Growth paths in polycrystalline thin films

The polycrystalline grain microstructure of metallic thin films coarsens during grain growth in a unique way when the initial grain structure contains multiple grains in the film thickness. A regime with fast coarsening is followed by a regime of slow coarsening. At the same time, the grain structure itself undergoes clear structural changes from a bulk-like to a bamboo-like structure. The overall coarsening process evolves continuously, whereas the growth paths of individual grains do not follow the ones observed and predicted in either two- or three-dimensional grain growth.


Introduction
Microstructural coarsening, which occurs during grain growth, is of the utmost importance for polycrystalline materials.Any change in the grain microstructure leads to changes in material properties due to the fact that, e.g.mechanical properties such as strength or toughness depend in addition to the chemical composition of the material also directly on the grain size.Hence, whenever the grain size changes during grain growth the associated properties change.Grain growth has thus been a crucial area of research in materials science for more than 70 yr.One of the earliest physically motivated grain growth hypotheses was developed by Smith [1] as early as the early 1950s.The essential equations characterizing the growth rate of grains as a function of their number of surrounding grains and a self-similar scaled grain size distribution, respectively, were derived by pioneers like Mullins and Hillert [2,3].Of course, these theories included already the knowledge that grains of different sizes within a specimen show different Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.behaviors.In average, small grains have fewer neighbors than large grains and show a tendency to shrink, whereas large grains-larger than average-grow.This makes grain growth a complex interplay of local grain boundary migration, local equilibrium at triple junctions, and the global demand for space filling.
Over the years, new experimental approaches, computer simulation tools, conceptual frameworks, and analytical models have been developed.In particular, the growing significance and application of computer modeling to processes such as recrystallization and grain growth has been one significant development during the past few decades.Many different methods have been created and applied to a wide variety of problems [4][5][6][7][8][9][10].One may even say that many of present-day issues, particularly those relating to normal grain growth, are debated nowadays in light of the outcomes of computer simulations.For instance, mesoscopic computer simulations allow statistical evaluations of huge grain ensembles over prolonged annealing times, and these results are frequently compared to new analytical models or even fuel the development of new analytic theories.
The wide spread use of computer-assisted three-dimensional reconstruction techniques to characterize polycrystalline microstructures in terms of grain size distributions, for example, has undoubtedly been a just as significant experimental achievement [11,12].Diffraction methods based on synchrotron sources are often employed as new opportunities for studying the three-dimensional grain structures and crystallographic orientations of polycrystalline materials without causing damage [13][14][15].Hence, traditional techniques for getting 3D grain size distributions from planar sections are used less and less.Nevertheless, in many projects aimed at investigating grain growth in thin films or foils two-dimensional considerations are used still today [16,17].For example, Hallberg and Olsson [16] investigated grain growth in copper films by using two-dimensional mesoscale level set and ab-initio modeling.They focused on the influence of grain boundary energy and mobility and found, e.g. that abnormal grain growth depends mainly on variations of stored energy.Also using two-dimensional modeling, Barmak et al [17] investigated grain growth in comparison with experiments performed on Al and Cu films.The experimental data on grain size distributions and topology were not mirrored by the 2D simulations.Then again, using a phase field crystal model reproduces some grain metrics such as grain sizes rather well [18,19].In particular, La Boissonnère et al [19] showed the importance of considering more than one metrics when comparing experiments with grain growth models.They observed in addition to the phase field crystal model yielding a good agreement with experimental results regarding grain size that the Mullins model recreates the grain geometry regarding, e.g. the distribution of number of edged correctly.
In recent years, additional advances in grain growth simulations have been achieved.One very noteworthy development is the introduction of the threshold dynamics, which can be used among others for computing the motion of grain boundaries by mean curvature.Introduced by Esedoglu and Otto [20] for N-phase mean curvature motion it allows the simulation of arbitrary surface tensions and arbitrary (isotropic) mobilities.It the meantime, it has been shown that is can be applied, e.g. to grain ′ boundary networks with isotropic as well as with anisotropic surface energies [21,22].
In contrast to two-dimensional considerations, Mullins [23] postulated already in 1958 that grain growth would become stagnant because thermal grooves would form in those locations, where the grain boundaries meet the surface, when the average grain size reaches the order of the film thickness.As atoms can migrate more easily at high temperatures, grain boundary grooves typically occur at elevated temperatures.While atoms can move in a variety of ways, including diffusion in lattices, diffusion at surfaces, and even evaporation from open surfaces, atomic diffusion along surfaces is especially interesting for grain boundary grooving.Driven by surface and boundary energies, atoms move away from the intersection of the grain boundary with the open surface.The removed material will deposit elsewhere on the surface, creating a depression along the grain boundary.This creates a local balance between the open surfaces and the grain boundary.
Such processes can, of course, only be modeled and understood in three dimensions.Recently, Zöllner and Pantleon [24] modeled grain growth in three-dimensional films by Potts model simulations taking grain boundary grooves into account.It was shown how the mere presence of grain boundary grooves reduces the driving force of boundary migration at surfaces immensely at any stage of coarsening and can indeed lead to full stagnation.This outcome cannot be obtained by two-dimensional simulations and shows the importance of taking the third dimensional into account even for thin films.
In the present work, we model normal grain growth in thin films by three-dimensional Potts model simulations.It is shown that the microstructure changes tremendously over time from bulk-like to columnar.This process progresses continuously.However, the growth paths that are analyzed do not follow the ones observed and predicted in either 2D or 3D grain growth.It is found that it is of utmost importance to take the grain type (interior grain, surface grain, or columnar grain) into account.

Method
For modeling processes such as recrystallization and grain growth different simulation approaches are available such as the Vertex method, the phase-field methods or the Monte Carlo Potts model.As they all work on the mesoscopic size scale, they enable simulations of large grain microstructures over long time spans allowing detailed statistical analyses of the simulated grain structures.
For the current investigation, the lattice-based Monte Carlo Potts model [8][9][10][25][26][27][28][29][30] is employed.In particular, a three-dimensional cubic primitive lattice is used to map a given polycrystalline grain structure.Each point of the lattice is called a Monte Carlo unit (MCU), which is also the smallest size unit in the simulation, i.e. the length of a lattice point.Each MCU represents a certain volume of a grain and has, hence, the local crystallographic orientation of that grain.Therewith, each grain consists of a certain number of lattice points of the same orientation.The number of lattice points per grain should be large enough to permit morphological characterizations of each grain.As each lattice point has a well-defined orientation, the Potts model is a sharp-interface model, which means that the grain boundaries are defined in an implicit manner that is between two neighboring lattice points of unlike orientation.
The smallest time unit of the simulation is one Monte Carlo step (MCS) and comprises N reorientation attempts, where N is the total number of lattice points in the digital sample.
During each reorientation attempt one lattice point is selected at random and then tested for potential orientation changes based on the configurational energy given by a Hamiltonian.For more details please see [8][9][10][25][26][27][28][29][30].For a physically correct modeling it is of utmost importance that the underlying lattice does not affect the simulated growth kinetics in any way.To this aim, a so-called simulation temperature is used [10].It is not related to the real temperature as used experimentally but influences solely the roughness of all grain boundary segments as well as the angles between boundaries as they form at triple junctions.Therewith, only two material's parameters enter the simulations: the grain boundary mobility m and the grain boundary energy γ, which both depend on the crystallographic misorientation between the two neighboring grains at a boundary segment under consideration.As m and γ are taken into account solely relative to their maximum values, and we are considering in the current case only high angle boundaries, m mmax = 1 and γ γmax = 1 results.Hence, in this paper we are considering solely the case of isotropic grain boundary energies and mobilities.Naturally, the problem is more complex when introducing grain boundary anisotropy.
While for a standard 3D simulation usually periodic boundary conditions are used, in the current case we want to model grain growth in a thin film or foil.Hence, following previous investigations [31][32][33] the top as well as the bottom layer of the simulation lattice is regarded as a free or open surface.Hence, all lattice points/MCUs at the surfaces have 17 nearest neighbor lattice points (five first nearest, eight second nearest, and four third nearest neighbors), whereas all inner MCUs have 26 nearest neighbors.
This type of specification has been applied successfully to various problems of grain growth in thin films in the past such as the investigation of texture controlled grain growth [32], of grain growth under strong temperature gradients [33] as well as of grain growth including the formation of boundary grooves [24].
In the current case, we focus on an investigation of the growth behavior of individual grains and show the influence of the grain type (inner grain, surface grain, columnar grain) on the individual growth behavior.

Results and discussion
For the current simulations, the initial grain microstructure is selected such that the film thickness is large enough with respect to the average grain size that multiple grains fill the film thickness.When the simulation is started coarsening sets in and small grains shrink and vanish while large grains grow.This process transforms the grain structure.Initially, it looks like a thin three-dimensional sample.For later annealing time, almost exclusively columnar grains can be observed (figure 1).
For normal grain growth in a 3D bulk material we expect the average grain size in terms of the average grain radius ⟨R⟩ (i.e.linear grain size) to follow [34] where α is a geometrical factor and t is the annealing time.The constant m describes the grain boundary mobility and γ the grain boundary energy.After integration we yield the well-known growth law [35] with the initial grain size ⟨R⟩ 0 .However, while simulations of normal grain growth in 3D follow indeed equation ( 2), such simulations are also based on the idea of periodic boundary conditions of the digital sample [10].In contrast, in the current case we consider the surfaces of the thin film as entities, where the grain boundaries can move freely using open boundary conditions at top and bottom.As a result, we get three different types of grains: interior grains that have no connection to either surface (just as in a corresponding 3D bilk model), surface grains that do connect with one of the surfaces, and columnar grains that span the total film thickness.

How the grain size develops
Seeing in figure 1 that the grain microstructure not only coarsens over time but also changes tremendously into a bamboo-like structure, the question arises how this change influences the grain sizes and relations between the grains and whether the grain type is of any importance.Figure 2(a) shows, in particular, how the average grain area measured at the surface of the thin film develops.A short initial period of coarsening (i) can be spotted clearly that depends on the initial grain microstructure.It is followed by a rather fast coarsening regime (I), where the average grain area increases linearly with time.Therewith, equation ( 2) is fulfilled as expected for normal grain growth in either 2D or 3D.However, after about t = 400 MCS first deviations from this linear relation can be detected that lead ultimately to a transition (T) into a new, slow coarsening regime (II), where the relation between average grain size and annealing time is again a linear one, but with a clearly smaller slope.This is in full agreement with previous observations [31].
While the grain microstructure increases its average grain size more and more surviving grains come in contact with first one and then the other surface forming for long annealing times a columnar, i.e. bamboo-like structure.This can be seen also in figure 2(b), where the percentage of columnar grains is plotted as a function of annealing time.The fast coarsening regime (I) as observed in figure 2(a) is associated with a more or less linear increase in percentage of columnar grains.In contrast, the slow coarsening regime (II) is associated with an almost constant percentage of columnar grains close to 100%.However, full stagnation is not reached as individual grains still tend to shrink and disconnect from one surface leading to their disappearance.We will see this kind of behavior later on in figure 5(b).
Discussing the temporal development of the average grain area at the surface, one may wonder if the location of measurement i.e. the location of the section parallel to the surface influences the results as shown in figure 2(a), since a section may likely yield different results compared to a surface.To that aim, we see in figure 3(a) the average grain area as a function of the location within the film that is the distance of the section from the top surface, z, for different annealing times.In particular, the measurement is taken each 100 MCS.The temporal development can be followed from bottom to the top curve: the lowest curve resembles the initial state.During the first coarsening regime, I, a bathtub-like shape develops as shown by the blue curves, even though due to the scale in the figure the bathtub does not seem to be too deep.When reaching the transition, T, between the coarsening regimes I and II, the curve becomes flat again (in green) and stays flat for long annealing times in regime II (in red).Hence, apart from a slightly retarded coarsening in regime I for the inner sections, the location of measurement when analyzing the temporal development of the average area can almost be neglected.
In contrast, it matters strongly what individual grains we analyze.For normal grain growth in two-and three-dimensional polycrystalline microstructures it is known that for example • the average linear grain size such as the average grain radius, ⟨R⟩, increases as the squareroot of time, t, (see equation ( 2)), • the growth rate of individual grains depends on the number of edges n of each grain, • the number of edges in turn relates to the scaled linear grain size i.e. the scaled grain radius x = R/⟨R⟩, and • the number of edges relates also to the average number of edges of the neighboring grains.
In general, it can be said that larger grains have more neighboring grains compared to smaller grains and they also show a different growth behavior.Grains larger than the average size are likely to grow-and the larger they are, the faster they grow.On the other hand, if a grain is smaller than average, it is likely to shrink and the smaller the grain the faster in shrinks [2,[36][37][38][39].
Based on a mean-field approach a three-dimensional grain growth model enables the prediction of the growth path of individual grains [40].While it was shown that the theory agrees well with simulations of grain growth for many grains, other grains of the same polycrystalline microstructure show clear deviations from the theory.A possible reason can be attributed to the fact that the theory is based on a mean-field description that does not take statistical deviations in grain ensembles into account.Nevertheless, it was shown in [40] that for many well-behaving grains the relation between grain size and annealing time can be predicted for three-dimensional grain microstructures forward and backwards in time.
However, in the current case the grain structures are limited by the surfaces and undergo as seen before a transition from bulk-like to bamboo-like.This changes, in particular, for long annealing times the growth paths visibly for the surviving grains as it is shown in figure 3(b).The growth paths for eight randomly selected grains show indeed a systematic behavior: Small grains have a limited growth period if any at all.Grains no. 1 and 2 vanish very fast, whereas grains no. 3 and 4 increase in size for a certain period of time before showing a continuous shrinkage.All other grains (no.5-8) increase in size.The larger those grains the more they grow to larger final grain size.In addition, the transition from growth regime I to II can be detected.

How individual grains behave
The question of how grains develop is more complicated in such a thin film compared to a full 3D microstructure considering that there is a number of grains in contact with the surfaces that are, hence, not fully enclosed by other grains.If we determine the grain type after the initial period, i.e. at t = 60 MCS, and then follow the growth paths for interior grains, for surface grains, and for columnar grains separately, we observe in figure 4 indeed different growth categories.All randomly selected interior grains shrink and vanish within regime I (figure 4(a)).Also, a number of the surface grains vanish very fast, while others-namely initially slightly larger ones-vanish somewhat slower (figure 4(b)).Apparently only few surface grains survive for long annealing times by transitioning to a columnar shape as we will see below.In contrast, most grains that are columnar already at the beginning of growth regime I survive at least within the analyzed time frame (figure 4(c)).Only few such grains vanish.
It can be concluded from figure 4 that the grain type gives a growth advantage resp.disadvantage.However, as not all grains that were columnar at the beginning of growth regime I survive the observed time span, the question arises: what triggers the different behavior?To that aim, in the following we analyze the growth path in combination with the grain type in figure 5.Here we can see immediately that the grain type changes clearly over time.The first grain starts off as an interior grain but connects already during the initial growth regime with one surface and then even with the second surface becoming columnar (figure 5(a)).Being a columnar grain, its volume increases steadily (even though with fluctuations) during regime II.The second grain, whose growth path is shown in figure 5(b), starts off also as an interior grain but with a slightly larger grain size.However, while its grain type changes also to surface and then columnar at later annealing times, its size does not increase as strongly as the previously analyzed grain.Hence, as a result of competitive growth during regime II the then smaller second grain starts to decrease in size again and finally vanishes.Therefore, we can conclude that the growth paths are controlled by the complex interplay of grain type, grain size and, of course, local grain environment as we will see below.

A comment on possible calculations of the grain size
Usually the local grain environment is described, e.g. by the relation between number of neighboring grains and scaled linear grain size.While this is straight forward in two-as well as threedimensional coarsening, where we can calculate the grain radius for each grain for example as the radius of a grain-volume-equivalent sphere and, hence, the average grain radius of the whole grain ensemble, in case of thin films we encounter an additional problem.In figure 3 we have analyzed, on one hand, the average grain area at different sections through the film and, on the other hand, the grain volume of individual grains.Here the question arises, how to calculate the scaled linear grain size.If we consider a full three-dimensional case of a polycrystalline grain microstructure undergoing normal grain growth and following equation (2), we may assume that any grain can be approximated as sphere and its volume relates to its grain radius as V = 4  3 π R 3 .Of course, there is a geometrical factor between the polyhedral grains and a sphere.In contrast, we have observed here that for long annealing times the vast majority of grains takes a bamboo-like shape.With a fixed film thickness, h, such grains can be described as columns and their volume relates then more precisely to the radius as V = π R 2 h.Hence, a grain with a given volume yields clearly different radii for the two different assumptions, and it becomes even more complex, when we consider again the surfaces and calculate the grain radius from the area measured at the surface from A = π R 2 , where the interior grains will not be considered at all.In order to illustrate this problem, two-dimensional sections through a thin film are presented as schemes in figure 6(a).In each scheme one grain is highlighted in light grey.On the left hand side, we have a polyhedral/polygonal grain that is located fully in the film.Its visible shape deviates slightly from a circle (lilac line as a section through a sphere).Nevertheless such an interior grain can be described well by a sphere regarding its size.On the right hand side, we have a grain that is in contact with both film surfaces.Its shape can be approximated by a cylinder resp. in the 2D section by a rectangle (green curve).While in the first case, the grain shares no contact area with any of the two surfaces, in the second case the grain touches the surfaces clearly and the corresponding contact region (in black at top surface) can be used to approximate the grain size.Regarding our simulation, we can calculate the temporal development of the grain radius from the grain surface area as well as from the grain volume assuming the grain on one hand as a sphere and on the other hand as a column.This is done for the grain as analyzed in figure 5(b) and the results are shown in figure 6(b).The black curve resembles the result when calculating the radius from the grain area at the surface of the film.The initial size is zero as the grain is an interior grain initially.After the grain hits the surface, the values are close to the values if we calculate the grain radius from the grain's volume assuming a spherical shape (lilac curve).When the grain becomes columnar, those two curves grow apart and the black curve gets closer to the green one, which shows the radius calculated from the volume but assuming indeed a columnar shape.
While there is no simple solution for calculating the grain radius, i.e. the linear grain size, for the current problem of grain growth in thin films or foils for all grains, this problem should be kept in mind for further analyses.

On individual growth paths
Naturally, the size of a grain is not the only descriptor that changes over time.Seeing that it is known that smaller grains have fewer neighbors compared to their larger counterparts, it is easy to understand that changes in grain size will lead to changes in the local topology.For an analysis we can follow the growth paths in terms of the temporal development of the relation between number of edges, n, of an individual grain and its grain size for randomly selected grains.This is shown in figure 7 for the three different grain types: interior grains, surface grains, and columnar grains.The grain types are determined after the initial period.
In figure 7(a) the growth paths of 30 randomly selected interior grains are given.The initial values are on the right-hand side of the figure showing a spread in volume between approximately 400 and 1000 MCU 3 (vertical axis).Apart from clear statistical fluctuations, we can see that both, grain size as well as number of edges decrease with increasing annealing time (blue arrow).All 30 grains finally vanish.These grains do not change their grain type during this process.In contrast, the surface grains in figure 7(b) show a deviating behavior: Many of these grains have clearly larger sizes as well as a larger number of neighboring grains initially (please note the different scaling of the axis).While here also the number of neighboring grains decreases over time for the individual grains (blue arrow), we see a significant increase in grain size during the process for most grains, before they shrink later on and vanish.Only one grain shows a different behavior and does not shrink in the observed time span.However, some of these grains undergo a change in grain type during the coarsening process: some become interior grains and the one increasing in size becomes even a columnar one.These different types of behavior as seen in figure 7(b) are even more visible for the initially columnar grains in figure 7(c).Here a large percentage of grains increases in size much more clearly and only few grains vanish within the observed time span.Hence, while the interior and surface grains disappear within the observed time frame to a rather large degree, most columnar grains survive.While it is nearly impossible to follow an individual growth path in figure 7 due to the large number of growth paths shown, it becomes clear that the respective grains in each image share some features meaning that there is a 'common' growth behavior in each of the three cases.Nevertheless, some few grains do show deviating behavior that can only be detected and evaluated by analyzing a large number of grains.

On topological relations
These different growth behaviors of the different types of grains influence also well-known topological relations such as the relation between grain size and number of edges.During normal grain growth we expect a time-independent relation between number of neighboring grains resp.edges, n, and the scaled grain size, x, defined as individual grain radius divided by average grain radius of the ensemble [41][42][43][44].However, as discussed above, a unique scaled linear grain size is complicated to calculate since the grain type of the individual grains may change over time, therewith changing the morphology of the grains.And there is more, we are observing a transition from bulk-like to columnar coarsening, where self-similarity [45,46] cannot be expected.Nevertheless, in the following we analyze the relation between number of edges and scaled grain size x = R ⟨R⟩ in order to keep the results comparable with previous investigations [41][42][43][44].
It is evident that this relation is not time-independent (see figure 8).Therewith the overall coarsening process is not self-similar.On the contrary, early on during the first growth regime the relation n (x) is quadratic and spans a rather large range of scaled grain size as well as numbers of edges (figure 8(a)).The largest grains have almost a volume 2.5 times larger than average, and the number of edges reaches almost 25.At the same time, it can be noted immediately that here also the grain type has clear influence on the outcome.Most of the high values are associated with a columnar grain shape (marked by blue squares), whereas surface grains (lilac dots) have overall lower values.Nevertheless, the width of the distributions is clearly larger for surface grains.It is interesting to note that the interior grains (red circles) have mostly grain sizes smaller than the ensemble average, while their associated number of edges lie above the quadratic least-squares fit, even though the latter describes the overall data rather well.Compared to surface grains, interior grains have statistically a higher number of edges for comparable grain volumes.This is due to the nature of the surface grains, which have a sometimes rather large facet at the sample surface, where no neighboring grains exist.
As growth regime I progresses the relation n (x) changes over time forming a more and more linear correlation (figure 8(b)).The interior grains vanish, and the surface grains show sizes clearly smaller than the average volume in most cases in contrast to the columnar grains that have usually larger grain sizes.During the transition from regime I to II almost all remaining surface grains vanish and the remaining columnar grains show again a quadratic relation between number of edges and scaled grain size (figure 8(c)).However, their size as well as number of edges range is rather limited.Only few grains grow larger than 1.5 times the size average, and only few have more than ten neighbors.
This strong contrasting behavior of the columnar grains in figures 8(a) and (c), respectively, can only be attributed to the existence of smaller non-columnar neighboring grains in growth regime I and the lack thereof during regime II, respectively.Hence, in the late coarsening regime the almost fully columnar grain structure has not only a reduced driving force due to a reduced boundary curvature but also a reduced number of neighboring grains as all of their neighbors are also of columnar shape.
In addition to the relation between grain size and number of edges the well-known Aboav-Weaire-law gives a purely topological description of a polycrystalline or cellular microstructure.Already more than 50 yr ago, Aboav found a solution to the problem on how the grains of a typical polycrystal resp.the cells in a cellular material are arranged in space [47].Over the years Weaire and Aboav himself extended this consideration [48,49] yielding the today well-known Aboav-Weaire-law Here n is the number of edges or neighboring grains of a grain under observation, ⟨n⟩ is the average number of edges of all grains of the ensemble, and n is the average number of edges of all neighboring grains, i.e. the local average.Naturally, for any cellular pattern, where three edges meet in a node, the average number of edges of all grains of the ensemble is equal to six.In addition, from the neighbor distribution the second moment µ 2 is needed, and α is a constant, whose value depends on the type of the cellular arrangement [11,[50][51][52][53].For microstructures undergoing ideal coarsening, that is normal grain growth, the relation is time-independent and the constant α is supposed to be close to unity [54].
While this relation has been found to be applicable to a broad variety of problems in two dimensions, the 3D case is more complex [6,55,56].Nevertheless, also in 3D self-similarity and therewith time-independence is always expected for normal grain growth.In clear contrast, we observe in the present study a strong change in the relation with progressing annealing time (see figure 9).This is, of course, consistent with our previous observations in figure 8 and absolutely to be expected as we are observing here a transient state of coarsening.Hence, in figure 9 the spread of the number of neighboring grains decreases visibly over time, while the interior and surface grains vanish.For early coarsening, where we have a grain microstructure similar as in a 3D bulk material, the average number of neighboring grains is ⟨n⟩ = 9.30 and the associated second moment of the neighbor distribution is µ 2 = 16.22 (figure 9(a)).Using these values we can derive the constant α form the first and from the second term of equation ( 3) to 1.07 and 1.03, respectively.In contrast, for late coarsening ⟨n⟩ = 6.05 is the same value as expected for 2D normal grain growth even though we are considering the three-dimensional coarsening of a thin film.Together with µ 2 = 1.95 for long time annealing (figure 9(c)) we can also derive α from equation (3), which yields 0.90 and 0.89, respectively.Hence, in both cases we get indeed values close to unity even though the relation changes tremendously.It is also interesting to note that the value of ⟨n⟩ = 9.30 is clearly smaller for early coarsening than expected for a full 3D microstructure.In between early and late coarsening, at t = 260 MCS, which is clearly within growth regime I, the average number of edges is with 7.97 still far from 6 and the resulting values for α are far from one.This is very notable as it shows that the microstructure is clearly in a transition between the three-dimensional and quasi-two-dimensional, columnar state, and this transition starts far before the change in growth rate becomes visible in figure 2(a).
The relation between local average number of neighboring grains, n, and number of edges, n, changes strongly in such a way that, in particular, columnar grains (blue crosses in figure 9) have initially more than ten neighbors, but for late coarsening less than ten.Hence, the question arises how the remaining grains in figure 9(c) changed their morphology over time.To that aim, we analyze the temporal change in nn (n) for a few selected grains as shown in figure 10.In the initial regime (i) the values for all three grain decrease visibly.In the subsequent growth regime I they decrease further.However, only one grain (marked in green) shows a rather smooth change with time, the other two show strong fluctuations.Nevertheless, the values when reaching the transition, T, are for all three grains clearly smaller than at the beginning of regime I.This is consistent with the finding that the transition of the microstructure is a more or less continuous process and does not suddenly set in with reaching T. In contrast, for late annealing (regime II) the values barely change.
Here it should be pointed out that a reduction of number of faces or edges of a grain is a common process in grain growth.On the contrary, grain growth cannot occur without topological transitions because from a mathematical point of view it is impossible that a decrease in number of grains per unit volume may take place without the topological transition that corresponds to the grain disappearing.This is a well-known problem [57][58][59][60][61] and a very interesting one as most grains of a polycrystalline structure will lose faces during normal grain growth, while few grains gain faces.This keeps the grain face distribution time-independent.However, in the current investigation we find among others that the width of the distribution of number of neighboring grains and therewith number of faces decreases during the transient coarsening (see horizontal axis in figure 9).

Conclusions
All in all, a three-dimensional simulation of grain growth shows that the grain microstructural coarsening in a thin film progresses clearly different compared to the case of grain growth in a 3D bulk material: 1.The average grain size follows two distinct growth regimes: an early regime, where the average grain area increases linearly, but strongly with annealing time, and a second regime, where the linear increase progresses much slower.However, while the transition from the first to the second regime seems quite fast regarding average size, the analysis of the Aboav-Weaire-law shows that the structural changes set in clearly earlier.2. The growth paths of the individual grains are not only controlled by the initial size of the grains (and their local topology) but also strongly by the grain type.Initially columnar grains and those that change their type to columnar early on have a higher chance to survive longterm compared to all other grains.This is clearly different to purely two-dimensional or three-dimensional simulations and experiments.3. Due to the microstructural transformation from bulk-like to columnar the classical laws of normal grain growth such as the Aboav-Weaire-law are not self-similar, i.e. timeindependent.This was to be expected due to the transient nature of the observed coarsening.Of course, this has further implications.For example, the well-known von Neumann-Mullins-law [2,36,37] yields a universal relation for two-dimensional coarsening and also 3D generalizations [39,62] should be time-independent for normal grain growth.However, for the current transient coarsening, where the spread, e.g. in number of edges changes drastically over time self-similarity cannot be expected.
Hence, when analyzing processes such as grain growth in thin films or foils it is of great importance what stage of coarsening the microstructure is in.In addition, the individual grains' behaviors do not follow the ones observed and predicted in either two-or three-dimensional grain growth.Their kinetics is influenced strongly by their grain type.

Figure 1 .
Figure 1.Temporal development of the grain microstructure in the simulated thin film.

Figure 2 .
Figure 2. Temporal development of: (a)-average grain area measured at the surface of the thin film showing initial regime as well as fast growth regime I and slow coarsening regime II; (b)-percentage of columnar grains.

Figure 3 .
Figure 3. (a)-Temporal development of the relation between average grain area and location of section with the thin film, where the sections are taken parallel to the surfaces (more details in description); (b)-growth paths of individual grains in terms of temporal development of grain size (volume) for eight randomly selected grains.

Figure 5 .
Figure 5. Growth behavior for two randomly selected grains that develop from interior grain via surface grains to columnar grains showing clearly different behaviors: (a)surviving grain for long annealing times; (b)-grain vanishing for long time annealing.

Figure 6 .
Figure 6.(a)-Scheme of two 2D sections through 3D films showing a more spherical grain shape for interior grains and a cylindrical grain shape for columnar grains.(b)-Growth path of one grain (as shown in figure 5(b)) showing different linear grain sizes calculated from surface area (black curve), from volume assuming a spherical shape (lilac) and from volume assuming a cylindrical shape (green).

Figure 7 .
Figure 7. Growth paths in terms of temporal development of relation between number of edges and grain volume for randomly selected: (a)-interior grains; (b)-surface grains; (c)-columnar grains.The temporal directions of the growth paths are given by blue arrows.

Figure 8 .
Figure 8. Relation between number of edges and scaled grain size for: (a)-t = 60 MCS; (b)-t = 260 MCS; (a)-t = 660 MCS.Grains are marked according to their type as interior grains (red circles), surface grains (lilac dots), and columnar grains (blue squares).In addition, quadratic least-squares fits are given by dashed curves.

Figure 10 .
Figure 10.Temporal development of 3D Aboav-Weaire-relation for three randomly selected grains that survive for the total observed time frame showing the different coarsening regimes.The temporal direction is indicated by the blue arrow.