Phase Field Modelling of Abnormal Grain Growth

materials

Article Phase Field Modelling of Abnormal Grain Growth

Ying Liu 1, Matthias Militzer 1,* and Michel Perez 2 1 The Centre for Metallurgical Process Engineering, The University of British Columbia, Vancouver, BC V6T1Z4, Canada; [email protected] 2 MATEIS, UMR CNRS 5510, INSA Lyon, Univ. Lyon, F69621 Villeurbanne, France; [email protected] * Correspondence: [email protected]; Tel.: +1-604-822-3676

Received: 20 October 2019; Accepted: 27 November 2019; Published: 5 December 2019

Abstract: Heterogeneous grain structures may develop due to abnormal grain growth during processing of polycrystalline materials ranging from metals and alloys to ceramics. The phenomenon must be controlled in practical applications where typically homogeneous grain structures are desired. Recent advances in experimental and computational techniques have, thus, stimulated the need to revisit the underlying growth mechanisms. Here, phase field modelling is used to systematically evaluate conditions for initiation of abnormal grain growth. Grain boundaries are classified into two classes, i.e., high- and low-mobility boundaries. Three different approaches are considered for having high- and low-mobility boundaries: (i) critical threshold angle of grain boundary disorientation above which boundaries are highly mobile, (ii) two grain types A and B with the A–B boundaries being highly mobile, and (iii) three grain types, A, B and C with the A–B boundaries being fast. For these different scenarios, 2D simulations have been performed to quantify the effect of variations in the mobility ratio, threshold angle and fractions of grain types, respectively, on the potential onset of abnormal grain growth and the degree of heterogeneity in the resulting grain structures. The required mobility ratios to observe abnormal grain growth are quantified as a function of the fraction of high-mobility boundaries. The scenario with three grain types (A, B, C) has been identified as one that promotes strongly irregular abnormal grains including island grains, as observed experimentally.

Keywords: abnormal grain growth; phase ﬁeld modelling; high mobility boundaries; disorientation; texture components

1. Introduction Abnormal grain growth (AGG) refers to a subset of grains that will grow excessively at the expense of surrounding normal grains leading to an obvious size advantage of the abnormal grains [1,2]. The AGG phenomenon has been observed in many materials, including steels [3–11], aluminum alloys [12,13], super alloys [14–16], ceramics [17–21], nanocrystalline materials [22–25] as well as thin films [25–28]. AGG is of important technological relevance and must be controlled in the thermal processing of polycrystalline materials. It is often undesired since it may lead to heterogeneous microstructures that result in unacceptable material properties. For example, Furnish et al. [23] showed that AGG leads to fatigue failure in nanocrystalline Ni–Fe. In some cases, however, AGG is a useful microstructure engineering concept, also known as secondary recrystallization, e.g., for the development of Goss or cube textures in electrical steels [4,9–11,29]. Similarly, AGG is beneficial for magnetostrictive Fe–Ga alloys as well as Fe-based shape memory alloys [30,31]. Further, Kusama et al. [32] demonstrated recently the formation of ultra-large single crystals by AGG. Thus, AGG studies remain an area of active research with an emphasis on electrical steels [9–11]. Furthermore, recent investigations deal with AGG during sintering [33–36] where, however, porosity plays a critical role [37]. Other studies consider the role of local plastic strain on the formation of abnormal grains [24,38,39]. In the context of the present study, AGG will be considered for polycrystalline materials without any external driving pressures or porosity.

Materials 2019, 12, 4048; doi:10.3390/ma12244048 www.mdpi.com/journal/materials Materials 2019, 12, 4048 2 of 20

The physical mechanisms for AGG may vary between different materials and they are still widely debated. There is, however, a general agreement that for a grain to grow abnormally it must have a persistent growth advantage over its neighbouring grains. The mechanisms of persistent AGG include anisotropy in grain boundary energy (wetting phenomenon), anisotropy in grain boundary mobility, selective unpinning of grain boundaries due to dissolution of precipitates and transitions in grain boundary structures with associated mobility changes [40–45]. In essence, all these mechanisms produce highly mobile boundaries that promote AGG. Important characteristics of AGG are complex grain shapes and island grain formation in addition to the continued grain size advantage [46]. The resulting grain size distribution is typically bimodal and can be used to characterize the grain growth process [47]. Recently a variety of computational methods have been used to simulate AGG, such as Monte Carlo (MC) models [2,48–50], the vertex model [51,52] and phase field modelling (PFM) [53–56]. An emphasis of these simulations has been to analyze the role of sub-boundaries, pinning particles and texture on AGG. Suwa et al. [55] included in their 3D-PFM grain growth simulations the role of two texture components, A and B, with the A–B boundary having a five times larger mobility than all other boundaries. They found that when a minority texture component is present, e.g., with a fraction of 0.03 in the initial grain structure, AGG features are predicted due to the rapid growth of these minority grains to comparatively large sizes prior to their impingement. More recently, DeCost and Holm [2] analyzed the phenomenology of AGG using a 2D MC Potts model in a system with three grain types (or three texture components) by simulating a single initially sufficiently large candidate grain of one grain type that is embedded into a matrix of grains which are randomly assigned one of the two other grain types according to a pre-set fraction of each type. The boundary of the candidate grain with grains of one of the other grain types has a mobility advantage of 1000 in these simulations resulting in AGG when sufficient high-mobility boundaries are present. In a few selected cases, as low as 20% of high-mobility boundaries are sufficient for AGG but only when the percolation threshold of 50% is reached do all candidate grains continue to grow abnormally. The initiation stage of AGG is, however, not considered in detail in these simulations. In this work, PFM is employed for a systematic parametric study to identify mobility advantage conditions that are required for AGG to be initiated. Those mobility advantages are introduced through three different scenarios, which are disorientation angles, two and three grain types, respectively. In the absence of a clear bimodal grain size distribution a grain that is at least five times larger than the average grain size can typically be considered as an abnormal grain.

2. Methodology During the past few decades, PFM has emerged as a versatile tool to simulate microstructure evolution. PFM has several advantages compared to other computational methods as there is no need to track interfaces explicitly. Therefore, PFM is a powerful methodology to deal with complex morphological features that are often observed for abnormal grains. In addition, phase field models provide outputs in a physical time rather than that of a numerical time as in MC models. In particular, multi-phase field modelling (MPFM) is frequently used as a computational method to simulate the microstructure evolution of metallurgical phenomena, including phase transformation [57,58], recrystallization [59,60] and grain growth [61–63]. As a diffuse-interface model, MPFM uses a finite thickness to describe the interfaces, within which the physical properties change continuously. A series of phase field (or order) parameters φi is used to describe the microstructure where φi equals 1 within grain i and its value changes continuously from 1 to 0 across the grain boundary. To identify abnormal grain growth conditions, we employ the commercial software MICRESS where for grain growth simulations the MPFM formulation according to Eiken et al. [64] is used such that the evolution of φi for isotropic grain boundary energies, σ, is governed by

XN µ σ 2 ! ∂φi ij ij 2 2 π = φi φj + (φi φj) (1) ∂t N ∇ − ∇ η2 − j=1 Materials 2019, 12, x FOR PEER REVIEW 3 of 20

푁 2 휕휙푖 휇푖푗휎푖푗 휋 = ∑ (∇2휙 − ∇2휙 + (휙 − 휙 )) (1) 휕푡 푁 푖 푗 휂2 푖 푗 푗=1 here 휇푖푗 and 휎푖푗 = 휎 represent boundary mobility and interfacial energy, respectively,  is the grain Materials 2019, 12, 4048 3 of 20 boundary thickness and N is the local number of phase field parameters that are not zero. Boundary mobility advantages are essential for AGG to occur. Here, we consider three different = ways tohere introduceµij and σij highσ -representmobility boundary boundaries. mobility In the and first interfacial approach, energy, mobility respectively, advantagesη is the grain are defined boundary thickness and N is the local number of phase field parameters that are not zero. through a Boundarycritical disorientation mobility advantages angle. are Grains essential in for the AGG initial to occur. grain Here, structure we consider are threeassigned different randomly a crystallographicways to introduce orientation high-mobility such that boundaries. each individual In the first approach,boundary mobility is characterized advantages areby defineda disorientation angle through. Then athe critical grain disorientation boundary angle.mobility Grains can in be the formulated initial grain structureas a function are assigned of disorientation randomly angle, 휇(휃) =a푓 crystallographic(휃). For the sake orientation of convenience, such that each a individual simplified boundary model is is characterized used as shown by a disorientation in Figure 1. Here, a angle θ. Then the grain boundary mobility can be formulated as a function of disorientation angle, critical disorientation angle (or threshold angle) 휃퐶 is introduced to select a portion of grain µ(θ) = f (θ). For the sake of convenience, a simplified model is used as shown in Figure1. Here, boundaries to be fast boundaries. When the disorientation angle 휃 is larger than 휃퐶, the boundary a critical disorientation angle (or threshold angle) θC is introduced to select a portion of grain boundaries mobilityto beis fast휇2. boundaries. If the disorientation When the disorientation angle 휃 is angle lessθ thanis larger 휃퐶, than theθ boundaryC, the boundary mobility mobility is is휇µ1.2 .The value of 휇2⁄If휇1 the (>1) disorientation is the mobility angle θ ratio.is less than θC, the boundary mobility is µ1. The value of µ2/µ1 (>1) is the mobility ratio.

Figure 1. The relation of boundary mobility and disorientation angle.

Another methodFigure to1. introduceThe relation boundary of boundary mobility mobility advantages and is disorientation through different angle. grain types or texture components. Both two and three grain type systems are considered in our simulations. When two grain types A and B are present, there are three types of grain boundaries, i.e., A–A, B–B and A–B Another method to introduce boundary mobility advantages is through different grain types or boundaries. If one more grain type C is added then there are six types of grain boundaries, i.e., A–A, textureA–B, components. A–C, B–B, B–C Both and two C–C and boundaries. three grain In both type scenarios, systems the are A–B considered boundaries in are our selected simulations. to be When two grainhigh-mobility types A boundariesand B are (withpresent, mobility thereµ2 )are and three all the types remaining of grain grain boundaries, boundaries are i.e. low-mobility, A–A, B–B and A– B boundaries.boundaries If (withone more mobility grainµ1). Intype the C present is added MICRESS then simulations, there are disixfferent types grain of grain types (i.e., boundaries, A, B, C) i.e., A– A, A–B,are A approximated–C, B–B, B–C as and different C–C phases bound thataries. have In the both same scenarios, Gibbs free energy, the A i.e.,–B theboundaries entropy of fusionare selected is to be infinitively small (10 7 J cm 3 K 1) such that there is no additional driving pressure due to the presence high-mobility boundaries− ·(with− · mobility− 휇 ) and all the remaining grain boundaries are low-mobility of these hypothetical phases. 2 boundariesAll (with simulations mobility of this 휇1 study). In arethe 2D present simulations MICRESS with periodic simulations boundary, conditions.different Forgrain the types critical (i.e., A, B, C) are disorientationapproximated angle as scenario, different it is phases crucial to that have have a statistically the same relevant Gibbs size free of the energy, simulation i.e. domain., the entropy of Thus, a square domain with 800−7 800−3 grid−1 points is used within which 1800 grains are positioned through fusion is infinitively small (10 J×cm K ) such that there is no additional driving pressure due to the Voronoi tessellation. Similarly, for simulations with different grain types a domain of 850 850 grid presence of these hypothetical phases. × points and 1800 grains in the initial structure is used. However, to increase computational efficiency, All simulations of this study are 2D simulations with periodic boundary conditions. For the a smaller domain of 300 300 grid points and 120 grains including one candidate grain for abnormal × criticalgrowth disorientation in the initial angle structure scenario, is employed it is crucial for selected to have simulations a statistically with three relevant grain types.size of For the all simulation domain.simulations, Thus, a the square grid spacing domain is taken with to be 800 1 µ m× and800 the grid high-mobility points is value usedµ is within set to 5 which10 2 cm 18004/(J s) . grains are 2 × − positionedThe low-mobility through value Voronoi is then tessellation. obtained by multiplying Similarly, the above for simulations value with the reciprocalwith different of the selected grain types a domainmobility of 850 ratio. × 850 For grid instance, points if and the mobility 1800 grains ratio is in 100, the then initial the mobilitystructure is reducedis used. by However, 0.01 for the to increase low-mobility boundaries. computational efficiency, a smaller domain of 300 × 300 grid points and 120 grains including one candidate grain for abnormal growth in the initial structure is employed for selected simulations with three grain types. For all simulations, the grid spacing is taken to be 1 µm and the high-mobility value −2 4 휇2 is set to 5 × 10 cm /(J s). The low-mobility value is then obtained by multiplying the above value

Materials 2019, 12, x FOR PEER REVIEW 4 of 20 with the reciprocal of the selected mobility ratio. For instance, if the mobility ratio is 100, then the mobility is reduced by 0.01 for the low-mobility boundaries. Sensitivity tests were performed to analyze the role of the selection of numerical parameters, i.e., interface thickness, time step etc. For instance, to test the sensitivity of time step selection, a simulation using automatic time stepping was compared to simulations with constant time steps of 0.002 s and 0.004 s, respectively, while keeping all other settings the same. Similarly, the interface thickness was selected to be 4 and 6 grid points. Based on the sensitivity analysis it was concluded that simulationMaterials 2019 results, 12, 4048 are not affected by the selection of these numerical parameters.4 of 20 Thus, all simulations for the parametric study were conducted with automatic time stepping and an interface thickness of 4Sensitivity grid points tests to were minimize performed computational to analyze the role co ofst. the selection of numerical parameters, i.e., Theinterface simulated thickness, grain time structure step etc. evolution For instance, is to quantified test the sensitivity by measuring, of time step selection,at selected a simulation times, the area of each grainusing automatici based timeon itsstepping phase was field compared parameter to simulations whereby with grid constant points time in steps the of grain0.002 s andboundaries 0.004 s, respectively, while keeping all other settings the same. Similarly, the interface thickness was contribute with a fraction that is given by the value of the phase field parameter 휙 . From the grain selected to be 4 and 6 grid points. Based on the sensitivity analysis it was concluded that simulation푖 area, theresults equivalent are not area aﬀected diameter by the selection (EQAD) of of these each numerical grain is parameters. determined. Thus, The all mean simulations EQAD for is the obtained from theparametric mean grain study area were and conducted the normalized with automatic diameter time stepping of a grain and is an introduced interface thickness as its ofEQAD 4 grid divided by the meanpoints EQAD. to minimize computational cost. The simulated grain structure evolution is quantified by measuring, at selected times, the area of each 3. Resultsgrain i based on its phase field parameter whereby grid points in the grain boundaries contribute with a fraction that is given by the value of the phase field parameter φi. From the grain area, the equivalent area diameter (EQAD) of each grain is determined. The mean EQAD is obtained from the mean grain 3.1. Mobilityarea andAdvantage the normalized through diameter Critical of Disorientation a grain is introduced Angle as its EQAD divided by the mean EQAD.

The3. 1800 Results grains in the initial structure are assigned randomly a cubic crystallographic orientation such that the disorientation distribution follows the Mackenzie distribution [65]. Then the fraction of3.1. fast Mobility boundaries Advantage is through a function Critical of Disorientation the threshold Angle angle, as shown in Figure 2. The potential for AGG will beThe maximized 1800 grains by in thehaving initial a structure good mix are assignedbetween randomly fast and a cubicslow crystallographic boundaries, e.g. orientation, for threshold angles betweensuch that 40 the° disorientationand 45° the distributionpercentage follows of highly the Mackenzie mobile distributionboundaries [65 falls]. Then in thethe fraction range ofof 40% to 58%. As fastan example, boundaries Figure is a function 3 shows of the the threshold evolution angle, asof shownthe grain in Figure structure2. The potential with a threshold for AGG will angle be of 40° maximized by having a good mix between fast and slow boundaries, e.g., for threshold angles between and a mobility ratio of 1000. Initially all grains are of a similar size as a result of Voronoi tessellation. 40◦ and 45◦ the percentage of highly mobile boundaries falls in the range of 40% to 58%. As an example, GraduallyFigure the3 grain shows size the evolution distribution of the widens grain structure and after with aabout threshold 50 s angle grains of 40 M◦ andand a N mobility are examples ratio of of the somewhat1000. larger Initially grains all grains in the are distribution. of a similar size At as 190 a result s, the of Voronoisize advantage tessellation. of Gradually grains M the and grain N sizehas further increaseddistribution as these widensgrains and have after comparatively about 50 s grains Mrapidly and N areconsumed examples oftheir the somewhat much smaller larger grains neighbouring in grains. Inthe grain distribution. M an At island 190 s, the grain size advantage has formed of grains which M and is N has a characteristic further increased sign as these of grains AGG. After have comparatively rapidly consumed their much smaller neighbouring grains. In grain M an island approximatelygrain has 400 formed s, however, which is agrains characteristic M and sign N ofhave, AGG. at After least approximately to some extent, 400 s, lost however, their grains size ad M vantage and the grainand N structure have, at least starts to some to extent,approach lost their that size of a advantage normal andgrain the size grain distribution. structure starts to approach that of a normal grain size distribution.

1.0

0.8

0.6

0.4

0.2

Fraction of High Mobility Boundaries of High Mobility Fraction

0.0 0 10 20 30 40 50 60 Threshold Angle (o)

Figure 2. Fraction of high-mobility boundaries as a function of threshold angle in a Mackenzie distribution.

Materials 2019, 12, x FOR PEER REVIEW 5 of 20

To more quantitatively analyze the changes in the grain structure with time including AGG stages, the evolution of the cumulative grain area distribution is shown in Figure 4 as a function of the normalized diameter. The red curve provides as reference the cumulative area distribution of normal grain growth that is obtained when all boundaries have the same mobility (mobility ratio of 1). Since normal grain growth is a self-similar process, the red curve represents the resulting scaling distribution where the maximum EQAD is about twice as large as the mean EQAD. The scaling distribution is used as a benchmark to measure the abnormality of grain growth. Initially, the grain area distribution resulting from Voronoi tessellation is much narrower than the scaling distribution but quickly broadens and its width surpasses that of the scaling distribution. After 50 s the maximum EQAD (i.e., that of grain N) is about 3.5 times larger than the mean EQAD. The maximum normalized diameter increases further to 6 at 190 s before it starts to decrease towards a value of 5 for larger times. Thus, there is a particular time (or time period) where the grain area distribution has its broadest range and the maximum normalized diameter obtained for this situation may be taken to assess the abnormality of grain growth for the selected grain growth parameters, i.e., threshold angle and mobility ratio. Therefore, one can find the widest distribution curve for each scenario to quantify the degree ofMaterials AGG.2019 , 12, 4048 5 of 20

o Figure 3.Figure Evolution 3. Evolution of of grain grain structure structure for µ for2/µ 1 휇=2⁄1000휇1 =and1000θC = 40and◦ where 휃퐶 high-mobility= 40 where boundaries high-mobility boundariesare shownare shown in black in and black low-mobility and low boundaries-mobility in boundaries blue: (a) t = 5in s, blue (b) t =: (50a) s, t (=c) 5 t =s,190 (b) s, t (=d )50 t = s,400 (c) t = 190 s. M and N refer to examples of larger grains in the distribution. s, (d) t = 400 s. M and N refer to examples of larger grains in the distribution. To more quantitatively analyze the changes in the grain structure with time including AGG stages, the evolution of the cumulative grain area distribution is shown in Figure4 as a function of the normalized diameter. The red curve provides as reference the cumulative area distribution of normal grain growth that is obtained when all boundaries have the same mobility (mobility ratio of 1). Since normal grain growth is a self-similar process, the red curve represents the resulting scaling distribution where the maximum EQAD is about twice as large as the mean EQAD. The scaling distribution is used as a benchmark to measure the abnormality of grain growth. Initially, the grain area distribution resulting from Voronoi tessellation is much narrower than the scaling distribution but quickly broadens and its width surpasses that of the scaling distribution. After 50 s the maximum EQAD (i.e., that of grain N) is about 3.5 times larger than the mean EQAD. The maximum normalized diameter increases further to 6 at 190 s before it starts to decrease towards a value of 5 for larger times. Thus, there is a particular time (or time period) where the grain area distribution has its broadest range and the maximum normalized diameter obtained for this situation may be taken to assess the abnormality of grain growth for the selected grain growth parameters, i.e., threshold angle and mobility ratio. Therefore, one can ﬁnd the widest distribution curve for each scenario to quantify the degree of AGG. Materials 2019, 12, x FOR PEER REVIEW 6 of 20

Materials 2019, 12, x FOR PEER REVIEW 6 of 20 Materials 2019, 12, 4048 6 of 20

o FigureFigure 4.4.Time Time evolution evolution of cumulative of cumulative grain grain area distributionarea distribution (µ2/µ1 (=휇21000⁄휇1 =and1000θC =and40◦ ).휃퐶 For= 40 reference,). For

thereference, scaling the distribution scaling distribution for 2D ideal for grain 2D growth ideal grain is shown growth as well. is shown as well. o Figure 4. Time evolution of cumulative grain area distribution (휇2⁄휇1 = 1000 and 휃퐶 = 40 ). For reference, the scaling distribution for 2D ideal grain growth is shown as well. TheThe scenarioscenario describeddescribed above above does does show show some some indication indication of of AGG AGG that that is is based based on on the the presence presenc ofe one or a few fast-growing grains. It is thus important to re-evaluate AGG events starting from different of oneThe or scenario a few fast described-growing above grains. does It showis thus some important indication to re of- evaluateAGG that AGG is based events on startingthe presenc frome initialdifferent structures. initial structures. The widest The distribution widest distribution curves resulting curves fromresulting a range from of initiala range structures of initial arestructures shown of one or a few fast-growing grains. It is thus important to re-evaluate AGG events starting from in Figure5 where the maximum EQAD is varying from 3.5 to 6. With the same amount of highly differentare shown initial in Figure structures. 5 where The the widest maximum distribution EQAD curvesis varying resulting from 3.5from to a6. range With ofthe initial same structuresamount of mobilehighly boundaries,mobile boundaries, their relative their positionsrelative positions significantly significantly affect the affect extent the of extent AGG. Theof AGG. highest The value highest for are shown in Figure 5 where the maximum EQAD is varying from 3.5 to 6. With the same amount of thevalue maximum for the maximum EQAD has EQAD indeed has been indeed observed been for observed the case for shown the case in Figures shown3 andin Fig4 andures this 3 and has 4 also and highly mobile boundaries, their relative positions significantly affect the extent of AGG. The highest been the only case with the formation of an island grain illustrating that AGG is a rather rare event for valuethis has for also the beenmaximum the only EQAD case haswith indeed the formation been observed of an island for the grain case illustrating shown in Figthatures AGG 3 and is a 4rather and therare present event thresholdfor the present angle scenario. threshold For angle a grain scenario. to grow For abnormally, a grain to a grow sustained abnormally mobility, a advantage sustained this has also been the only case with the formation of an island grain illustrating that AGG is a rather is required and for the investigated threshold angle case the required mobility advantage can only be raremobility event advantage for the present is required threshold and for angle the scenario.investigated For threshold a grain to an growgle case abnormally the required, a sustained mobility attainedadvantage for can very only few be grains attained and for a limited very few time grains period. and a limited time period. mobility advantage is required and for the investigated threshold angle case the required mobility advantage can only be attained for very few grains and a limited time period.

Figure 5. Widest cumulative grain area distribution for five different initial structures with θC = 40◦ o andFigureµ2/ 5.µ 1Widest= 1000. cumulative grain area distribution for five different initial structures with 휃퐶 = 40 and 휇 ⁄휇 = 1000. Figure 25. Widest1 cumulative grain area distribution for five different initial structures with 휃 = 40o To quantify the role of threshold angle and mobility ratio on AGG, we select the initial퐶 structure and 휇 ⁄휇 = 1000. of theTo above quantify2 1 case the with role the of mostthreshold pronounced angle and AGG mobility structure ratio on (see AGG, Figure we3 )select for a the parametric initial structure study. Theof the widest above distribution case with the curves most of pronounced each scenario AGG are presentedstructure (see in Figure Figure6 to 3) showfor a parametric the e ffect of study. threshold The To quantify the role of threshold angle and mobility ratio on AGG, we select the initial structure angle and mobility ratio, respectively, on growth abnormality. When changing the threshold angle ofwidest the above distribution case with curves the most of each pronounced scenario AGGare presented structure in (see Fig Figureure 6 to 3) showfor a parametric the effect of study. threshold The from 20◦ to 50◦, it is confirmed that growth abnormality is restricted to the threshold angle range of widest distribution curves of each scenario are presented in Figure 6 to show the effect of threshold

Materials 2019, 12, x FOR PEER REVIEW 7 of 20 Materials 2019, 12, x FOR PEER REVIEW 7 of 20 angle and mobility ratio, respectively, on growth abnormality. When changing the threshold angle angleMaterials and2019 mobility, 12, 4048 ratio, respectively, on growth abnormality. When changing the threshold angle7 of 20 from 20° to 50°, it is confirmed that growth abnormality is restricted to the threshold angle range of from 20° to 50°, it is confirmed that growth abnormality is restricted to the threshold angle range of 40–44° where the percentage of the fast boundaries is around 50% (i.e., here 58% and 45%, 40–44° where the percentage of the fast boundaries is around 50% (i.e., here 58% and 45%, respectively). For threshold angles of 30° and 50°, on the other hand, 82% and 22% of the boundaries respectively).40–44◦ where theFor percentagethreshold angles of the fastof 30 boundaries° and 50°, on isaround the other 50% hand, (i.e., 82% here and 58% 22% and of 45%, the respectively). boundaries are highly mobile, essentially eliminating the conditions for individual grains to have a sufficient areFor highly threshold mobile angles, essentially of 30◦ and 50eliminating◦, on the other the hand,conditions 82% and for 22%individual of the boundaries grains to have are highly a sufficient mobile, growth advantage and restricting the maximum normalized diameter to about 3, i.e., a grain structure growthessentially advantage eliminating and restricting the conditions the maximum for individual normalized grains to diameter have a su toﬃ aboutcient 3, growth i.e., a grain advantage structure and that approaches that of normal grain growth. Similarly, when changing the mobility ratio from 5 to thatrestricting approaches the maximum that of normal normalized grain diametergrowth. Similarly, to about 3, when i.e., a changing grain structure the mobility that approaches ratio from that 5 to of 1000 for a threshold angle of 40°, one can conclude from Figure 6b that when the mobility ratio is 10 1000normal for graina threshold growth. angle Similarly, of 40° when, one can changing conclude the from mobility Figure ratio 6b from that 5when to 1000 the for mobility a threshold ratio angleis 10 or less, grain growth does not occur abnormally since the distribution curves are sufficiently close to orof less, 40◦, onegrain can growth conclude does from not Figureoccur 6abnormallyb that when since the mobility the distribution ratio is 10 curves or less, are grain sufficiently growth does close not to that for normal grain growth. Mobility ratios above 50 lead to AGG where the severity of AGG is thatoccur for abnormally normal grain since growth. the distribution Mobility ratios curves above are su 50ﬃ cientlylead to close AGG to where that for the normal severi grainty of AGG growth. is augmented when increasing the mobility ratio from 100 to 1000 as the associated maximum augmentedMobility ratios when above increasing 50 lead to the AGG mobility where ratio the severity from 100 of AGG to 1000 is augmented as the associated when increasing maximum the normalized diameter increases from 5 to 6. normalizedmobility ratio diameter from 100 increases to 1000 asfrom the 5 associated to 6. maximum normalized diameter increases from 5 to 6.

Figure 6. Cumulative grain area distribution for the widest grain size distributions: (a) effect of FigureFigure 6. CumulativeCumulative grain grain area area distribution distribution for the for widest the widest grain sizegrain distributions: size distributions: (a) effect ( ofa) thresholdeffect of o threshold angle when= 휇2⁄휇1 = 100 ; (b) effect of mobility= ratio when 휃퐶 = 40o . For reference, the thresholdangle when angleµ2/µ when1 100 휇;(2⁄b휇)1 effect= 100 of mobility; (b) effect ratio of whenmobilityθC ratio40◦. whenFor reference, 휃퐶 = 40 the. scalingFor reference, distribution the scaling distribution for 2D ideal grain growth is shown as well. scalingfor 2D idealdistribution grain growth for 2D is ideal shown grain as well.growth is shown as well.

The influence of mobility ratio is further illustrated in Figure 7 by comparing the grain structure TheThe influence inﬂuence of of mobility mobility ratio ratio is is further further illustrated illustrated in in Figure Figure 77 byby comparing the graingrain structure images with the widest size distribution. For a mobility ratio of 10, there are no obvious grain size imagesimages with with the the widest widest size size distribution. distribution. For For a a mobility mobility ratio ratio of of 10, 10, there there are are no no obvious obvious grain grain size size advantages and no tendency of island grain formation. However, if the mobility ratio is increased to advantagesadvantages and and no no tendency tendency of of island island grain grain formation. formation. However, However, if the if the mobility mobility ratio ratio is increased is increased to 100, grain N has a clear size advantage over the other grains and the size advantage of grain N 100,to 100, grain grain N Nhas has a clear a clear size size advantag advantagee over over the the other other grains grains and and the the size size advantage advantage of of grain grain N N becomes even more pronounced when the mobility ratio is further increased to 1000. Meanwhile, becomesbecomes even even more more pronounced pronounced when when the the mobility mobility ratio ratio is is further further increased increased to to 1000. 1000. Meanwhile, Meanwhile, with a mobility ratio of 100, grain M shows a tendency to embrace one of its small neighbour grains with a mobility ratio ofof 100,100, graingrain M M shows shows a a tendency tendency to to embrace embrace one one of of its its small small neighbour neighbour grains grains but but an island grain has not yet formed. If the mobility ratio is further increased to 1000, there is an butan islandan island grain grain has nothas yetnot formed.yet formed. If the If mobility the mobility ratio ratio is further is further increased increased to 1000, to there1000, isthere an island is an island grain formed within grain M. islandgrain formedgrain formed within within grain M.grain M.

Figure 7. GrainGrain structure structure for for the the widest widest grain grain size size distribution distribution when when 휃θC==4040o◦ andand ( (aa)) 휇µ2⁄/휇µ1==1010 ;; Figure 7. Grain structure for the widest grain size distribution when 휃퐶 = 40o and (a) 휇 2⁄휇1 = 10 ; (b) µ /µ = 100; (c) µ /µ = 1000. 퐶 2 1 (b) 휇22⁄휇1 = 100 ; (c) 휇22⁄휇11 = 1000. (b) 휇2⁄휇1 = 100 ; (c) 휇2⁄휇1 = 1000. Whilst in these simulations with a random distribution of the highly mobile boundaries the

potential of at least mild AGG is conﬁrmed, it is a rather rare event. Extensive AGG appears to be limited Materials 2019, 12, 4048 8 of 20 as the very few grains that acquire temporarily an appreciable size advantage stop to grow rapidly as soon as they attain a situation where they are entirely surrounded by low-mobility boundaries.

3.2. Mobility Advantage in Two-Grain Type Systems Another method to introduce a mobility advantage is through diﬀerent grain types or texture components. Here, we start with a two-component system consisting of A and B grains where the A–B boundaries are taken as highly mobile and their fraction is then determined by the fraction of B grains in the initial microstructure. An example of evolution of the grain structure in the A–B system is shown in Figure8 for a mobility ratio of 1000 and 2% of B grains randomly distributed in the initial structure with a narrow size distribution resulting from Voronoi tessellation. Because of the small fraction of B grains, all of these B grains have initially only A grains as their neighbours and are, thus, surrounded entirely by highly mobile grain boundaries. As a result, there is a rapid evolution of the B grains. Whether or not they grow or shrink depends on their size with respect to their neighbours, i.e., smaller B grains will shrink and larger B grains will grow. After a short time (t = 25 s), about 1/3 of the B grains (e.g., grain III) have shrunk and disappeared very quickly in the scenario shown in Figure8. Approximately 1 /3 of B grains (e.g., grain I) grow rapidly at the expense of the surrounding A grains, thereby forming a bimodal grain size distribution that is characteristic for abnormal growth. The remaining 1/3 of B grains with a hexagonal structure (e.g., grain II) form, at least for some time, a stable grain structure with their neighbours that is a speciﬁc feature of 2D grain growth which does not exist in 3D. The maximum normalized diameter depends, then, primarily on the spacing of the growing B grains which, in the present case, constitute 0.67% of all grains in the initial structure.

ImpingementMaterials 2019 of, 12 growing, x FOR PEER grains REVIEW occurs later, see e.g., images at 50 and 75 s in Figure8. 9 of 20

Figure 8. Microstructure evolution of A–B system with fast A–B boundaries when 휇2⁄휇1 = 1000 and Figure 8. Microstructure evolution of A–B system with fast A–B boundaries when µ2/µ1 = 1000 and 0.67% growing B grains in the initial structure: (a) t = 0 s; (b) t = 25 s; (c) t =50 s; (d) t = 75 s. B grains 0.67% growing B grains in the initial structure: (a) t = 0 s; (b) t = 25 s; (c) t =50 s; (d) t = 75 s. B grains shown in orange and A grains in white. shown in orange and A grains in white. Even though a clearly bimodal grain size distribution can be obtained with the above mobility scenario, it does not show the development of any irregular grains with complex morphologies that are frequently observed in AGG. The growing B grains show a circular evolution in the 2D simulations, which would translate into spherical growth in 3D, i.e., the grain structure while consisting of two grain types with different sizes remains morphologically equiaxed. Even so, similarly to the threshold angle approach we performed a systematic parameter study to explore the influence of mobility ratio and initial percentage of growing B grains on the extent of the bimodality of the grain structure. Taking the scenario with 0.44% of growing B grains in the initial structure the mobility ratio is varied from 5 to 1000 and the widest distribution curves are summarized in Figure 10. For mobility ratios larger than 50, the distribution curves are very close to each other with a maximum normalized diameter of about 14, i.e., the degree of abnormality is not sensitive to mobility ratios larger than 50. For a mobility ratio of 5 and 10 the bimodality is less severe with a maximum normalized diameter of 7 and 9, respectively. Changing the percentage of growing B grains for a given mobility ratio (here 1000) affects the average distance between these grains and thus the maximum normalized diameter that can be attained decreases from 14 for 0.44% of the initial growing B grains to about 5 when 3.33% of B grains in the initial structure grow. The latter may be taken as threshold for AGG.

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As discussed above for the threshold angle method, one can plot the evolution of the cumulative grain area distribution as a function of normalized diameter to further evaluate the abnormality of grain growth, as presented in Figure9. The red curve is the scaling distribution of normal grain growth as a reference. The initial grain size distribution resulting from Voronoi tessellation is very narrow before a bimodal distribution emerges quickly due to the rapid growth of a few of the B grains. The bimodality of the distribution is represented by the plateau in the cumulative grain area distribution, i.e., there are two populations of grains in the structure consisting of small grains (here the A grains and B grains that do not grow) and large grains (i.e., the growing B grains) with virtually no size overlap. With time the size of the growing B grains and their area fraction rapidly increases. For example, after 30 s the maximum normalized diameter is 5 and the area fraction of the large grains is about 0.1. After 70 s the largest normalized diameter of about 11 is reached and the area fraction of the large grains is increased to about 0.7. Eventually, the growing B grains impinge (see Figure8) and have consumed all A grains. As a result, the overall grain structure approaches that of a normal grain size distribution of the coarse B grains and the maximum normalized diameter starts to decrease at longer times. As a speciﬁc feature of the 2D grain growth simulations, the initially stable hexagonal B grains are incorporated into the B grain microstructure as shown in Figure8d. These smaller B grains remain almost frozen for some time as their shrinkage requires migration of the low-mobility B–B grain boundaries thereby retaining technically a bimodal structure for some time with, however, an increasingly marginal area fraction of the small grains, e.g., after 90 s the area fraction of the small grainsMaterials is 2019 reduced, 12, x FOR to aboutPEER REVIEW 0.1 for the case shown in Figure9. The residual bimodality may, however,10 of 20 be considered as an artefact of 2D simulations.

Figure 9. Time evolution of cumulative grain area distribution in the A–B system (µ2/µ1 = 1000, growing B = 0.67%). For reference, the scaling distribution for 2D ideal grain growth is shown as well. Figure 9. Time evolution of cumulative grain area distribution in the A–B system (휇2⁄휇1 = 1000, growing B = 0.67%). For reference, the scaling distribution for 2D ideal grain growth is shown as well. Even though a clearly bimodal grain size distribution can be obtained with the above mobility scenario, it does not show the development of any irregular grains with complex morphologies that are frequently observed in AGG. The growing B grains show a circular evolution in the 2D simulations, which would translate into spherical growth in 3D, i.e., the grain structure while consisting of two grain types with diﬀerent sizes remains morphologically equiaxed. Even so, similarly to the threshold angle approach we performed a systematic parameter study to explore the inﬂuence of mobility ratio and initial percentage of growing B grains on the extent of the bimodality of the grain structure. Taking the scenario with 0.44% of growing B grains in the initial structure the mobility ratio is varied from 5 to 1000 and the widest distribution curves are summarized in Figure 10. For mobility ratios larger than 50, the distribution curves are very close to each other with a maximum normalized diameter of about 14, i.e., the degree of abnormality is not sensitive to mobility ratios larger than 50. For a mobility ratio of 5 and 10 the bimodality is less severe with a maximum

Figure 10. Effect of mobility ratio on cumulative grain area distribution for widest grain size distributions in the A–B system with 0.44% growing B grains in the initial structure. For reference, the scaling distribution for 2D ideal grain growth is shown as well.

3.3. Mobility Advantage in Three-Grain Type Systems In the two-grain component systems, most (and for sufficiently small fractions all) growing B grains are entirely surrounded by highly mobile A–B grain boundaries and grow rapidly in a regular, equiaxed fashion. In contrast, much more irregular shapes of abnormal grains are observed in many experimental AGG scenarios, including for 2D grain growth. To add complexity into our simulations, a third grain type C is introduced as a result of which growing B grains will have a mixture of fast A–B boundaries and slow B–C boundaries. The grain types are randomly assigned in the initial structure to match a pre-scribed fraction of each component. Figure 11 shows a typical example of the grain structure obtained in the three-grain type simulations with a mobility ratio of 1000. Here, the red grains represent the B grains; and orange and white grains are defined as A and C grains, respectively. Initially all grains have a similar size and 1% B grains are randomly distributed in the A–C grain matrix with equal fractions of A and C grains. Similar to the two-grain type systems, some B grains with a smaller size will shrink and disappear quickly. The second group of B grains with a hexagonal structure will neither shrink nor grow and

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Materials 2019, 12, 4048 10 of 20 normalized diameter of 7 and 9, respectively. Changing the percentage of growing B grains for a given mobility ratio (here 1000) aﬀects the average distance between these grains and thus the maximum normalized diameter that can be attained decreases from 14 for 0.44% of the initial growing B grains Figure 9. Time evolution of cumulative grain area distribution in the A–B system (휇2⁄휇1 = 1000, to aboutgrowing 5 when B = 3.33%0.67%). of For B reference, grains in the scaling initial distribution structure grow. for 2D The ideal latter grain maygrowth be is taken shown as as threshold well. for AGG.

Figure 10. Effect of mobility ratio on cumulative grain area distribution for widest grain size distributions inFigure the A–B 10. system Effect with of mobility 0.44% growing ratio on B cumulative grains in the grain initial area structure. distribution For reference, for widest the gr scalingain size distributiondistributions for in 2D the ideal A–B grain system growth with 0.44% is shown growing as well. B grains in the initial structure. For reference, the scaling distribution for 2D ideal grain growth is shown as well. 3.3. Mobility Advantage in Three-Grain Type Systems 3.3. InMobility the two-grain Advantage component in Three-Grain systems, Type mostSystems (and for sufficiently small fractions all) growing B grainsIn are the entirely two-grain surrounded component by highly system mobiles, most A–B (and grain for boundariessufficiently andsmall grow fractions rapidly all) in growing a regular, B equiaxedgrains are fashion. entirely In surrounded contrast, much by highly more mobile irregular A– shapesB grain of boundaries abnormal and grains grow are rapidly observed in ina regular, many experimentalequiaxed fashion. AGG scenarios,In contrast, including much more for 2Dirregular grain growth.shapes of To abnormal add complexity grains are into observed our simulations, in many aexperimental third grain type AGG C is scenarios, introduced including as a result for of 2D which grain growing growth. BTo grains add complexity will have a into mixture our simulations, of fast A–B boundariesa third grain and type slow C B–C is introduced boundaries. as Thea result grain of types which are growing randomly B assignedgrains will in thehave initial a mixture structure of fast to matchA–B boundaries a pre-scribed and fraction slow of B– eachC boundaries. component. The grain types are randomly assigned in the initial structureFigure to 11 match shows a a pre typical-scri examplebed fraction of the of grain each structurecomponent. obtained in the three-grain type simulations with aFigure mobility 11 ratio shows of 1000. a typical Here, example the red grains of the represent grain structure the B grains; obtained and orange in the and three white-grain grains type aresimulations defined as with A and a mobility C grains, ratio respectively. of 1000. Here Initially, the red all grains grains represent have a similar the B grains; size and and 1% orange B grains and arewhite randomly grains are distributed defined inas theA and A–C C graingrains, matrix respectively. with equal Initially fractions all grains of A andhave Ca grains.similar Similarsize and to1% the B two-graingrains are typerandomly systems, distributed some B in grains the A with–C grain a smaller matrix size with will equal shrink fractions and disappear of A and C quickly. grains. TheSimilar second to groupthe two of-grain B grains type with systems, a hexagonal some B structure grains with will a neither smaller shrink size will nor growshrink and and only disappear those Bquickly. grains withThe second a size advantage group of B over grains their with neighbours a hexagonal will structure grow. A will few neither of these shrink growing nor grow B grains and show clear signs of abnormal grains, i.e., complex grain morphologies including the formation of island grains. For computational efficiency, subsequent simulations were performed in a smaller domain to focus on the behavior of one candidate-growing B grain. The percentage of A grains will influence the fraction of the highly mobile A–B boundaries that will change with time depending on the local environment of the growing B grain. Figure 12 illustrates examples of the obtained grain morphologies of the B grain for different A grain percentages in the initial microstructure and a mobility ratio of 1000. When there are 40% A grains in the matrix, the B grain grows in a regular, equiaxed fashion with a polygonal shape that is more realistic than the circular shapes in the two texture component systems (Figure 12a). Because some of the grain boundaries are fast A–B boundaries the B grain becomes the largest grain in the structure but its size advantage is rather modest such that the overall grain structure appearance is close to that for normal grain growth. Increasing the percentage of A grains Materials 2019, 12, x FOR PEER REVIEW 12 of 20

Materials 2019 12 of A percentage, , 4048 and mobility ratio (here 1000). The evolution of grain B is shown as its normalized11 of 20 grain diameter defined by: leads to a higher fraction of fast A–B boundaries, thereby푑퐵 promoting the growth of the B grain into an 휌 = (2) abnormally large grain. For an A grain percentage푑퐴퐶 of 50%, the B grain evolves into an abnormally large grain with a more complex shape (Figure 12b). Increasing the A grain percentages to 60%, 70% where 푑 is the EQAD of the B grain and 푑 represents the average EQAD of the matrix grains and 80%,퐵 the B grain becomes an even more prominent퐴퐶 abnormal grain that also includes island grains consisting of A and C grains. As expected, there is some variation in detail due to the initial structure that belong to grain type C (Figure 12c–e). These island grains form when the growing B grain locally but the general conclusions are less affected. The percolation threshold of 50% A grains is required reaches a situation where A grains completely surround a C grain. As the B grain can consume these A to consistently reach the 5 grain diameter threshold for AGG. The B growth rate increases with the A grains rapidly an island grain will result within the B grain. These island grains are unstable but will grain percentage and the associated increase of fast A–B boundaries and the maximum size of B remain for some time as they can only be eliminated by migration of the low-mobility B–C boundaries. grains (here about 9) is found for A grain percentages of 70% and higher. It must be noted, however, The probability of the island formation hinges on a combination of a suﬃciently high percentage of A that the size advantages are here limited by the smaller domain size of the simulations. As discussed grainsMaterials with 2019, a12 still, x FOR sizeable PEER REVIEW percentage of C grains. Increasing the A grain percentage to 90%, the B11 grain of 20 for the A–B scenario, the size advantage is also critically dependent on the spacing between approaches a more equiaxed, circular shape but with some local inlet type features due to pinning by onlyabnormally those B growing grains with grains. a size advantage over their neighbours will grow. A few of these growing B some small C grains (Figure 12f). When increasing the A percentage further, the evolution of the B grainsIn show addition clear to signs size ofadvantage, abnormal complex grains, i.e. grain, complex shapes grain are an morphologies important characteristic including the of formation AGG. To grain becomes increasingly similar to the situation of the two-grain type system discussed above in ofquantify island thegrains. grain morphology, the circularity, 휀, defined by Section 3.2. 4휋퐴 휀 = (3) 푃2 may be used where A is the area of the B grain and P its perimeter. In the most pronounced AGG cases, i.e., for 60%, 70% and 80% A, the circularities of the fully developed abnormal B grains fall into the 0.45–0.70 range. For cases with comparable grain size the number of grain neighbours is an alternative way to evaluate the morphological complexity. Taking the examples of Figure 12, the number of B grain neighbours increases from 13 for 40% A to 22 for 50% A, 25 for 60% A and 33 for 70% A but then decreases to 26 and 27 for 80% and 90% A, respectively. The larger grain number of neighbours for 70% A is consistent with the more complex grain morphology in this case that includes an island grain. Interestingly, the majority of the neighbours are small C grains that form low-mobility B–C boundaries such that these grains are obstacles for the progression of the faster moving A–B boundaries leading to the formation of peninsula-type grains or even island grains. All these abnormally large grains have still a fraction of highly mobile boundaries that increases from about 1/3 for 40%, 50% and 60% A to 0.4 for 70% A, 0.6 for 80% A and 0.8 for 90% A. The above simulations in the three-grain type system have been carried out with a mobility ratio of 1000. To identify regions of AGG it is also of interest to consider the role of mobility ratio in addition to A percentage that provides a measure of the fraction of highly mobile A–B boundaries. Figure 11. Typical grain structure in a three-grain type system (initial composition of B% = 1%, TakingFigure the 11.5 grain Typical diameter grain structure size advantage in a three -asgrain a threshold type system a mobility(initial composition ratio of 30 of is B% required = 1%, A% for = 50% A% = C% = 49.5%) with a mobility ratio of 1000. Grains of type A, B and C are shown in orange, A whereasC% = 49.5%) mobility with ratios a mobility of 10 ratioand of5, respectively,1000. Grains of are type sufficient A, B and for C are70% shown and 100%in orange, A, respectively. red and redwhite, and respectively. white, respectively.

For computational efficiency, subsequent simulations were performed in a smaller domain to focus on the behavior of one candidate-growing B grain. The percentage of A grains will influence the fraction of the highly mobile A–B boundaries that will change with time depending on the local environment of the growing B grain. Figure 12 illustrates examples of the obtained grain morphologies of the B grain for different A grain percentages in the initial microstructure and a mobility ratio of 1000. When there are 40% A grains in the matrix, the B grain grows in a regular, equiaxed fashion with a polygonal shape that is more realistic than the circular shapes in the two texture component systems (Figure 12a). Because some of the grain boundaries are fast A–B boundaries the B grain becomes the largest grain in the structure but its size advantage is rather modest such that the overall grain structure appearance is close to that for normal grain growth. Increasing the percentage of A grains leads to a higher fraction of fast A–B boundaries, thereby promoting the growth of the B grain into an abnormally large grain. For an A grain percentage of 50%, the B grain evolves into an abnormally large grain with a more complex shape (Figure 12b). Increasing the A grain percentages to 60%, 70 % and 80%, the B grain becomes an even more prominent abnormal grain that also includesFigure island 12. grainsCont. that belong to grain type C (Figure 12c–e). These island grains form when the growing B grain locally reaches a situation where A grains completely surround a C grain. As the B grain can consume these A grains rapidly an island grain will result within the B grain. These island grains are unstable but will remain for some time as they can only be eliminated by migration of the low-mobility B–C boundaries. The probability of the island formation hinges on a combination of a sufficiently high percentage of A grains with a still sizeable percentage of C grains. Increasing the A grain percentage to 90%, the B grain approaches a more equiaxed, circular shape but with some local inlet type features due to pinning by some small C grains (Figure 12f). When increasing the A percentage further, the evolution of the B grain becomes increasingly similar to the situation of the two-grain type system discussed above in Section 3.2. The details of the evolution of grain B depend also on the initial grain structure. Figure 13 compares two runs from different initial structures with otherwise unchanged parameters in terms

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Figure 12. StructureStructure of of rapidly rapidly growing growing B Bgrain grain (shown (shown in in red) red) in in three three grain grain type type systems systems with with a mobilitya mobility ratio ratio of of1000 1000 when when (a) (Aa)A = 40%;= 40%; (b) A (b =)A 50%;= 50%; (c) A (=c)A 60%;= (60%;d) A (=d 70%;)A = (e70%;) A = (80%,e)A =(f)80%, A = 90%(f)A where= 90% A where grains A are grains in orange are in orangeand C grains and C in grains white. in white.

The details of the evolution of grain B depend also on the initial grain structure. Figure 13 compares two runs from different initial structures with otherwise unchanged parameters in terms of A percentage and mobility ratio (here 1000). The evolution of grain B is shown as its normalized grain diameter defined by: d ρ = B (2) dAC where dB is the EQAD of the B grain and dAC represents the average EQAD of the matrix grains consisting of A and C grains. As expected, there is some variation in detail due to the initial structure but the general conclusions are less affected. The percolation threshold of 50% A grains is required to consistently reach the 5 grain diameter threshold for AGG. The B growth rate increases with the A grain percentage and the associated increase of fast A–B boundaries and the maximum size of B grains (here about 9) is found for A grain percentages of 70% and higher. It must be noted, however, that the size advantages are here limited by the smaller domain size of the simulations. As discussed for the A–B scenario, the size advantage is also critically dependent on the spacing between abnormally growing grains. In addition to size advantage, complex grain shapes are an important characteristic of AGG. To quantifyFigure 13. the Effect grain of morphology,percentage of A the grains circularity, on the sizeε, defined evolution by of grain B for a mobility ratio of 1000. The size of grain B is shown in units of the average grain diameter of A and C grains.

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4πA ε = (3) P2 may be usedFigure where 12. StructureA is the of area rapidly of the growing B grain B grain and (shownP its perimeter. in red) in three In the grain most type pronounced systems with AGGa cases, i.e., for 60%,mobility 70% ratio and of 80% 1000 A,when the (a circularities) A = 40%; (b) A of = the50%; fully (c) A developed= 60%; (d) A = abnormal 70%; (e) A = B 80%, grains (f) A fall = into the 0.45–0.7090% range. where A grains are in orange and C grains in white.

Figure 13. Eﬀect of percentage of A grains on the size evolution of grain B for a mobility ratio of 1000. Figure 13. Effect of percentage of A grains on the size evolution of grain B for a mobility ratio of 1000. The size of grain B is shown in units of the average grain diameter of A and C grains. The size of grain B is shown in units of the average grain diameter of A and C grains. For cases with comparable grain size the number of grain neighbours is an alternative way to evaluate the morphological complexity. Taking the examples of Figure 12, the number of B grain neighbours increases from 13 for 40% A to 22 for 50% A, 25 for 60% A and 33 for 70% A but then decreases to 26 and 27 for 80% and 90% A, respectively. The larger grain number of neighbours for 70% A is consistent with the more complex grain morphology in this case that includes an island grain. Interestingly, the majority of the neighbours are small C grains that form low-mobility B–C boundaries such that these grains are obstacles for the progression of the faster moving A–B boundaries leading to the formation of peninsula-type grains or even island grains. All these abnormally large grains have still a fraction of highly mobile boundaries that increases from about 1/3 for 40%, 50% and 60% A to 0.4 for 70% A, 0.6 for 80% A and 0.8 for 90% A. The above simulations in the three-grain type system have been carried out with a mobility ratio of 1000. To identify regions of AGG it is also of interest to consider the role of mobility ratio in addition to A percentage that provides a measure of the fraction of highly mobile A–B boundaries. Taking the 5 grain diameter size advantage as a threshold a mobility ratio of 30 is required for 50% A whereas mobility ratios of 10 and 5, respectively, are suﬃcient for 70% and 100% A, respectively.

3.4. Comparison with Monte Carlo Simulation PFM and MC simulations are effective and versatile methods to study AGG. However, whether the simulation methods will influence the simulation results is of great interest. Figure 14 compares the results of PFM and MC simulations. The MC simulation results are taken from DeCost and Holm [2]. Both simulations start from the same initial structure with 68% A grains, as shown in Figure 14a,d, and the mobility ratio is set to be 1000. In PFM, the B grain is in red while A and C grains are shown in white and orange, respectively. In the MC method [2], the white grain represents the B grain; red and blue grains are defined as A and C grains, respectively. Comparing Figure 14b,e, a small island grain is found in Figure 14e [2] while at the same position, this small C grain has already been consumed in the PFM simulation. After running the simulation for a longer time, some island grains also start to form in the PFM simulation, as shown in Figure 14c. However, in the MC model, there are many more island grains found as presented in Figure 14f. [2] The difference may be caused by the straightening Materials 2019, 12, x FOR PEER REVIEW 14 of 20

3.4. Comparison with Monte Carlo Simulation PFM and MC simulations are effective and versatile methods to study AGG. However, whether the simulation methods will influence the simulation results is of great interest. Figure 14 compares the results of PFM and MC simulations. The MC simulation results are taken from DeCost and Holm [2]. Both simulations start from the same initial structure with 68% A grains, as shown in Figure 14a,d, and the mobility ratio is set to be 1000. In PFM, the B grain is in red while A and C grains are shown in white and orange, respectively. In the MC method [2], the white grain represents the B grain; red and blue grains are defined as A and C grains, respectively. Comparing Figure 14b,e, a small island grain is found in Figure 14e [2] while at the same position, this small C grain has already been consumed in the PFM simulation. After running the simulation for a longer time, some island grains also start to form in the PFM simulation, as shown in Figure 14c. However, in the MC model, there are many more island grains found as presented in Figure 14f. [2] The difference may be caused by the straightening effect of grain boundaries in PFM. In PFM, in order to reduce the total free energy, the grain boundaries tend to be straight lines thus losing the driving pressure due to curvature. Meanwhile, in the MC simulation, there is no such straightening effect. Nevertheless, apart from these minor differences, the overall coarsening mode of the abnormal grain is in both simulations very close to each other. Since the time scales in PFM and MC simulations are different, it is of interest to match the two time scales and develop a more quantitative way to analyze the consistency of the two computational methods. In PFM, the simulation time is the real phase field time (PFT), and the unit is second. In the MC simulation, the time scale is measured by the Monte Carlo step (MCS). To match the two time Materialsscales,2019 a constant, 12, 4048 parameter 휅 is introduced, where: 14 of 20 푡(푀퐶푆) = 휅 ∙ 푡(푃퐹푇) (4) effect ofHere grain 휅 boundaries= 5.1 is obtained in PFM. to Inbest PFM, match in orderthe two to simulation reduce the results total free in energy,terms of the the grain size evolution boundaries tendof the to beabnormal straight grain. lines The thus increase losing of the the driving candidate pressure grain area due is to plotted curvature. in Figure Meanwhile, 15 as a function in the of MC simulation,simulation there time. is From no such this straighteninggraph, despite eofffect. little Nevertheless, fluctuations, the apart overall from grain these growth minor paths differences, are thealmost overall the coarsening same in both mode simulations. of the abnormal grain is in both simulations very close to each other.

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FigureFigure 14. 14.Comparison Comparison between between phasephase fieldfield modelling ( (PFM,PFM, left) left) and and Monte Monte Carlo Carlo (MC (MC,, right) right) [2] [2] simulationssimulations with with the the samesame initial microstructure microstructure:: (a) ( ainitial) initial structure structure for PFM, for PFM, (b) initial (b) initial structure structure for forMC, MC, (c ()c intermediate) intermediate structure structure for PFM, for PFM, (d) intermediate (d) intermediate structure structure for MC, for (e MC,) final ( estructure) final structure for PFM, for PFM,(f) final (f) final structure structure for MC. for MC.The B The grain B grain is shown is shown in red in for red PFM for and PFM in and white in for white MC for simulations. MC simulations. The TheA Agrains grains are are in white in white (PFM) (PFM) and anddark darkred (MC) red (MC)and the and C grains the C grainsare in orange are in (PFM) orange and (PFM) grey and(MC), grey (MC),respectively. respectively.

Figure 15. Size evolution of the abnormal grain obtained in PFM and MC simulations in units of the initial grain area of the candidate grain.

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Since the time scales in PFM and MC simulations are different, it is of interest to match the two time scales and develop a more quantitative way to analyze the consistency of the two computational methods. In PFM, the simulation time is the real phase field time (PFT), and the unit is second. In the MC simulation, the time scale is measured by the Monte Carlo step (MCS). To match the two time scales, a constant parameter κ is introduced, where: Figure 14. Comparison between phase field modelling (PFM, left) and Monte Carlo (MC, right) [2] simulations with the same initial microstructuret(MCS: )(a =) initialκ t(PFT structure) for PFM, (b) initial structure for (4) MC, (c) intermediate structure for PFM, (d) intermediate· structure for MC, (e) final structure for PFM, (f)Here final structureκ = 5.1 is for obtained MC. The toB grain best matchis shown the in twored for simulation PFM and resultsin white infor terms MC simulations. of the size The evolution of theA grains abnormal are in grain.white (PFM) The increaseand dark ofred the (MC) candidate and the C grain grains area are isin plottedorange (PFM) in Figure and grey 15 as (MC), a function of simulationrespectively. time. From this graph, despite of little fluctuations, the overall grain growth paths are almost the same in both simulations.

Figure 15. Size evolution of the abnormal grain obtained in PFM and MC simulations in units of the Figureinitial 15. grain Size area evolution of the candidateof the abnormal grain. grain obtained in PFM and MC simulations in units of the 4. Discussioninitial grain area of the candidate grain.

Even though there is remarkable agreement between PFM and MC simulations in the above benchmark case, there are in detail some apparently diﬀerent conclusions that may be drawn from the present PFM simulations as compared to the MC study of DeCost and Holm [2]. For example, they present also a case of initiation of abnormal growth for a situation with 30% A grains (red grains in their paper). Figure 16 shows the local starting environment for this grain indicating that there is a clear percolation path due to the arrangement of A grains such that the white grain can grow into an abnormal grain. Furthermore, when considering growth of an initially large circular candidate grain (see e.g., Figure 14b) they performed 120 independent simulations for each A (red) grain fraction scenario. For 50% A grains, the candidate grain grows in all cases abnormally, for 40% A grains in the majority of the cases abnormal growth occurs whereas for 30% A abnormal growth is recorded in about half of the cases. This provides further evidence that the local A grain environment is critical rather than the global A grain fraction. Furthermore, the initial size advantage promotes the chances for abnormal growth due to the larger numbers of grain neighbours including those providing high-mobility boundaries. In the present PFM study the focus has been, in addition to the fraction of highly mobile boundaries, to also assess the role of the mobility ratio on the initiation of abnormal growth. Starting from an initial grain structure obtained by Voronoi tessellation the eventually abnormally growing grains had initially only a marginal size advantage (usually with a normalized grain diameter of 1.5 or less). Thus, the growth potential of these grains is limited compared to the cases considered by DeCost and Holm [2] such that the percolation limit is more restrictively enforced. Furthermore, it is not only the local grain environment Materials 2019, 12, x FOR PEER REVIEW 16 of 20

4. Discussion

Even though there is remarkable agreement between PFM and MC simulations in the above benchmark case, there are in detail some apparently different conclusions that may be drawn from the present PFM simulations as compared to the MC study of DeCost and Holm [2]. For example, they present also a case of initiation of abnormal growth for a situation with 30% A grains (red grains in their paper). Figure 16 shows the local starting environment for this grain indicating that there is a clear percolation path due to the arrangement of A grains such that the white grain can grow into an abnormal grain. Furthermore, when considering growth of an initially large circular candidate grain (see e.g., Figure 14b) they performed 120 independent simulations for each A (red) grain fraction scenario. For 50% A grains, the candidate grain grows in all cases abnormally, for 40% A Materialsgrains in 2019 the, 12 majority, x FOR PEER of REVIEWthe cases abnormal growth occurs whereas for 30% A abnormal growth16 of 20is recorded in about half of the cases. This provides further evidence that the local A grain environment 4.is criticalDiscussion rather than the global A grain fraction. Furthermore, the initial size advantage promotes the chances for abnormal growth due to the larger numbers of grain neighbours including those Even though there is remarkable agreement between PFM and MC simulations in the above providing high-mobility boundaries. benchmark case, there are in detail some apparently different conclusions that may be drawn from the present PFM simulations as compared to the MC study of DeCost and Holm [2]. For example, they present also a case of initiation of abnormal growth for a situation with 30% A grains (red grains in their paper). Figure 16 shows the local starting environment for this grain indicating that there is a clear percolation path due to the arrangement of A grains such that the white grain can grow into Materialsan abnormal2019, 12 grain., 4048 Furthermore, when considering growth of an initially large circular candidate16 of 20 grain (see e.g., Figure 14b) they performed 120 independent simulations for each A (red) grain fraction scenario. For 50% A grains, the candidate grain grows in all cases abnormally, for 40% A but also the mobility ratio that determines whether the candidate grain will grow abnormally or even grains in the majority of the cases abnormal growth occurs whereas for 30% A abnormal growth is shrink, as illustrated in Figure 17. Here, a scenario is shown where the candidate grain shrinks for recorded in about half of the cases. This provides further evidence that the local A grain environment a mobilityFigure ratio 16. Initial of 5 and structure will eventually for initiation disappear of abnormal as itsgrowth A grain of aneighbours white candidate have grain sufficiently with a fraction grown (see is critical rather than the global A grain fraction. Furthermore, the initial size advantage promotes the Figureof 170.3b) for whereas A (red) grains for a mobilitythat provide ratio the of high 10ly the mobile candidate boundaries grain with develops the white into grain; an abnormally bottom half large chances for abnormal growth due to the larger numbers of grain neighbours including those grainof (see Figure Figure 8a in 17 ref.c). [2]. providing high-mobility boundaries. In the present PFM study the focus has been, in addition to the fraction of highly mobile boundaries, to also assess the role of the mobility ratio on the initiation of abnormal growth. Starting from an initial grain structure obtained by Voronoi tessellation the eventually abnormally growing grains had initially only a marginal size advantage (usually with a normalized grain diameter of 1.5 or less). Thus, the growth potential of these grains is limited compared to the cases considered by DeCost and Holm [2] such that the percolation limit is more restrictively enforced. Furthermore, it is not only the local grain environment but also the mobility ratio that determines whether the candidate grain will grow abnormally or even shrink, as illustrated in Figure 17. Here, a scenario is shownFigure where 16. theInitial candidate structure grain for initiation shrinks of abnormalfor a mobility growth ratio of a whiteof 5 and candidate will eventually grain with adisappear fraction as Figure 16. Initial structure for initiation of abnormal growth of a white candidate grain with a fraction its A ofgrain 0.3 for neighbours A (red) grains have that sufficiently provide the grown highly mobile(see Figure boundaries 17b) whereas with the white for a grain;mobility bottom ratio half of of10 the of 0.3 for A (red) grains that provide the highly mobile boundaries with the white grain; bottom half candidFigureate grain8a in ref. develops [2]. into an abnormally large grain (see Figure 17c). of Figure 8a in ref. [2].

In the present PFM study the focus has been, in addition to the fraction of highly mobile boundaries, to also assess the role of the mobility ratio on the initiation of abnormal growth. Starting from an initial grain structure obtained by Voronoi tessellation the eventually abnormally growing grains had initially only a marginal size advantage (usually with a normalized grain diameter of 1.5 or less). Thus, the growth potential of these grains is limited compared to the cases considered by DeCost and Holm [2] such that the percolation limit is more restrictively enforced. Furthermore, it is not only the local grain environment but also the mobility ratio that determines whether the candidate grain will grow abnormally or even shrink, as illustrated in Figure 17. Here, a scenario is shown where the candidate grain shrinks for a mobility ratio of 5 and will eventually disappear as its A grain neighbours have sufficiently grown (see Figure 17b) whereas for a mobility ratio of 10 the Figure 17. Effect of mobility ratio on growth behavior of candidate B grain (shown in red) for 50% A candidate grain develops into an abnormally large grain (see Figure 17c). grains (shown in orange): (a) initial grain structure, (b) grain structure after 10 s for a mobility ratio of 5, (c) grain structure after 30 s for a mobility ratio of 10.

5. Conclusions The onset conditions for abnormal grain growth (AGG) have been quantified for the first time with systematic phase field simulations. For AGG to be initiated, selected individual grains must have some grain boundaries with a sufficient mobility advantage compared to other boundaries. Two-dimensional MPFM simulations have been conducted using different ways to introduce highly mobile grain boundaries, i.e., a critical disorientation angle above which the boundaries are highly mobile as well as systems with two (A–B) and three (A–B–C) grain types where the A–B boundaries are assumed to be highly mobile. Systematic parametric studies havebeen performed for these systems by changing the fraction and distribution of the highly mobile boundaries and the mobility ratio between high and low-mobility boundaries to identify conditions for AGG. The high-mobility boundaries are randomly distributed in the disorientation angle approach and modest AGG scenarios with maximum grain sizes of 5–6 times larger than the mean grain size are obtained for a narrow range of threshold angles where about 40%–60% of the boundaries are highly mobile with mobility ratios of at least 100. AGG with a well-developed bimodal grain structure can readily occur for mobility ratios as low as 5 in the two-grain type system when the fraction of growing B grains with high-mobility boundaries is sufficiently small. The maximum grain size depends primarily on the distance between the growing grains. In the present simulations, a maximum grain size of 14 times larger than the apparent mean grain size has been obtained when 0.4% of the initial grains can grow abnormally. The two-grain type simulations lead, however, to rather unrealistic circular shapes of the abnormally large grains. Materials 2019, 12, 4048 17 of 20

This aspect can be mitigated when introducing a third grain type in the system where the growing grains are not completely surrounded by highly mobile boundaries. Realistic shapes of abnormally growing grains can be obtained including the formation of island grains for a range of conditions with a sufficiently high fraction of highly mobile boundaries while maintaining a critical amount of the third grain type. Furthermore, the PFM simulations leading to truly abnormal grains are quantitatively similar to MC simulation results by DeCost and Holm [2], as verified with a benchmark simulation. To establish in more detail AGG conditions, it may be useful to apply machine-learning techniques to the analysis of simulation data bases for the three-grain type systems. So far, all the simulations have been conducted in 2D and provide an important insight into 2D abnormal grain growth observed e.g., in thin films [25–28]. The simulation results also provide guidance for future 3D simulations to specify in more detail AGG conditions for bulk materials. Furthermore, the above studies apply to pure systems but can be further extended to include the role of precipitate pinning as well as solute drag, which will add complexity to the analysis.

Author Contributions: The conceptualization of this work was provided by M.M. and M.P. The methodology was proposed by M.M. and Y.L. conducted all simulations including data analysis, validation and visualization. The original draft of the paper was prepared by Y.L. The manuscript was reviewed and edited by M.M. and M.P. The work was supervised by M.M. who administered the project and facilitated the funding acquisition. Funding: The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants program (Grant Number: RGPIN-2015-04259) is greatly acknowledged. Acknowledgments: We thank Brian DeCost and Elizabeth Holm for providing the initial structures of the MC simulations and many helpful discussions. We also express our gratitude to Ali Khajezade for his help in data analysis and finalizing editorial details. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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