Grain growth in thin film under strong temperature gradients

Grain growth in thin films has attracted lots of attention due to the numerous applications of polycrystalline films, on one hand, and modified properties due to changes in the microstructure caused by thermal effects, on the other hand. While the phenomenon of grain growth is well understood in general, effects of e.g. temperature gradients leave questions open still today. In the present study, we investigate the influence of strong temperature gradients on grain growth in thin films. To that aim, a modified three-dimensional Potts model algorithm is employed, where the annealing temperature depends on the position within the sample leading to spatial heterogeneities in the grain boundary network. As a consequence of different mobilities, a drag effect occurs on the boundary network evolution that has serious consequences for the microstructures evolving during grain growth.


Introduction
Grain boundaries in polycrystalline materials are inherent structural features usually on the micrometer size scale.They are of particular interest as almost all mechanical properties, such as hardness or strength, depend on the distance between grain boundaries, i.e. the grain size [1,2].Changes in the boundary network lead to a change in properties, which may render materials unsuitable for certain applications.Understanding how such changes in the boundary network progress during deformation, recrystallization or grain growth is of utmost importance to predict the materials behavior and materials life spans during operation at elevated temperatures.At first glance, grain growth appears to be a rather simple process: at elevated temperatures, grain boundaries move in such a manner that larger grains grow at the expense of smaller neighbors.This leads to a reduction of the total grain boundary interfacial area and therewith of the Gibbs free energy of the microstructure.
On a more detailed level, this coarsening process is very complex as there are a number of influential factors.One of them are open surfaces, which gain importance in thin films.It has been observed in relatively thick [3] as well as very thin [4] metal layers that when the average grain size reaches the size of the layer thickness, grain growth slows down and comes to a halt [5,6].Because of the rather large number of applications of thin films, e.g., as protective coatings or in semiconductor technology, understanding this is of importance as an increasing number of overview publications on the topic of polycrystalline grain microstructures and grain growth shows [7][8][9][10].Quite a large number of these investigations analyze the problem in two dimensions (2D).Even though this is in accordance with traditional metallography, i.e. a two-dimensional characterization of materials microstructures by optical microscopy, and a number of important findings have been achieved, inherent problems of 1249 (2022) 012010 IOP Publishing doi:10.1088/1757-899X/1249/1/012010 2 such 2D investigations are well known.Any two-dimensional section through an object in three dimensions (3D) may yield only a poor idea about true size or shape of the object.This becomes even more important for polycrystalline materials in the presence of heterogeneities within the sample.
In thin films, coarsening during grain growth may become even more complicated when temperature gradients are considered.While temperature gradients may be very small and therewith negligible in some applications, in others -such as in nuclear reactors [11] or in the heat affected zone during welding [12] -strong spatial temperature gradients have to be taken into account.In a recent paper, Zöllner [13] modeled the impact of a strong temperature gradient on grain growth in films using a three-dimensional Potts model.She found that the growth of grains at the hot open surface, where the boundary mobility is higher, is restricted by the need to maintain contiguity with smaller grains in colder regions with lower grain boundary mobility, whereas the growth of grains at a cold open surface is enhanced due to required contiguity with larger grains in warmer regions.In compliance with the temperature gradient, a visible gradient in the microstructure is observed.
In the present investigation, we model a strong temperature gradient, which influences the threedimensional grain growth via the grain boundary mobility.Additionally, we construct an artificial microstructural gradient from the three-dimensional microstructure evolution during isothermal annealing to distinguish effects of locally different temperatures from that of a temperature gradient.

Theoretical Background
Changes in a grain boundary network develop thermally activated and gradually in time by grain boundary migration.The higher the temperature T, the faster the progress occurs, provided, that the driving force from a reduction in stored energy is the same.The temperature dependence of the grain boundary mobility m follows at realistic temperatures an Arrhenius dependence [14] determined by a quasi temperature-independent pre-exponential factor m0, the Boltzmann constant kB and the activation energy Q* for thermally activated boundary migration.For achieving the same progress at two different constant temperatures T and T´, different corresponding annealing times t and t´ are required, which are related to each other according to () = (′)′.
(2) This relation can be utilized for predicting the progress of grain growth at a certain temperature T based on the progress at another temperature T´ taking Eq.(1) into account.

Simulations
For about 40 years, computer algorithms have enabled researchers to simulate recrystallization, grain growth, and similar phenomena such as Ostwald ripening in polycrystalline materials.In particular, mesoscopic simulation methods work well on a microstructural level and allow a selective tailoring of materials parameters.Different models were developed over the years, such as the Monte Carlo Potts model and the phase-field method (cf., [15][16][17][18][19]).In particular, the Potts model has a relative simple algorithm that can be modified to tailor and model complex problems [13,[20][21][22].
In the Potts model all grain boundaries are sharp interfaces with zero width.Only two materials parameters characterize the boundaries, the boundary mobility m and the boundary energy γ.Commonly, both depend on the misorientation between two adjacent grains, but also on the annealing temperature T.Where the boundary mobility depends strongly on temperature following the Arrhenius relation of Eq. (1), the boundary energy has only a minor influence within the investigated temperature range and, therefore, is considered constant in the current investigation.
The Potts model is utilized here for simulation of grain growth in a strong linear temperature gradient through a thin film (see figure 1) for up to 1000 Monte Carlo time steps (MCS).The simulation box (see scheme in figure 1a) contains  = 250 × 250 × 75 voxels on a 3D cubic lattice.The length of each voxel along the coordinate axis is called Monte Carlo Unit (MCU).Specifically, the layer thickness, i.e. the size in z-direction, has been selected such as to simulate a statistically relevant number of grains in the gradient direction and to circumvent simulation effects of extremely thin layers ( ≫ 1 MCU).A linear relation between temperature and position along the z-direction in the material assuming steady-state thermal conduction as in [13] is applied (figure 1b).It is assumed that the top surface at z = 1 MCU is kept at a constant temperature Ttop = 300 °C whereas the bottom at z = 75 MCU is kept at room temperature Tbottom = 25 °C.The activation energy of grain boundary motion is selected such that the mobility, which is highest at the top surface z = 1 MCU, decreases throughout the layer to mbottom = mtop/10 at the bottom in an exponential manner (figure 1c).For any given voxel within the lattice, a neighborhood of the size 3 × 3 × 3, i.e. the point of interest and its 26 nearest neighbors, is considered for local interactions.For the margins of the simulation box, specific boundary conditions are presumed: periodic boundary conditions for the surfaces parallel to the temperature gradient and free boundary conditions at the top and bottom surface (parallel to the xy-plane perpendicular to the temperature gradient).An initial microstructure is generated and projected on the simulation box with 25000 grains following a normal grain size distribution.
Grain growth under the temperature gradient is simulated up to 1000 MCS.The grain structure resulting after 800 MCS is illustrated in figure 2e.Obviously, due to the temperature gradient, the progress of grain growth is different in each of the layers (at different depth).The coarsening has proceeded most at the highest temperatures towards the top and least at the lowest temperatures at the bottom (at least as long as surface effects do not play a role).
In an alternative approach to achieve the microstructure resulting from the linear temperature gradient, an artificial microstructural gradient is constructed from a separate also three-dimensional, but isothermal simulation of grain growth at a constant temperature T´ = T top = 300 °C.In order to mimic the depth-dependent differences in evolution under a temperature gradient, layers from corresponding different isothermal annealing times t´(z) are assembled.Each layer in depth z in the temperature gradient T(z) is represented by a layer of the isothermal simulation at the same height after an effective isothermal annealing time which follows directly from Eqs. ( 1) and (2).For this effective annealing time, the progress in microstructural evolution in the layer in the isothermal simulation is the same as for the simulation under the temperature gradient after t = 800 MCS.This artificially reconstructed microstructural gradient takes into account the different thermal activation in each layer depending on its depths, but not the same spatial coupling as occurring in the grain growth simulation under the temperature gradient.Nevertheless, this approach has not only the advantage that the underlying isothermal simulations are generally faster than real temperature gradient models, but also it enables the reuse of older simulation data and, hence, calculating microstructures with different temperature profiles from one and the same isothermal simulation.

Results
In a three-dimensional grain structure, where the grain size is comparably small in contrast to the film thickness, the grain structure within the film evolves mostly just as within a bulk sample.The average grain size -in terms of the average radius R -follows the standard parabolic growth during normal grain growth independent of dimensionality [23,24].Analogously, the average apparent grain area 〈〉 measured from a 2D section increases linearly with time t.Of course, this is only true for normal grain growth, where all boundaries have the same properties.Under this condition, the parabolic growth law is also obeyed in thin films, when the grain size is indeed smaller than the film thickness.In general, even for a mostly columnar grain microstructure the parabolic growth law holds as well with a smaller growth rate at the surfaces [25,26].
In the current case, the evolution of the simulated grain microstructure under isothermal conditions is shown in figure 2 (a to c) for three different annealing times.Grain coarsening is clearly visible and no preferred direction of growth is detected.The latter is even true for the simulation of grain growth in the linear temperature gradient (figure 2e).The grains still appear equiaxed without preferred growth direction, but a strong microstructural gradient develops throughout the thickness with large grains close to the top surface and much smaller grains at the bottom.Applying the above idea of creating a microstructural gradient artificially, 75 layers each in its own height are taken from 75 different annealing times given by Eq. ( 3) of the isothermal simulation to construct the 3D microstructure such that the result is supposed to describe a microstructure having evolved under the temperature gradient after simulation for t = 800 MCS.The respective image is shown in figure 2d.As the different sections of the artificial microstructural gradient are taken from different annealing times of the isothermal simulation, here also a clear microstructural gradient becomes visible.The top of the artificial microstructure has a clearly larger average grain size as it is taken from a later annealing time (′ = 800 MCS) compared to the bottom of the gradient sample gathered from (′ = 78 MCS).At a first glance, the artificially created microstructural gradient has a very similar appearance compared to the simulated microstructure under the temperature gradient.
For a more detailed analysis, the simulated microstructures have been characterized regarding the evolution of the average grain size.To that aim, the average grain areas 〈〉 were followed over time for different sections (z = {1, 5, 10, … 75} MCU).The results are shown in the top row of figure 3. The case of isothermal annealing at 300 °C is illustrated in figure 3a.The data for the two surfaces (z = {1, 75} MCU) are colored in red, the next nearest analyzed sections (z = {5, 70} MCU) in magenta, and all other inner sections from the bulk in black.The presence of a surface effect on top and bottom is apparent.While all inner sections yield similar results (in particular up to 500 MCS) with some statistical fluctuations, both open surfaces evolve faster.This is a result of the fact that the sample terminates at these surfaces (which is not so for the faces of the simulation box along the temperature gradient where periodic boundary conditions maintain continuity).Grain boundaries align themselves perpendicular to the open surfaces and no grains from regions outside the film can grow into these.As an additional result, the sections close to the surface experience a slight drag effect: Their average area increases faster compared to those further inside; but not as fast as that at the surfaces.Consequently, the resulting average grain area plotted as a function of film thickness z for t = 400 MCS in figure 3c shows a statistically homogeneous microstructure in the film center with an average grain area 〈〉 = 142 MCU 2 .Towards both surfaces, regions up to about 15 layers show enhanced coarsening.For the simulation with the linear temperature gradient as in figure 1b, the evolution of the mean average grain area in the different layers (figure 3b) shows that there is indeed a microstructural gradient resulting from the underlying mobility gradient.The hot surface associated with the highest mobility evolves fastest (plotted in red).The nearest analyzed section is given in magenta and has a slightly retarded growth.All inner sections (z = {10, 15, … 65} MCU) follow a systematic, where the lower the temperature, the smaller the growth rate.This is even true for z = 70 MCU shown in cyan.The cold surface (plotted in blue) shows again faster coarsening, however, faster with respect to the slow coarsening of the nearest sections (plotted in cyan).The resulting dependence of the average grain area on the depth z within the film is given in figure 3e for t = 400 MCS.
Finally, the obtained average sizes are compared to that of the artificial microstructural gradient.Here, a special feature arises: As the microstructure of the artificial gradient is calculated from different time instants of an isothermal simulation, one cannot follow its temporal evolution easily.Nevertheless, one can obtain grain microstructures for a specific time .In particular, for the artificial microstructural gradient constructed for t = 400 MCS, the average grain sizes from the sections z = {1, 5, 10, 15, … 75} MCU each taken at their respective t´ are added to figure 3b (green circles) comparing the artificial microstructural gradient directly to the microstructural gradient simulated for the temperature gradient.The small deviations visible are fully within the range of statistical fluctuations.The good correspondence between both microstructures becomes even more apparent when comparing the depth-dependent average grain areas 〈〉() for both gradients as shown in figure 3d and 3e.Further comparison between the microstructure caused by the temperature gradient and the artificial microstructural gradient created to resemble the different progress in growth requires additional quantitative analysis.Figure 4 presents the results of a point-to-point comparison between both microstructures after 400 MCS.From the 25000 grains in the initial configuration, only 5502 survived in the simulated gradient and 5566 in the artificial one.The non-matching grains are illustrated in figure 4a in blue, if present in the proper simulation with the temperature gradient, but not in the artificially created one, and red in the opposite case.Apparently, there are a non-negligible number of grains in both simulations that evolve differently on the individual grain level, even when the general results agree well on the microstructural level according to figure 3.As shown in figure 4b, the average fraction of differently assigned voxels in the entire volume is 15% with largest deviations at the hot surface.Despite these differently assigned points, the averaged grain size defined as equal spherical radius derived from the volumes of all grains differs only slightly: 9.93 MCU (10.17 MCU) for the simulated gradient vs. 9.83 MCU (10.04 MCU) for the artificially created one.The numbers in brackets refer to bulk grains only.The corresponding distributions for the equivalent spherical radii are matching closely (cf.figure 4c).Kolmogorov-Smirnov testing allows quantifying similarities between the obtained distributions: Where the distributions from the artificial gradient microstructure and the simulated one do not differ significantly, the hypothesis that the distribution of all grains and bulk grains only originate from the same distribution is rejected at a significance level of 5%.The artificially created gradient assembling layers corresponding to different annealing times of an isothermal simulation resembles closely the real microstructural gradient from simulating a temperature gradient.Consequently, the microstructure established in a temperature gradient is a sole consequence of the locally different mobilities and no effects from growing in a temperature or mobility gradient are discerned.In both microstructural gradients, the grains do not exhibit any preferred growth direction and maintain equiaxiality.
Still the question remains, how does the microstructural gradient develop?The first effect that comes to mind is the fact that at higher temperatures grain boundaries move faster due to their higher mobility.Hence, one is inclined to presume that a growing grain may grow faster into the hotter than into the colder region.One may even expect a certain skewness of the grain shape developing.This is shown exemplarily for the simulation of three grains undergoing recrystallization in a temperature gradient in figure 5.Here recrystallization is used as it allows an easy tracking of the boundaries of the growing grains.Three seeds are put into a three-dimensional box of size 120 × 120 × 350 MCU 3 on a straight line along the temperature gradient.In figure 5 the temporal development of the section through the center of the three grains is shown.All three seeds grow into small grains.If one focusses on the central green grain, one realizes that within the first 10 MCS of annealing, the seed grows and the center of mass zc does not change significantly along the z-direction, that is the direction of the temperature gradient.When the grain has reached a substantial size, its center moves to lower z, i.e. upwards and therewith into the hotter region.This indicates clearly that recrystallizing grains grow indeed faster into the hotter region than into the colder, which is in agreement with the above expectations.However, when the simulation continues, at about 25 MCS the growing grain starts to impinge on the upper grain.At this point, the center stops moving upwards.As the grain can not continue to grow further into the hot region, the center starts even to move downwards.After 45 MCS, the shift downward exceeds the maximum shift upwards.While this is easy to understand for such a simplified case, for a more complex 3D distribution of many seeds, the individual behavior of the center of mass of a grain during recrystallization will depend not only on its location and therewith on temperature and local temperature gradient, but also on the neighboring seeds.This is also the case, for the above simulated case of grain growth in a temperature gradient.In average, the grains develop neither a pronounced skewness along the direction of the temperature gradient, nor a systematic shift in their center of mass position.Their coarsening depends rather, on one hand, on the local temperature and local grain boundary mobility, respectively.On the other hand, due the complex boundary network, effects of the grain sizes and their number of edges are non-negligible for the evolution and an equiaxed microstructure is maintained.

Summary and Discussion
While in general the phenomenon of grain growth in polycrystalline materials is well understood, effects of temperature gradients leave questions open still today.Therefore, in the present study, we investigated the influence of strong temperature gradients on grain growth in thin films.To that aim, we modified a three-dimensional Potts model algorithm such that the annealing temperature varies within the thickness of the sample producing spatial heterogeneities in grain boundary mobility.
We have shown that in contrast to isothermal grain growth, under a temperature gradient the grain microstructure evolves from a spatially homogenously distributed structure with equiaxed grains to a distinct microstructure with visibly larger grains at the hot surface compared to the colder surface.In addition, the grain boundaries at the open surfaces align perpendicular to the surface, while the average areas at the surfaces evolve slightly faster compared to the inner regions close by.In the bulk of the volume, the grains remain equiaxed despite the temperature gradient.
Furthermore, we constructed an artificial model for the evolving microstructure gradient, where the specific microstructure of each layers originates from the time evolution of isothermal grain growth.While this does not predict the actual evolution of a certain grain, the temporal evolution of the average grain size as well as the resulting grain size distributions agree quite well.

Figure 2 .
Figure 2. Simulated grain growth in thin films: the top row shows the 3D microstructural evolution under isothermal conditions.The grain structures represent annealing at 300 °C for (a) 78 MCS, (b) 402 MCS and (c) 800 MCS.The bottom row shows the (d) constructed artificial microstructural gradient in comparison to (e) the microstructural gradient evolving under the temperature gradient simulated for 800 MCS.

Figure 3 .
Figure 3. Average grain size: In the upper row, the temporal development of the average grain area is shown for (a) isothermal coarsening and (b) coarsening in a linear temperature gradient, both for different sections along the z-direction through the films.In the lower row, the average areas after 400 MCS are given as a function of the layer depth z within the films for all three analyzed cases: (c) the isothermal simulation, (d) the artificially constructed microstructure, and (e) the simulation with linear temperature gradient.

Figure 4 .
Figure 4. Comparison between simulated and artificial microstructural gradient: (a) The voxels of non-matching grains are highlighted; grains in the microstructure gradient simulated under the temperature gradient, not present in the artificially constructed gradient are shown in blue.Red grains are present solely in the artificial microstructural gradient.(b) Fraction of differently assigned voxels in the volume.(c) Distribution of equivalent spherical radii from real and artificial gradient independently for all grains and bulk grains only.

Figure 5 .
Figure 5. Three-dimensional simulation of recrystallization originating form three seeds under a linear temperature gradient along z.(a) On the left, the temporal development of a section is shown.The top of the samples is hot, the bottom cold.(b) On the right, the shift of the center of mass of the green grain along z-direction resp.along the temperature gradient is given.