, the segrega-tion level. To identify such a shape, the following technique is adopted.At a given time, grain boundary atoms are identified using the order parametercriteria discussed in Section These boundary atoms are then projected aspoint objects into a two-dimensional map, (X, Y), see Figure 4.5 as an example. Theshape of shrinking grain can be obtained by constructing a convex hull of thesepoints, i.e. the smallest convex polygon that contains every point in this map. A57simple analogy of a two-dimensional convex hull is an elastic rubber band beingplaced to confine a given set of points.Figure 4.5: An illustration of convex-hull technique to determine the shape of shrinkinggrain in a type-1 bicrystal.This work will employ the monotone chain algorithm proposed by Andrew[224]. A detailed discussion about the algorithm can be found in [225]. The algo-rithm sorts the boundary points lexicographically, i.e. first by X-coordinate, and incase of a tie, by Z-coordinate. It then constructs an upper and lower hull of thesepoints, see Figure 4.5. The shape of shrinking grain is defined as the area enclosedby the vertices generated from the convex-hull algorithm. Average curvature of a migrating curved boundaryIn addition to identifying the shape of a migrating curved boundary in a type-1bicrystal, it is also important to calculate the average curvature of the same bound-ary, 〈κ〉. This is especially useful for the case of curved boundary migration inan alloy system where solute atoms are expected to affect the overall shape of theboundary, and consequently, the average curvature. Since the average curvature〈κ〉 is proportional to the average driving pressure P, via Eq. (2.7), 〈κ〉 can be usedto estimate the change in the driving pressure due to the presence of solute atoms.The technique adapted to calculate κ(t) is illustrated in Figure 4.6, where a two-dimensional map of boundary atoms is presented. For a given atom i in the curvedboundary, a circle is drawn to fit the atom i and two other atoms to its left and right(total of 5 atoms). The Pratt algorithm [226] is employed to determine the centerA and radius κ−1A of the fitting circle. The fit quality is represented by ∆2, i.e. the58sum of the squared distance between the fitting circle and the set of atoms. Thecircle is then gradually enlarged by expanding the set of atoms being fit to a totalof 5 + 2n atoms, where n = 1, 2, ..., 13. The circle that fits the most number of atomsand has the lowest ∆2 is chosen to represent the local curvature at atom i, κ(zi),e.g. the circle centered at B with a radius of κ−1B in Figure 4.6. This procedure isrepeated for all atoms in the curved boundary, Ncgb, to obtain the instantaneousaverage curvature κ(t), i.e.κ(t) =1NcgbNcgb∑i=1κ(zi) (4.22)κ(t) is calculated for every time interval τ during the migration (i.e. t = 0, τ, . . . Ntτ)to obtain the average curvature 〈κ〉, i.e.〈κ〉 = 1Nt + 1Nt∑k=0κ(t = kτ) (4.23)Figure 4.6: A technique implemented in this work to calculate the average curvature κ(t)of a migrating curved boundary at a given time t, see text. Formation energy of He in Fe matrixIn a bcc Fe crystal, a He atom can occupy a tetrahedral site, an octahedral site, ora substitutional site (thus replacing a Fe atom). The energy required to bring aHe atom into any of these sites is referred to as the formation energy, FEbulk,t/0/swhere indices t, o and s indicating tetrahedral, octahedral and substitutional site,59respectively. The formation energy is determined from the difference between Etot,the potential energy of a relaxed bcc Fe crystal containing NFe atoms and a He atomat 0 K, and that of the same bcc crystal that contains no He atoms, via [211]FEbulk,t/0/s = Etot − (NFe − Nvac)e0,Fe − e0,He (4.24)where Nvac is the number of vacancies (0 for site t or o, 1 for site s), e0,Fe is thepotential energy of a Fe atom at 0 K, i.e. -4.013 eV [205] and e0,He is the potentialenergy of a He atom at 0 K, i.e. -7 meV [209].Table 4.4 presents the formation energy of a He atom in different types of sitein the bulk crystal of bcc Fe, which are obtained by performing MS calculations us-ing the Ackland04 Fe-Fe potential [205], the Aziz95 He-He potential [209] and theGao11 Fe-He potential [211]. These energies are consistent with previous work, asshown in Table 4.4. Unless stated otherwise, subsequent simulations will be per-formed with He atoms on tetrahedral sites. This is in agreement with experimentsthat identified He atoms as an interstitial solute in Fe matrix [27, 81].Table 4.4: Formation energy of a single He atom in the bulk crystal of bcc Fe.SiteFormation energy [eV]MS calculation† DFT calculation [78]Tetrahedral FEbulk,t 4.38 4.37Octahedral FEbulk,o 4.47 4.60Substitutional FEbulk,s 3.76 4.08†The values in this column are obtained from MS calculations performed in this work and areconsistent with the values reported in [211].Table 4.4 further shows that a He atom has the lowest formation energy as asubstitutional solute although its atomic size relative to the size of an Fe atom in-dicate it is an interstitial solute. If there is a vacancy site in the Fe matrix, a He atominitially at a tetrahedral interstitial site can identify the vacancy and become a sub-stitutional solute. There is an energy change, Et→s, that accompanies the transitionof a He atom from a tetrahedral to a substitutional solute in bcc Fe crystal, i.e.Et→s = FEbulk,s − (EV + FEbulk,t) (4.25)60where EV is the formation energy of a vacancy in a bulk bcc Fe (1.71 eV [205]).Substituting the relevant values from Table 4.4 to Eq. 4.25 yields Et→s of -2.33 eV.The negative sign indicates that the transition from a tetrahedral to a substitutionalsite is energetically preferred. Binding energy of He to Fe grain boundariesThe interaction between planar grain boundaries and a single He atom occupyinga tetrahedral site at a distance z from the grain boundary plane is represented bythe single-atom binding energy profile, Eb(z). The binding energy Eb(z) is definedas the difference between the average formation energy of a tetrahedral He atomat a distance z from the grain boundary in a type-2 bicrystal cell (Figure 4.4(d)) at 0K, FEgb(z), and the average formation energy of a tetrahedral He atom in a singlebcc crystal, FEbulk,t, see Table 4.4. This is given by [227]Eb(z) = FEgb(z)− FEbulk,t= [Etot − NFee0,Fe − e0,He − 2γA]− FEbulk,t (4.26)where Etot is the total potential energy of the bicrystal cell after being subjected toMS relaxation, A is the boundary area, γ is the boundary energy from Eq. (4.14). Diffusivity of HeTo determine the diffusivity of a single He atom at a finite temperature, its trajec-tory is monitored as a function of time, r (t), including its image due to periodicboundary conditions. By applying an analysis similar to the RW technique earlier,the MSD of the He atom,〈∆r2〉, can be determined as a function of time interval τ.Based on Einstein’s random walk theory [195], the diffusivity D is given by〈∆r2〉= 6Dτ (4.27)The bulk diffusivity Dbulk can be determined by placing a He atom on a tetra-hedral site of a bulk Fe crystal and performing the MSD analysis. Similarly, the61grain boundary diffusivity Dgb can be obtained from the same type of analysis us-ing a bicrystal of type-2 that contains a He atom at the boundary. The correct grainboundary diffusivity can be computed provided that the He atom remains at theboundary throughout the entire simulation time. Cluster size distributionTo characterize and quantify the distribution of cluster size, this work employs theWard technique [228] implemented via Lance-Williams algorithm [229]. This tech-nique is chosen since the number of clusters are not known prior to the analyses.A brief overview is discussed here; details are available elsewhere [230].A cluster k of size n is an object containing n number of He atoms. The mini-mum and maximum possible size of cluster k are 1 (a single He atom) and NHe (allHe atoms). The in-cluster variance of cluster k, σ2k , is defined as σ2k = ∑ni=1 |ri − rk|2where i is the index of atom that makes up cluster k, ri is the position of atom i,rk is the center-of-mass of cluster k. A cluster of size n = 1 (individual atoms), bydefinition, has σ2k = 0. The analysis is guided by the total variance σ2 = ∑k σ2k .Each atom is initially labelled as a cluster of size one (n = 1). On each iteration,all possible scenarios of merger between any two clusters k and l are tabulated. Thepair (k, l) that leads to the lowest increase in the total variance, ∆σ2, will be mergedinto a larger cluster j whose size nj is equal to nk + nl. Due to periodic boundaryconditions, only images of k and l that produce the minimum ∆σ2 are considered.Figure 4.7: An illustration of dendrogram, showing the progress in ∆σ2 (see text) as clustersgrow with the number of iteration. λc is the clustering cut-off parameter.62As illustrated in the dendrogram in Figure 4.7, the merger of any two clusterscontinues so long as ∆σ2 is less than λc, a cut-off parameter. In this work, severalvalues of λc have been tested. The cluster distribution is found insensitive to λcwhen λc is between 1.2 to 2.2 nm2. Subsequent analyses use λc = 1.5 nm2. Cluster radiusClusters that are characterized using the technique described in Section be further analyzed to determine the volume of space that they occupy and,consequently, their effective radius. For this purpose, the convex-hull conceptintroduced in Section is extended to a three-dimensional space. A three-dimensional (3D) convex-hull can be considered equivalent to a paper wrap thatencloses a 3D object which contains n number of vertices, see Figure 4.8.Figure 4.8: (a) The position of atoms that constitute a cluster, (b) the convex-hull construc-tion to determine the volume and radius of a cluster.This work will employ the quickhull algorithm [231] for constructing the convex-hull of each cluster and subsequently for computing their volume vc. The effectiveradius of a cluster rc is determined fromrc = (3vc/4pi)13 (4.28)where spherical shapes have been assumed. The quickhull algorithm is compatiblefor clusters that have more than 4 non-coplanar vertices, i.e. tetramers or larger.The radius of clusters smaller than a tetramer, i.e. monomers, dimers and trimers,are ill-defined since no 3D objects can be constructed from these clusters.63Chapter 5Kinetic Interplay between Ideal SoluteAtoms and a Migrating Grain Boundary5.1 IntroductionThis chapter examines the interaction between a migrating planar boundary andsolute atoms in an ideal solution using kinetic Monte Carlo simulations. The mainobjective is to explore the parameter space of a solute drag model from an atomisticperspective. An emphasis is placed upon studying the effect of solute diffusion,and its kinetics relative to that of the boundary migration, on the magnitude of so-lute drag. The findings are compared with predictions from the solute drag modelby Cahn [31]. Accordingly, a revised model is proposed and assessed against thesimulation results as well as previous work.5.2 Simulation setupKMC simulations are performed using cells illustrated in Figure 4.1(a). The celldimensions are chosen as follows. The height NZ is 200 unit cells, set such that asteady state migration can be observed. To set the dimensions of the grain bound-ary, i.e. NX and NY, the finite size effect on grain boundary structure is investi-gated. This is achieved by calculating the grain boundary roughness rgb [69] fordifferent temperatures T and grain boundary area NXNY,rgb =[1NXNY − 1NXNY∑i=1(hi − h)2]1/2(5.1)where h is the average grain boundary position. The finite size effect is deemedinsignificant if the roughness between two boundary dimensions for a given tem-64perature differ at most by 5%.The parameter space of the solute drag model is explored in KMC simulationsby studying the boundary velocity V as a function of a number of simulation vari-ables, i.e. the driving pressure P, the activation barrier for bulk solute diffusionQd, the absolute temperature T and the solute concentration C0. The effect of thesevariables on the boundary velocity is obtained for a given set of material proper-ties, the order of magnitude of which is well established for metallic alloys. A set ofmaterial properties in this range is selected as the simulation input and presentedin Table 5.1. The definition of the simulation variables and the simulation inputhave been presented in Section 4.1.Table 5.1: Material properties, which act as the input for KMC simulations.Properties Values Unit Relevant Equationsa0 0.3 nm (4.1), (4.3) via NX, NY and NZγ 0.53 J m−2 (4.1), (4.3)Eb0 16 kJ mol−1 (4.3)ν 1013 s−1 (4.4), (4.5), (4.6)Ω 2.7×10−29 m3 (4.5Qm 10 kJ mol−1 (4.4), (4.5)5.3 Results5.3.1 Grain boundary migration in a pure bicrystalAs a starting point, the finite size scaling technique is applied to determine theroughening temperature of the grain boundary [232]. In this technique, a numberof boundary dimensions are considered. For a given dimension, the boundaryroughness is calculated as a function of temperature.The roughening temperature is defined as the temperature above which theroughness varies with the boundary dimension for small dimension [232, 233].Identifying this transition is important since the temperature at which the grainboundary operates relative to Tc determines its structure, thus its dynamics. Sucha transition is only expected in three-dimensional systems because the boundary is65a two-dimensional surface [69, 233]. Systems with a lower dimensionality roughenspontaneously; its roughness increases with increasing boundary dimensions forall dimensions [196], e.g. there is no Tc in a two-dimensional KMC model devel-oped in [193].Figure 5.1: The boundary roughness as a function of temperature for different boundarydimensions, NXNY. The roughening temperature Tc is indicated.Figure 5.1 shows the finite-size scaling technique for the KMC boundary de-fined in Section 4.1.2. The roughening temperature Tc is found to be approximatelykBTc / a20γ ~ 2.42, corresponding to Tc = 833 K for the boundary energy and latticeconstant in Table 5.1. Below Tc, the boundary remains relatively flat, consistingpredominantly of terrace sites, see Table 4.1. Above Tc, the variation of boundaryroughness at a given temperature with boundary dimensions is less than 5% for di-mensions larger than 120× 120 unit cells. The cell size for subsequent simulationsis thus set as NX = NY = 120 unit cells.Next, the boundary is driven at several driving pressures and the velocity de-termined. Figure 5.2 presents the dependence of grain boundary velocity on thedriving pressure P for temperatures above and below Tc. The grain boundary ve-locity is obtained at a given temperature by taking the slope of a linear fit to the66average boundary position (h) as a function of time. Each point on this plot rep-resents the average of five different simulations, the variation between runs beingsmaller than the size of the symbols in this figure.Figure 5.2: The velocity of the grain boundary as a function of driving pressure for a purebicrystal at different temperatures.At T < Tc, the grain boundary migration occurs in a two-step process: thenucleation of an island, e.g. the atom i in Figure 4.1(c), followed by the lateralpropagation of ledges surrounding the nucleated island. Owing to the high barrierfor island nucleation relative to kBT, the rate of grain boundary migration at lowtemperatures and low driving pressures is negligibly small. In order to have asignificant migration of the boundary, a critical driving pressure has to be applied,its magnitude increasing with decreasing temperatures. A similar trend has alsobeen observed in a recent MD study where the ADF technique was applied to drivethe boundary at temperatures lower than the roughening temperature [234].At T ≥ Tc, the barrier for island nucleation is reduced relative to kBT andthe boundary migrates predominantly by spatially uncorrelated island nucleation.The boundary roughens and the velocity-driving pressure relationship obeys theBurke-Turnbull model, where for sufficiently small driving pressures the relation-ship is approximately linear [109].67Based on the above results, all subsequent simulations will be performed at1000 K (T > Tc) so as to allow for a direct comparison with predictions fromcontinuum models, where a linear velocity-driving pressure relation is typicallyassumed in the low driving pressure limit.5.3.2 Grain boundary migration in the presence of diffusing solutes5.3.2.1 Effect of solute concentrationIn subsequent simulations, the bicrystal is populated with a random distributionof solute at 1000 K (T > Tc). Table 5.2 provides the investigated solute contentsand the corresponding number of atoms Nsolute. A driving pressure is imposedand the grain boundary is observed to migrate. A transient regime ensues until asteady-state distribution of solute segregating to the boundary is achieved. Uponreaching steady state, the average boundary displacement is found to vary linearlywith time, similar to the case of pure bicrystals described above.Table 5.2: The number of solute atoms introduced to KMC simulations.C0 [at%] 3 4.5 6 15Nsolute [atoms] 89070 135070 183830 508230Both solute concentration and solute diffusivity are varied to study their effecton the rate of grain boundary migration. The effect of solute concentration on theboundary migration is shown in Figure 5.3 for solute diffusion with an activationenergy of Qd = 26 kJ mol−1. At low driving pressures, the boundary velocity de-creases with increasing solute concentration for a given driving pressure. As thedriving pressure is increased, the boundary velocity approaches that of the puresystem, indicating a breaking-away of the boundary from its solute cloud. Thisqualitatively obeys the classical description of solute drag from the continuummodels discussed earlier (see e.g. [31]). It is also of interest to note that such abreak-away transition was observed in the previous work only when regular solu-tion thermodynamics was assumed [235].Following Cahn [31], and based on the results shown in Figure 5.3(a), the dragpressure is computed from Eq. (2.16). Here, Pint is the driving pressure required to68achieve the same magnitude of velocity in a pure bicrystal, see Figure 5.3(a). Thedata-set for the pure boundary is fit to a general form of Burke-Turnbull model,V = VT (1− exp (−Pint/PT)) [109] where (VT, PT) are the temperature-dependentmodel parameters. These parameters are VT = 553 µm/s and PT = 623 MPa forresults obtained at 1000 K. A linear regime of velocity-driving pressure is observedfor sufficiently low driving pressures, i.e. P ≤ 0.3PT ≈ 200 MPa.Figure 5.3: (a) Grain boundary velocity as a function of driving pressure for bicrystalscontaining different solute contents, the diffusivity of which has an activation barrier Qdof 26 kJ mol−1, (b) The calculated drag pressure calculated and plotted versus velocitybased on data in (a). The solid curves were drawn using Eq. (5.2).The calculated drag pressure Pd is shown in Figure 5.3(b), having been fit to amodel that is given by,Pd =αC0V(1+ βV)2(5.2)where (α, β) are model parameters that are independent of solute concentrations,α = 3.5× 1014 m−4 J s and β = 1.6× 10−4 m−1 s. The model is shown to satisfacto-rily describe the simulation results, indicating a linear relationship between solutedrag pressure Pd and bulk solute content C0. The linear Pd-C0 is also consistentwith the prediction from the original Cahn model [31].It is emphasized that the mathematical form of the model in Eq. (5.2) is differentfrom the form of the model proposed by Cahn [31], i.e. Eq. (2.13). It is also noted69that in the Cahn model, the explicit forms of parameters (α, β) are proposed as afunction of the diffusivity profile and the binding energy profile (see Eqs. (2.14)and (2.15)), whereas the modified model in Eq. (5.2) considers (α, β) as adjustableparameters, treated in a manner similar to how they are treated in the fitting ofexperimental results, e.g. [124]. Further discussion that compares the modifiedmodel with the Cahn model is outlined in Section Effect of solute diffusivityThe effect of solute diffusivity on solute drag pressure is examined for a fixed so-lute concentration C0 of 3.0at%. Figure 5.4(a) presents the simulation results fora number of solute diffusivities, as indicated by the magnitude of the diffusionactivation energy Qd given in Table 5.3.Table 5.3: Solute diffusivities and the parameters for drawing fitting curves in Figure 5.4(a).DesignationQd Model parameters, see Eq. (5.2)[kJ mol−1] α [m−4J s] β [m−1s]Fast diffusivity 17 1.5× 1014 0.5× 10−4Moderate diffusivity 26 3.5× 1014 1.6× 10−4Slow diffusivity 35 9.9× 1014 5.5× 10−4It is important to acknowledge that the modified model in Eq. (5.2) remainsvalid and is able to provide a good fit to the results in Figure 5.4(a), see Table 5.3for the model parameters. The trend demonstrated by Figure 5.4(a) can be assessedin terms of predictions from the classical solute drag theory by Cahn. The theorypredicts that these curves will collapse into one curve when the boundary veloc-ity is scaled by the solute diffusivity and a characteristic length-scale, such as theboundary width or the lattice parameter [31]. Scaling the data in this way (Figure5.4(b)) does not appear to collapse the data into a single trend. The Cahn theoryalso predicts that parameters (α, β) depend linearly on the reciprocal of diffusivity.Additionally, it also postulates that the peak drag pressure is proportional to theα/β ratio and thus independent of the solute diffusivity. These predictions are not70consistent with the results in Figure 5.4(b) where it is revealed that the peak dragpressure increases with increasing solute diffusivity.Figure 5.4: Drag pressure plotted as a function of (a) velocity, and (b) normalized veloc-ity for bicrystals with a constant C0 of 3at%, but with different solute diffusivities. Thenormalized velocity is the velocity multiplied by the ratio of lattice parameter and diffu-sivity. Dashed lines indicate the fit between simulation results and the modified solutedrag model, Eq. (5.2), the fit parameters being given in Table 5.3.These deviations can be attributed to the fluctuation of the grain boundarytopology during the course of its migration. Observing the boundary structure atthe velocity corresponding to the peak drag pressure reveals significant differencesbetween systems containing slow diffusing and fast diffusing solute atoms. Figure5.5 illustrates snapshots of the boundary plane under these conditions, particularlyhighlighting the roughness of the interface plane. While both boundaries exhibitsimilarities, it can be seen that the boundary interacting with fast diffusing soluteexhibits a higher degree of spatial correlation in the boundary ‘height’ leading tolocal bulging in several locations. Such bulges require the coordinated motion of alarge number of neighbouring boundary segments, suggesting a correlation of thelocal behaviour of grain boundary atoms.71Figure 5.5: Snapshots of grain boundary cross-sectional view during its steady-state mi-gration at the velocity corresponding to maximum drag pressure and interacting with (a)slow-diffusing solute and (b) fast-diffusing solute. Yellow pixels indicate interfacial sol-vent atoms i whose height is equal to the average position of the boundary (hi = h). White,orange and salmon pixels indicate hi = h− a0, hi = h + a0 and hi > h + a0, respectively.Regions corresponding to bulges in (b) are highlighted.This bulging behaviour can be further quantified by examining the boundaryroughness rgb from each data point in Figure 5.4(b). As illustrated in Figure 5.6(a),the interface roughness is shown to vary depending on the solute diffusivity andthe boundary velocity. It is of interest to first note that previous KMC simulations[235] also reported a velocity-dependent behaviour of the boundary roughness.The trend reported in that work is, however, opposite to that reported in this work.In the previous work [235], if non-ideal solutes are present, the roughness wasmaximum when the boundary was stationary, its magnitude decreasing slowlywith increasing velocity. In the same work [235], the boundary roughness of a purebicrystal was reported to increase with increasing velocity, the opposite to the trendobserved in their enriched boundaries. No explanation was provided in [235] torationalize the solute effect on the boundary roughness, although it is possible thatthe solute-dependent behaviour of the boundary roughness is associated with theregular solution assumption employed in the simulations [235].72Figure 5.6: (a) The time-average boundary roughness (Eq. (5.1)) plotted against the nor-malized boundary velocity, i.e. velocity × lattice parameter/bulk diffusivity, for varyingsolute diffusivity; (b) A time-resolved trace of boundary roughness corresponding to thepeak drag pressure for interface interacting with fast-diffusing and slow-diffusing solutes,i.e. points P1 and P2 in (a).Figure 5.6(a) shows that, at low and high velocities, the boundary remains rel-atively flat irrespective of solute diffusivity, its roughness being similar to that of anon-driven boundary. In the case of systems containing slow diffusing solutes, theboundary roughness increases by less than half its stationary value as the velocityapproaches its value at the peak drag pressure. In the case of fast-diffusing solute,the boundary roughness nearly doubles relative to its stationary value. A grainboundary that migrates in the presence of fast-diffusing solute dissipates more en-ergy via solute drag than the boundary interacting with slow-diffusing solute. Asa result, the former has a higher peak drag pressure than the latter.5.4 Effect of solute diffusivity on the structure of grain boundaryWhile Figure 5.6 and the discussion that follows provide an explanation for theobserved increase of maximum drag pressure with increasing solute diffusivity,they do not explain the relationship between solute diffusivity and grain boundarytopology. The dependence of grain boundary topology on solute diffusivity can beexplained from the perspective of the rate of solute jumping into and away from amigrating boundary.73For a detailed look at this phenomenon, one may start by considering a flatboundary containing a steady-state distribution of segregated solute. At some lo-cation along the flat interface plane, a small boundary segment will advance intothe adjacent crystal due to the applied driving pressure. The next atomistic eventdepends sensitively on the diffusivity of the solute atoms surrounding this ad-vanced boundary segment.At low diffusivity (high Qd), the rate of solute hopping is predominantly deter-mined by the activation barrier Qd. The rate of hopping to and from the interfaceis approximately the same, i.e. Γsolbulk→int ≈ Γsolint→bulk = νD exp (−Qd/kBT) sinceQd is large relative to Eb0. The next event in the simulation is also likely to bethe advance of an adjacent boundary segment. This is due to the imposed driv-ing pressure that favours this event as well as the fact that the system energy canbe lowered by preferentially having the neighbouring segments to advance. Thisoperation is expected to be repeated along the boundary plane to reduce the totalinterface energy (via Eq. (4.3)), occuring more frequently than solute atom diffu-sion into the sites available in the advanced boundary segments. The distributionof grain boundary height is thus expected to be largely uncorrelated as seen in Fig-ure 5.5(a). Moreover, it is unlikely that an advanced boundary segment will fur-ther proceed before the neighbouring segments catch up because of the increasedroughening penalty, as specified in Eq. (4.1).If the solute diffusivity is very high (low Qd), however, then immediately af-ter the boundary segment advances the solute atom that is left behind has a highprobability to jump back to a location in the (now advanced) boundary since thebarrier for jumps into the interface is significantly lowered by the binding energyof the solute, Γsolbulk→int ≈ νD exp (−(Qd − 0.5Eb0)/kBT). The rates of solute jump-ing out of the boundary, on the other hand, are low owing again to the bindingenergy of solute, i.e. Γsolint→bulk ≈ νD exp (−(Qd + 0.5Eb0)/kBT). Segregation ofsolute to a bulge on the boundary will result in a reduction in the total system en-ergy, partially compensating for the increased energy due to the larger total areaof the boundary. In this situation, the next event can include a further extensionof the already advanced boundary segments under the imposed driving pressuresince the energy penalty associated with this event is partially compensated by the74segregated solute. This is evident in Figure 5.5(b) where the height distributionappears more spatially correlated compared to Figure 5.5(a). The advance of a sin-gle boundary atom by more than two atomic positions ahead of its neighbours inthe same grain has a low probability owing to the rapid increase in grain boundaryenergy (via Eq. (4.1)). When a bulge extends beyond this point, its growth will stalland the boundary will flatten so as to reduce the total energy of the system. Basedon this explanation and as observed in simulations (Figure 5.6(b)), bulges will ap-pear, grow then slow down and finally disappear as the whole boundary advanceswith new bulges appearing at other locations on the grain boundary plane.5.5 Modified solute drag modelThe original solute drag model by Cahn, Eq. (2.13), provides an estimate to bridgethe velocity-drag pressure trend from the slow and fast velocity regime usingmodel parameters, i.e. α and β. For the slow velocity regime, the drag pressureincreases with increasing velocity. For the fast velocity regime, the drag pressurediminishes as the velocity increases.In the original Cahn model, the task of determining the parameters (α, β), givena data set of Pd as a function of V, is completed by performing a linear regressionof V2 against V/Pd. If the model is a good fit for the data, the plot of V2 versusV/Pd is linear, the slope b and the ordinate intercept −k being related to the modelparameters via α = [(b/k)]/C0 and β = 1/√k. The parameters (α, β)The plot of V2 versus C0V/Pd from the simulation results presented in Figures5.3 and 5.4 are closer to parabolic than linear. The original model by Cahn, Eq.(2.13), is thus deemed a poor fit for the simulation results. On the other hand,when the simulation results are presented in the form of V versus√C0V/Pd, alinear trend is apparent, i.e. V = b√V/Pd−k where b is the slope and −k is theordinate intercept. Such a relationship is equivalent to the modified model in Eq.(5.2), where parameters (α, β) are related to b and −k via α = [(b/k)2]/C0 andβ = 1/k.The modified model has also been implemented in other applications. Figure5.7(a) and (b) show the application of the modified model to fit the numerical so-75lution of the solute-drag diffusion equation (see Figure 2.8(c)) and the results fromother atomistic simulations, i.e. the phase-field crystals method presented in [161],respectively. The close alignment between the modified model and the data set, asdemonstrated in Figure 5.7(a) and (b), suggest that the modified model providesa better description of the solute drag dependence on the boundary velocity thanthe original model.Figure 5.7: (a) The drag pressure data from Figure 2.8(c), now being fit to the modifiedsolute drag model. (b) The PFC simulation results from the literature [161] on the dragpressure variation with velocity for different solute diffusivities, being fit to both the orig-inal and the modified solute drag models.A further potential application is related to the development of models on com-plex microstructural evolution, e.g. via classical phase field techniques [236]. Inthese models, the effect of impurities on the kinetics of boundary migration is con-sidered by implementing continuum expressions such as Eqs. (2.13) or (2.19) [236].These expressions are used to estimate the local drag pressure of a boundary seg-ment migrating at a certain velocity due to its local curvature [236]. Microstruc-tural features predicted by these models, e.g. the average grain size, are thus de-pendent upon the type of solute drag expressions being assumed.765.6 SummaryThe dynamic interaction between a migrating boundary and diffusing solute atomsof an ideal solution has been investigated using kinetic Monte Carlo simulations.The simulation results demonstrate trends that are generally consistent with thepredictions from the Cahn solute drag model, e.g. the drag effect scales linearlywith increasing solute content.In other situations, departure from the classical model is observed. In contrastto the Cahn model, which predict the peak drag pressure to be independent of so-lute diffusivity, the simulation results show an increasing peak drag pressure withsolute diffusivity. Such a trend has been attributed to the change in the bound-ary roughness at the peak drag pressure, the magnitude of which depends on thesolute diffusivity. Fast diffusing solutes allow for local bulges to form and then dis-appear on the boundary plane during the boundary migration. The formation ofthese bulges results in a more tortuous path for boundary migration and thereforea higher rate of energy dissipation.To provide a quantitative description of solute drag pressure on the bound-ary velocity, a modification to the Cahn model has been proposed. The proposedmodel has been demonstrated to produce a better fit to the KMC results, as well asto other applications; however, it should be considered strictly as a phenomenolog-ical expression with a set of adjustable parameters. This is because an estimationof these parameters from relevant material properties, e.g. binding energy pro-file, solute diffusivity and interface width, has no physical grounds. Extractionof these properties via simulations are ill-defined because the model material de-signed for this type of simulations only admits two atomistic events, i.e. the spinflip representing the boundary migration and the atomic jump representing solutediffusion. Such a restriction is acknowledged as an inherent limitation of the KMCframework that has been developed in this work.77Chapter 6Grain Boundary Migration in Pure BCC IronBicrystals6.1 IntroductionThis chapter investigates grain boundary migration in pure bcc iron using MD sim-ulations. It aims to provide a frame of reference for the discussion in the Chapters7 and 8 where helium atoms are present. A particular emphasis is placed uponestablishing the correlation between the migration of curved and planar bound-aries. The migration of〈110〉curved twin boundaries in pure bcc Fe is first ex-amined in terms of their shapes and reduced mobilities. The properties of planarinclined segments that make up the curved boundaries, i.e. their absolute mobilityand energy, are computed independently. These properties are incorporated intocontinuum models correlating the migration of curved and planar boundaries. Fi-nally, atomistic events governing the migration of both types of boundaries areidentified, the similarities of which providing further evidence of the correlationbetween their migration behaviour.6.2 Simulation setupThere are three sets of simulations performed for the purpose of this chapter. Inthe first set of simulations, the type-0 cell is constructed to calculate the latticeparameters of bcc Fe crystal at finite temperatures, aT, see Figure 4.4(a). The celldimension L0 is 50a0, where a0 is the lattice parameter at 0 K, i.e. 2.8553 Å [205].The cell is simulated under a zero pressure and constant temperature condition for1 to 2 ns from 800 to 1200 K. The average lattice parameter at temperature T, aT, isobtained from the average volume of the cell, vT.78In the second set of simulations, the type-1 cells are employed to extract re-duced mobilities of curved twin boundaries, see Figure 4.4(b) and Table 4.2. Bench-marking simulations will be initially performed by carrying out simulations onseveral type-1 cells of varying dimensions to verify that the cell size has no effecton the simulation results. Figure 4.4(b) is referred to for notations. The reducedmobilities from the simulations are found to be independent of the cell size whenthe length LU is at least 10 atomic planes, the thickness LX is at least 10 atomicplanes and the loop diameter DU is at least 25 nm and 35 nm for the coherent andincoherent loop, respectively.In summary, the minimum dimensions of the type-1 cells are LX = 4.1 nm, LY =58 nm and LZ = 40 nm, corresponding to at least 750,000 Fe atoms. The boundarymigration is simulated under a constant volume condition for 1 to 2.5 ns from 800to 1200 K. The reduced mobility at a given temperature is computed by averagingover the results taken from three half-loop diameters DU: 25, 30 and 40 nm for thecoherent loop and 35, 42 and 56 nm for the incoherent loop.Table 6.1: Dimensions of the type-2 bicrystals used for determining the energy and abso-lute mobility of planar boundaries. Refer to Figure 4.4(d) and Table 4.3 for the illustrationand the crystallographic axes of type-2 bicrystals, respectively.Inclination Dimensions [nm] NFeϕ [◦] LX LY LZ [atoms]0.0*12.214.9 33.7 5184008.1 15.1 28.3 43704015.8 13.0 29.3 38820019.5 12.7 29.9 38808026.8 15.7 44.2 70728035.3 12.2 29.2 37680044.7 14.0 39.6 56856054.7 13.0 30.1 38700063.2 11.1 62.5 70680070.5 12.0 42.2 51648074.2 14.7 20.6 30864081.9 14.2 40.2 58512090.0** 14.0 19.7 285600*planar coherent twin, **planar incoherent twin79In the third set of simulations, the type-2 cells (Figure 4.4 (d)) are constructedfor computing the absolute mobility and energy of planar boundaries at 800, 1000,and 1200 K. At a given temperature, each cell is simulated in four runs. The celldimensions and the corresponding number of Fe atoms NFe, having been checkedfor finite size effect, are given in Table 6.1. The boundary conditions for the type-1and type-2 cells have been discussed in Section Results6.3.1 Lattice parameter of bulk bcc Fe at finite temperaturesThe lattice parameter of bcc Fe crystal at temperatures between 800 and 1200 K,aT, is obtained from the average volume of the type-0 cell, vT, via aT = (v1/3T )/50.Figure 6.1 presents the simulation results along with the lattice parameters fromthe literature, both from experiments [237, 238] and simulations [206, 207].Figure 6.1: The lattice parameter of bcc Fe crystal, aT, as a function of temperature between800 and 1200 K, showing good agreement with experiments [237, 238] and molecular dy-namics simulations that employed the same potential used in this work (the Ackland04potential) [206] and a different type of potential (the Dudarev07 potential) [207].The lattice parameters obtained from this work underestimate the experimental80values by 0.3% [237] to 0.5% [238], indicating good agreement. Further, the thermalcoefficient of expansion calculated from the simulation is 1.5×10−5 K−1, about 90%of the reported values from experiments [237, 238]. The values of aT will be usedin subsequent MD simulations, e.g. to determine the order parameter of bicrystalsfor mobility calculations, see Section Reduced mobilities of curved boundariesThe kinetics of a migrating curved boundary is observed by monitoring the evo-lution of the shrinking grain volume, see Figure 6.2(a) for an example. Within agiven period, the grain shrinkage is said to have occurred at steady state if the0.1-ns moving linear regression in that period has a slope that varies by less than20%. The reduced mobility M∗ is obtained from the steady-state shrinkage rate ata given temperature and DU, see Figure 6.2(a), via Eq. (4.18).Figure 6.2: (a) An example of the grain shrinkage due to curvature-driven boundary mi-gration, data set being taken from a 25-nm coherent loop at 900 K (b) Arrhenius plot ofreduced mobilities, dashed lines indicating the Arrhenius fit (see text).The reduced mobilities of coherent and incoherent curved boundaries are de-termined in the temperature range of 800 - 1200 K and presented in Figure 6.2(b).Each data point and its error bar are the average and the standard deviations fromthree simulation runs, respectively, over results taken from three half-loop diame-81ters: DU of 25, 30 and 40 nm for the coherent loop and DU of 35, 42 and 56 nm forthe incoherent loop. The reduced mobilities show an Arrhenius relationship, i.e.M∗ = M∗0 exp (−Qm/RT) where M∗0 is the prefactor, Qm is the migration energy,and R is the gas constant. Table 6.2 presents the Arrhenius parameters for bothcurved boundaries.Table 6.2: Arrhenius parameters of the reduced mobilities of curved twin boundaries,shown in Figure 6.2(b).Curved boundaries M∗0 [× 10−7 m2s−1] Qm [kJ mol−1]Coherent twin 4.2 ± 0.9 10.5 ± 0.1Incoherent twin 4.4 ± 1.1 12.0 ± 0.1The coherent loop has a reduced mobility that is ~10-20% higher than that ofthe incoherent loop in the investigated temperature range. More importantly, bothloops developed significantly different shapes, as illustrated in Figure 6.3. Theirmorphological difference suggests that the kinetics of migration varies as a func-tion of inclination. In order to verify such a hypothesis, the absolute mobilities ofplanar segments composing the curved boundaries are investigated, starting withthe segment at the loop tip, i.e. the planar symmetric coherent and incoherent twinboundaries, respectively.Figure 6.3: The shape of (a) 25-nm diameter coherent loop and (b) 35-nm diameter inco-herent loop during their steady-state migration (> 1 ns) at 1000 K, atoms coloured basedon their potential energy.826.3.3 Absolute mobilities of planar symmetric twin boundaries6.3.3.1 The artificial driving force (ADF) techniqueThe absolute mobilities of boundary segments at the loop tip, i.e. the planar sym-metric incoherent and coherent twin boundaries (ϕ = 90◦ and 0◦ in Table 6.1, re-spectively), will first be examined using the artificial driving force technique. Thetechnique, discussed in Section, has the advantage of being conventionaland intuitive since the absolute mobility can be obtained from the slope of velocityvs. driving pressure in the small driving pressure regime.Figure 6.4: The velocity of planar incoherent twin boundaries driven via ADF techniqueby 53.7 MPa at 900 K. (a) Average order parameter profile η(z) at 1.5 ns and (b) the cor-responding profile of order parameter gradient ∆η(z)∆z at 1.5 ns, (c) the evolution of grainboundary positions, i.e. position of the two peaks in panel (b).To calculate the boundary velocity, the technique described is implemented.The profiles of order parameters along the boundary normal at a given time duringthe migration, ηA(z) and ηB(z), are shown in Figure 6.4(a). From these profiles,the gradient of order parameter differences can be calculated, as shown in Figure6.4(b), the two peaks indicating the position of grain boundaries. By repeating thecalculation for every 0.05-ns interval, the evolution of grain boundary position is83obtained, see Figure 6.4(c). The boundary velocity is equal to the slope of linearregression from Figure 6.4(c).Figure 6.5(a) presents the simulation results for the planar incoherent twin bound-ary, each data point and its error bar indicating the average and the standard de-viation from four runs. A linear trend intercepting the origin is apparent below80 MPa, the absolute mobility at a given temperature being equal to the slope oflinear fit within such a regime. These results are fit following M = 2.3×10−5 exp(−50 kJ mol−1/RT) m4 J−1 s−1, see Figure 6.5(b).Figure 6.5: (a) The velocity of planar incoherent twin as a function of driving pressure atseveral temperatures, linear regression being applied for P < 80 MPa, (b) Arrhenius plotof the absolute mobilities of the planar incoherent twin, obtained from the ADF technique.The ADF technique has also been applied to drive the planar coherent twinboundary. At 1200 K, extra energies umax ranging from 5 to 150 meV (equivalent todriving pressures of 66 MPa to 2 GPa, see Eq. (4.20)) are introduced to the bicrystalcell. The coherent twin remains stationary in all cases, even when being driven bythe highest driving pressure. The absolute mobility of planar coherent twin is thusconcluded as zero.846.3.3.2 The random walk (RW) techniqueWhile intuitive, the ADF technique is associated with a number of issues, includingits inability to properly discriminate grains adjacent to low-angle grain boundaries,causing the effective driving pressure to be smaller than the input value [171, 175].This issue may affect the computation of the absolute mobilities of inclined planarboundaries required for the analysis of the shapes and kinetics of curved boundarymigration. Furthermore, the ADF technique is inherently incompatible for alloys,e.g. when He atoms are present, since it fails to discriminate between boundaryatoms and bulk atoms that have been displaced by a repulsive force by soluteatoms via their short-ranged interaction.An alternative technique to extract the absolute mobility of grain boundary isthe random-walk technique, discussed in detail in Section As a bench-marking study, the RW technique is first used to compute the absolute mobilitiesof the planar incoherent and coherent twin boundaries. This allows a comparisonbetween the absolute mobilities from the RW and ADF techniques.Figure 6.6: (a) The average position of the incoherent and coherent twin as a function oftime, (b) the mean-squared displacement (MSD) obtained from the evolution shown in (a),error bars and lines representing the standard deviation and linear regression, respectively.Figure 6.6(a) shows an example of the evolution of the average grain bound-85ary position, h(t), where the random walk of the planar coherent twin displaysa less prominent fluctuation compared to that of the planar incoherent twin. Theamplitude of the latter is approximately 4 Å.Figure 6.7: The distribution of boundary displacement at 1000 K for several time intervalsτ, evaluated from the h(t) of (a) the planar coherent twin boundary, and (b) the planarincoherent twin boundary, shown in Figure 6.6(a).Figure 6.7(a) and (b) shows the displacement distribution for the planar coher-ent twin and the planar incoherent twin at 1000 K, respectively. Both are observedto be Gaussian, consistent with the random walk nature of the boundary motion.The former has a width that shows little variation with τ while the latter exhibitsan increasing width as τ increases, indicating a diffusive random walk.The mean-squared displacement (MSD) of both boundaries can be computedfrom these distributions, see Figure 6.6(b). The planar incoherent twin has an MSDthat increases with increasing time-intervals where the deviation from a perfectlystraight line is a signature of the thermal noise from the displacement distributionshown in Figure 6.7(b). Despite these undulations, the data is well fit by a linear re-gression, allowing the absolute mobility to be calculated by Eq. (4.21). The planarcoherent twin, in contrast, shows an insignificant MSD evolution. This indicatesan absolute mobility that is close to zero.The same procedure is repeated for other temperatures (800 to 1200 K) to extractthe absolute mobilities of the planar incoherent twin boundary. Figure 6.8 presentsthe results, obtained from averaging four simulation runs and whose error barsrepresenting the standard deviation. The absolute mobility results are fit to an86Arrhenius relationship, M = M0 exp (−Qm/RT) where the parameters (M0, Qm)are given in Table 6.3.Figure 6.8: Arrhenius plot of the absolute mobility of the planar incoherent twin, computedfrom the RW technique and the ADF technique. Fitting parameters are given in Table 6.3.The migration energies of the curved boundaries (Table 6.2) can be comparedwith those of the planar boundaries (Table 6.3) since both cases employed bound-aries of the same misorientation, albeit different geometry. The migration energyof the planar incoherent twin (Qm = 51 kJ mol−1) is four times larger than that ofthe curved incoherent twin (Qm = 12 kJ mol−1). On the other hand, the planar co-herent twin is effectively immobile this being in contrast with the mobile curvedcoherent twin (Qm = 10 kJ mol−1).Table 6.3: Arrhenius parameters of the absolute mobilities of planar symmetric twinboundaries, shown in Figure 6.8.Planar boundaries M0 [× 10−5 m4 J−1 s−1] Qm [kJ mol−1]Coherent twin∗ - -Incoherent twin (ADF technique) 2.3 ± 0.2 50.0 ± 0.5Incoherent twin (RW technique) 2.7 ± 0.1 51.0 ± 0.1∗ the boundary is flat at all temperatures, resulting in zero absolute mobilities.87Such a discrepancy in the migration energies has also been observed in the ex-perimental study on Zn grain boundaries, and was attributed to the possibilityof curvature driven boundaries migrating under a different mechanism than pla-nar boundaries [9]. In order to verify whether the migration mechanism is indeedshape-dependent, the kinetic behaviour of the planar segments that make up thecurved boundary is considered next.6.3.4 Absolute mobilities and energies of planar inclined boundariesThe RW technique is selected as the preferred method for extracting the absolutemobilities of planar inclined boundaries (ϕ 6= 0◦, 90◦ in Table 6.1). The MSD evolu-tion of the inclined boundaries is monitored at 800, 1000 and 1200 K. Each bound-ary is found to perform a diffusive random walk, permitting the use of Eq. (4.21)to compute their absolute mobilities M.Figure 6.9: (a) Grain boundary energy γ and (b) absolute mobility M as a function ofinclination ϕ computed from the simulations at several temperatures; dashed lines in (a)and (b) are fitting curves for each temperature.To compute the grain boundary energy at a given temperature T, γ via Eq.(4.14), a separate set of MD simulations has been performed where a bulk bcccrystal with a dimension of 20a3T is maintained at zero-pressure and constant-88temperature condition. The potential energy per bulk Fe atom eT is obtained byaveraging the potential energy of the bulk crystal, yielding eT of −3.90, −3.87 and−3.84 eV, which correspond to 800, 1000 and 1200 K, respectively.The variation of γ and M with the inclination ϕ from the simulations are pre-sented in Figure 6.9(a) and (b), respectively. Each data point and its error bar rep-resent the average and the standard deviation from four runs respectively.For a given temperature, the grain boundary energy increases as the inclinationgoes from the planar symmetric coherent twin (ϕ = 0◦) to the planar symmetricincoherent twin (ϕ = 90◦), see Figure 6.9(a). For the purpose of the continuummodelling to be performed later, the relationship between γ and ϕ is further fit to,γ(ϕ) = (γmax − γmin)[1− exp (−B1ϕ)] + γmin (6.1)where parameters B1, γmax, and γmin are given in Table 6.4. Eq. (6.1) is chosen asthe fitting function because its mathematical form provides a compromise betweenprecision and the number of fitting parameters. The relationship between stiffnessand inclination is obtained by substituting Eq. (6.1) to Eq. (2.8), yieldingΓ (ϕ) = γmax −(1+ B21)(γmax − γmin) exp (−B1ϕ) (6.2)The boundary energy also shows a variation with temperature, i.e.γ decreases astemperature increases. The temperature effect on γ is second order and more pro-nounced in the inclined boundaries compared to both symmetric twin boundaries.A similar trend of γ with temperature has been reported, from experiments [62]and simulations [239, 240].Figure 6.9(b) shows the variation of absolute mobilities with the inclination an-gle as obtained from the RW technique, including the earlier results for the planarcoherent and incoherent twin boundaries. The boundaries having inclinations be-tween 15◦ and 55◦ have the highest mobilities. While there is no clear monotonictrend of absolute mobilities M with the inclination angle ϕ, the dependence of Mon ϕ for a given temperature has been fit to.M(ϕ) = B2ϕ−p exp(−B3ϕ−q) (6.3)89where fitting parameters B2, B3, p and q are given in Table 6.4. The extent of scatterbetween the data and the model varies with ϕ. The absolute mobility of the 44.7◦-inclination boundary from simulations, for example, coincides well with the fittingcurves for 800 and 1000 K, but is about twice as high as the predicted value at 1200K. Such scatter emphasizes that Eq. (6.3) is only a first order approximation todescribe the absolute mobility from simulations. This must be kept in mind whenthe results of the continuum models that make use of these analytical expressionsare compared to the simulation results in the discussion below.Table 6.4: Fitting parameters in Eqs. (6.1) and (6.3) to draw the continuous approximationof γ (ϕ) and M (ϕ) in Figure 6.9.Parameters 800 K 1000 K 1200 KEq.(6.1)γmin 0.25 0.20 0.19[J m−2]γmax 1.36 1.32 1.30[J m−2]B1 [deg −1] 4.04×10−2 3.53×10−2 2.26×10−2Eq.(6.3)4.46×104 5.32×100 1.49×101B2 [m4J−1s−1degp]B3 [degq] 83.6 78.6 69.5p 6.0 4.0 4.0q 0.8 1.0 0.96.4 Correlation between the migration of curved and planarboundaries6.4.1 ShapeAs a starting point for discussing the migration of the coherent and incoherentloop, an approach similar to that in [191] is employed to predict the steady-stateloop shapes via a continuum model [1, 241]. The model asserts that the boundary90shape at steady state, (x, y), can be obtained via [1]x (φ) = v−1ˆ φ0M (φ) Γ (φ) sin φdφy (φ) = v−1ˆ φ0M (φ) Γ (φ) cos φdφ (6.4)where φ is the inclination relative to the segment at the loop tip, i.e. φ = ϕ forthe coherent loop and φ = (90◦ − ϕ) for the incoherent loop, and v is a scalingfactor with a unit of velocity determined from imposing the boundary conditions,i.e. y(φ = ±90◦) = ∓DU/2. An isotropic half-loop boundary, i.e. M, Γ 6= f (φ),will maintain its initial shape throughout its migration. The analytical fits to Γ (φ)and M (φ) from Eqs. (6.2) and (6.3), respectively, are substituted into Eq. (6.4). Thesteady-state shapes (x, y) are then solved for −90◦ ≤ φ ≤ 90◦.Figure 6.10: Superimposition of the predicted shapes of curved boundaries [solid red lines,see Eq. (6.4)] into the shapes observed from simulations (i.e. Figure 6.3) for (a) the coherentloop and (b) the incoherent loop at 1000 K. The Γ (φ) and M (φ) used in the continuummodel are shown on the right side of the figure. Note that the Γ (φ) and M (φ) data set arethe same for both boundaries, but shifted along the φ-axis by 90◦.91The predicted shapes are in reasonable agreement with the simulation results,as shown in Figure 6.10. Some discrepancies, primarily observed in the case ofthe coherent loop (Figure 6.10(a)), can be attributed to the fact that the simple Eqs.(6.1) and (6.3) do not capture all of the details of the γ and M dependencies on φfrom simulations, see Figure 6.9(a) and (b).6.4.2 MobilitiesThe model in Eq. (6.4) can be extended to estimate the reduced mobility of thecurved boundaries. Consider the half-loop boundary migrating at steady state,shown in Figure 6.11. The maximum inclination of the curved boundary, φmax,may be less than 90◦. Beyond φmax, the curved portion is terminated.Figure 6.11: Geometrical features of a curved boundary as a visual for Eq. (6.5).A segment ds at the curved portion is associated with its properties, M andΓ. The segment ds, which lies at an inclination φ from the tip, has a velocity ofVφ = |−→Vφ| = M (φ) Γ (φ) κ (φ) where κ (φ) is the local curvature of segment ds.Each segment migrates at a velocity of |−→Vx | = Vφ cos φ along a direction that formsan angle φ with the segment’s normal. At steady state, the magnitude of |−→Vx | isuniform for all segments, and is equal to the velocity V in Eq. (4.16). The curvatureis defined as κ (φ) = d|−→Tφ | / ds where −→Tφ is the unit tangent vector for segment ds[242]. By substituting Eq. (4.16) to Vφ, the reduced mobility from the model M∗modisM∗mod =DUpi´M (φ) Γ (φ) cos φdTφ´ds(6.5)92where the integration is performed over the entire loop, i.e. −φmax ≤ φ ≤ φmax.The magnitude of ds and dTφ can be expressed in φ as ds = [(x′)2 + (y′)2]1/2 dφ anddTφ = [(T′x)2 + (T′y)2]1/2 dφ, where [′] is the first-order derivative with respect to φ.Using the steady-state shapes produced by Eq. (6.4) and the fits to Γ (φ) andM (φ) from Eqs. (6.2) and (6.3), Eq. (6.5) is used to determine the reduced mobil-ities of the coherent and incoherent loop. Table 6.5 compares the mobilities com-puted from this continuum model M∗mod to those found from simulations M∗, seeFigure 6.2(b), showing that good agreement is found. The largest deviation (28%)belongs to the incoherent loop at 800 K, this most likely being due to the continu-ous fits to M and γ being a first order approximation.Additionally, M∗mod values are fitted to an Arrhenius relationship. The obtainedQm of the coherent loop is 14 kJ mol−1, in good agreement with the simulation re-sult (10 kJ mol−1 in Table 6.2). On the other hand, the obtained Qm of the incoher-ent loop is 22 kJ mol−1, which is about a factor of two larger than the simulationresult (12 kJ mol −1 in Table 6.2). The discrepancy is attributed to the monotonictemperature trend of the mobility deviations, ∆, see Table 6.5.Table 6.5: Reduced mobilities extracted from simulations [Table 6.2] and a continuummodel [Eq. (6.5)] where ∆ = (M∗mod - M∗) / M∗.Coherent loop Incoherent loopT M∗ M∗mod ∆ M∗ M∗mod ∆[K] [×10−8 m2s−1] [%] [×10−8 m2s−1] [%]800 8.64 8.56 -1 7.37 5.30 -281000 12.0 14.3 19 10.3 10.4 11200 14.4 17.0 18 13.3 15.8 196.4.3 Atomistic mechanismsThe continuum model discussed above aims to explain the shape and reduced mo-bilities of curved boundaries using the properties of planar boundaries. While itis evident from Figure 6.10 and Table 6.5 that the migration of planar and curvedboundaries are correlated, it is important to substantiate this correlation with thenature of the underlying atomistic mechanisms controlling both processes. To in-vestigate these mechanisms, the following analysis was performed.93Figure 6.12: Snapshots taken from the migration of (a) the curved incoherent twin, (b)the curved coherent twin, (c) the planar boundary of 27◦ inclination, and (d) the planarincoherent twin at 1000 K. Arrows indicate displacement vectors of each atom from 0.5 to0.9 ns after the steady-state migration started. The planar boundaries in (c) and (d) weredriven using the ADF technique. Dashed boxes indicate the random shuffling of atoms,while dotted ovals indicate the cooperative atomic motion (see text).A layer of atoms in the (110)-plane is extracted from a number of migratingboundaries, both curved and planar. The initial and final position of each atom94are monitored for a given time interval. Vectors representing each atom’s displace-ment are then mapped.In the migration of the incoherent loop, Figure 6.12(a), atoms located aroundthe loop tip undergo a random shuffling, see the dashed box in Figure 6.12(a1).The shuffling event is different from the event occuring near the planar portion ofthe boundary, see the dotted oval in Figure 6.12(a2), where a cooperative motionof atoms along the 〈1 1 1〉 direction is operational. This cooperative motion canalso be identified in the coherent loop migration, the dotted oval in Figure 6.12(b),where it appears to the predominant mechanism throughout the curved portion ofthe boundary. Additionally, the random shuffling is apparent in the planar portionof the boundary.Both atomistic events, i.e. the cooperative motion and the random shuffling,are also present in the migration of planar boundaries, see Figure 6.12(c) and (d).These planar boundaries are driven using the ADF technique at 1000 K with adriving pressure of 80 MPa. As noted earlier, the ADF technique was used insteadof the RW technique as the atomic mechanisms underlying the migration can notbe analyzed from the stationary boundary used in the RW analysis.In Figure 6.12(c), the net boundary migration occurs in the〈2 2 19〉// 〈4 4 3¯〉 di-rection but the underlying event is the cooperative motion along the 〈1 1 1〉 direc-tion, e.g. the dotted oval. The random shuffling is also apparent in Figure 6.12(c),but not as prevalent as the cooperative motion. Figure 6.12(d) shows the oppositesituation, i.e. the migration of the planar incoherent twin occurs predominantly bythe random shuffling event.The above analyses demonstrate that the migration of curved and planar bound-aries share a number of similarities in terms of their mechanisms, both involvinga combination of the random shuffling and the cooperative atomic motion. It canbe inferred from the mobility values shown in Figure 6.2(b) and 6.9(b) that bound-aries that migrate predominantly via the cooperative motion, e.g. boundaries inFigure 6.12(a) and (c), are more mobile than those advancing primarily via the ran-dom shuffling, e.g. boundaries in Figure 6.12(b) and (d). This suggests that thecooperative motion event is faster than the random shuffling event.These analyses can further be applied to reanalyze recent work published in95the literature. An MD study on the migration of curved twin boundaries in Fe-Cralloys [191] determined that the curved coherent twin was highly mobile due tothe fast cooperative motion being responsible for its migration. Based on this, theauthors suggested that the planar coherent twin must be highly mobile too [191].It was acknowledged that this deduction differed from the classical view whichpostulates that the planar coherent twin has a low mobility due to the absence ofline defects [2]. In this work, it has been shown (Figure 6.6(b)) that the planar co-herent twin has a mobility close to zero. It is proposed here that the high mobilityof the curved coherent twin is attributed to the fast cooperative motion along the〈1 1 1〉 direction, see Figure 6.12(b). The cooperative motion is operational only iftriggered by the random shuffling that takes place along the planar segments thatconnect to the endpoints of the curved portion. The random shuffling is availableon these segments because they consist of the planar incoherent twin, see Figure6.12(d). On the other hand, no random shuffling is present to trigger the coopera-tive motion in the planar coherent twin due to the entire boundary being coherent.As a result, the planar coherent twin has a zero mobility.6.5 SummaryThe migration of curved and planar grain boundaries have been analyzed in termsof their shapes, their mobilities and the atomic-scale mechanisms underlying theirmigration. The shape that curved boundaries assume during their migration canbe explained by a continuum model that incorporates the stiffness and absolutemobility of boundary segments that make up the curved boundaries. The modelwas extended to provide a quantitative estimation of the reduced mobilities ofcurved boundaries, which is in good agreement with the simulation results. Fi-nally, analyses at the atomic-scale have determined that, while the detailed mech-anisms for the migration of both curved and planar boundaries may not be exactlythe same, their similarities provide further evidence of the correlation betweentheir migration behaviour. This further implies that the mechanism of boundarymigration is independent of the type of driving pressure, a seemingly intuitiveconcept that has recently been called into question.96The absolute mobilities of planar inclined boundaries discussed in this chapterwill be used as a reference for the discussion in Chapter 7 where the effect of seg-regation and clustering of helium atoms on the absolute mobilities of a number ofinclined boundaries will be investigated. The reduced mobilities of curved bound-aries will be used as a reference for the discussion in Chapter 8 where the dynamicinteraction between clusters and a migrating curved boundary will be analyzed.97Chapter 7Interaction of Helium Clusters withNon-driven Grain Boundaries in BCC Iron7.1 IntroductionThis chapter explores the clustering behaviour of He atoms and their interactionwith non-driven planar grain boundaries in Fe bicrystals. The main objective isto quantify the effect of segregated He clusters on the absolute mobilities of non-driven planar grain boundaries. The diffusion of single helium atoms in the bulkcrystal and grain boundaries of iron are first simulated and their difference ad-dressed. Dilute concentrations of He atoms are next introduced to Fe bicrystals toobserve the formation of He clusters. Subsequently, segregation of clusters to theboundaries and their size distribution are characterized. The spatial fluctuation ofcluster-enriched grain boundaries are monitored and the absolute mobilities of theboundaries determined using the random walk method. The effect of clusters onthe boundary mobilities is quantified and then discussed in terms of the propertiesof He atoms evaluated earlier. Finally, the simulation results are assessed againstclassical models on solute-grain boundary interaction.7.2 Simulation setupThe diffusivity of a single He atom in the bulk of an Fe crystal is computed byplacing a He atom in a tetrahedral site of the type-0 cell with dimensions of (50a0)3,see Figure 4.4(a). The cell is brought to constant temperatures between 300 to 1200K and the trajectory of the He atom monitored. This allows the computation ofthe bulk diffusivity of a single He atom using the technique described in Section4. examine the interaction energy between a single He atom and planar grainboundaries in Fe, the type-2 bicrystals are used, see Figure 4.4(d). Table 7.1 pro-vides a list of planar boundaries investigated here. They are inclined boundarieswhose inclination angle ϕ lies between the coherent and incoherent twin bound-aries. A He atom is placed in tetrahedral interstitial sites located at varying dis-tances from the grain boundary. The variation of the binding energy as a functionof distance to the boundaries is computed using the procedure described in Section4.2.5.7. Similarly, the grain boundary diffusivity of a single He atom is also calcu-lated at 800-1100 K by monitoring the random walk motion of He atom initiallyplaced in the grain boundary for 6 ns.Table 7.1: Geometrical setup of simulation boxes and their solute contents. Refer to Figure4.4(d) and Table 4.3 for the illustration and the crystallographic axes of type-2 bicrsytals,respectively.Cell type ϕ [◦]Dimensions [nm] NFe NHe [atoms]LX LZ LZ [atoms] D† 0.1at% 0.5at% 1.0at%Type-0 cells- 14.3 14.3 14.3 2500001250 1256 2525(Figure 4.4(a))19.520.220.0 17.8 646000 647 3256 6543Type-2 cells 44.7 20.9 19.7 712200 709 3560 7157(Figure 4.4(d)) 70.5 22.2 16.8 642000 648 3230 648590.0 21.0 17.8 642000 645 3230 6485† A single He atom is placed on a tetrahedral site in the bulk of bulk crystal and the grain boundaryof type-2 bicrystals to compute the bulk and grain boundary diffusivity, respectively.To study the clustering behaviour of He atoms, multiple He atoms are placedrandomly in tetrahedral interstitial sites of the type-0 and the type-2 cells. Bulksolute concentrations, C0, of 0.1 at%, 0.5 at% and 1.0 at% are used. Table 7.1 liststhe number of He atoms that correspond to each concentration. Periodic boundaryconditions are applied in all directions. Simulations are run at 1000 K under a zero-pressure condition for a duration of 5 ns for the bulk crystal cells and 20-40 ns forthe type-2 bicrystal cells. Atomic positions are recorded every 1 ns during whichtime the cluster distribution is analyzed using the Ward’s technique, see Section4.2.5.9. Clustering is deemed to have reached steady state at a given time if thedifference between the size distribution taken from that time and those from the99next 3 and 6 ns differ by less than 20%.Additionally, to investigate the effect of grain boundaries on clustering, clustersare classified as to whether they are located at the boundary or in the bulk. Clus-ters are considered to belong to the boundary if 60% of the He atoms in a clusterare within δ/2 distance away from the average boundary position, on each side.Here, δ is the boundary width determined from the binding energy profile. Once asteady-state cluster size distribution is achieved in the bulk and boundary, the av-erage position of the boundary, h, will be monitored over time and the boundarymobility determined using the random-walk (RW) technique.7.3 Results7.3.1 Properties of He in the bulk Fe crystal7.3.1.1 Bulk diffusivity of a single He atomFigure 7.1: (a) The evolution of mean-squared displacement of a single He atom at tetra-hedral sites in bulk Fe at several temperatures, (b) the bulk diffusivity of He atom as afunction of temperature, calculated from (a) via Eq. (4.27).The interstitial diffusion of He atom is computed by tracking the evolution of themean-squared displacement of a He atom in the bulk, as shown in Figure 7.1(a).The diffusivity at a given temperature can be obtained from the slope of the MSD100plot by linear regression via Eq. (4.27). Figure 7.1(b) shows an Arrhenius plotof bulk diffusivity. Fitting these results to D = D0,bulk exp (−Qd,bulk/RT) givesD0,bulk is 2.1 × 10−8 m2s−1 and Qd,bulk is 3.9 kJ mol−1. This high diffusivity isqualitatively consistent with experimental observation [27] and quantitatively inagreement with previous MD simulations, i.e. D0,bulk = 1.6 × 10−8 m2s−1 andQd,bulk = 3.8 kJ mol−1 [80]. Size distribution of bulk He clustersFigure 7.2 shows snapshots of the type-0 cell containing 0.5at% He captured af-ter 3 ns at 1000 K where He clusters are shown to have formed in the matrix.Additionally, several Fe atoms in the vicinity of He clusters appear to have beendisplaced from their lattice positions. The displaced Fe atoms are referred to asself-interstitial atoms (SIA), i.e. secondary point defects which accompany clus-ter formation [243, 244]. The presence of SIAs has been identified experimentallyas the origin of cluster-induced swelling in irradiated steels [23]. He clusters canthus be considered as substitutional defects since they now occupy the main latticesites originally inhabited by SIAs. Clusters in substitutional sites are immobile andstable, i.e. no dissociation was observed within the simulation time.Figure 7.2: Snapshots of a portion of the type-0 cell with 1.0at% He at 1000 K captured at 3ns showing (a) both Fe (black) and He (magenta) atoms, (b) only He atoms.After 3 ns, the cluster size distribution is deemed to have reached steady state.The rapid formation of clusters is consistent with the fast bulk diffusivity of He101discussed in Section Figure 7.3 presents the size distributions at steady statefor different bulk concentrations C0 and temperatures. The area under the curvefor each data set is equal to the total number of He atoms. For all concentrations,the most prevalent clusters are those containing 4 to 8 atoms. No systematic effectof temperature on the size distribution of clusters is observed (Figure 7.3) for thetemperature range under investigation. The height of distributions increases withincreasing C0.Figure 7.3: The cluster size distribution in the bulk crystal of Fe captured after 3 ns simu-lations for bulk concentration C0 of (a) 0.1at%, (b) 0.5at%, and (c) 1.0at%. Volume and radius of He clustersClusters that have been characterized, see Figure 7.3, are further analyzed to deter-mine their volume using the convex-hull technique discussed in Section,and subsequently their radius using Eq. (4.28). The cluster radius rc is plotted as afunction of the number of He atoms in the cluster, c, in Figure 7.4(a). In this figure,data from the simulation at 1000 K is shown for illustration. Only clusters largerthan or equal to tetramers can be analyzed since the convex-hull technique is de-fined only when there are at least four non-coplanar points in space. Figure 7.4(a)suggests that the cluster radius is proportional to the squared root of cluster size c,i.e. rc ∝ c1/2. This yields r2c = krc where kr = 2.1 ×10−21 m2, as presented in Figure7.4(b).102Figure 7.4: (a) Average cluster radius rc and (b) r2c as a function of cluster size c obtainedfrom cluster characterization at 1000 K.To rationalize the linear r2c - c trend in Figure 7.4(b), a model that describes theequation of state for inert gases at high pressure proposed by Brearly and MacInnes[245] will be used. The Brearly-MacInnes model has been demonstrated to providean accurate description of He clusters in an Al matrix from experiments [246]. Inthis model, the number of He atoms in a cluster, c, which is embedded in a metallicmatrix, can be estimated from the cluster radius rc via [245]c =PcvczkBT=(2γpm/rc)(43pir3c )zkBT=(8piγpm3zkBT)r2c =r2ckr(7.1)where Pc is the Laplace pressure inside the cluster, i.e. Pc = 2γpm / rc [18]. Here, γpmis the interface energy between He clusters and α-Fe matrix, z is a compressibilityfactor that depends on temperature and pressure [245], and kr is a constant.The model in Eq. (7.1) predicts a linear r2c - c trend, which is consistent with thetrend observed in Figure 7.4(b). The model can further be applied to estimate thevalue of γpm using Eq. (7.1) and assuming, as a first-order approximation, that z =2 [245]. At 1000 K, the model predicts γpm = 1.56 J m−2, which is about 0.9 timesthe interface energy between α-Fe matrix and He clusters reported from previouscalculations, i.e. 1.82 J m−2 [247] and 1.75 J m−2 [248]. The consistency betweenthe calculated γpm and those from the literature suggests that the behaviour of Heclusters embedded in α-Fe matrix departs from the ideal gas behaviour, which isconsistent with earlier work [245, 248].1037.3.2 Properties of He in grain boundaries of Fe7.3.2.1 Binding energy profile of a single He atomThe interaction between planar grain boundaries in Fe and a single He atom at 0K is represented by the single-atom binding energy profile, which has been calcu-lated following the technique discussed on Section Figure 7.5 shows thebinding energy profile of a He atom for each investigated boundary. The bound-ary inclination appears to have little effect on the binding energy, as is evident fromthe maximum value of single-atom binding energy, Eb0 which varies between 1.3to 1.4 eV for the boundary inclination being considered here. The boundary widthδ can be determined from the binding energy profile, as indicated in Figure 7.5(c).Here, δ is found to be between 7 and 9 Å.Figure 7.5: The profile of the binding energy of a single He atom to several planar bound-aries calculated by molecular statics at 0 K, the regime of non-zero binding energy beingdefined as the boundary width δ. Grain boundary diffusivity of a single He atomGrain boundaries also affect the diffusivity and clustering of He atoms. Between800 and 1100 K, the He atom at the boundary performs a random walk motion butremains within the boundary width at all times for each bicrystal. At temperatureslower than 800 K, the random walk motion is limited to a few Å, i.e. the He atom iseffectively immobile within the 6-ns simulation time. Above 1100 K, the He atom104is able to escape from the boundary and diffuse to the bulk portion of the bicrystal.In this case, the motion of the He atom is influenced by both the bulk and grainboundary diffusivity.Figure 7.6 shows the Arrhenius plot of the single-atom grain boundary diffu-sivity, Dgb, for temperatures between 800 and 1100 K, the parameters being givenin Table 7.2. It is noted from this figure that Dgb is 2 to 4 orders of magnitudelower than Dbulk, i.e. the grain boundary is not a fast diffusion path for interstitialHe atoms. This observation is consistent with simulations [80] and experiments[249] for interstitial diffusion of H, C and He in Fe grain boundaries.Figure 7.6: Grain boundary diffusivities of a single He atom as a function of tempera-ture for grain boundaries of different inclinations. Bulk diffusivities from Figure 7.1(b)have been included for comparison. Dashed lines are the Arrhenius relationship, D =D0 exp (−Qd/RT), with parameters given in Table 7.2.Table 7.2: Arrhenius parameters of the single-atom diffusivity in Figure 7.6.DiffusivityBulkGrain boundary inclination (ϕ)parameters 19.5◦ 44.7◦ 70.5◦ 90.0◦D0 [m2s−1] 2.1× 10−8 9.8× 10−6 3.5× 10−6 1.3× 10−8 2.1× 10−8Qd [kJ mol−1] 4 110 95 37 421057.3.2.3 Size distribution of segregated He clustersThe behaviour of multiple He atoms in the type-2 bicrystals at 1000 K is now pre-sented. The formation of He clusters in the bulk bcc crystal occurs within a fewns, as is evident from Figure 7.7(a), and is consistent with the high diffusivity ofHe atom in the bulk Fe matrix. The clustering behaviour in the bulk and in theboundary, however, are markedly different, both in terms of the size distributionand their evolution. Figure 7.7(b) and (c) present the cluster distribution in the bulkand boundary, respectively, captured at different times. The similarity between thedistribution in the bulk portion of the bicrystal, see Figure 7.7(b), and that in thesingle crystal, see Figure 7.3(b), is noted.Figure 7.7: (a) A snapshot of a 44.7◦-bicrystal containing 0.5 at% He taken at 3 ns, showingonly He atoms coloured according to cluster size; (b) evolution of cluster size distributionin the bulk, and (c) at the boundary portion of the bicrystal shown in (a).The cluster size distribution in the bulk, see Figure 7.7(b), shows a negligiblevariation over a 36-ns simulation time, suggesting a steady state size distribution.Additionally, these bulk clusters are immobile. This implies that, within the sameperiod of time, no clusters formed in the bulk migrate to the boundary. Clustersfound at the boundary form during the initial transient time (< 3 ns), due to thesegregation of single fast-diffusing He atoms. Segregation takes place rapidly andreaches steady state at approximately the same time as the bulk clustering does.In contrast to the bulk clusters, the distribution of clusters at the boundary ex-hibits a notable evolution for times > 3 ns, see Figure 7.7(c). Segregated individual106He atoms diffuse along the boundary to form larger clusters, thereby changing thecluster size distribution at the boundary. The slow rate of formation of clustersat the boundary compared to that of bulk clusters is consistent with the single-atom boundary diffusivity being 2 to 4 orders of magnitude lower than the bulkdiffusivity, see Figure 7.6. The boundary diffusivity of a single He atom can thusbe considered as the rate-controlling process for cluster formation at the bound-ary. While the cluster distribution at the boundary continues to change over time,the variation becomes negligible at longer times. The boundary distribution wasdeemed to have reached steady state at 12 ns, for the 44.7◦ boundary containing0.5 at% He, because the distribution captured at that time differs only by 15% com-pared to that captured at 15 ns, see Figure 7.7(c). Similar observations can be madefor the other three boundaries studied here.Figure 7.8: (a) Average segregation for different boundaries as a function of C0, error barsare the standard deviation (b) the effective binding energy according to the Langmuir-McLean model, Eq. (7.2).The segregation level is quantified by normalizing the number of segregatedHe atoms, i.e. the area under the graphs in Figure 7.7(c), with the total number ofboundary sites, i.e. the number of tetrahedral sites available in a perfectly planargrain boundary. The segregation level was computed in intervals of 3 ns to obtainthe average segregation level, Cgb. Figure 7.8(a) shows the segregation Cgb as afunction of bulk content C0. The minor variation in Cgb with inclination ϕ indicates107that the sink strength of these four boundaries is similar.The effective binding energy E∗b0 of these boundaries can be estimated from Cgbusing the Langmuir-McLean model in Eq. (2.3) [14]. Assuming that all boundarysites are available for segregation (C0gb = 1), the effective binding energy is givenbyE∗b0kBT= ln[C0(1− Cgb)Cgb(1− C0)](7.2)and is plotted as a function of C0 in Figure 7.8(b). The effective binding energyvaries moderately with C0, indicating that the assumption of non-interacting soluteatoms [14] is an oversimplification for the Fe-He system. The magnitude of thisvariation is about 5% (or 0.3kBT) for all boundaries. The model is thus considereda reasonable first order approximation for E∗b0, which is taken as the average ofthe values obtained from three different C0. Table 7.3 summarizes the effectivebinding energy calculated from the cluster segregation in Figure 7.8(b), which arecompared to the single-atom binding energy from the binding energy profile inFigure 7.5. Additionally, the boundary width and the He diffusivity at 1000 K arealso presented in Table 7.3.Table 7.3: Properties describing the interaction between He and Fe grain boundaries.Property Reference BulkGrain boundary inclination (ϕ)19.5◦ 44.7◦ 70.5◦ 90.0◦E∗b0 [eV] Figure 7.8(b) 0 0.48 0.50 0.53 0.52Eb0 [eV] Figure 7.50 1.31 1.34 1.34 1.38δ [Å] - 6.8 7.7 7.8 8.8D1000 K Figure 7.61.2 1.5 3.1 1.6 1.4[m2s−1] ×10−8 ×10−11 ×10−11 ×10−10 ×10−10The effective binding energies, E∗b0, are about three times smaller than the single-atom binding energies, Eb0, presented in Section The single-atom bindingenergy, Eb0, has been obtained from 0 K simulations and would lead to saturatedsegregation, i.e. Cgb = 1, at 1000 K, overestimating the actual segregation in Fig-ure 7.8(a). E∗b0 can thus be considered a more representative parameter than Eb0 todescribe the interaction between grain boundaries and He clusters.1087.3.3 Mobilities of cluster-enriched grain boundariesUpon clusters reaching a steady-state distribution at the boundary, simulationswere run for another 20 to 30 ns. The boundary position, h, was recorded every0.1 ns. An example of the evolution of average boundary position, h(t), for thecluster-enriched (C0 = 0.5at%) 44.7◦-boundary is shown in Figure 7.9(a). The h(t)evolution for the case of pure boundary is shown in Figure 7.9(b) for comparison.The fluctuation of h(t) for the cluster-enriched boundary case, Figure 7.9(a), showsan attenuation trend for t < t?, where t? is the time at which the cluster size distri-bution have reached steady state, e.g. t? = 12 ns as determined from Figure 7.7(c).At t > t?, the evolution of h(t) shows a fluctuation with a consistent amplitude ofabout 0.2 Å. As a comparison, the evolution of h(t) for the pure 44.7◦-boundary isshown in Figure 7.9(b), its amplitude of fluctuation being 1.5 Å.Figure 7.9: (a) The evolution of average boundary position, h(t), for the cluster-enriched(C0 = 0.5at%) 44.7◦-boundary at 1000 K, t? indicating the time at which cluster size distri-bution at the boundary reached steady state (see text), (b) h(t) for the pure boundary ofthe same inclination.The h(t) evolution for t > t? is used to compute the boundary displacement fora number of time-intervals τ. Figure 7.10(a) shows an example of the distributionof boundary displacement ∆h(τ = 0.2 ns) for the pure and cluster-enriched (C0= 0.5at%) 44.7◦-boundaries. Both distributions are Gaussian, their width indicat-109ing the extent of boundary fluctuation. The average mean-squared displacement(MSD) at different τ for enriched and pure boundaries are plotted in Figure 7.10(b)and (c), respectively. The boundary mobility is obtained from the slope of linearregression of the MSD via Eq. (4.21).Figure 7.10: (a) Distribution of boundary displacement at an interval τ of 0.2 ns, for thepure and cluster-enriched (C0 = 0.5at%) 44.7◦-boundary at 1000 K; the mean-square dis-placement as a function of τ of (b) the pure and (c) the cluster enriched boundaries.The same procedure is repeated for different boundaries and solute contents tocompute the mobilities of these boundaries, see Figure 7.11(a). These mobilities arelabelled as an effective quantity, Meff, because they arise from the intrinsic mobil-ity of the boundary Mpure and the contribution of cluster segregation Mclust. Thepresence of clusters reduces the mobility by about 3 to 4 orders of magnitude.The effective mobility can be further decoupled into its intrinsic and clustercontribution component following, for example, Cahn [31], i.e.M−1eff = M−1pure + M−1clust (7.3)The cluster drag, i.e. the inverse of cluster contribution to the boundary mobility,is plotted in Figure 7.11(b) as a function of bulk concentrations C0. Within therange of solute content investigated here, the cluster drag appears to show a linearincrease with increasing C0, which is consistent with the Cahn solute drag model110[31]. The linear trend, i.e.M−1clust = αC0 (7.4)is given by the dashed lines in Figure 7.11(b) where α denotes the drag coefficient,its values presented in Table 7.4.Figure 7.11: (a) Effective mobility of pure and enriched grain boundaries, (b) Cluster dragcontribution, M−1clust in Eq. (7.3), as a function of bulk solute content.7.4 Cluster drag coefficientsTo further analyze the cluster drag coefficient α in Figure 7.11(b), the solute dragmodel of Cahn [31] is adopted. The model postulates that, in the limit of lowvelocity, the drag coefficient α can be obtained from Eq. (2.14). The bulk atomicvolume, Ω, at 1000 K is given by Ω = 12 a3T = 1.2×10−29 m3 where aT = 2.882 Å at1000 K, see Figure 6.1.The parameters Eb(z) and D(z) in the Cahn model correspond to propertiesrepresenting the interaction between a grain boundary and a solute atom. In orderto adapt this model for the drag effect of He clusters, the effective binding energyof clusters is employed instead of the single-atom binding energy so as to obtainan enrichment level consistent with the simulation results. The single-atom grainboundary diffusivity is also used since it controls the cluster size distribution at theboundary. Both parameters are integrated into an assumed triangular profile, see111Figure 7.12(a) and (b), to obtain a closed-form solution for α via Eq. (2.14). Table7.3 provides the numerical values used for the Eb(z) and D(z) profiles given byFigure 7.12(a) and (b).Figure 7.12: Assumed profile of (a) binding energy and (b) diffusivity to estimate α fromEq. (2.14).Table 7.4 compares the drag coefficients estimated from Eq. (2.14) with thosefrom simulations, see Figure 7.11(b). Both drag coefficients are of the same orderof magnitude. The drag coefficients obtained from the model are smaller by afactor of 0.5 to 0.8, but the trends with boundary inclination are similar in bothcases. Given the simplifications employed by the model, the simulation resultsdemonstrate good agreement with the model.Table 7.4: Cluster drag coefficients α obtained from the simulations and the Cahn model.α Grain boundary inclination (ϕ)[×1011 m−4 J s] 19.5◦ 44.7◦ 70.5◦ 90.0◦Simulations [Fig. 7.11(b)] 5.1 4.2 2.9 2.3Cahn model [Eq. (2.14)] 4.1 2.4 1.8 1.1In order to rationalize the use of the Cahn model for assessing the cluster dragcoefficients from the simulations, it is useful to first review the assumptions em-ployed in computing the effective mobility, Meff, from the simulations. The effec-tive mobilities have been computed using the random-walk technique [170], i.e.Eq. (4.21), under the condition that the cluster size distribution at the boundaryhas reached steady state. Such an assumption, i.e. the steady-state cluster size dis-tribution, is key to applying the random-walk analysis because the observed MSDof the boundary in that time period shows a linear progression with time-intervalτ, e.g. see Figure 7.10(b).112Prior to the cluster size distribution reaching steady state, e.g. for t < t? on Fig-ure 7.9(a), the amplitude of h(t) attenuates as segregated He atoms diffuse alongthe boundary and form clusters. The MSD of the boundary calculated within suchtime period does not follow a linear progression with time-interval τ. A similarnon-linear behaviour of MSD vs τ has been reported in experiments, e.g. H atomdiffusion in amorphous metals [250], and has been attributed to the random walkof particles being coupled to each other [251, 252]. In the present work, the non-linear MSD-τ trend corresponding to the h(t) data for t < t? can be attributed tothe coupling between the intrinsic random walk of the boundary and the randomwalk of segregated He atoms.After the steady-state cluster size distribution is reached, e.g. for t ≥ t? onFigure 7.9(a), the boundary fluctuation, h(t), exhibits a consistent fluctuation, i.e.no attenuation with time is observed. Consequently, the MSD of the boundaryshows a diffusive behaviour, i.e. the MSD-τ relationship is linear. The linear MSD-τ trend suggests that the observed random walk from which Meff is computed isno longer coupled.Furthermore, Meff is 3 to 4 orders of magnitude smaller than the pure boundarymobility, Mpure, see Figure 7.11(a). Such orders of magnitude difference suggeststhat the observed random walk of the boundary is no longer controlled by theintrinsic fluctuation of the boundary, but rather by the random walk of segregatedHe atoms along the boundary. The kinetics of the latter event is described by Dgb,i.e. the grain boundary diffusion of He atoms.The solute drag model by Cahn is able to provide a good estimate of Meff fromsimulations because in the limit of zero velocity, the model predicts that the contri-bution of solute clusters to the effective mobility, Mclust, is a function of the bound-ary diffusivity Dgb, via Eq. (7.4) and Eq. (2.14). This is consistent with the rate-controlling process determined from the simulations, i.e. the random walk of theboundary after the cluster size distribution reaches steady state is controlled by thediffusion of segregated He atoms, Dgb.1137.5 SummaryThe effect of solute clusters on the mobility of planar grain boundaries in the zerovelocity limit for dilute solutions of He in bcc Fe bicrystals has been investigatedvia atomistic simulations. The cluster drag effect observed from simulations can,to a first order approximation, be rationalized using the Cahn solute drag model.Properties obtained from these simulations have been shown to provide valu-able insight in analyzing the kinetic cluster-boundary interaction. The single-atomboundary diffusivity has been identified as the rate-controlling process for clusterformation at grain boundaries. The cluster drag model presented here, however, isonly applicable in the zero velocity limit. From a practical perspective, it is of inter-est to investigate the case of a migrating boundary and this is the focus of the nextchapter where He clusters interact with a capillarity driven migrating boundary.114Chapter 8Effect of Helium Clusters on Grain BoundaryMigration in BCC Iron8.1 IntroductionThis chapter explores the dynamic interplay among three kinetic events, namelygrain boundary migration, solute clustering and solute segregation. The main ob-jective is to develop a phenomenological model that describes the interaction be-tween He clusters and grain boundary migration in Fe bicrystals. The migrationof a curved boundary in the presence of segregating interstitial He atoms is firstsimulated. Multiple clusters of different size are expected to form and reduce themigration rate. To establish a correlation between the size distribution of clustersat the boundary and the boundary velocity, additional simulations are performedin which monosized clusters of varying size are made to interact with a migrat-ing boundary. The magnitude of boundary retardation due to monosized clustersis determined from this set of simulations and subsequently incorporated into amodel on the boundary pinning. The model is used to explain the relationshipbetween the boundary velocity and clusters of varying size.8.2 Simulation setupThe simulation box is a bicrystal cell containing a half-loop U-shaped curved bound-ary, i.e. the type-1 cell in Figure 4.4(b). He atoms are introduced to the simulationbox following three configurations, see Figure 8.1 for illustrations.Type-i He atoms are placed randomly in tetrahedral interstitial sites of the bicrys-tals. He atoms may segregate to the migrating boundary or form clusters inthe bulk.115Figure 8.1: Different cases of solute configuration investigated in this chapter.Type-s A given number of randomly chosen Fe atoms at the boundary are re-moved and replaced with the same number of He atoms. The pre-segregatedsubstitutional He atoms will not form clusters due to their low diffusivity.Type-c A given number of randomly chosen Fe atoms in the bulk are removed andreplaced with an object that consists of c He atoms. In this set of simulations,the object size, c, ranges from 1 to 6. Objects with 2 ≤ c ≤ 6 are morpholog-ically classified as clusters. For nomenclature purposes, they are referred toas (in ascending order of their size c) dimers, trimers, tetramers, pentamersand hexamers. Objects with c = 1 are substitutional He atoms and referredto as monomers. For the type-c simulations, the discussion about monomersis carried out in the same manner as the discussion about objects with c ≥ 2,i.e. the quantity of monomers is expressed in clusters, instead of atoms. It isemphasized that such a practice is merely for brevity in nomenclature.For a given cluster size, the initial structure of premade clusters will be iden-tical, following that of the He clusters generated from bulk cluster simula-tions in Chapter 7, see Figure 7.2(b). All premade clusters will be placed in asection of length LC = 0.65LY along the height of the bicrystal, LY, see Figure8.1(c). The boundary will initially contain no clusters.116Table 8.1 below lists the number of He atoms or clusters prepared for each case ofconfiguration. In the type-i configuration, one He atom is equal to a concentrationC0 of 1.03 ppm since there are 967,000 Fe atoms. For labelling convenience, how-ever, one He atom is treated as equivalent to C0 of 1 ppm. In the type-s and type-cconfigurations, C0 is expressed in the units of atoms and clusters, respectively.Table 8.1: Types of initial configuration and quantities of He atoms being introduced to atype-1 bicrystal that contains the incoherent curved twin boundary.LabelInitial configuration Cluster size Number of atoms/clusters NHeof He atoms c [atoms] introduced to the bulk C0 [atoms]type-i 1run #150, 100, 150, 200,NHe=C0×cBulk interstitial atoms 250, 300, 400, 500,600, 700, 1000, 2000run #2† 400, 500, 600, 700type-s Boundary substitutional atoms 1 10, 30, 50, 60, 70, 80, 100type-cmonomers 1 100, 200, 300, 500Bulk dimers 2 100, 200, 300substitutional trimers 3 100, 200, 300clusters tetramers 4 100, 200, 250pentamers 5 50, 100, 200hexamers 6 50, 100, 200†Different initial placement of He atoms.The type-i cells permit a simultaneous occurrence of grain boundary migration,solute segregation and bulk clustering. The type-s and type-c bicrystals representidealized situations where the boundary migration is affected only by solute seg-regation or bulk clustering, respectively. Results from the type-s and type-c simu-lations will be used to explain observations from the type-i simulations.The cell dimensions are LX = 4.1 nm, LY = 60 nm, LZ = 50 nm, and DU = 35nm, see Figure 4.4(b) for symbol notation. For the type-c cells, the effective heightof the simulation box that contains premade clusters is LC = 39 nm, see Figure8.1(c). The curved tip is the incoherent twin boundary. Each cell contains 965,000to 967,000 Fe atoms. All simulations are run at 1000 K under a constant-volumecondition for up to 17 ns. The boundary conditions for the type-1 cells have beendiscussed in Section Results8.3.1 Trapping of segregated He atomsFigure 8.2 shows snapshots of the type-i bicrystal with 100 He ppm taken at differ-ent times during the boundary migration. Segregation of He atoms to the migrat-ing boundary is evident, e.g. the dashed oval at 0 ns is compared to that at 0.3 ns.He atoms that had been at the boundary at 0.3 ns (dashed oval) were no longer atthe boundary at 2.7 ns, as evident by the solute-free portion in the dashed box at2.7 ns. He atoms in the dashed oval regime remained at their positions prior to theboundary detachment, even at 6 ns. This suggests that the formerly segregated Heatoms are trapped in the bulk of the growing grain. The same trapping situation isalso observed for He atoms in the solid oval at 2.7 and 6 ns.Figure 8.2: A series of snapshots of the boundary migration in the type-i bicrystal contain-ing 100 ppm He. The oval regimes indicate the trapping of segregated He atoms, see text.118To explain the trapping behaviour, the local structure of trapped He atoms is ex-amined. The trapped He atoms are found to have an average coordination numberof eight, which is equal to the coordination number of bulk bcc sites. The averagenearest neighbour distance of the trapped He atoms is found to be 0.9aT, about4% higher than the nearest neighbour distance of a bulk Fe atom. These findingssuggest that the trapped He atoms are part of the substitutional sub-lattice.Figure 8.3: Atomistic mechanism of the trapping of segregated He atoms.The mechanism by which segregated interstitial He atoms become bulk substi-tutional atoms is illustrated in Figure 8.3. In this figure, a schematic of the atomicshuffling that underlies the curved boundary migration is included, see Section6.4.3 for details. The shuffling events may leave behind empty lattice sites, i.e. va-cancies [253]. Segregated He atoms that are near the vacant sites may be able toidentify and diffuse to these sites. As a result, interstitial He atoms formerly at theboundary now occupy substitutional sites after the boundary unpins itself awayfrom these He atoms. Substitutional He atoms are less mobile than interstitial Heatoms since the binding energy of He atom to a vacancy is 2.33 eV from molecularstatics (MS) calculations, see Eq. (4.25), in agreement with the value reported froma recent atomistic study [83].1198.3.2 Effect of He clustering and segregation on grain boundary migration8.3.2.1 Steady-state grain boundary velocityThe evolution of shrinking grain volume of the type-i bicrystals for various Heconcentrations C0 is presented in Figure 8.4. While there is no clear universal trendof shrinkage rate with C0, three regimes can be distinguished: (i) 50-300 ppm, (ii)400-700 ppm, and (iii) ≥ 1000 ppm. Between 50 and 300 ppm, the shrinkage ratedecreases with increasing C0. The trend in shrinkage evolution with C0 is not asobvious for C0 between 400 and 700 ppm. Above 1000 ppm, the boundary becomesessentially immobile for times in the nanosecond scale.Figure 8.4: Evolution of the shrinking grain volume in the type-i bicrystals for differentconcentrations.For the 50-300 ppm concentration range, most He atoms segregate to the bound-ary; only few form clusters in the bulk. The migration rate for this concentra-tion range is dominantly determined by segregation. The lowest migration rate isshown for the 300 ppm case, corresponding to the highest segregation achieved forthis concentration range.Between 400 and 700 ppm, there are a sufficient number of He atoms initiallyin the matrix such that larger bulk clusters are able to form. The formation of bulk120clusters means fewer He atoms are available for segregation. The boundary retar-dation due to segregation for this concentration range is thus not as significant asthe 300 ppm case. On the other hand, the boundary retardation due to bulk clustersis higher for the 400-700 ppm range than it is for the 50-300 ppm range. Addition-ally, while the overall grain boundary migration for the 400-700 ppm case showsa certain degree of fluctuation, there is no systematic trend between the boundarymigration and the bulk content. The variation of the boundary migration in the400-700 ppm range seems to be of statistical nature, as indicated by the variationof shrinkage evolution for a given C0 in this range, see Figure 8.5(a)-(d).Figure 8.5: Evolution of the shrinking grain volume in the type-i bicrystals for C0 of 400 to700 ppm from two simulation runs, differing in the initial placement of He atoms.To explore the coupled effect of clustering and segregation on grain boundarymigration more quantitatively, the following analysis is performed. For a given C0,the steady-state boundary velocity will be extracted by determining a time periodfrom Figures 8.4 and 8.5 during which the grain shrinkage occurs at a constantrate. A steady state migration is considered to have occurred in a certain timeperiod if the 1-ns moving linear regression within that time period has a slope thatvaries by less than 20%. Figure 8.6(a) shows the steady-state portion of the data121set presented in Figures 8.4 and 8.5. The boundary velocity is proportional to theslope of linear regression of the data set in Figure 8.6(a) via Eq. (4.17) and is plottedagainst C0 in Figure 8.6(b).Figure 8.6: (a) The steady state portion of the shrinkage evolution from Figures 8.4 and 8.5,(b) the steady state velocity as a function of C0 for the type-i bicrystals. Shapes of a grain boundary during its steady state migrationThe shape of a migrating boundary changes as the boundary interacts with Heclusters. It is important to quantify the effect of He clusters on the boundary shapesince the shape determines the effective driving pressure. Figure 8.7 shows theboundary snapshots for different C0 taken at a time during steady-state migration.The elongated boundary shape of the pure bicrystal (Figure 8.7(a)) has been dis-cussed in Section 6.4.1. To quantify the variation of local boundary curvature, thetechnique discussed in Section is implemented.Figure 8.8(a) presents the profile of calculated local curvature κ(z) along the z-axis, i.e. perpendicular to the migration direction, for the boundary shape shownin Figure 8.7. The curvature profile has a maximum around the loop tip, its max-imum values varying with C0, and approaches zero as it goes to the planar sidesof the boundary (z = 7 and 42 nm). The instantaneous average curvature κ(t) iscalculated from the spatial average of the local curvature according to Eq. (4.22).122By repeating the procedure every 0.1 ns for all concentrations, the evolution ofinstantaneous average curvature κ(t) can be obtained.Figure 8.7: Snapshots of the type-i boundary for different C0 at a time during their steady-state migration.Figure 8.8(b) shows examples of such an evolution for the case of 0 (pure) and500 ppm. The variation of κ(t) with time for the pure boundary is within 2% ofits average value, indicating a shape that is self-similar throughout the migration.On the other hand, the trend of κ(t) vs time for the 500 ppm case varies within6% of its average value, indicating a more pronounced change in shape comparedto the pure case. Figure 8.8(b) also indicates that the pure boundary has a higheraverage curvature than the 500 ppm bicrystal since the former has an elongatedshape while the latter is more rounded.123Figure 8.8: (a) The local boundary curvature profile κ(z) for the shapes presented in Figure8.7, (b) the evolution of average curvature κ(t) for the case of 0 (pure) and 500 ppm.To compare the average curvature among different C0, the time average of κ(t),〈κ〉eff, is calculated from Eq. (4.23); the subscript ’eff’ is to distinguish between theeffective average curvature when solute atoms are present, 〈κ〉eff, and that of thepure boundary, 〈κ〉pure. The effective average curvature 〈κ〉eff, after being normal-ized against 〈κ〉pure, is presented as a function of C0 in Figure 8.9, showing that〈κ〉eff varies within ±15% of 〈κ〉pure. The effect of bulk clusters and segregatedatoms on the boundary curvature, thus, appears to be rather minor.Figure 8.9: The effective average curvature of the boundary from the type-i simulations,〈κ〉eff, normalized by the average curvature of the pure boundary, 〈κ〉pure.1248.3.2.3 Population of boundary clusters at steady state migrationThe number of He clusters that are present at the boundary during the steady-statemigration will be determined using a procedure illustrated in Figure 8.10.Figure 8.10: An illustration of the procedure to determine the number of clusters in thecurved boundary: (a) a snapshot of type-i bicrystal with 500 ppm He (run #1) at 4 ns, (b)Fe atoms that make up the boundary, (c) He clusters classified based on their location inthe bicrystal, (d) He clusters in the curved boundary, characterized based on their size, seeSection for details. The size of atoms is exaggerated for clarity.125First, the boundary shape is constructed from the position of Fe atoms thatmake up the boundary, see Figure 8.10(b), using the technique discussed in Section4.2.5.4. He clusters whose center of mass is within 0.5 nm away from the curvedportion of the boundary are then counted, i.e. Figure 8.10(c) and (d).The procedure is repeated every 0.1 ns for all concentrations. An example ofthe evolution of number of He atoms at the curved boundary during the steady-state migration of type-i bicrystal with 500 ppm He atoms is presented in Figure8.11(a). The size distribution of boundary atoms at different times during such aperiod are shown in Figure 8.11(b)-(g).Figure 8.11: (a) Total number of He atoms during the period of steady-state migrationfor the type-i bicrystal with 500 ppm He, (b)-(g) the cluster size distribution taken fromdifferent times during such a period.The same analysis is performed for all concentrations C0 to obtain the averagesize distribution of boundary clusters, i.e. Nc vs c, during the steady-state migra-126tion of the boundary. Figure 8.12 presents the average size distribution of bound-ary clusters for different C0, showing two trends: (i) the 50-300 ppm case and (ii)the 400-1000 ppm case. For the case of 50-300 ppm, the population of monomers(c = 1) increases with increasing C0, while no systematic trend is apparent for thecase of 400-1000 ppm.Figure 8.12: The average number of boundary clusters Nc vs. cluster size c, taken byaveraging the size distribution captured every 0.1 ns during the period of steady-statemigration, the error bars representing the standard deviation over the same time period.127The trends from Figure 8.12 are consistent with the trends in the shrinkage evo-lution discussed in Section The boundary velocity in the 50-300 ppm casedecreases with increasing C0 because the migration is mainly retarded by segre-gated He atoms. For the case of 400-1000 ppm, no monotonic relationship betweenthe velocity and C0 can be identified since the boundary cluster population has noobvious dependence on C0.The average total number of He atoms in the boundary is obtained for a givenC0 from the area under the distribution plot in Figure 8.12 and is plotted as a func-tion of C0 in Figure 8.13(a). Above 300 ppm, the average number of boundaryatoms shows a negligible variation with C0, indicating a saturation trend around70 atoms. Figure 8.13(b) presents the relationship between the average numberof boundary atoms (the ordinate of Figure 8.13(a)) and the boundary velocity (theordinate of Figure 8.6(b)). While this figure summarizes the overall behaviour ofboundary migration under the presence of solute atoms, it couples the effect ofsegregation and clustering on the boundary velocity into a single parameter, i.e.the average number of boundary He atoms.Figure 8.13: (a) The average number of boundary He atoms as a function of C0 in type-ibicrystals, (b) the boundary velocity as a function of the average number of He atoms, i.e.the ordinate from Figure 8.6(b) vs. the ordinate from part (a) of this figure.1288.3.3 Effect of pre-segregated He atoms on grain boundary migrationTo explain the results in Figure 8.13(b), an isolated situation where only segrega-tion takes place and no bulk clustering occurs, i.e. the type-s simulations, is sim-ulated. Figure 8.14(a) shows the shrinkage evolution of the type-s bicrystals fordifferent number of boundary He atoms, C0. The boundary migration occurs at adelayed time due to the presence of pre-segregated He atoms.When there are less than 80 He atoms at the boundary, the migration is delayedby a fraction of the total simulation time (10 ns). The unpinning time for C0 below80 atoms increases with increasing C0. For C0 higher than or equal to 80 atoms,the delay is longer than the total simulation time and the boundary is effectivelypinned. The highest C0 at which the boundary is able to unpin is 70 atoms. This isconsistent with the saturation trend observed in Figure 8.13(a) and (b).Figure 8.14: (a) The shrinking grain volume evolution in the type-s bicrystals for differentconcentrations, (b) the velocity of unpinned boundaries at steady state.After the unpinning occurs, all He atoms remain at their initial pre-segregatedposition and the boundary continues to migrate at steady state. The shape of theunpinned boundaries during their migration, as illustrated in Figure 8.15, is con-sistent with that of the pure boundary case, i.e. Figure 8.7(a). The velocity of theunpinned boundary is calculated using the technique described in Section presented in Figure 8.14(b) as a function of the He content before unpinning.129The velocity of the unpinned boundary decreases as the initial He content in-creases, despite the fact that the unpinned boundary is clean, interacting with nosolute or clusters during its migration. Simultaneously, the shape of the unpinnedclean boundaries is consistent with that of the pure boundary, the average curva-ture in both cases, i.e. 〈κ〉eff and 〈κ〉pure, being less than 2% different from eachother. This implies that the driving pressure is equal in both cases, i.e. Peff = Ppure.Since Veff < Vpure and Peff = Ppure, it follows that Meff < Mpure. In other words,pre-segregation decreases the effective mobility of an unpinned clean boundary.Figure 8.15: Snapshots of the type-s boundary that initially contained (a) 30 and (b) 50 Heatoms, taken at a time during steady-state migration after unpinning occurs.8.3.4 Effect of monosized clusters on grain boundary migrationThe effect of monosized clusters on the boundary migration for the type-c simula-tions is now discussed. Six sets of type-c simulations have been performed, corre-sponding to monosized clusters of size c of 1 (monomers) to 6 (hexamers). For eachc, a number of bulk clusters C0 from 50 to 500 clusters have been introduced to thebicrystal, see Table 8.1. For example, a bicrystal with tetramers (c = 4) of concen-tration C0 = 100 clusters means that there are 100 tetramers, resulting in a totalof 400 He atoms in the bicrystal. For a given c and C0, the steady-state effectivevelocity Veff and the average number of clusters at the boundary Nc are obtainedusing the procedure discussed in Sections and, respectively. Figure1308.16(a) and (b) presents the results, i.e. Veff and Nc as a function of C0, respectively.Figure 8.16(c) presents the effective boundary velocity Veff as a function of theaverage number of monosized clusters at the boundary Nc for different clustersizes c, i.e. monomers to hexamers. The effective velocity decreases as more clus-ters are present at the boundary. Additionally, the decline in velocity for a givenincrease in the number of boundary clusters is higher for larger cluster size c.Figure 8.16: The simulation results of the type-c boundary. (a) The steady-state effectiveboundary velocity, Veff, and (b) the average number of monosized clusters that are presentat the boundary, Nc, as a function of the bulk cluster concentration C0, dotted lines drawnas a visual guide, (c) the ordinate from (a) plotted as a function of the ordinate from (b),dashed lines indicating the linear fit following Eq. (8.1), see text.The data set in Figure 8.16(c) can be used to determine the pinning pressureof a single cluster with a given size c, ρc. This is obtained by performing a linearregression to the data set, following Veff = V0 − bNc where V0 is the vertical axisintercept and b is the slope. The Zener model [36], for example, considers thatV0 = Vpure for any cluster size c, where Vpure is the boundary velocity in a puresystem, see Eq. (2.20). Additionally, the Zener model obtains the pinning pressureof a single cluster, ρc, from the slope b, via b = Mpureρc where Mpure is the average131mobility of the boundary in a pure system.The original Zener approach, however, is not consistent with the simulationresults presented in Figure 8.15(b). The velocity of a clean boundary that has pre-viously interacted with solute atoms, V?, is lower than that of a clean boundaryin a pure system, Vpure, i.e. V? < Vpure. This has been attributed to the effectivemobility being lower than the mobility of a clean boundary in a pure system, i.e.Meff < Mpure. A cluster pinning model relevant for the present work is proposedin which V?, instead of Vpure, is set as the vertical axis intercept and the regressionslope b is related to the single-cluster pinning pressure ρc via b = Meffρc, i.e.Veff = V? −MeffρcNc (8.1)As a first order approximation, it is assumed that V? = kMVpure and Meff =kM Mpure where kM is a constant between 0 and 1 that is independent of C0 orcluster size c. Based on the results in Figure 8.15(b), kM is set as kM = 0.9. Furtherdiscussion of the constant kM will be presented in Section 8.5. Additionally, Vpureis 9 m s−1 (see Figure 8.6(b)) and Mpure is obtained from Vpure/Ppure = 1.7× 10−7m4 J−1 s−1 since Ppure = Γpi/DU = 53 MPa (see Eqs. (4.15) and (6.2)). As aresult, the model parameters, V? and Meff, are 8.1 m s−1 and 1.5×10−7 m4 J−1 s−1,respectively. These parameters are employed in Eq. (8.1) to produce the linear fitof each data set, as shown in Figure 8.16(c).Figure 8.17: Pinning pressure of a single cluster ρc as a function of cluster size.132Furthermore, for each data set in Figure 8.16(c), the slope of the linear fit canbe divided with Meff to obtain the single-cluster pinning pressure ρc. Figure 8.17presents ρc as a function of cluster size c, indicating the pinning pressure that in-creases linearly with increasing cluster size. A linear fit through the origin is drawnin Figure 8.17 followingρc = Kc (8.2)where K is the cluster pinning coefficient that is independent of cluster size. Basedon the data in Figure 8.17, K = 0.73 MPa. The cluster pinning coefficient K willnow be quantitatively assessed using the framework of the Zener model. It willultimately be employed to explain the simulation results from the type-i simula-tions, see Figure 8.6(b).8.4 Cluster pinning pressureThe Zener model on the particle pinning pressure [36] will be employed to ratio-nalize the pinning pressure of a single cluster, ρc, presented in Figure 8.17. Beforeapplying such a model, it is useful to mention a few comments to address the dif-ference between the simulation results and the inherent assumption in the model.The original model by Zener is derived by considering the restraining force inthe boundary due to three-dimensional stationary objects, i.e. particles. Machlinconsidered the restraining force of objects the size of an individual atom and pro-posed that the Zener pinning pressure model can be extended to individual atoms[135]. The Machlin extension has been applied recently by Hersent et al [136, 137]to revisit the analysis of experiments performed by Aust and Rutter [10, 11].In this work, only clusters larger or equal to tetramers can be said to occupy athree-dimensional space since atoms are treated as zero-dimensional objects. Thesingle-cluster pinning pressure ρc from Figure 8.17, nevertheless, shows a lineartrend with the cluster size c for all c, including monomers (i.e. individual atoms).This suggests that the use of a model, e.g. the pinning pressure model by Zener[36] and its extension by Machlin [135], to rationalize the ρc-c trend from the simu-lations can be extrapolated to clusters smaller than tetramers.It is also useful to distinguish between the analysis presented in Section 7.4 and133the pinning pressure of monomers presented in this chapter, see Figure 8.17. Inthe former case, the effect of solute atoms on the boundary mobilities have beendiscussed using the framework of the solute drag model in the limit of zero ve-locity. In the latter case, the monomers are essentially stationary solute atoms.The boundary-monomers interaction thus belongs to the high velocity regime ofthe solute drag model, where the restraining force due to solute atoms, i.e. thepinning pressure of monomers, is independent of the velocity [33, 34]. A similarvelocity-independent restraining force has been reported in the high-velocity limitof the solute drag analysis from PFC simulations [161]. The difference betweenthe approach employed in [161] and the analysis performed in this chapter is that,in the former case, the effective boundary mobility, Meff, is assumed to be equalto the mobility of a pure boundary, Mpure, while here, it is a fraction of the pureboundary mobility, i.e. Meff = 0.9Mpure, see Section 8.3.4.The linear trend between the pinning pressure of a single cluster ρc and clustersize c, see Eq. (8.2), will now be assessed against the Zener model, i.e. Eq. (2.19).In the Zener model, the size-dependent parameter that determines the pinningpressure of a single cluster ρc is the cluster radius rc. The model further postulatesthat ρc has a quadratic dependence on rc via ρc = kcrr2c , i.e. [36]kcr = 2piγ/vs (8.3)where γ is the average grain boundary energy, vs is the swept volume of the bicrys-tal (see Figure 2.10(b)) and kcr is the geometric pinning coefficient. It will be shownthat the linear ρc vs. c trend observed from the simulations is consistent with thequadratic ρc vs. rc from the model.It is useful to recall from Section that the cluster radius rc has been de-termined as a function of cluster size c from simulation results. A linear trendbetween the squared radius of clusters r2c and the cluster size c is apparent fromthe results presented in Figure 7.4(b), i.e. r2c = krc and kr = 2.1× 10−21 m2. Sucha relationship is substituted to Eq. (8.2) to yield a linear trend between ρc and r2c ,i.e. ρc = kcrr2c where kcr = K/kr = 2.9×1026 Pa m−2 is the geometric pinning coeffi-cient. The linear ρc vs. r2c trend from the simulations is consistent with the classical134Zener model.A further assessment between the simulations and the Zener model can bemade by estimating the geometric pinning coefficient via Eq. (8.3) and comparingit to the simulation value, i.e. kcr = 2.9×1026 Pa m−2. The average grain boundaryenergy γ is 0.8 J m−2, from Figure 6.9(a). The swept volume vs here is defined asthe interaction zone that extends to a distance rc outside of the boundary widthδ, resulting in the total width of the interaction zone of δ + 2rc. This definitiondiffers from the assumption originally employed by the Zener model [36], e.g. Fig-ure 2.10(b), where the boundary width is assumed to be sharp (δ = 0). The grainboundary investigated in this work has a finite width δ, which has been deter-mined from Figure 7.5, and its average value is 0.7 nm.The swept volume can thus be approximated as vs = (δ+ 2rc)LXsU where rc =√krc (Eq. (7.1)), LX is the bicrystal thickness (4.1 nm) and sU is the length of thecurved portion of the boundary. The length sU is computed by performing a lineintegration over the curve that fits the shape generated by the boundary atoms inFigure 8.7, giving sU of 68±2 nm. As a result, vs is 2.4±0.2×10−25 m3. Substitutingvs and γ into Eq. (8.3) yields kcr of 2.1±0.2×1025 Pa m−2.Figure 8.18: Average number of boundary clusters, Nc, as compared to the expected num-ber of boundary clusters for a given number of bulk clusters (dashed lines), i.e. C0vs/vbulk.135The coefficient kcr estimated from the Zener model is about 10 times larger thanthe coefficient kcr calculated from the simulations. The discrepancy between thesimulations and the model is partly attributed to the fact that the curved boundaryis flexible and has a finite width. As a result, the boundary captures more clus-ters than the sharp boundary in the Zener model where the number density ofboundary clusters is assumed to be equal to the number density of bulk clusters.This is demonstrated in Figure 8.18, where the actual average number of clustersat the boundary from the simulations, Nc (see Figure 8.16(b)), is compared withthe expected number of boundary clusters for a given number of bulk clusters, i.e.C0vs/vbulk. The normalizing parameter vbulk is the bicrystal volume that containsthe total number of monosized clusters, i.e. vbulk = LX LZLC = 7.35× 10−24 m3, seeFigure 8.1(c).The average number of boundary clusters from the simulations, Nc, is about afactor of two larger than the expected number of boundary clusters from the modelassumption. Taking this fact into consideration, the pinning pressure obtainedfrom the simulations is thus 5 times larger than the Zener model. A similar trendhas been reported in previous experiments [130, 131] and Monte Carlo simulations[132, 133], where the Zener model is shown to underestimate the pinning effect bya factor of 2 to 10.8.5 The effective cluster pinning modelThe pinning pressure coefficient K will be used to estimate the boundary velocityfrom the type-i bicrystal simulations for a given C0, i.e. Figure 8.13(b), based onthe corresponding size distribution of boundary clusters, i.e. Figure 8.12. Sincemultiple clusters of varying size are present, the approach introduced by Fullman[138] is adopted here, i.e. the Zener model is extended to consider particles with asize distribution. The total pinning pressure PZ is thus obtained from the sum ofindividual contribution from monosized clusters of a given size c viaPZ =∑cρcNc = K∑ccNc (8.4)136where ρc is the pinning pressure of a cluster of size c from Eq. (8.2), Nc is theaverage number of boundary clusters with a size c as presented in Figure 8.12(a)-(k), and K is the cluster pinning coefficient from Eq. (8.2), i.e. K = 0.73 MPa.Figure 8.19: Comparison between the boundary velocities from the type-i simulations (Fig-ure 8.6(b)) and those predicted by the effective cluster pinning model (Eq. (8.5)).The pinning pressure PZ in Eq. (8.4) replaces the monosized cluster pinningterm ρcNc in Eq. (8.1), yielding a prediction on the effective boundary velocity,Veff, given the size distribution of boundary clusters, Nc, i.e.Veff = V? −MeffK∑ccNc (8.5)where V? and Meff are the same model parameters discussed earlier, see Section8.3.4. Figure 8.19 compares the predicted boundary velocity based on Eq. (8.5)with the effective velocity extracted from the simulations, i.e. the ordinate fromFigure 8.6(b). The model is shown to be in good agreement with the simulationresults, the discrepancy being less than 20% of the simulation results.A key component of the model is the accounting of the effect of boundary clus-ters Nc on the effective mobility of the boundary, Meff. Such an effect has been137considered by employing a single parameter kM, independent of Nc, to accountfor the change in the boundary mobility due to boundary clusters, i.e. via Meff =kM Mpure, where 0 < kM < 1. The assumption of kM being independent of Nc isreasonable for highly dilute systems (C0 ≤ 1000 ppm), as indicated by Figure 8.19.At higher C0, it is possible that Meff varies with Nc, and consequently with C0.An evidence that demonstrates the variation of Meff with Nc has been pre-sented in the previous chapter. For example, the simulation results from Figure7.11(a) has shown that the effective mobility Meff of boundaries that contains seg-regated clusters with a steady-state size distribution decreases with increasing C0.Furthermore, the results has been rationalized using the framework of the Cahnsolute drag model where an analytical expression of Meff = f (C0) has been demon-strated to be in good agreement with the simulation results, see Eqs. (7.3) and (7.4)as well as Table 7.4.Such an analytical approach is not applicable for the present discussion sincethere is an insufficient amount of time for segregated atoms to diffuse along theboundary and form clusters with a steady-state size distribution. The size distribu-tion of clusters at a migrating boundary, Nc, is different from the size distributionof clusters at a stationary boundary, e.g. Figure 8.12(a)-(k) is compared to Figure7.7(c). This suggests that the size distribution of boundary clusters Nc not onlyvaries with the bulk concentration C0, but is also a function of the relative kineticsbetween solute diffusion at the boundary, Dgb, and the effective boundary velocity,Veff. In other words, Nc = f (C0, Veff, Dgb).Since the effective boundary mobility Meff depends on the boundary clustersNc, it can be written as Meff = f (C0, Veff, Dgb). While such a relationship is thedomain of solute drag model, its analytical expression is not known for the caseof clustering solutes. Furthermore, along with Eq. (8.5), this relationship indicatesthat Meff and Veff are implicitly dependent on each other. An analytical closed-form expression of the effective mobility of a migrating boundary Meff is thus dif-ficult to be made. Having said that, it is important to emphasize that the initialconclusion remains valid, namely, that the effective boundary mobility of an en-riched boundary is lower than the mobility of a pure boundary, i.e. Meff < Mpure.It is of interest to note further that the effective cluster pinning model in Eq.138(8.5) has a similar form as the solute-loaded (low-mobility) regime of the mobileparticle pinning theory by Gottstein and Shvindlerman [37]. There is, however, anumber of differences between the Gottstein-Shvindlerman theory and Eq. (8.5).The former considers Meff as a parameter in the order of particle mobility, whilehere, Meff reflects the effective boundary mobility due to segregated clusters thatchange the boundary structure. The Gottstein-Shvindlerman theory also recog-nizes no complete pinning, i.e. there is no critical driving pressure below whichthe boundary is not able to migrate. The effective cluster pinning model in Eq.(8.5), on the other hand, allows for a complete pinning to occur.8.6 SummaryThe kinetic interplay between grain boundary migration, solute segregation andcluster formation in the Fe-He system has been investigated. The framework ofclassical particle pinning model by Zener has been used to explain the trend in theboundary velocity with bulk solute concentrations since He clusters are effectivelyimmobile, their behaviour is similar to particles in the Zener model. It has alsobeen identified that He atoms that segregate to the boundary become trapped inthe matrix upon the boundary detaching itself from segregated He atoms. Thetrapping of segregated He atoms has been associated with the vacancies generatedduring the boundary migration.When a migrating boundary sweeps across clusters of varying size, the totalcluster pinning contribution is proportional to the size and population of clus-ters. Furthermore, the pinning pressure of a single cluster is also found to dependquadratically with the cluster radius, which is consistent with the classical Zenermodel.A cluster pinning model that takes into account the size distribution of clustershas been developed. The model provides a description that is consistent with thesimulation results. A number of assumptions employed in the Zener model havebeen revisited during the development of the present model. In the Zener model,the driving pressure was assumed to be independent of particles. Such an assump-tion has been found reasonable for discussing the simulation results, at least for the139range of concentrations investigated here.On the other hand, the assumption that considers the boundary mobility beingindependent of the presence of impurities at the boundary has been demonstratedto be inconsistent with the simulation results. For example, the effective mobilityof clean grain boundaries that have previously interacted with pre-segregated so-lute atoms (the type-s simulations) has been found to be lower than the mobilityof a clean boundary in a pure system. The reduction of the boundary mobilitydue to the presence of clusters at the boundary has also been deduced from thecase of interstitial atoms that segregate to a migrating boundary and form clusters(the type-i simulations) and from the case of bulk clusters that are captured by amigrating boundary (the type-c simulations). Such a mobility reduction due to so-lute clusters has also been observed and discussed in Chapter 7. In summary, thestudy finds that clusters affect the grain boundary migration not only by reduc-ing the net available driving pressure via pinning pressure but also by altering theeffective mobility of grain boundary via the change in boundary structure.140Chapter 9Conclusions and Future Work9.1 ConclusionsThis work has developed an atomistic modelling approach to simulate the inter-play between solute clusters and a migrating grain boundary. The parameter spacegoverning the effect of non-clustering solute atoms on grain boundary migrationhas been explored via kinetic Monte Carlo simulations to develop an atomisticallyinformed solute drag model. The dynamic interaction between solute clusters anda grain boundary has been investigated systematically using molecular dynamicssimulations of the iron-helium (Fe-He) system, starting with the identification ofatomistic events that control the grain boundary migration in pure bcc Fe. Theclustering and segregation behaviour of He atoms in stationary Fe grain bound-aries have been investigated. Finally, a model that describes the effect of He clus-ters on the migration of Fe grain boundaries has been developed. The overallachievements of this work are summarized as follows.• The atomistic kinetic Monte Carlo (KMC) simulations developed in this workis able to reproduce the thermal behaviour of grain boundary migration thatis consistent with the trends observed from experiments. Above the rough-ening temperature, the boundary migration occurs by random island nucle-ation and the boundary velocity depends linearly on the driving pressure.Below the roughening temperature, the migration occurs via a two-step pro-cess, i.e. island nucleation followed by ledges propagation, and there is aminimum driving pressure below which the boundary remains stationary.• The effect of solute atoms in an ideal binary system on grain boundary mi-gration has been examined using the same KMC framework. To provide aquantitative description on the relationship between the drag pressure and141grain boundary velocity observed from KMC simulations, a phenomenologi-cal model adapted from the Cahn model was developed. The adapted modelhas also been shown to produce a better fit to describe the solute drag be-haviour reported from previous work in the literature.• The maximum solute drag effect extracted from KMC simulations was foundto increase with higher solute diffusivity, indicating a departure from theclassical Cahn model. Such a discrepancy was attributed to the solute-inducedchange in the boundary structure. Fast diffusing solute atoms allow for localbulges to form in the boundary, resulting in a more tortuous path for bound-ary migration. Classical models are inherently incapable of accessing suchatomistic information.• Using the migration of planar and curved twin boundaries in Fe, this workhas produced a number of evidence that suggest the correlation between themigration of curved and planar grain boundaries. It has been shown thatthe shape and the reduced mobility of curved boundaries can be determinedfrom the mobility and energy of planar inclined segments along the curvedboundary. Furthermore, atomic-scale analysis has determined that, while thedetailed mechanisms for the migration of both curved and planar boundariesmay not be exactly the same, the similarities among these mechanisms indi-cate that there is a correlation between their migration behaviour.• The size distribution of He clusters in the bulk of bcc Fe crystals has beencharacterized at high temperatures. It was found that, for the temperaturerange being considered, there is no significant effect of temperature on thesize distribution. Additionally, clusters with the most population are thosecontaining 4 to 7 atoms. Clusters have significantly reduced diffusivitiesrelative to interstitial He atoms. This is attributed to the formation of self-interstitials that accompany cluster formation. Self-interstitials are formedwhen Fe atoms are ejected from their main lattice sites into interstitial sitesleaving behind vacancies. The binding energy of He atom-vacancy complexis in the order of 2 eV, thus no apparent diffusion of He clusters in the bulkFe crystals can be observed in the time-scale of MD simulations.142• He atoms segregate to grain boundaries in Fe. The maximum single-atombinding energy obtained from molecular statics calculation is found to be 1.3eV while the effective binding energy extracted from the overall segregationlevel is approximately 0.5 eV. The low effective binding energy relative to themaximum single-atom binding energy has been attributed to the clusteringof segregated He atoms. The size distribution of boundary clusters is foundto reach steady state at a much later time than the steady-state clustering inthe bulk. Such a kinetic difference has been attributed to the low diffusiv-ity of He atoms in grain boundary relative to the bulk diffusivity. The slowboundary diffusivity is further associated with the presence of energeticallyfavourable vacancy-like sites at the boundary. He atoms that occupy thesesites must overcome a high energy barrier to diffuse along the boundary.• There is an appreciable difference between the structure of cluster-enrichedgrain boundaries and that of clean boundaries. Further analysis shows thatthe spatial fluctuation of non-driven cluster-enriched boundaries occurs at amuch lower amplitude relative to that of pure boundaries. This has beenintepreted as the absolute mobilities of cluster-enriched boundaries beinglower than the absolute mobilities of pure boundaries. The cluster drag coef-ficient, i.e. the ratio between the inverse of cluster contribution to the abso-lute mobilities of enriched boundaries and the bulk solute concentration, hasfurther been analyzed in the framework of Cahn solute drag model. Thereis satisfactory agreement between the cluster drag coefficients obtained fromthe simulations and those predicted by the model.• In addition to reducing the effective absolute mobilities of grain boundaries,segregated He atoms are also found to have a pinning effect on the bound-ary migration, i.e. they delay or completely inhibit the migration. Whenthe boundary is able to unpin itself, the formerly segregated He atoms thatare now left behind appear to be trapped in their segregated positions. Thetrapping of segregated He atoms has been associated with the vacant sub-stitutional sites generated during the atomic shuffling motion that underliethe grain boundary migration. Segregated He atoms may identify these va-143cancies and thus become a substitutional solute. As a result, the formerlysegregated He atoms are now trapped in the bulk matrix of Fe.• Bulk He clusters are found to have a pinning effect on the boundary migra-tion. The magnitude of cluster pinning pressure is proportional to the clustersize, i.e. the number of atoms contained in a cluster. The pinning pressuredue to clusters has been evaluated against the classical Zener pinning model.It has been shown that the total pinning pressure from the simulations is pro-portional to the size and population of clusters, which is in agreement withthe Zener model. The magnitude of pinning pressure from the simulations,however, is an order of magnitude larger than the predicted value and ispartly attributed to the boundary being flexible and having a finite width.• Finally, the kinetic interplay between the clustering of segregated He atomsand grain boundary migration in Fe has also been investigated. A phe-nomenological cluster pinning model adapted from the Zener model hasbeen developed to explain the simulation results. The model considers thatclusters do not only decrease the available driving pressure via the clusterpinning pressure but they also reduce the effective mobility of the boundary.9.2 Future workTo expand upon the present understanding on the dynamic interaction betweensolute clusters and a grain boundary, several suggestions can be made for furtherinvestigations. It would be interesting, for example, to introduce the anisotropyin the grain boundary mobility and grain boundary energy into mesoscale sim-ulations of grain growth, e.g. phase-field models, and analyze the change in theoverall microstructure. Such study would be an extension to this work where ithas been demonstrated that the inclination dependence of absolute mobility andthe grain boundary energy affect the shape and kinetics of curved grain bound-ary migration. Present literature on phase-field models for grain growth considersonly the local variation of curvature while maintaining the mobility and the grainboundary energy isotropic.144It is also interesting for further study to address the computational issues per-taining to molecular dynamics simulations of the migration of planar grain bound-aries in an alloy system via the ADF technique. The issue with the ADF techniquebeing unable to distinguish grains adjacent to low-angle grain boundaries has beenresolved in two papers published recently [174, 175]). The solute-related issue,however, remains unresolved, i.e. the ADF technique is unable to discriminatethe lattice local disorder due to solute atoms from the lattice local disorder due tograin boundaries. A technique that resolves such an issue is desirable since it willbe compatible for alloy systems. It is a logical progression of this work, therefore,to develop an improved technique which permits a flat grain boundary in an alloysystem to be driven by a given driving pressure. Subsequent study can be dedi-cated to continue on molecular dynamics simulations to investigate the migrationof cluster-enriched planar grain boundaries. From such study, the effect of clus-ters on the driving pressure can be decoupled from the effect of clusters on theboundary mobility. A systematic understanding of the latter can be used to im-prove the assumption on the boundary mobility employed by the cluster pinningmodel model developed in this work.The cluster pinning model can be further parameterized using atomistic kineticMonte Carlo simulations. The KMC framework developed in this work for an idealbinary system can, in principle, be extended to consider the clustering behaviourof solute atoms. 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Eng. 18, 015012(2010). doi:10.1088/0965-0393/18/1/015012 (page: 189)172Appendix ARunning MD simulations in LAMMPSThis chapter aims to present an example of molecular dynamics simulations per-formed in LAMMPS [203]. Applications discussed in this chapter will be takenfrom the simulations that have been carried out during the completion of thiswork. Interested readers are referred to LAMMPS website, lammps.sandia.gov[254], for a more comprehensive discussion on features available in LAMMPS.A.1 BackgroundIt is assumed that a working LAMMPS is already installed in a computer. If this isnot the case, the website [254] provides pre-built executables that can be run with-out installation. The version of LAMMPS used in this work is the lammps-26May-13version. There are three stages that LAMMPS users may find themselves be in-volved when running an MD simulation:1. Pre-processing2. Analysis3. Post-processingThere is a clear end to each stage. Users must therefore anticipate the type of datathat they need in Stage 2 and 3 (the analysis and post-processing stages) beforecarrying out Stage 1 (the pre-processing stage). This will allow the simulation toproduce useful information at the end of its run, and prevent the need to redo thesame simulation.173A.2 Pre-processingA.2.1 File managementFor archival purpose, different sets of simulations with different objectives willbe placed in separate folders. The simulations for extracting equilibrium latticeparameter at finite temperatures, for example, are contained in one folder, separatefrom the simulations for calculating the diffusivity of a solute atom. Within eachfolder, several sub-folders can be created for storing different types of files, namely:• Sub-folder INPUT. It contains the input files and the PBS (portable batch sys-tem) files, i.e. script files to submit simulation jobs to a scheduler in a parallelcomputing cluster.• Sub-folder POTENT. It contains the empirical interatomic potential files.• Sub-folder LOG. It contains the thermodynamic output of simulations (or thelog files), e.g. total potential energy, temperature and pressure of the simu-lation cell. Typically, a simulation run will only produce 1 log file upon itscompletion.• Sub-folder DUMP. It contains the per-atom output of simulations (or the dumpfiles), e.g. atomic coordinates, velocities. Dump files can be produced forevery given time steps during the simulation. Depending on the size of sim-ulation cell, dump files can be quite large. From disk storage perspective, itmay be unwise to create dump files at a high frequency.• Sub-folder RESTART. It contains restart files that can be used to continue run-ning past simulations from a particular time step. Restart files are especiallyuseful for long simulation runs that are submitted to computing clusterswhere a limit on the job duration is imposed by the scheduling system (PBS).In other words, the submitted job may be terminated even if the simulationhas yet to finish. Restart files can only be opened and read by LAMMPS,partly for storage management purpose. If one requires access to a partic-ular information from a restart file, s/he would need to write an input file174that asks for the said restart file and produce the required information in theform of either log file or dump file. Similar to dump files, restart files easilyconsume space if not managed properly.A.2.2 Molecular statics (MS) and molecular dynamics (MD)In this work, MD simulations at T > 0 K are run after performing MS calculationsthat produce the equilibrium configuration of the simulation cell at 0 K. Two typesof input files are created. The first type is an input file for an MS calculation wherea simulation cell is set up and atoms initialized according to the desired config-uration, e.g. atoms in a single crystal or a bicrystal cell. The minimum energyconfiguration of the cell at zero pressure is obtained by performing the MS calcu-lation. A restart file is written at the end of the first simulation. The second typeis an input file that runs an MD simulation by first reading the configuration ofthe equilibriated cell from the restart file produced by the MS calculation. To dis-cuss the input files, these files are divided into several blocks. Each block will bedescribed independently, starting from the topmost one.A.2.3 Input file for MS calculationBlock 1: Declaring identifiersLine No Content1 # This is INPUT.bcc.Fe.0K.in .2 # == BLOCK 1 : Declaring identifiers ==34 variable ID equal 15 variable SIZE equal 506 variable TEMPERATURE equal 07 variable latparam equal 3.089 log ../LOG/LOG.bcc.Fe.size${SIZE}.${TEMPERATURE}K.id${ID}.txtFigure A.1: Block 1 of an input file for MS calculation.175A line preceded by ’#’ sign will be ignored by LAMMPS. In line 4 to 7, variablesare declared. These variables will be used mainly for setting up the simulationparameters, and also for naming the output files (the log file, the dump files andthe restart files).In line 6, declaring a variable TEMPERATURE with a given value (i.e. 0) does notmean that the temperature has been set to that value. It only means when thevariable TEMPERATURE is called later in the script, it will present a value of 0 (seeline 8). The same applies to line 7 that does nothing but declaring and assigning avalue to the variable latparam, representing lattice parameter (in Ångstrom). Anarbitrary value of 3.0 is set as the initial value of latparam, which is close to thezero-pressure lattice parameter of bcc Fe from the potential file, i.e. 2.8553 Å.In line 8, the command log assigns a name to the log file. LAMMPS, by default,uses a generic name for its log file, i.e. log.lammps if no specific name is assigned.Without line 8, the log file from a simulation run, i.e. ID#1 in Figure above, isnamed log.lammps and will be erased and overwritten when another simulation,e.g. ID#2, is run. If line 8 is added, the log file will be stored in LOG sub-folder, andnamed LOG.bcc.Fe.size50.0K.id1.txt. Notice the use of ${...} syntax to printthe value of a variable, e.g. ${TEMPERATURE}Kwill be printed as 0K.Block 2: Constructing simulation cellLine No Content9 # == BLOCK 2 : Constructing simulation cell ==1011 units metal12 dimension 313 boundary p p p14 atom_style atomic1516 lattice bcc ${latparam}17 region box block 0 ${SIZE} 0 0.5*${SIZE}0 1.5*${SIZE} units lattice18 create_box 2 boxFigure A.2: Block 2 of an input file for MS calculation.176Lines 11 to 14 are necessary for LAMMPS to determine the type of calculation willbe performed. The units command sets the default units for quantities such asenergy, distance and time, which are expressed in eV, Å and ps, respectively, forunits metal. The dimension command sets the dimensionality, i.e. 1 to 3. Theboundary command sets the boundary condition of simulation, e.g. p indicatingperiodic. The atom_style command assigns the type of attributes that are associ-ated to each atom. The default is atomic where the atomic velocities and forcesare the basic attributes of each atom. If other properties such as charges or dipolemoments are important features of each atom, the atom_style command can beset as dipole. See [254] for detailsLine 16 declares the type of lattice crystal being used (e.g. bcc). The string${latparam} assigns the value of variable latparam as the lattice constant of thebcc crystal.Line 17 declares the shape and size of simulation box. The shape is determinedby the third string (block = rectangular box), while the size is determined by thenumbers that follow. The numbers that follow indicate the lower and upper limitof the dimension in x, y and z direction, respectively. The last two keywords, unitslattice, indicate the unit of preceding numbers. For example, in Figure A.2, thedimension in x-, y- and z-direction will be equal to 50, 25 and 75 lattice constants,or equivalently, 150, 75, and 225 Å. The other option is to use units box, insteadof units lattice. In this way, the unit of box dimension is Å.Line 18 requests LAMMPS to construct a simulation box based on the require-ments (i.e. dimensions, boundary conditions, lattice structure) specified earlier.The simulation box can only be created once in a simulation. This can be achievedby calling the command create_box or by reading a restart file from previous simu-lations (see Figure). The number (e.g. “2”) indicates the number of atomic speciesthat will be introduced to the box, e.g. “2” indicates that the box will be a binary so-lution. This number must be equal to the number of atomic species that are statedin the potential file.177Block 3: Filling the simulation cell with atomsLine 21 requests LAMMPS to set the orientation of atoms with respect to the box. Inthis example, the standard Cartesian orientation is used. Line 22 requests LAMMPSto fill the region box with atoms type-1 (i.e. Fe atoms) according to the orientationin line 21.Line No Content19 # == BLOCK 3 : Filling the simulation cell with atoms ==2021 lattice bcc ${latparam} orient x 1 0 0orient y 0 1 0 orient z 0 0 122 create_atoms 1 region box2324 pair_style eam/fs25 pair_coeff * * ../POTENT/Fe-He.eam.fs2627 neighbor 2.0 bin28 neigh_modify delay 10 check yesFigure A.3: Block 3 of an input file for MS calculation.Line 24 declares the type of atomic potential that will be used, e.g. the EAM po-tential. Line 25 declares the coefficients of atomic potential. Here, the coefficientsare taken from a potential file named Fe-He.eam.fs which consists of the Fe-Fepotential, the Fe-He potential, and the He-He potential.Line 27 sets the neighbour list of each atom, i.e. “2.0” means that neighboursthat are within extra “2.0” distance unit from the force cutoff-distance (stated inthe potential file) will be tabulated in the neighbour list. Line 28 states how oftenthe neighbour list is updated, i.e. “10” means that the list is checked for an updateevery 10 simulation time steps.Block 4: Setting the type of output filesLine 31 states the frequency of the thermodynamic output of the simulation, e.g.the number “100” means that the thermodynamic output is printed every 100 timesteps. Line 32 declares the type of thermodynamic output that will be printed in178the log file. In this example, the log file will contain the time step, the total potentialenergy (pe), the total kinetic energy (ke), the total energy (etotal), temperature(temp), pressure (pxx, pyy, pzz, pressure) and volume (vol).Line No Content29 # == BLOCK 4 : Setting the type of output files ==3031 thermo 10032 thermo_style custom step pe ke etotal temppxx pyy pzz press vol3334 compute poteng all pe/atom35 dump dump1 all cfg 100 ../DUMP/SNAPSHOT.bcc.Fe.size${SIZE}.${TEMPERATURE}K.id${ID}.txtid type xs ys zs c_potengFigure A.4: Block 4 of an input file for MS calculation.Line 34 declares a dummy string poteng that requests LAMMPS to computethe potential energy per atom. The proper commands to compute other per-atomattributes, e.g. the kinetic energy, the coordination number, etc, are discussed inLAMMPS website. Line 35 declares the name and the type of information printedin the dump files. The string cfg denotes a particular format of the dump file.The number “100” means that the dump file is printed every 100 time steps. Eachdump file will contain lines of per-atom attributes, namely their positions (xs yszs) and their potential energy (c_poteng, the prefix c_ indicates that the string isof the compute type).Block 5: Performing energy minimization iterationLine 38 states the type of constraint (fix) that must be satisfied when the calcula-tion is carried out. Here, the command box/relax requests that the pressure in x,y and z direction be set to 0 during the energy minimization iteration and that theoverall fractional volume change between two iterations should be less than 0.001(or 0.1%).179Line No Content36 # == BLOCK 5 : Performing energy minimization ==3738 fix 1 all box/relax x 0.00 y 0.00 z 0.00 vmax 0.00139 min_style cg40 minimize 1e-25 1e-20 50000 50000041 write_restart ../RESTART/RESTART.bcc.Fe.size${SIZE}.${TEMPERATURE}K.id${ID}.resFigure A.5: Block 5 of an input file for MS calculation.Line 39 declares the type of energy minimization algorithm, e.g. cgmeans con-jugate gradient. Line 40 states the condition of the iteration, i.e. the iteration isstopped if the relative change in energy is less than 10−25, or the relative changein total force is less than 10−20, or the number of iterations exceeds 50,000 or thenumber of total force evaluations exceeds 500,000.Line 41 requests LAMMPS to write a restart file at the end of MS calculation.The restart file will contain the minimum energy cell that satisfied the relaxationcondition, i.e. zero pressure in all directions.A.2.4 Input file for MD simulationBlock 1: Declaring identifiersLine No Content1 # This is INPUT.bcc.Fe.800K.in .2 # == BLOCK 1 : Declaring identifiers ==34 variable ID equal 15 variable SIZE equal 506 variable TEMPERATURE equal 80078 log ../LOG/LOG.bcc.Fe.size${SIZE}.${TEMPERATURE}K.id${ID}.txtFigure A.6: Block 1 of an input file for MD simulation.180The lines 1-8 in Figure A.6 are similar to those in Figure A.1.Block 2: Loading the Restart fileLine No Content9 # == BLOCK 2 : Loading the Restart file ==1011 units metal12 dimension 313 boundary p p p14 atom_style atomic1516 read_restart ../RESTART/RESTART.bcc.Fe.size${SIZE}.0K.id${ID}.res17 reset_timestep 0Figure A.7: Block 2 of an input file for MD simulation.Refer to Figure A.2 for the description of lines 11-14 in Figure A.7. Line 16 requestsLAMMPS to load the simulation cell obtained from the MS calculation earlier, seeLine 41 in Figure A.5. Line 17 requests LAMMPS to reset the time step such thatthe initial time step of MD simulation is labelled with 0.Block 3: Assigning velocities to atoms in the simulation cellLine No Content18 # == BLOCK 3 : Assigning atomic velocities ==1920 pair_style eam/fs21 pair_coeff * * ../POTENT/Fe-He.eam.fs2223 neighbor 2.0 bin24 neigh_modify delay 10 check yes2526 velocity all create ${TEMPERATURE} 1234Figure A.8: Block 3 of an input file for MD simulation.181Refer to Figure A.3 for the description of lines 20 to 24 in Figure A.8. Line 26requests LAMMPS to assign velocities to each atom using a random distributionwith a seed random number 1234.Block 4: Equilibriation stage of MD simulationLine No Content27 # == BLOCK 4 : Equilibriation stage of MD simulation ==2829 thermo 100030 thermo_style custom step pe ke etotal temppxx pyy pzz press vol3132 restart 50000 ../RESTART/RESTART.bcc.Fe.size${SIZE}.${TEMPERATURE}K.id${ID}.*.res3334 fix 1 all npt temp ${TEMPERATURE} ${TEMPERATURE}1.0 aniso 0.0 0.0 1.035 run 100000036 unfix 1Figure A.9: Block 4 of an input file for MD simulation.Refer to Figure A.4 for the description of lines 29 to 30 in Figure A.9. Line 32requests LAMMPS to produce a restart file every 50,000 time steps. This is differentfrom the command write_restart in line 41 of Figure A.5, which requests therestart file to be produced only once when the command is executed.Line 34 is a command of the type fix. The string npt requests that the MDsimulation is run under the condition that the number of atoms, the pressureand temperature are maintained as constant. The strings temp ${TEMPERATURE}${TEMPERATURE} declares that the temperature must be fixed between the valuespecified by the variable TEMPERATURE, i.e. 800 in line 6, Figure A.6. The number“1.0” after ${TEMPERATURE} is the damping parameter that determines how rapidly(in the time unit) the temperature is adjusted, i.e. “1.0” means the temperatureis adjusted every 1.0 ps (or 1000 timesteps). The strings aniso 0.0 0.0 declares182that the pressure must be fixed as 0.0 anisotropically (i.e. the cell dimensions canchange independently along the x-, y- and z-direction). The number “1.0” is thedamping parameter for the pressure control.Line 35 asks LAMMPS to run the MD simulation for 1000,000 time steps (a totalof 1 ns since 1 time step = 1 fs). Line 36 requests LAMMPS to undeclare the com-mand fix named “1”. At the end of the simulation run, the log file will be used todetermine if the cell has reached an equilibriated state. This is done by checking ifthe average temperature and pressure data from the log file have reached the de-sired values. A degree of fluctuation in the temperature and pressure is expected.In this work, the equilibriated state can be said to have occurred if the fluctuationis less than 2 to 5%.Block 5: Data collection stage of MD simulationLine No Content37 # == BLOCK 5 : Data collection stage of MD simulation ==3839 compute poteng all pe/atom40 compute kineng all ke/atom41 dump dump1 all cfg 100 ../DUMP/SNAPSHOT.bcc.Fe.size${SIZE}.${TEMPERATURE}K.id${ID}.txtid type xs ys zs c_poteng c_kineng4243 fix 1 all npt temp ${TEMPERATURE} ${TEMPERATURE}1.0 aniso 0.0 0.0 1.044 run 100000045 unfix 1Figure A.10: Block 5 of an input file for MD simulation.Refer to Figure A.4 for the description of lines 39 to 41 in Figure A.10. Refer toFigure A.9 for the description of Line 43 to 45.183A.2.5 Supplementary filesA.2.5.1 Potential fileIn addition to input files, there are a number of supplementary files that are re-quired to run the MD simulations. The most important one is the potential file.In this work, the potential file for the Fe-He system follows the Embedded AtomModel obtained from the literature [205, 209, 211]. The procedure for translatingthe EAM force fields from the literature into files that are readable by LAMMPS(i.e. a setfl format) has been discussed in [255]. Briefly, the anatomy of a LAMMPS-compatible EAM file is described in the following table.Table A.1: Line-by-line description of a setfl file.Line # Description1 A non-value statement. I included citation to [205] here.2 A non-value statement. I included citation to [209] here.3 A non-value statement. I included citation to [211] here.4 (int number_of_elem) (Name of 1st element) (Name of 2nd element)/* Implementation example: 2 Fe He5 (int NPTS_RHO) (double RHO_increment) (int NPTS_DIST)←↩(double DIST_increment) (double r_cutoff)/*• NPTS_RHO = number of data points of rho in the file,–e.g. [205] uses 10,000 data points;• RHO_increment = increment of rho–e.g. 0.05;• NPTS_DIST = number of data points for r_distance;• DIST_increment = r_cutoff/NPTS_DIST;• r_cutoff = the greatest max. range of repuls. potential from the 2 elements–e.g. in the case of Fe-He system, r_cutoff is from Fe, i.e. 5.3 Å*//* Implementation example: 10000 5E-02 10000 5.3E-04 5.3E+001846 (int Z1) (double Ar_Z1) (double lattparam_Z1) (stringcrystalstruc_Z1)/*• Z1 = atomic number of elem 1;• Ar_Z1 = atomic mass of elem 1;• lattparam_Z1 = lattice parameter of elem 1;• crystal_struc_Z1 = crystal structure of element1*//* Implementation example: 26 5.585E+01 2.855312E+00 bcc7 to each line contains 5 values, each value is calculated(NPTS_RHO/5 + 6) from the embedded function of element 1.Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the electron density of element 1 - element 1.(NPTS_DIST/5 - 1)Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the electron density of element 1 - element 2.(NPTS_DIST/5 - 1)Cont. from prev. line (int Z2) (double Ar_Z2) (double lattparam_Z2) ←↩(string crystalstruc_Z2)/*• Z2 = atomic number of elem 2;• Ar_Z2 = atomic mass of elem 2,• lattparam_Z2 = lattice parameter of elem 2;• crystal_struc_Z2 = crystal structure of element2*//* Implementation example: 2 4.0026E+00 3.570E+00 hcpCont. from prev. line each line contains 5 values, each value is calculateduntil the next from the embedded function of element 2.(NPTS_RHO/5 - 1)Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the electron density of element 1 - element 2.(NPTS_DIST/5 - 1)Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the electron density of element 2 - element 2.(NPTS_DIST/5 - 1)185Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the REPULSIVE potential of element 1 - element 1.(NPTS_DIST/5 - 1)Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the REPULSIVE potential of element 1 - element 2.(NPTS_DIST/5 - 1)Cont. from prev. line each line contains 5 values, each value is calculateduntil the next from the REPULSIVE potential of element 2 - element 2.(NPTS_DIST/5 - 1)A.2.5.2 Neighbour list for order parameter calculationAnother supplementary files that are relevant for the type of simulations per-formed in this work are the files that complement the calculation of order param-eter η in LAMMPS, see Section In order to calculate η for a crystal with agiven orientation, LAMMPS requires a list of the position of nearest neighbours ofan atom in the bulk of the crystal.In a bcc single crystal, an atom i is expected to have 8 nearest neighbours j, eachat a distance aT√3/2 from i where aT is the lattice constant at temperature T. IfR0ij is the reference list for an atom i in the bulk of the standard frame, i.e. a framewhose orthogonal axes (X, Y, Z) are ([100], [010], [001]), the list R0ij contains thefollowing vectors[ujvjwj],R0ij=[ aT2aT2aT2] [ aT2aT2aT2] [aT2aT2aT2] [aT2aT2aT2][aT2aT2aT2] [aT2aT2aT2] [aT2aT2aT2] [aT2aT2aT2] (A.1)where j ∈ [1, 8] for bcc.If crystal A is a grain with a non-standard frame of orientation, its orthogo-nal axes (XA, YA, ZA) are ([hxAkxAlxA], [hyAkyAlyA], [hzAkzAlzA]). The positionof nearest neighbours in grain A, i.e. RAij ={[u′jv′jw′j]}, can be obtained fromR0ij ={[ujvjwj]}via the following transformation,186u′j =[ujvjwj] • [hxAkxAlxA]√h2xA + k2xA + l2xA, v′j =[ujvjwj] • [hyAkyAlyA]√h2yA + k2yA + l2yA, w′j =[ujvjwj] • [hzAkzAlzA]√h2zA + k2zA + l2zA(A.2)for all j (8 for bcc). The transformation described by Eq. (A.2) is useful whenworking with bicrystal cells, where the orthogonal axes of grain A and B are (XA,YA, ZA) and (XB, YB, ZB), respectively.As an example, the positions of nearest neighbours of atoms that belong to thebulk of either side of the planar coherent twin will be given. Referring to Table 4.3,the orthogonal axes that define the orientations of grains that are adjacent to theplanar coherent twin are (XA, YA, ZA) = (〈1 1 0〉,〈1 1 2〉, 〈1 1 1〉) and (XB, YB, ZB)= (〈1 1 0〉,〈1 1 2〉,〈1 1 1〉). Table A.2 presents the nearest neighbour lists of grainA and B, which have been obtained by applying the transformation given in Eq.(A.2) to these axes.Table A.2: The position vectors of the 8 nearest neighbours of atoms that belong to the bulkgrain of either side of the planar coherent twin boundary.Standard Grains adjacent to the coherent twinframe, R0ij A BaT[121212]aT[00√32]aT[0√2√31√12]aT[121212]aT[0√2√31√12]aT[00√32]aT[121212]aT[1√21√61√12]aT[1√21√61√12]aT[121212]aT[1√21√61√12]aT[1√21√61√12]aT[121212]aT[00√32]aT[0√2√31√12]aT[121212]aT[0√2√31√12]aT[00√32]aT[121212]aT[1√21√61√12]aT[1√21√61√12]aT[121212]aT[1√21√61√12]aT[1√21√61√12]Since half of the neighbour lists given in Table A.2 is the inverse of the otherhalf, LAMMPS only requires the tabulation of the first half. Figure A.11 shows anexample of the neighbour list files required for computing the order parameters in187a bicrystal that contains a coherent twin boundary at 1000 K.FILE #1: gb_nn_coh_twin1.txt0.000000000 0.000000000 2.4727739750.000000000 2.331353661 0.8242579922.019011496 -1.165676831 0.824257992-2.019011496 -1.165676831 0.824257992FILE #2: gb_nn_coh_twin2.txt0.000000000 -2.331353661 0.8242579920.000000000 0.000000000 2.4727739752.019011496 -1.165676831 -0.824257992-2.019011496 -1.165676831 -0.824257992Figure A.11: The content of the neighbour list files for computing the order parameter inLAMMPS, based on Table A.2, at 1000 K (aT at 1000 K is 2.882Å, see Figure 6.1).A.3 AnalysisA.3.1 Thermodynamic output (log file)Figure A.12 shows an example of the first 10 lines of the thermodynamic outputfrom the MD simulation discussed in Section A.2.4. The fluctuation of temperatureare about±10 K from the target temperature (800 K) while that of pressure is±200bar from the target pressure (0 bar). The magnitude of fluctuation indicates theequilibriation stage is not yet reached.188Time [fs] PotEng [eV] KinEng [eV] TotEng [eV] Temp [K] Pxx [bar] Pyy [bar] Pzz [bar] Press [bar] Volume [Å^3]0 -64166.835 1654.4264 -62512.408 800 -25089.55 -25224.793 -25082.634 -25132.326 189984.511000 -62421.996 1683.8237 -60738.172 814.21508 28.941359 294.50641 80.635034 134.69427 189918.412000 -62410.629 1644.0102 -60766.618 794.96322 -265.80862 -58.559804 564.26583 79.965802 189921.493000 -62386.815 1660.0432 -60726.772 802.71597 -31.201578 -44.253317 3.2817293 -24.057722 190039.894000 -62424.383 1666.707 -60757.676 805.93829 403.59989 -147.11616 410.86941 222.45105 189881.115000 -62389.177 1660.0427 -60729.134 802.71573 -528.69585 471.58024 -175.97018 -77.695263 190022.576000 -62409.357 1648.4826 -60760.874 797.12586 -190.06395 -380.79293 887.05266 105.39859 189941.177000 -62412.297 1663.5642 -60748.733 804.41857 -582.16477 -250.40679 92.720378 -246.61706 189993.088000 -62401.63 1646.2441 -60755.386 796.04344 -205.28019 155.93331 -1.4474722 -16.931452 190006.439000 -62400.691 1649.5671 -60751.124 797.65027 -537.39609 -82.835442 659.41774 13.06207 189976.5910000 -62413.536 1654.0907 -60759.445 799.83763 -350.38624 7.0333157 229.15984 -38.064361 189958.73Figure A.12: An example of the log file from an MD simulation, based on the input filediscussed in Section A.2.4.A.3.2 Per-atom output (dump files)The per atom output files are often used to create snapshots of the simulation cellat a number of times during the simulation. The snapshots can be used as a visualguide to inspect if the simulation run is consistent with its objective. In this work,the snapshots of MD simulations have been produced using AtomEye [256] andOVITO [257].A.4 Post-processingThe post-processing stage consists of analyzing the output files (the thermody-namic output and the per-atom output) using numerical algorithms to obtain rel-evant information. A number of algorithms employed in this work have beendiscussed in Section 4.2.5. As an example, the volume column from the thermody-namic output (Figure A.12) has been used to obtain the lattice parameter at finitetemperatures, see Figure 6.1. Another example is illustrated in Figure 7.3, wherethe per-atom output has been used to obtain the size distribution of helium clustersin the bulk bcc Fe crystal.189


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