Elsevier

Acta Materialia

Volume 78, 1 October 2014, Pages 125-134
Acta Materialia

Grain growth in four dimensions: A comparison between simulation and experiment

https://doi.org/10.1016/j.actamat.2014.06.028Get rights and content

Abstract

A 3-D isotropic phase field simulation was used to predict the morphology of individual grains during grain growth. The simulation employed a polycrystalline array of titanium alloy Ti-β-21S experimentally characterized by X-ray tomography as an initial condition. The non-destructive nature of X-ray tomography allowed for a second characterization of the same sample following coarsening induced by a heat treatment. Thus, direct comparisons of individual grains between simulation and experiment could be made. Although the experimental system appeared isotropic from a statistical standpoint, direct examination of individual grains revealed very distinct anisotropy in the grain boundaries on the local scale. The comparison between experiment and phase-field simulations revealed regions with excellent agreement, despite the complex topological changes grains may undergo during grain growth. Thus, the sequence of topological transitions that occurred experimentally is correctly captured by the phase-field model. We therefore conclude that this phase-field model for isotropic systems has been verified experimentally.

Introduction

The dynamics of grain boundaries in polycrystalline structures has been a topic of great interest for many years. There has been a great deal of study on grain growth in two dimensions both theoretically [1], [2], [3] and experimentally [3], [4]. Recently, there has been a resurgence of interest in the dynamics of 3-D grain growth, with the generalization of the 2-D von Neumann–Mullins relation to three dimensions [5], examination of the topology of ideal grain systems [6] and large-scale computer simulations of grain growth [7], [8]. Although the growth rate of a grain in an isotropic system is known as a function of its interface morphology, predicting the evolution of even an isotropic polycrystalline material remains challenging due to the topological changes associated with face creation and elimination that occur during grain growth [9], [10], [11]. Experimental investigations of 3-D grain growth have typically focused on static structures, since the methods used to interrogate the samples were destructive. As a result, comparisons between theory, simulation and experiment were made using statistically averaged properties. To avoid the effects of initial conditions, studies of the grain structure are usually carried out in the self-similar coarsening regime. Because comparisons with the predictions of theory and simulation are, by necessity, statistical in nature, typical quantities measured tend to be limited to metrics such as number of faces per grain, grain size distribution and other averaged topological quantities. Thus, it is necessary to collect large experimental data sets that contain many thousands of grains. These experimental studies have clearly illustrated the importance of grain boundary energy anisotropy in the evolution of grain structures [12], [13], [14] and grain topology [15] during grain growth.

By contrast, X-ray-based techniques are non-destructive in nature, and as such provide opportunities for direct studies of microstructure evolution during, for example, phase transformation, recrystallization or grain growth. For example, Offerman et al. examined the nucleation and growth of individual ferrite grains during the austenite-to-ferrite phase transformation in a carbon steel. The X-ray technique employed allowed the dynamics of individual grains to be measured, thereby enabling a direct comparison to theory without recourse to statistical measures [16]. Using 3-D X-ray diffraction microscopy it was possible to examine the growth of individual grains during recrystallization [17], [18]. This has more recently been extended by Hefferan et al. to a full 3-D characterization of recovery and recrystallization in high-purity aluminum using high-energy X-ray diffraction microscopy [19]. 3-D X-ray diffraction microscopy has also been applied to studies of grain growth comparing the evolution of the same 3-D structure at two different annealing times in an aluminum alloy [20]. Syha et al. extended such non-destructive 3-D characterization to studies of grain growth in ceramics using diffraction contrast tomography [21], [22]. These works clearly show the potential of grain-resolved X-ray diffraction techniques for studies of microstructural evolution in polycrystalline materials. However, although the spatial resolution of such diffraction-based techniques continues to improve, the currently attainable spatial resolution is lower than what can be achieved with direct X-ray imaging methods such as, for example, absorption-contrast tomography or phase-contrast tomography. Thus, in order to acquire the high-fidelity grain shape information necessary for modeling polycrystalline morphologies, the current study makes use of edge-enhanced tomography combined with special heat-treatment procedures to allow for detailed, yet non-destructive characterization of 3-D grain structures [23].

In addition to experimental investigations, insight into grain growth has also been sought through theory and simulation with a variety of different approaches. One such approach to modeling grain growth that allows the morphology of individual grains to be determined involves explicitly tracking the grain boundaries that separate each grain. Assuming a system with an isotropic grain boundary energy and mobility, the velocity at a point on the boundary is proportional to the local mean curvature of the grain boundary:V=-MHwhere V is the interfacial velocity in the interface normal direction, M is the orientation-independent reduced mobility (the product of the grain boundary energy and mobility), and H is the local mean curvature. This approach requires that the location and morphology of the boundaries be tracked explicitly. To reduce the complexity of meshing each boundary in the system, some approaches only follow the location of the grain boundary vertices, e.g. triple junctions in two dimensions or allow for some curvature in the boundary by introducing a point on the grain boundary in the center of a face [24], [25], [26]. Relatively few simulations have discretized the entire grain boundary surface due to the need to repair the mesh when a grain loses or gains a face. This rendered most of these methods computationally inefficient and was limited to systems with at most some hundreds of grains [27]. Recently, however, these limitations have been overcome allowing fully resolved 3-D calculations with hundreds of thousands of grains [8], [11], [28].

Phase-field methods also permit the curvatures and shapes of individual grains to be determined. In this case, an interface is described implicitly using an order parameter that varies smoothly across the boundary between grains. Thus there is no need to track the location of a sharp interface. Instead partial differential equations are solved at all positions in the system to determine the time evolution of the order parameters. There are three broad categories of phase-field methods used to model grain growth. The first was inspired by models of order–disorder processes in materials [29]. In this case, a non-conserved order parameter, ηi is assigned to grain i, where i=1,2,N for a system with N distinct grain orientations. The free energy of the system, F, is then written as:F=Vf(η1,η2,,ηN)+i=1Nκiηi2dVwhere V is the volume of the system, f is the bulk free energy, and κi is the gradient energy coefficient of grain i. The bulk free energy f is chosen to have minima of equal depth for each grain orientation, i.e. for ηi=1, and ηji=0. These order parameters are then evolved in time by the Allen–Cahn equation [30]:ηit=-LδFδηiwhere L is the mobility of the order parameter. There are no constraints on the value of the sum of the order parameters at grain boundaries or trijunctions, although the parameters can be chosen such that the sum of the order parameters is 1 across grain boundaries [31]. This method has been extended to systems with orientation-dependent and mobility-dependent grain boundary properties [31], [32], [33], [34]. A second class of methods involves order parameters that are interpreted as the volume fraction of phases [35]. The sum of the order parameters must thus be one at all points by definition. An advantage of this approach is that they can easily be extended to the complexities of multiphase multicomponent alloys. This model also recovers the classical motion by mean curvature result. Consistent with the constraint on the sum of order parameters, the sum must also be one at grain boundaries, trijunctions and quadrijunctions. A third model uses a non-analytic form of the gradient free energy to create grain boundaries [36]. This method has the appeal of satisfying the rotational invariance of the bulk free energy, as must be the case in reality.

A major advantage of the phase-field method is that it is straightforward to use measured 3-D microstructures as initial conditions in a calculation. This is because experimental data is naturally voxelized and thus provides a clear path to assign an order parameter to each grain in the structure [37], [38]. Through this approach it is possible to compare the predicted dynamics for a particular structure with those measured experimentally. With the advent of experimental methods to record the 3-D morphology of materials and even its evolution in time, see Ref. [39], it is likely that this approach will be used more often in the future, and thus it is important to ensure the validity of the phase-field method.

Essential to grain growth are topological singularities. In this case, as grains grow or shrink, grain faces are added and removed when trijunctions impinge. Grains can also disappear from the system. Trijunction-impingement processes must thus be modeled in a physically realistic manner. The interfacial widths employed in these calculations are typically much larger than the width of a real grain boundary in order to ensure computational tractability. However, this model has been shown to yield sharp interface dynamics, Eq. (1), when the simulated grain boundary width is judiciously chosen [40]. Thus, the trijunctions, whose widths normal to the trijunction line are unphysical, must also be handled carefully in order to obtain the correct evolution of topological singularities. A recent paper by Moelans et al. [41] comparing two phase-field models for grain growth indicates that the manner in which the trijunctions are modeled may affect the morphological evolution of the grain structure. In this case, the evolution of a grain using the phase-field method where the order parameters are not constrained to sum to 1 was compared to a phase-field method with this constraint. While both methods yielded similar statistically averaged results for the evolution of polycrystalline arrays, they yielded a different topological evolution pathway for some grains. Since both methods recover motion by mean curvature, Eq. (1), but treat trijunctions differently, the subtleties of trijunction evolution during topological singularities may be the cause of this difference. Thus, there is a need to verify the morphological and topological pathways in various simulation implementations using experimental results.

Currently, all comparisons between simulations of grain growth in three dimensions and experimental results rely on being in the self-similar regime of grain growth and employ statistical measures of the structures such as average grain size and number of faces per grain. In contrast, this is not the case during grain growth in two dimensions where direct comparisons have been made [42]. Traditionally, a statistical approach has been taken in 3-D studies because the available characterization techniques required destroying the sample in order to measure the grain shapes. One advantage of performing a statistical comparison between structures is that the initial and boundary conditions of the simulation are significantly less important. Here we take a different approach using a nondestructive technique to measure grain morphologies. We can thus compare simulated and experimentally characterized morphologies of individual grains in a polycrystalline material. A phase-field method is employed since, as mentioned above, it is relatively straightforward to use 3-D experimental results as an initial condition. The structure is then computationally evolved and the morphologies of a chosen set of grains are compared to those measured experimentally. Thus, we avoid the need to ensure that we are in the self-similar grain-coarsening regime and the complications of making comparisons using statistically averaged results. Finally, the comparison between simulation and experiment will allow us to determine the validity of the employed phase field model.

The rest of this work is structured as follows: in Section 2 the experimental procedure is outlined. Section 3 introduces the phase-field method and computational approach. Section 4 compares the results of the simulation to experiment, before the paper is concluded in Section 5.

Section snippets

Experiment

The sample used in this study was a cylindrical single-phase polycrystal of Ti-β-21S with a diameter of approximately 378 μm. Ti-β-21S is a β stabilized Ti alloy produced by TIMET with a composition of Ti–15.4Mo–2.9Nb–2.9Al–0.17Si–0.29Fe–0.12O. This alloy has the beneficial feature that it allows for controlled, preferential precipitation of α phase to the grain boundaries, thus providing a contrast mechanism for X-ray imaging. The as-received sample was annealed for 2 h at 725 °C and air-cooled

Simulation

As a simple approximation, we employ a phase-field model with an isotropic grain boundary energy and isotropic grain boundary mobility. This is reasonable for Ti-β-21S since recent results by Rowenhorst et al. suggest anisotropic effects of the grain boundary energy and mobility do not appear to have a significant role in the growth rate of grains [15]. Additionally, Rowenhorst et al. found that the grain size distribution and average number of faces per grain is close to those obtained using

Grain morphology

Fig. 1 shows the experimentally measured structure at time 1, which was used as an initial condition in the phase-field simulation, along with the simulated structure at time 2. The reconstructed region of the initial experimental sample contained over 1200 grains, with an average number of faces per grain of 13.8. In comparison, a number of theories suggest that the number of grain faces required for an average growth rate of zero for grains with isotropic grain boundary energy is between 13

Conclusions

We have compared the morphology of grains calculated using a phase-field model assuming isotropic grain boundary properties to those measured experimentally through edge-enhanced X-ray tomography. To allow us to directly compare the morphologies of individual grains computed using simulation with that of experimentally measured grains, we performed two experimental characterizations of the same sample. A coarsening stage induced by heat treatment is carried out between the characterizations.

Acknowledgments

We thank David J. Rowenhorst for useful discussions, Richard W. Fonda for valuable assistance with sample preparation and the Office of Naval Research (ONR-CNV0044048) for financial support.

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