Distribution of Topological Types in Grain-Growth Microstructures
Emanuel A. Lazar, Jeremy K. Mason, Robert D. MacPherson, David J. Srolovitz
DDistribution of Topological Types in Grain-Growth Microstructures
Emanuel A. Lazar , Jeremy K. Mason , Robert D. MacPherson , David J. Srolovitz Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel Department of Materials Science and Engineering,University of California, Davis, California 95616, USA School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA Department of Materials Science and Engineering,City University of Hong Kong, Hong Kong SAR China (Dated: July 7, 2020)An open question in studying normal grain growth concerns the asymptotic state to which mi-crostructures converge. In particular, the distribution of grain topologies is unknown. We introducea thermodynamic-like theory to explain these distributions in two- and three-dimensional systems.In particular, a bending-like energy E i is associated to each grain topology t i , and the probabilityof observing that particular topology is proportional to s ( t i ) e − βE i , where s ( t i ) is the order of anassociated symmetry group and β is a thermodynamic-like constant. We explain the physical originsof this approach, and provide numerical evidence in support. Introduction.
Theory, simulation, and experimentalwork have shown that during normal grain growth, poly-crystalline microstructures evolve toward an asymptoticstate in which scale-invariant properties become constant[1, 2]. It has also been observed that this state is reachedlargely independently of initial conditions [3, 4]. A majorgoal in this field has therefore been to characterize andunderstand this universal grain-growth microstructure.In addition to its geometric features [5–7], its topologicalfeatures have also been carefully studied. In two dimen-sions, grains can be classified by their number of edges[8–10]. An analogous approach is insufficient in three di-mensions, as grains with the same number of faces canhave distinct topologies. Recent work has focused oncharacterizing the types of grain faces [11], the arrange-ments of those faces [12–16], and the manner in whichedges are arranged in the grain boundary network [17].Previous studies characterizing the distribution ofgrain topologies is limited in two important ways. First,little connection has been made between two- and three-dimensional systems; a general theory explaining bothis desirable. Second, despite careful characterization ofgrain types that appear and their relative frequencies, anexplanation of these observations remains elusive.This letter introduces a novel, thermodynamic-like ap-proach to explain the observed distributions of topolog-ical types in two- and three-dimensional grain-growthsystems. In particular, we associate a bending-like en-ergy to each grain that depends only on its topology, andshow that this energy can help predict the distributionof topologies in these cellular microstructures.
Theory.
The most basic topological property of a grainis its number of neighbors. In this letter, we use the term neighbors to refer to pairs of grains that share a commonedge or face, in two or three dimensions, respectively.In two dimensions, the topology of a grain is fully de-scribed by its number of edges, which in most cases isequal to its number of neighbors (in exceptional cases,a pair of neighboring grains can share multiple edges).The arrangement of neighbors in three dimensions, how- ever, is more complicated. Consider, for example, Fig. 1,which illustrates two grains, each with eight faces. Al-though the grains have identical numbers and types offaces, differences in the arrangements of those faces indi-cate differences in the arrangements of their neighbors.We say that two grains have the same topological type ,or topology, if their neighbors can be paired so that neigh-bors of one grain are themselves neighbors if correspond-ing neighbors of the other grain are also neighbors. Intwo dimensions, each topological type is identified with anatural number. In three dimensions, each type is iden-tified with a graph isomorphism class [12, 18].
Thermodynamics.
A central goal of statistical thermo-dynamics is understanding the distribution of microstatesof a system when only macrostate features are known. Asystem of N identical particles confined to a fixed vol-ume in thermal equilibrium with a surrounding fixed-temperature heat bath is a classic example, the canoni-cal ensemble [19]. What are its possible microstates andwhat are the probabilities of observing them? In thisexample, the probability density p ( ω ) of observing thesystem in microstate ω depends only on its energy E ω and a constant β , commonly understood as an inversetemperature: p ( ω ) = 1 Z e − βE ω . (1)The partition function Z = Z ( β ) is a normalizing con-stant which ensures that p is a probability distributionon Ω, the set of all possible microstates. The exponentialdependence of probability on energy results from treat-ing the system and its significantly larger surroundings FIG. 1. Two grains with eight faces but different topologies. a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l as an isolated system with fixed energy, and in which allpossible microstates are equally probable [20].These concepts may not initially appear relevant tograin growth for several reasons. First, unlike in clas-sical statistical mechanics, the energy defined below isnot a conserved quantity—the total energy of an isolatedsystem changes with time. Second, whereas β is tradi-tionally interpreted as an inverse temperature, temper-ature has no obvious physical interpretation in studyingthe scale-invariant statistical properties of steady-stategrain growth microstructures. We nevertheless suggestthat grain growth be considered in this thermodynamicspirit. In particular, we treat each grain as a separatethermodynamic system whose microstate is described byan energy written solely in terms of its topology. Theprobability of a grain having a given topology is thenpostulated to depend on this energy in a form similar toEq. (1). We ask that lack of a priori justification for thisapproach be momentarily ignored in light of its successin describing the relevant probability distributions. Two dimensions.
Although grains in two-dimensionalsystems are not regular polygons, we consider them so asa first-order approximation. Adjacent edges of a regular n -sided polygon meet at internal angles of α n = π − π/n .Energetic factors in isotropic grain growth, however,cause edges to meet at angles of θ = 2 π/
3. We thereforedefine an energy associated with each vertex of an n -sidedface as the square of the difference between α n and θ ,in a manner analogous to a conventional elastic energy.The total energy associated with an n -sided grain is thesum of these energies over its n vertices: E ( n ) = n ( α n − θ ) . (2)Since each angle of a regular hexagon is α = θ , theenergy associated with the n = 6 topology is zero.Although microstate energies largely determine theirprobabilities, the manner in which microstates arecounted must also be considered. In particular, if neigh-bors of a grain are cyclically permuted, or else their or-der is reversed, then its topology is unchanged, as pairsof grains are neighbors after this transformation only ifthey were neighbors before it. This identification leadsto a corrective factor of 1 /s ( t i ), where s ( t i ) is the orderof the symmetry group of grain topology t i . This factoris analogous to the more familiar 1 /N ! factor that arisesin systems of N indistinguishable particles, described byEq. (1), which are invariant under the N ! permutationsbelonging to the symmetric group of degree N .In two dimensions, the symmetry group of each regular n -gon is the dihedral group with order 2 n , suggesting thefollowing probability distribution of n -sided grains: p ( n ) = 1 Z e − βE ( n ) n , (3)for some constant β . As mentioned before, we are notaware of any physical interpretation of β , and regard itas a fitting parameter. . . . . Sides ( n ) p ( n ) Grains in 2DFaces in 3D
FIG. 2. Distribution of n -sided grains and faces in two- andthree-dimensional steady-state grain growth [22], respectively,compared with Eqs. (3) and (5), with β = 1 .
62 and β = 1 . Figure 2 compares the distribution of grain topologiesin steady-state, two-dimensional normal grain-growthmicrostructures as described by Eq. (3) with data ob-tained from prior front-tracking simulations [21, 22]. Aweighted least-squares method finds that the data fit theproposed theory best when β = 1 .
62 ( χ = 0 . p ( n ) do not agreeexactly, their similarity in shape suggests that the pro-posed thermodynamic approach might provide a valuablefirst-order approximation of the distribution. Three dimensions.
Our earlier simulations of three-dimensional grain growth suggested that certain topolo-gies appear more frequently than others, even amongthose with the same number and types of faces [12]. Weobserved that “just as curvature flow drives towards ge-ometrically symmetric spheres . . . it also drives towardstopologically symmetric polyhedra.” We now extend theapproach introduced above to analyze three-dimensionalsystems and to quantify this topological symmetry.
Distribution of faces.
We first consider grain facesin three dimensions. Whereas edges in isotropic, two-dimensional grain growth meet at angles 2 π/
3, in threedimensions they meet at angles θ = cos − ( − / ≈ . ◦ . This suggests defining a bending-like energy as-sociated with an n -sided face in three dimensions: E ( n ) = n ( α n − θ ) , (4)analogous to the energy defined in Eq. (2); as before, α n = π − π/n . This energy can be used to estimate thedistribution of faces with n sides in three dimensions: p ( n ) = 1 Z e − βE ( n ) n . (5)Figure 2 shows steady-state data collected fromisotropic grain growth simulations with over 250,000grains [22, 23], compared with Eq. (5). A weighted − − − − − E f ( t i ) s ( t i ) p ( t i ) (a) − − − − − E v ( t i ) s ( t i ) p ( t i ) (b) FIG. 3. The product of the observed probability p ( t i ) and the symmetry group order s ( t i ) as a function of energies (a) E f and(b) E v for each observed topology t i , as suggested by Eq. (7). Data are taken from three-dimensional front-tracking simulationsof steady-state grain growth [22, 23]. least-squares method finds this equation describes theobserved data best when β = 1 .
29 ( χ = 0 . E (6) = 0, in three dimensions, E ( n ) > n , and is minimal when n = 5. Distribution of grain topologies.
Topologically-definedenergies can also be used to estimate the distributionof topological types in three dimensions. We define twosuch energies for each grain topology t i . The first is asum of Eq. (4) over all F faces of a grain: E f ( t i ) = F (cid:88) j =1 E ( n j ) , (6)where n j is the number of sides of face j . This energy ex-tends the one defined for polygonal faces to entire grains.The probability of a grain with topology t i can then beestimated by p ( t i ) = 1 Z e − βE ( t i ) s ( t i ) , (7)where E ( t i ) = E f ( t i ), and where s ( t i ) is the order ofthe associated symmetry group; more details about thissymmetry group and the algorithm used to calculate itsorder can be found in Ref. [24]. The product s ( t i ) p ( t i )is generally reported in the following to emphasize itsexponential dependence on energy.Figure 3(a) shows the product s ( t i ) p ( t i ) as a functionof E f for topologies observed in simulations. Those withlarge E f appear infrequently, while those with small E f may appear frequently or infrequently. These data sug-gest that Eq. (7) reasonably approximates the distribu-tion of grain topologies.Although E f quantifies the energetic favorability ofeach grain topology, it depends only on the types of facesof a grain, but not on how those faces are arranged. Suchinformation, however, might yield a more accurate esti-mate of the distribution of topologies. For example, the two topologies illustrated in Fig. 1 have the same numberand types of faces, and hence E f values, yet the topol-ogy illustrated in Fig. 1(a) appears nearly 100 times morefrequently than that illustrated in Fig. 1(b).We therefore define a second energy to quantify howcurvature is distributed over grain vertices. If threeregular n -sided polygons meet at a vertex v , then theGaussian curvature concentrated at that vertex is K v =2 π − ( α n + α n + α n ), where n j is the number of sidesof face j . If we approximate each face as a regular poly-gon, then K v approximates the actual curvature in purelytopological terms. In isotropic grain growth, however,the Gaussian curvature at each vertex is ˆ K = 2 π − θ ,where θ = cos − ( − / E v ( t i ) of topology t i as a sum of these energies over its V vertices: E v ( t i ) = V (cid:88) j =1 ( K v j − ˆ K ) (8)Two grains with the same number and types of faces willgenerally have identical E f but different E v .Figure 3(b) shows s ( t i ) p ( t i ) as a function of E v foreach observed grain topology. Grains with large E v ap-pear infrequently, while those with low E v can appearfrequently or infrequently. In contrast to Fig. 3(a), thepredicted probabilities are more scattered.We next consider the relationship between E f , E v , and s ( t i ) p ( t i ) when restricted to grains with fixed numbers offaces. For each fixed number of faces, we use a weightedleast-squares method to fit data to a curve of the form s ( t i ) p ( t i ) = Z e − βE ( t i ) . Figure 4 shows data for typeswith 12, 13, and 14 faces. These data suggest that E f , E v , and s ( t i ) can be used to more accurately estimatethe distribution of types when restricted to fixed numberof faces. The only notable outlier appears in Fig. 4(f)for a point with E v ≈
0, which appears less frequentlythan predicted. This point represents the truncated oc- − − − − (a) E f ( t i ) s ( t i ) p ( t i )
12 faces . e − . E f (b) E f ( t i )
13 faces . e − . E f (c) E f ( t i )
14 faces . e − . E f − − − − (d) E v ( t i ) s ( t i ) p ( t i )
12 faces . e − . E v (e) E v ( t i )
13 faces . e − . E v (f) E v ( t i )
14 faces . e − . E v FIG. 4. The product of the observed probability p ( t i ) and the symmetry group order s ( t i ) as a function of (a-c) E f and of(d-f) E v for each observed topology t i with fixed numbers of faces. Probabilities p ( t i ) are normalized so that they sum to 1 foreach number of faces. Dashed curves show five standard deviations of the sample mean for the relevant sample size. Data aretaken from simulations of three-dimensional steady-state grain growth [22, 23]. tahedron, which appears only once in the grain-growthsimulation dataset.Finally, we consider sets of grain topologies with iden-tical numbers and types of faces, but in which those facesare arranged differently, thus providing multiple E v val-ues for fixed E f . Figure 5 shows three such datasets,chosen because of their high number of samples of mul-tiple topological types. In each set, increasing values of E v are clearly associated with an exponential decreasein s ( t i ) p ( t i ), suggesting that E v and E f together providea more accurate prediction of probability than does E f alone. Specifically, grain topologies in which faces meetin unfavorable ways, as characterized by E v , appear or-ders of magnitude less frequently than other topologiesconstructed from identical sets of polygonal faces. Conclusions.
The most surprising finding of this workis the ability of a topologically-defined “energy” to pre-dict the distribution of grain topologies in steady-state,isotropic grain growth. The similarity between the formsof the energies and distributions in two and three dimen-sions suggests a common factor governing their behavior.These energies can be understood as measuring the de-viation of realistic grains and their geometries from idealones in topological terms.The relationship between topologically-defined energy,symmetry, and probability is reminiscent of the classicalstatistical mechanics approach toward analyzing equilib-rium systems. Although grain-growth microstructuresare not equilibrium systems, their steady-state proper-ties provide a similar setting for this kind of analysis [25]. In particular, the existence of an asymptotic statein which scale-invariant properties are statistically con-stant implies that once dimensional factors are scaled out,microstructure is determined by an energy minimizationprinciple. This is not unusual in systems for which thereare large disparities in time scales of different processes;here, the overall coarsening of the microstructure canbe considered as “slow” while the topological or scale-free microstructural evolution is “fast”. Hence, late-timeevolution of grain growth can be described using a mi-crostructural Born-Oppenheimer approximation.While the energies suggested here can be thought ofas approximating bending energies, other topologically-defined energies might also be considered. For example, atwisting energy can be defined along grain edges to quan-tify the strain resulting from differences in face arrange-ments at alternate ends. Further, while the context of thecurrent study is grain growth in polycrystalline metals,the suggested approach may find application in under-standing data collected in studies of polyhedra-shapedcells in other systems, such as bubbles in soap foams [26]and polyhedrocytes in blood clots and thrombi [27].Finally, the approach introduced in this letter might becompared with that recently proposed by Lutz et al. [14].In both, an energy is defined in purely topological termsto capture the favorability of each topological type, andis then used to estimate its probability. One strength ofthe approach suggested here is its connection to classi-cal statistical mechanics and its exponential relationshipbetween energy, symmetry, and probability. − − − E f = 5 .
12 faces : 1 triangle,3 squares, 4 pentagons,3 hexagons, 1 heptagon E v ( t i ) s ( t i ) p ( t i ) . e − . E v E f = 1 .
14 faces : 2 squares,8 pentagons, 4 hexagons E v ( t i )1 . e − . E v E f = 2 .
15 faces : 2 squares,9 pentagons, 3 hexagons,1 heptagon E v ( t i )1 . e − . E v FIG. 5. Three examples for which a fixed set of faces provides a distribution of topological types. Probabilities are relative toother samples in the limited dataset; error bars show standard errors of the mean.
Acknowledgments.
E.A.L. and D.J.S. acknowledge thegenerous support of the U.S. National Science Founda-tion, through Award DMR-1507013. The research con-tribution of D. J. S. was also sponsored, in part, by theArmy Research Office and was accomplished under GrantNumber W911NF-19-1-0263. The views and conclusionscontained in this document are those of the authors and should not be interpreted as representing the official poli-cies, either expressed or implied, of the Army ResearchOffice or the U.S. Government. The U.S. Government isauthorized to reproduce and distribute reprints for Gov-ernment purposes notwithstanding any copyright nota-tion herein. [1] M. Hillert, “On the theory of normal and abnormal graingrowth,”
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