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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 [14].

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 [16] (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) [2]. 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 [010] edge of the precipitate


[1] R. Shi, Y. Zheng, D. Wang, H. Fraser, Y. Wang. Heterogenous Nucleation During β→ α+ β Transformation in Titanium Alloys.  Proceedings of the 13th World Conference on Titanium: John Wiley & Sons, Inc.; 2016. p. 1931-1936.

[2] R. Shi, V. Dixit, G. B. Viswanathan, H. L. Fraser, Y. Wang, Experimental assessment of variant selection rules for grain boundary alpha in titanium alloys, Acta Materialia, 102  (2016) 197-211.

[3] D. Qiu, P. Y. Zhao, R. P. Shi, Y. Z. Wang, W. J. Lu, Effect of autocatalysis on variant selection of a precipitates during phase transformation in Ti-6Al-4V alloy, Computational Materials Science, 124  (2016) 282-289.

[4] D. Qiu, R. Shi, P. Zhao, D. Zhang, W. Lu, Y. Wang, Effect of low-angle grain boundaries on morphology and variant selection of grain boundary allotriomorphs and Widmanstatten side-plates, Acta Materialia, 112  (2016) 347-360.

[5] D. Qiu, R. Shi, D. Zhang, W. Lu, Y. Wang, Variant selection by dislocations during alpha precipitation in alpha/beta titanium alloys, Acta Materialia, 88  (2015) 218-231.

[6] Y. Zheng, R. E. A. Williams, D. Wang, R. P. Shi, S. Nag, P. Kami, J. M. Sosa, R. Banerjee, Y. Z. Wang, H. L. Fraser, Role of omega phase in the formation of extremely refined intragranular a precipitates in metastable ss-titanium alloys, Acta Materialia, 103  (2016) 850-858.

[7] Y. Zheng, R. E. A. Williams, J. M. Sosa, Y. Wang, R. Banerjee, H. L. Fraser, The role of the ω phase on the non-classical precipitation of the α phase in metastable β-titanium alloys, Scripta Materialia, 111  (2016) 81-84.

[8] Y. Zheng, R. E. A. Williams, J. M. Sosa, T. Alam, Y. Z. Wang, R. Banerjee, H. L. Fraser, The indirect influence of the omega phase on the degree of refinement of distributions of the alpha phase in metastable beta-Titanium alloys, Acta Materialia, 103  (2016) 165-173.

[9] Y. Zheng, J. M. Sosa, R. E. Williams, Y. Wang, R. Banerjee, H. L. Fraser. Development of Ultrafine α Microstructures in a Metastable β Titanium Alloy.  Proceedings of the 13th World Conference on Titanium: John Wiley & Sons, Inc.; 2016. p. 521-527.

[10] Y. Zheng, J. M. Sosa, H. L. Fraser, On the Influence of Athermal omega and alpha Phase Instabilities on the Scale of Precipitation of the alpha Phase in Metastable beta-Ti Alloys, JOM, 68 (5) (2016) 1343-1349.

[11] Y. Zheng, D. Choudhuri, T. Alam, R. E. A. Williams, R. Banerjee, H. L. Fraser, The role of cuboidal omega precipitates on alpha precipitation in a Ti-20V alloy, Scripta Materialia, 123  (2016) 81-85.

[12] Y. Zheng, R. E. A. Williams, R. Shi, Y. Gao, D. Wang, S. Nag, Y. Wang, R. Banerjee, H. L. Fraser. Integrated Experimental and Computational Studies of Non-Conventional Transformation Pathways in Titanium Alloys.  Proceedings of the International Conference on Solid-Solid Phase Transformations in Inorganic Materials 20152015. p. 663-670.

[13] Y. Zheng, R. E. A. Williams, H. L. Fraser, Characterization of Various Interfaces Structure in a Titanium Alloy Using Aberration-Corrected Scanning Transmission Electron Microscope, Microscopy and Microanalysis, 21 (S3) (2015) 1517-1518.

[14] H. Song, R. Shi, Y. Wang, J. J. Hoyt, Simulation Study of Heterogeneous Nucleation at Grain Boundaries During the Austenite-Ferrite Phase Transformation: Comparing the Classical Model with the Multi-Phase Field Nudged Elastic Band Method, Metallurgical and Materials Transactions A,  (2016) 1-9.

[15] R. S. Yunzhi Wang. FINDING CRITICAL NUCLEUS FOR HETEROGENEOUS NUCLEATION AT GRAIN BOUNDARIES DURING SOLID-STATE PHASE TRANSFORMATIONS.  Proceedings of the International Conference on Solid-Solid Phase Transformations in Inorganic Materials 20152015. p. 837-844.

[16] J. K. Lee, H. I. Aaronson, Influence of faceting upon the equilibrium shape of nuclei at grain boundaries—II. Three-dimensions, Acta Metallurgica, 23 (7) (1975) 809-820.

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

Ti-5.4Al-4V 30C/s


Ti-5.8Al-4V 100C/s


Ti-6.4Al-4V 150C/s


Explicit Nucleation