Hesham Salama

In this study, we examine grain boundary energy as a function of both misorientation and inclination using a phase-field approach. Through systematic investigation of their independent and combined influence, we observe that misorientation appears to influence energetic preferences, with an increased frequency (approximately 35%) of low-angle boundaries (0–5^∘), while inclination tends to affect local boundary geometry, with boundary planes showing preference for 45–50^∘ inclination angles. Our simulations suggest that while inclination dependence influences boundary morphology, the combined effect leads to morphological features including elongated grains and distinctive growth kinetics. Notably, inclination-dominated cases show slower growth rates compared to isotropic or misorientation-dependent systems, though misorientation-dependent cases exhibit faster decrease in interface energy density. The model’s behavior is examined through studies of facet formation in cubic systems and configurations at multiple grain junctions. Statistical analysis of two-dimensional polycrystalline systems indicates that the interaction between misorientation and inclination may be more intricate than previously considered, suggesting potential value in incorporating both factors when studying anisotropic grain growth. These observations contribute to our developing understanding of microstructural evolution in polycrystalline materials.

Bainitic steels are extensively utilized across various sectors, such as the automotive and railway industries, owing to their impressive mechanical properties, including strength, hardness, and fatigue resistance. However, the pursuit of achieving the desired optimal mechanical properties presents considerable challenges due to the intricate bainitic microstructures consisting of multiple phases. To tackle these challenges, an automated workflow is used for extracting 2D and 3D microstructural features. The proposed method allows for a detailed examination of the correlations between microstructure characteristics and the processing parameters, specifically the holding temperature during transformation. In these findings, it is revealed that as the holding temperature decreases, there is a notable reduction in microstructural element size and carbon partitioning. Some of the observations are microstructural features such as area, perimeter, and thickness of the bainitic ferrite grains under two different holding temperatures. Phase‐field simulations results show that the microstructures at lower holding temperatures have finer grains. The distributions of grain areas and perimeters are uniform, with smaller grains dominating at low and high isothermal holding temperatures. While the grain thickness measurements from simulations and experiments at high temperature are qualitatively aligned, data from low temperatures show discrepancies.

Recent breakthroughs resolving open questions in materials science by phase-field simulations are reported. They relate to solidification structure formation in additive manufacturing, carbon redistribution during bainitic transformation, and the onset of damage during high-temperature creep of superalloys. The first example deals with the balance between epitaxial growth and nucleation in solidification. The second relates to the controversy regarding diffusion control and dominance of massive transformation in bainite transformation. The third relates to directional coarsening (rafting) in superalloys as a diffusion-controlled phase transformation: loss of coherency of precipitates marks the onset of damage associated with rotation of the crystal lattice and topological inversion. Technical details of the phase-field method are reviewed as necessary, and limitations of the approach are discussed. Graphical

We present an effective mechanical equilibrium solution algorithm suitable for finite strain consideration within the phase-field method. The proposed algorithm utilizes a Fourier space solution in its core. The performance of the proposed algorithm is demonstrated using the St. Venant–Kirchhoff hyperelastic model, but the algorithm is also applicable to other hyperelastic models. The use of the fast Fourier transformation routines and fast convergence within several iterations for most common simulation scenarios makes the proposed algorithm suitable for phase-field simulations of rapidly evolving microstructures. Additionally, the proposed algorithm allows using different strain measures depending on the requirements of the underlying problem. The algorithm is implemented in the OpenPhase phase-field simulation library. A set of example simulations ranging from simple geometries to complex microstructures is presented. The effect of different externally applied mechanical boundary conditions and internal forces is also demonstrated. The proposed algorithm can be considered a straightforward update to already existing small strain solvers based on Fourier space solutions.

For the purpose of numerically exploring different microstructure evolution processes, the phase-field approach has been developed. Large domains and adequate models that combine many physical fields in statistically relevant volume elements are needed to describe these complicated processes accurately. In order to facilitate such development for multi-physics simulations, OpenPhase was developed for application in material science and engineering. This chapter presents two examples simulated using the OpenPhase software, grain growth and dendritic solidification. The corresponding input files for OpenPhase are listed and explained. Also, some hints for postprocessing are given.

Many regions of the Earth’s mantle deform in grain size-sensitive creep regimes. The grain size right below the transition zone is believed to be very small, and the grain size should in subsequent depths be mainly controlled by normal grain growth. The grain size evolution is commonly predicted using either existing grain growth laws in combination with grain boundary diffusion coefficients or by extrapolating empirically determined grain growth laws. Effects of Zenner pinning and different ratios of second phases have been studied, while the role of anisotropic grain boundary properties is currently neglected1.  The grain boundary energy varies with the orientation of the grain boundary plane, as expressed through the typical crystal habitae (Wulff-shapes). Individual crystals in a polycrystalline material maintain a grain boundary energy anisotropy during grain growth. Here we study how grain boundary anisotropy impacts grain boundary migration and normal grain growth rates by three-dimensional phase-field simulations2. We imply grain boundary energy minimization by faceting/varying the grain boundary plane to minimize the grain boundary energy. The ideal grain boundary energy anisotropy for the solid-solid interface is taken from experimentally investigated grain boundary plane distributions and grain boundary energy distributions on periclase (MgO). We compare the grain size evolution in simulations with isotropic and anisotropic grain boundary energy of cubic crystal symmetry. We found that the grain boundary energy anisotropy has a significant influence on grain boundary migration and grain growth kinetics2. 2 - 2 = kt The change in grain size is given as variation between the initial and final average grain radius, r. The time is t, and a material-specific parameter that accounts for the grain boundary energy anisotropy is extracted from the simulations as k = A·µgb·σgb Where, the grain boundary energy, σgb varies with orientation, and the grain boundary mobility, µgb assumed to be isotropic. We found that the rate of grain growth for periclase, A is a factor of 3 smaller compared to an isotropic material. These results are of three-fold importance: The theory of grain growth is developed for purely isotropic materials1 and clearly fail to predict grain growth for materials with anisotropic grain boundary energy correctly. Predicting grain growth using existing rate-laws and grain boundary diffusion studies can lead to an incomplete conclusion and a distorted picture of the real phenomenon. Because energy reduction associated with grain growth is small, already small energy variations, such as those associated with grain boundary energy anisotropy, can have a significant effect. A better prediction of grain size evolution will need to develop an anisotropic theory for grain growth, that address our lake in knowledge regarding pressure effects on both grain boundary energy anisotropy and diffusion. 1. Rohrer, G. S. Annu. Rev. Mater. Res. 2005 2. Salama et al. Acta Materialia 2020