Examples of Microstructures Predicted by the Microstructure Simulator
In this program (DMR1435483 (OSU) and DMR1435611(UNT)), we have developed various modeling capabilities and computational tools such as a new modified embedded atom method (MEAM) potential for Ti, a combined NEB + multi-phase-field method to determine the properties of critical nuclei, an explicit nucleation algorithm in the phase field method that seeds these critical nuclei, and phase field codes for microstructural evolution during both cooling and heating in different Ti-alloy systems considered.
Major results delivered include:
(1) Alloy compositions and heat treatment schedules for achieving various refined and super-refined microstructures
(2) A microstructure Simulator
(3) Sources codes for new MEAM Ti model included in Open KIM system for use with LAMMPS
(4) Co-evolution of microstructure and texture during a precipitation in polycrystalline b matrix, a dissolution upon heating and effect of dislocations and GB dislocation network on a precipitation
(5) An artificial neural network (ANN) model for strength and ductility
(6) Free energy and interfacial energy data (database files available upon request)
(7) Synthetic microstructural data and animations
Finding Critical Nucleus for Heterogeneous Nucleation during Transformation in Titanium Alloys
A fine, uniform and microtexture-free a+b precipitate microstructure in Ti-alloys is difficult to achieve but highly desired for an optimal balance of mechanical properties. Depending on the b grain size and texture and cooling rate, nucleation of the a phase particles may occur predominantly inter- or intra-granularly. The former (nucleation at crystalline defects such as GBs and dislocations [1-5]) is most likely to lead to coarse and strongly textured precipitate microstructures (See Fig. 1(a)) while the latter is most likely to lead to fine and randomly textured precipitate microstructures (Figs. 1(b) and (c)). Recently ultra-fine a+b precipitate microstructures (Figs.1(b)-(c)) have been achieved in Ti-alloys via intra-granular nucleation under the influence of pre-existing compositional and/or structural non-uniformities in the parent phase grains that could arise, for example, from precursory precipitation in and spinodal decomposition of the phase [6-13].
Figure 1. Typical coarse alpha+beta microstructure (a) in Ti-alloys produced through intergranular nucleation and growth of precipitates at prior grain boundaries. Refined (b) and super-refined (c) alpha+beta microstructures produced through intragranular nucleation and growth of precipitate in grain interior via non-conventional transformation pathways via the pseudo-spinodal and precursory omega precipitation mechanisms, respectively.
A fundamental understanding of the mechanisms of inter- and intra-granular heterogeneous nucleation of a precipitates and their interplay is key to control the scale, uniformity and micro-texture of a+b microstructures in Ti-alloys. In this program, we strive to develop such understanding by integrating critical experimental characterization (Figs. 2(a) and (b)) with multi-scale computational modeling and simulation (Figs. 2(c) and (d)). The fundamental properties of a critical nucleus of the a phase, including their configurations and activation energies for heterogeneous nucleation at arbitrary existing concentration and structural non-uniformities (precursory w metastable precipitate Fig. 2(b)) as well as crystalline defects such as dislocations and grain boundaries (Figs. 2(a)) have been obtained using a fully variational approach based on multi-phase field theory and nudged elastic band method (MPF-NEB) [14, 15]. With the nucleus and GB being treated as a whole and forces and torques being balanced naturally at triple junctions, and without any a priori assumptions about the shapes of the critical nucleus and GB plane, we show that both the critical nucleus shape (Fig. 2(c)) and the activation energy (Fig. 2(d)) differ significantly from those obtained by previous approaches based on the graphical construction of the critical nucleus shape.
Taking advantages of available atomistic simulation results on heterogeneous nucleation of (ferrite) precipitates at prior (austenite) grain boundaries (GBs) in a pure Fe polycrystalline system, the MPF-NEB approach developed for quantitative prediction of properties of a critical nucleus (including size, shape and activation energy) at GB as a function of grain boundary energy, interfacial energy, relative orientation between a low-energy facet of the nucleus and the GB plane have been validated .
Figure 2. (a) A super-refined a+b microstructure in a Ti-alloy (a) produced by intragranular nucleation; (b) possible nucleation and growth of an a precipitate at an w precipitate; (c) predicted configuration of a critical nucleus of the a phase (red) at a prior b grain boundary (green); (d) the activation energy of nucleation as a function of facet inclination of the nucleus predicted by the MPF-NEB method (discrete symbols) and its comparison with Lee and Aaronson’s approach  (red curve).
Effect of Low-angle Grain Boundaries on Morphology and Variant Selection of Grain Boundary Allotriomorphs and Widmanstatten Side-Plates [3-5]
Morphology and variant selection (VS) behavior of GB allotriomorphs and Widmanstatten sideplates of phase in an titanium alloy, Ti-6Al-4V (wt%), are investigated using a three-dimensional phase field model (Fig. 3). The structures of low-angle GBs (misorientation ) are modeled as discrete dislocation networks using Frank-Bilby theory. It is shown that allotriomorphs and side-plates compete with each other during precipitation and the final morphology and selected variants exhibit a strong correlation with the GB dislocation structures. While the side-plate morphology is more preferred by a symmetrical tilt GB with , it can also be produced by heterogeneous nucleation at a pure twist GB with . Quantitative analysis indicates that precipitate morphology and VS are determined by the interplay among (i) elastic interaction between a nucleating precipitate and the GB dislocation network, (ii) growth anisotropy determined by the relative inclination of the habit plane with respect to the GB dislocations, (iii) density of nucleation sites for the same variant and coalescence during growth, and (iv) spatial confinement from simultaneously nucleated neighboring variants of dissimilar types. These findings may help to identify at what GBs (characterized by misorientation and inclination) discontinuous is preferred over a continuous layer of that would has a deleterious effects on tensile ductility.
Figure 3 (a) Morphologies and variant selection of precipitates at (a) a single straight edge dislocation in matrix; (b) symmetrical title GB; (c) pure tilt GBs. Different variants of are colored according to the color bar.
Experimental assessment of variant selection rules for GBa in Ti-Alloys
The applicability of all current empirical rules for VS of GB by prior GBs has been assessed systematically using experimental characterization of GB misorientation, GB plane inclination, and orientation relationships between the GB and adjacent grains in Ti-5553 (Fig. 4) . In particular, how a single or a combination of different rules contributes to VS of GB have been analyzed and evaluated systematically against the experimental observations. It is found that all the VS rules could be violated for a given grain boundary. Based on the frequencies of each of these empirical rules being violated or followed from the experimental observations, whether the arguments underlying each of these rules are physically sound and why rules are violated are analyze theoretically, and when a sound prediction could be made using these empirical VS rules is also discussed.
Figure 4 Schematic illustration of different empirical rules concerning the influence of grain boundary (GB) parameters, misorientation and grain boundary plane (GBP) inclination, on variant selection of grain boundary alpha, GB . is the disorientation angle associated with the deviation matrix that is a quantitative measure of the deviation of the orientation relationship (OR) between the GB and the non-Burgers grain from the Burgers OR. are the inclination angles between the GBP and one of the planes and are the inclination angles between the GBP and one of the directions. X, Y, Z represents the sample reference frame where orientations of matrix grains and grain boundary plane inclinations are expressed.
Quantitative Assessment of Competition between Inter- and Intra-Granular Nucleation of α upon Cyclic Cooling and Heating
In order to address the influence of heating/cooling rate on the refinement of microstructure as well as effect of variation of alloy composition with specification, a quantitative 3D phase field model of non-isothermal process for Ti-6Al-4V has been developed. The microstructural evolution processes during cooling and heating processes in a bi-crystal are investigated and the results are shown in Fig. 5. Upon a relatively slow cooling rate (50K/s) and small amount of -phase stabilizer (5.4wt%Al), only a few variants of -phase form both as an allotriomorph along prior GBs and as Widmanstatten plates, leading to a relatively coarse lamellar structure, as shown in Fig. 2(a). In this case, inter-granular nucleation at GB is dominant with little intragranular precipitation observed. In contrast, with increasing cooling rate ( up to 150K/s) and amount of stabilizer (up to 6.4wt%Al), intragranular nucleation mechanism starts to operate (b) and becomes more dominant over intergranular nucleation mechanism (c), leading to a relatively fine microstructure with basketweave morphologies. Build upon on such parametric studies, the ratio between homogeneous nucleation rate and heterogeneous nucleation rate in the composition-cooling rate is predicted.
Figure. 5: Influence of alloy composition and cooling rate on microstructure development during (red) precipitation in a bi-crystal matrix (blue and transparent): (a) Ti-5.4Al-4V at 50K/s; (b) Ti-5.8Al-4V at 100K/s; (c) Ti-6.4Al-4V at 150K; (d) Ratio between homogeneous nucleation rate and heterogeneous nucleation rate as a function of alloy composition and cooling rate. (a)-(c) show a clear transition from a relative coarse colony structure produced by inter-granular nucleation at GB to a relative refined basketweave microstructure produced by intra-granular nucleation.
Investigation of w precipitation and solute partitioning between w and b in binary Ti-Mo and Ti-V alloys
In order to understand the mechanism of the heating rates effect (i.e., why the -assisted nucleation mechanism was found only upon continuous heating) and hence optimize the heat treatment schedule, the individual influences from concentration non-uniformity produced by solute partitioning between isothermal and matrix (Fig. 6), and from the misfit stress field associated with coherent precipitate (Fig.7), have been quantified in Ti-20Mo and high misfit Ti-20V (wt%) system using Thermo-Calc / Pandat and microelasticity theory, respectively. In addition, based on the atom probe tomography (APT) analysis of solute concentration at interfaces, the depandance of the Gibbs free energy of the metastable phase on solute concentration for Ti-Mo and Ti-V systems at has been determined , which will serve as an input in the MPFM-NEB model to quantify the combined effect from both concentration and stress on nucleation.
Figure 6. Variation of chemical driving forces for (a) the nucleation of a and (b) the growth of a as a function of V composition in the b matrix
Figure. 7 Elastic interaction energy, , associated with a nucleating a nucleus around a pre-existing coherent w particle (a)-(b) large and small cuboidal w particle, and (c) ellipsoidal w particle. The size along <111>b of ω particles in (a)-(c) are around 50, 25 and 30 nm, respectively. The (010)β cross-section through the center shows the variation of around the w precipitate. The iso-surface in purple (corresponding to =-1250J/mol) indicates the most favorable nucleation site, i.e. along the  edge of the precipitate
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Alpha Dissolution upon Fast Heating
Dissolution at 1000°C (1831°F)
Bcc (β) region with alpha/near alpha (α) composition ← structural change leads to compositional one
Variant Selection in Polycrystalline Beta Matrix
Alpha Reprecipitation/Growth upon Fast Cooling