Sha W., Malinov S. Titanium Alloys: Modelling of Microstructure, Properties and Applications

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Titanium alloys: modelling of microstructure228

Temp 850 °C

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(d) at 800 °C

Temp 800 °C

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(e) at 730 °C

Temp 730 °C

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8.9

Continued

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Finite element method: morphology 229

The isothermal simulations are also performed at different temperatures,

in order to study the difference in the morphology of the α phase after the

transformation has completed. Fig. 8.10d,e and f show the final morphology

of the α colony after isothermal exposures at 900, 930 and 850 °C, respectively.

The model simulations showed that lower temperature of isothermal exposure

results in the following:

(i) Finer microstructure with thinner α lamellae (Fig. 8.10f). This is in

agreement with the results from metallographic examination of

(a) 900 °C, 2 min

Time 2 min

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(b) 900 °C, 5 min

Time 5 min

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8.10

FEM simulations for the microstructure evolution of the β to α

phase transformation in Ti-6Al-4V alloy under isothermal conditions.

The scales to the right of each figure represent vanadium

concentration. (a) 900 °C, 2 min; (b) 900 °C, 5 min; (c) 900 °C, 10 min;

(d) 900 °C, 30 min; (e) 930 °C, 30 min; (f) 850 °C, 30 min.

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Titanium alloys: modelling of microstructure230

Time 10 min

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(d) 900 °C, 30 min

Time 30 min

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(e) 930 °C, 30 min

Time 30 min

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8.10

Continued

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Finite element method: morphology 231

samples quenched after different temperatures of isothermal exposure

(Chapter 6).

(ii) Larger amount of the α phase. This is in agreement with the knowledge

of the phase equilibria in the Ti-6Al-4V alloy, and the calculations

performed on the equilibrium α phase volume fractions (see Fig. 5.1a).

We obtain good agreement between the FEM simulations and the

thermodynamic calculations in the amounts of α phase, especially at

the relatively high temperatures (Table 8.1).

Using the 1-D and 2-D models, we are able to predict the main features of

the Widmanstätten morphology formed during thermal treatment – the

distribution and geometry of α phase lamellae and their dependence on the

processing route. However, since the real microstructure formation problem

is a 3-D one, a precise description of the evolution of the α/β interface, α-

phase transformed fraction and vanadium concentration in the β phase can

be obtained only by solving the 3-D problem for nucleation and growth of

the Widmanstätten α plates. The development of a 3-D model, as well as

Time 30 min

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(f) 850 °C, 30 min

Table 8.1

Equilibrium (Thermo-Calc) and calculated (FEM)

amounts of the α phase

(°C) Thermo-Calc FEM (vol.%)

(vol.%)

950 23 22

930 36 35

900 53 45

850 73 59

8.10

Continued

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Titanium alloys: modelling of microstructure232

extension of the model to predict for larger domain with initial real β grain

and even group of β grains, however, requires a dramatic increase of

computational resources.

Finally, it should be mentioned that the simulation discussed above is

obtained either for cooling with constant cooling rate or for isothermal exposure.

However, the models developed and program packages are capable of 1-D

and 2-D simulations of the morphology of the β to α phase transformation in

Ti-6Al-4V alloy for continuous cooling with any cooling path (not necessarily

constant cooling rate) as well as for an arbitrary combination and mix between

continuous cooling and isothermal exposure. In this way, real production

time–temperature routes can be modelled.

8.7 Summary of the models for Ti-6Al-4V

The morphology, distribution and geometry of the fine microstructure of Ti-

6Al-4V alloy, formed during heat treatment, and their dependence on the

processing route are simulated in the framework of a mathematical model

describing the nucleation and growth processes of the new phase. The finite

element modelling and the volume of fluids method are used for obtaining a

numerical solution of the problem for localisation and evolution of the α/β

interface. Computer program packages are developed for solving the 1-D

and 2-D problems for nucleation and growth of the α phase Widmanstätten

plates in the β phase.

The packages are used to simulate the morphology of the microstructure

evolution during different heat treatment conditions. The effects of the cooling

rate and the temperature on the microstructure are predicted and analysed. A

good agreement between the model simulations and experimental data is

observed regarding (i) the α lamellae thickness obtained with different heat

treatment conditions; (ii) the kinetics of the β to α phase transformation; (iii)

the rate of nucleation of the α phase; and (iv) the simulated phase equilibria.

The model can be used for predictions of the morphology of the β to α

phase transformation in Ti-6Al-4V alloy in a real processing route.

8.8 Extending to other alloys

The values for the driving force obtained by thermodynamic calculations

(see Chapter 5) are used in deriving ∆g. The unknown parameters, such as

number of nucleation sites per unit volume, free interface energy of nucleation

and shape factor, are derived by fitting the model simulation to the experimental

data from the microstructural observations (Figs. 8.1 and 8.2b) and kinetics

of phase transformations under different conditions (Fig. 7.7). The calculated

barrier and rate of nucleation for the β to α transformation in Ti 6-2-4-2 alloy

are given in Fig. 8.11.

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Finite element method: morphology 233

The process of α phase growth is simulated based on the assumption that

the migration of the interface separating the two phases is a result of flux of

alloying element atoms across the interface. As the precipitate grows, the β

alloying element concentration adjacent to the interface C

increases above

the initial concentration C

. In this way, the process of diffusion redistribution

between the α and the β phases takes place, leading to enrichment of the β

phase with different alloying elements depending on the alloy composition

(see Chapter 5).

8.11

Calculated (a) barrier for nucleation and (b) rate of nucleation

for the β to α transformation in Ti-6Al-2Sn-4Zr-2Mo alloy as functions

of the molybdenum content and the temperature. In calculating

∆

in Kelvin is used.

2 wt.% Mo

6 wt.% Mo

10 wt.% Mo

14 wt.% Mo

18 wt.% Mo

2 wt.% Mo

6 wt.% Mo

10 wt.% Mo

14 wt.% Mo

18 wt.% Mo

∆

100

0.1

750 800 850 900 950 1000

(°C)

(a)

750 800 850 900 950 1000

(°C)

(b)

Rate of nucleation (nuclei/µm.s)

0.005

0.004

0.003

0.002

0.001

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Titanium alloys: modelling of microstructure234

8.12

FEM simulations for the microstructure evolution of the β to α

phase transformation in Ti-6Al-2Sn-4Zr-2Mo alloy at continuous

cooling with constant cooling rate of (a) – (e) 10 °C/min and

(f) 5 °C/min, assuming α colony growth in three different directions.

The scales to the right of each figure represent molybdenum

concentration. (a) 960 °C; (b) 940 °C; (c) 920 °C; (d) 880 °C; (e) 760 °C;

and (f) 5 °C/min, 760 °C.

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(d)

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(c)

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(b)

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(e)

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Finite element method: morphology 235

The model described in previous sections is adjusted and used to simulate

the phase transformation morphology in Ti 6-4 alloy only. In the Ti 6-4 alloy,

the β to α phase transformation is controlled by the diffusion redistribution

of vanadium between the two phases (see Fig. 5.1). The model can be further

developed and adjusted for simulation of other α, near-α and α + β alloys,

where the process of phase transformation may be controlled by diffusion

redistribution of other elements, e.g. molybdenum in the Ti 6-2-4-2 and Ti 8-

1-1 alloys.

One numerical simulation of the β to α phase transformation in the Ti 6-

2-4-2 alloy, when the grain boundary α phase colonies grow in different

directions, is presented in Fig. 8.12. The microstructure evolution at cooling

from β region is shown (Fig. 8.12a–e).

8.9 Summary

Models and modules for simulation of the microstructure evolution during

the course of the β to α phase transformation at cooling in α, near-α and α

+ β titanium alloys have been developed. The models are based on classical

theory for nucleation and diffusion controlled growth. Experimental data are

used to build and verify the models. The models and program packages

developed are capable of 1-D and 2-D simulations of the morphology of the

β to α phase transformation in titanium alloys for continuous cooling with

any cooling path and for an arbitrary combination between continuous cooling

and isothermal exposure, including that representing a real processing route.

The models are convenient and powerful tools for optimisation of the processing

parameters in titanium alloys. The programs have potential for direct practical

applications in solving various problems in titanium alloys and for a significant

reduction of the experimental work.

The model cannot be used straightforwardly to simulate the β to α + β

phase transformation in β or near-β alloys because these alloys are usually

quenched after solution treatment and the β to α + β phase transformation

takes place during additional ageing after quenching. The phase nucleation

and growth may have significantly different mechanisms.

8.10 References

Borgenstam A, Engström A, Hoglund L and Ågren J (2000), ‘DICTRA, a tool for simulation

of diffusional transformations in alloys’, J Phase Equilibria, 21 (3), 269–80.

Malinov S, Sha W and Voon C S (2002), ‘In situ high temperature microscopy study of

the surface oxidation and phase transformations in titanium alloys’, J Microscopy, 207

(3), 163–68.

Malinov S, Katzarov I and Sha W (2005), ‘Modelling, simulations and monitoring of

lamella structure formation in titanium alloys controlled by diffusion redistribution’,

Defect and Diffusion Forum, 237–240, 635–46.

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Titanium alloys: modelling of microstructure236

Shin S and Lee W I (2000), ‘Finite element analysis of incompressible viscous flow with

moving free surface by selective volume of fluid method’, Int J Heat Fluid Flow, 21

(2), 197–206.

Wilkinson D S (2000), Mass Transport in Solids and Fluids, Cambridge: Cambridge

University Press.

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237

Phase-field method: lamellar structure

formation in γ-TiAl

Abstract: This chapter describes a phase-field model for simulating the

formation evolution of the lamellar microstructure in γ-TiAl alloys. The

mechanism of formation of TiAl lamellae proposed by Denquin and Naka is

incorporated into the model. The model describes the formation and

evolution of the face-centred cubic (fcc) stacking lamellar zone followed by

the subsequent appearance and growth of the γ phase, involving both the

chemical composition change by atom transfer and the ordering of the fcc

lattice. This chapter describes the model in detail to enable the reader to

undertake similar modelling.

Key words: phase field models, titanium aluminides, phase transformation,

modelling, simulation.

9.1 Introduction

Alloys based on the aluminides of titanium that contain large volume fractions

of the γ-TiAl phase have attracted particular interest. For single-phase γ-TiAl

alloys, the ductility is limited, giving poor formability and fracture toughness.

These problems are ameliorated partially by adjusting the composition and

heat-treatment to give a lamellar mixture of γ-TiAl and α

-Ti

Al phases.

Alloys with such a microstructure usually exhibit slightly improved ductility

and even higher strength than single-phase alloys.

The lamellar structure is generally characterised by a set of several γ

lamellae alternating with a single α

lamella, following the orientation

relationship: (0001)

α2

//{111}γ, and

〈〉〈〉1120 / / 110

2α

. The lamellar

microstructure is formed by decomposition of either disordered α-phase

(hexagonal structure, A3) or ordered α

(hexagonal structure, DO

) matrix

into lamellar precipitates of γ-phase (ordered tetragonal structure, L1

following one of the two transformation paths: α → α

→ α

+ γ or α →

+ γ. Each γ lamella is divided into numerous order domains. These domains

are generated due to the fact that the

〈〉110

and

〈〉011

directions of the L1

structure are not equivalent to each other.

Transmission electron microscopy studies on the initial stages of the

precipitation have shown the existence of large stacking faults (SF), which

originate at prior grain boundaries or grain boundary γ allotriomorphs. The

mechanism of the formation of lamellar structure supposes the formation of

γ lamellae at these faults. The whole sequence of the lamellar structure

formation involves: (i) a crystal structure change from hexagonal close-

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