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

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Titanium alloys: modelling of microstructure198

= 0.998

0 0.2 0.4 0.6 0.8 1

Fraction experimental

(a) Ti-6Al-4V

Fraction calculated

0.8

0.6

0.4

0.2

Data points

Best linear fit

Exp. = calc.

= 0.998

0 0.2 0.4 0.6 0.8 1

Fraction experimental

Fraction calculated

0.8

0.6

0.4

0.2

Data points

Best linear fit

Exp. = calc.

R = 0.999

0 0.2 0.4 0.6 0.8 1

Fraction experimental

(b) Ti-6Al-2Sn-4Zr-2Mo-0.08Si

Fraction calculated

0.8

0.6

0.4

0.2

7.19

Regression analysis of calculated and measured degree of transformation during continuous cooling in different

titanium alloys. The data are for all cooling rates used.

Data points

Best linear fit

Exp. = calc.

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Johnson–Mehl–Avrami method adapted to continuous cooling 199

Data points

Best linear fit

Exp. = calc.

= 0.999

0 0.2 0.4 0.6 0.8 1

Fraction experimental

(d) IMI 834

Fraction calculated

0.8

0.6

0.4

0.2

Data points

Best linear fit

Exp. = calc.

= 0.997

0 0.2 0.4 0.6 0.8 1

Fraction experimental

(e) IMI 367

Fraction calculated

0.8

0.6

0.4

0.2

Data points

Best linear fit

Exp. = calc.

Fraction calculated

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1

Fraction experimental

(f) Ti-46Al-2Mn-2Nb

= 0.994

7.19

Continued

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Titanium alloys: modelling of microstructure200

An option is incorporated into the model to enable it to be linked to a real

furnace. In this case, the cooling path of the furnace (i.e. time–temperature

curve) is read into the model as an ASCII data file. Thus, the transformation

process is monitored in real-time. This kind of model could provide a useful

insight into such processes.

7.8 Summary

The DSC technique is suitable for quantitative description of transformations

in titanium alloys during continuous cooling. Using DSC data, CCT diagrams

with lines of iso-degree of transformation are proposed. The cooling rate is

shown to influence the lattice parameter of the α phase in conventional

titanium alloys.

The JMA theory can be adapted to describe and model the kinetics of

transformations in titanium alloys as functions of the temperature. The kinetic

parameters can be derived. Good agreement between the calculated and the

experimental transformed fractions is demonstrated. Using the derived kinetic

parameters, the transformations can be described for any cooling path and

conditions. A model has been developed to monitor the transformation process

for complex cooling paths.

Input Output

Continuous cooling

path (temperature–

time curve)

Kinetic

parameters

Thermodynamic

equilibria

Monitoring the kinetics

of transformation

(transformed fraction

as a function of the

time/temperature)

Temperature

Fraction

Time

Time/temperature

The model

From furnace or

arbitrary user

defined cooling

path

7.20

Block diagram of model for tracing transformation on

continuous cooling.

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Johnson–Mehl–Avrami method adapted to continuous cooling 201

7.21

Calculated kinetics of β to α transformation in Ti-6Al-2Sn-4Zr-

2Mo-0.08Si for continuous cooling with an arbitrary cooling path.

Input

Beta transus

Output

0 200 400 600 800 1000 1200 1400 1600

(°C)

1100

1050

1000

950

900

850

800

100

Amount of α phase (%)

0 200 400 600 800 1000 1200 1400 1600

Time (sec)

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Titanium alloys: modelling of microstructure202

7.9 References

Bein S and Bechet J (1996), ‘Phase transformation kinetics and mechanisms in titanium

alloys Ti-6.2.4.6, β-CEZ and Ti-10.2.3’, J Physique IV, 6 (1), C1-99–108.

Butler C J (1995), The Solidification Characteristics of Titanium Aluminides, PhD thesis,

University of Nottingham, UK.

Cias W W (1991), in: VanderVoort G F (ed), Atlas of Time–Temperature Diagrams for

Nonferrous Alloys, Materials Park, OH: ASM International, 376–80. [Source: Cias W

W, ‘Phase transformation kinetics, microstructures, and hardenability of the Ti-6Al-

2Sn-4Zr-2Mo titanium alloy’, Rp-27-72-03, Climax Molybdenum, 7 June 1972.]

Homberg D (1996), ‘A numerical simulation of the Jominy end-quench test’, Acta Mater,

44 (11), 4375–85.

Lütjering G and Williams J C (2003), Titanium, Berlin: Springer, 233.

Malinov S, Guo Z, Sha W and Wilson A (2001a), ‘Differential scanning calorimetry study

and computer modeling of β ⇒ α phase transformation in a Ti-6Al-4V alloy’, Metall

Mater Trans A, 32A (3A), 879–87.

Malinov S, Guo Z, Sha W, Guo Z X and Wilson A (2001b), ‘Differential scanning

calorimetry study and computer modelling of β ⇒ α phase transformation in Ti-6Al-

2Sn-4Zr-2Mo alloy’, in Winstone M R (ed), Titanium Alloys at Elevated Temperature:

Structural Development and Service Behaviour, London: IoM Communications, 69–

88.

Malinov S and Sha W (2001), ‘Computer modelling of the kinetics of phase transformation

in Ti-46Al-2Mn-2Nb titanium alloy’, in Proceedings of Annual Scientific Session,

Varna, Bulgaria, The Technical University of Varna, 29–34.

Mitchell D R (1991), in: VanderVoort G F (ed), Atlas of Time–Temperature Diagrams for

Nonferrous Alloys, Materials Park, OH: ASM International, 374–75. [Source: Mitchel

D R, ‘Welding evaluation of Ti-6Al-2Sn-4Zr-2Mo Sheet’, TMCA Project BW-10-1,

Final Report, June 1968, as published in Aerospace Structural Metals Handbook for

Titanium, June 1978.]

Mythili R, Saroja S, Vijayalakshmi M and Raghunathan V S (2005), ‘Selection of optimum

microstructure for improved corrosion resistance in a Ti–5%Ta–1.8%Nb alloy’, J

Nucl Mater, 345 (2–3), 167–83

Rosenberg H W, Vordahl M B and Hunter D B (1991), in: VanderVoort G F (ed), Atlas of

Time–Temperature Diagrams for Nonferrous Alloys, Materials Park, OH: ASM

International, 354.

Sieniawski J, Filip R, Ziaja W and Grosman F (1996), ‘Microstructure factors in fatigue

damage process of two-phase titanium alloys’, in Blenkinsop P A, Evans W J and

Flower H M (eds), Titanium ’95: Science and Technology. Proceedings of the Eighth

World Conference on Titanium, London: The Institute of Materials, 1411–18.

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203

Finite element method: morphology of

β to α phase transformation

Abstract: A mathematical model and computer programs are described for

numerical simulation of the processes of nucleation and growth of the α

phase Widmanstätten plates during the course of the β to α phase

transformation in titanium alloys. The α phase appearance at the grain

boundary of β phase is described by a numerical procedure for random

nucleation as a function of the β stabiliser concentration and the temperature.

The rate at which an interface moves depends both on the intrinsic mobility

and on the rate at which diffusion can remove the excess of β stabiliser

atoms ahead of the interface.

Key words: microstructure evolution, finite element method, diffusion,

thermodynamics, nucleation.

8.1 Introduction

The modelling of phase transformations in titanium alloys described in the

last three chapters was about the thermodynamics and the kinetics during

processing without taking into account their morphology and geometry.

The morphology, distribution and geometry of the fine microstructure

cannot be described in the framework of the Johnson–Mehl–Avrami theory.

The problem for predicting the morphology of titanium alloys, formed during

phase transformations, can be solved with the development of mathematical

models describing the nucleation and growth processes of the new phase. If

the mathematical formulation of the problem is known as a system of partial

differential equations, one of the most powerful numerical techniques for

obtaining its solution is the finite element method (FEM).

In this chapter, we extend the kinetics modelling in the last two chapters.

We will show a mathematical model and one-dimensional and two-dimensional

finite element computer simulation of the nucleation and growth processes

of the α phase Widmanstätten plates lamella structure during the β to α + β

phase transformation on cooling from homogeneous β range, and the

morphology, distribution, and geometry of the fine microstructure. This type

of cooling is the practical case of cooling after β solution treatment of α,

near-α and α + β alloys (see the alloy classification in Chapter 1). The model

has been adjusted and verified to simulate the microstructure evolution in Ti-

6Al-4V (Ti 6-4), Ti-6Al-2Sn-4Zr-2Mo (Ti 6-2-4-2) and Ti-8Al-1Mo-1V (Ti

8-1-1) alloys. It is generally based on the fundamental metallurgical theories

for phase nucleation and diffusion controlled growth. The FEM will be used

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Titanium alloys: modelling of microstructure204

to obtain numerical solution of the problem for localisation and evolution of

the α/β interface. The model is a general and powerful tool for optimisation

of the processing parameters in order to obtain pre-defined microstructure

and properties for different applications.

8.2 Experimental and modelling methodology

During the course of the model development, many unknown parameters

need to be derived or fitted. This can be done with the help of data from

experiential work and integrating different modelling techniques.

Experimental data are used to feed and verify the models developed.

Once developed, the models are capable of giving predictions and simulations

for new processing conditions for which experimental data do not exist. The

experimental and materials conditions used for developing the model presented

in this chapter are described in an original research paper (Malinov et al.,

2005).

The phase transformations and microstructure evolution taking place at

different processing conditions are studied after heat treatment of furnace

cooling from the β field. Optical, including high-temperature in situ, microscopy

is used to characterise the morphology of the phases present and to study the

mechanisms of phase nucleation and growth in different heat treatment

conditions. Experiments using room-temperature microscopy on alloys after

different heat treatments are useful for studying the correlations between the

processing parameters and the fine microstructure. Experiments with high-

temperature microscopy are useful for studying in situ the morphology of

the α ↔ β phase transformations in titanium alloys. Information on the

specimens and further details on the microscopy study are given by Malinov

et al. (2002).

Differential scanning calorimetry (DSC) is used to in situ study the kinetics

of the β to α phase transformation at various conditions of continuous cooling.

The data from the DSC can be used to compare and fit the numerical simulations

for microstructure evolution in terms of the amounts of phases at different

temperatures during a continuous cooling process. The following time–

temperature cycle has been found particularly useful: heating up to β-field

using a heating rate of 20 °C/min; isothermally holding for 20 min for

homogenisation; and thereafter cooling with varying cooling rates of 5, 10,

20, 30, 40 and 50

°C/min.

The α phase appearance at the grain boundary of β phase is described by

a numerical procedure for random nucleation as a function of the alloy

composition and the temperature. The rate at which an interface moves

depends both on the intrinsic mobility and on the rate at which diffusion can

remove the excess of β stabilising atoms ahead of the interface. The finite

element method is used for solving the diffusion equation on the domain

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Finite element method: morphology 205

occupied by β phase. The elements chosen have dimensions in both space

and time. A computer code based on finite element modelling and the volume

of fluids method is used.

8.3 Experimental observation of the morphology

of the phase transformation

Thermodynamics and kinetics of the phase transformation process consider

mainly the amounts of phases present. However, different morphologies of

the phases and characteristics of the microstructure determine completely

different combination of mechanical properties, and therefore the application

of the alloys, even if the amounts of the phases present are the same. The

microstructure of the different alloys is formed during manufacturing and

subsequent thermomechanical and/or heat treatment processes. Usually, any

change of the heat treatment parameters, such as temperature, time, heating/

cooling rates, results in significant change in the characteristics of the

microstructure. Considering these, we discuss in this chapter experimental

study and computer modelling of the morphology of the phase transformations

in titanium alloys at different processing conditions.

In Fig. 8.1, examples are given for microstructure from a same alloy (Ti

6-2-4-2) and same processing history, with only difference in the final cooling

rate. The difference in the morphology is obvious and would result in difference

in the properties. All cooling rates given in Fig. 8.1 are within the range of

usual cooling in industrial conditions, ranging between furnace and air cooling.

The final cooling rate plays an essential role on the microstructure formation.

A technique was developed for in situ microscopy study of titanium alloys

based on the effect of thermal etching, which can be used to study the

morphology of the α ↔ β phase transformation in different processing

conditions including isothermal treatment and continuous heating/cooling.

Figure 8.2 shows examples of in situ high temperature microscopy images

taken at 780 and 840 °C for Ti 6-4 and Ti 6-2-4-2 alloys, respectively, upon

cooling from 1100 °C. The data from this type of study are discussed in

detail in Chapter 2 and are used to verify the model for the microstructure

evolution in titanium alloys.

8.4 Mathematical formulation in the model for the

microstructure of Ti-6Al-4V

The microstructure of the Ti-6Al-4V alloy after heat or thermomechanical

treatment in the β phase field depends strongly on the cooling rate from the

β region. When the alloy is slowly cooled below the β-transus temperature,

the α phase is nucleated and grown in plate form, starting from β-grain

boundaries. The resulting lamellar structure is referred to as plate-like α.



Titanium alloys: modelling of microstructure206

Faster cooling (air cooling) results in a fine needle-like α phase referred to

as acicular α phase. Certain intermediate cooling rates develop Widmanstätten

structures, which become finer with increasing cooling rate. The β phase

remains as a thin layer between the Widmanstätten α plates. The α plates

grow with their long dimension parallel to {110}β plane.

0.1 mm 0.1 mm

(a) (b)

0.1 mm

(e)

8.1

Microstructure of the Ti-6Al-2Sn-4Zr-2Mo alloy after continuous

cooling at different cooling rates. (a) 5 °C/min; (b) 10 °C/min;



Finite element method: morphology 207

The β to α transformation in Ti-6Al-4V begins when a nucleus of α phase

forms preferentially at β grain boundary. Since the vanadium content in the

β phase is greater than in the α phase (Fig. 5.1) and because of the low

diffusivity of the vanadium in the β phase, partial enrichment of vanadium

is observed along the α phase boundary. This phenomenon produces a locally-

stabilised β phase, and results in the formation of plate-like morphology of

the α grain boundary. The α phase grain boundary presents a morphology

with concave parts rich in vanadium that hinder the growth and with convex

parts rich in aluminium where the Widmanstätten α plate can grow easily.

The observations by transmission electron microscopy and the data of energy

dispersive spectrometry analysis indicate that the β to α phase transformation

is controlled mainly by vanadium diffusion. The Widmanstätten α plates

obtained at slower cooling rates are thicker than the plates obtained at faster

cooling rates due to the longer diffusion periods (Chapter 7).

The β to α phase transformation in titanium alloys including Ti-6Al-4V,

as stated above, follows the classical way of phase transformations, namely

nucleation and growth of the new α phase. The first appearance of the α

phase at the grain boundary within the metastable β phase is followed by the

growth of this nucleus into the surrounding matrix. An interface is created

during the nucleation stage and then migrates into the surrounding parent

phase during the growth stage by the transfer of atoms across the interface.

8.4.1 Nucleation

The free energy change associated with the process of nucleation of the α

phase has the following three main contributions:

(a) (b)

0.1 mm

8.2

In situ

high temperature microscopy images of (a) the Ti-6Al-4V

alloy at 780 °C and (b) the Ti-6Al-2Sn-4Zr-2Mo alloy at 840 °C. The

images were taken at these temperatures during continuous cooling

from 1100 °C at a cooling rate of 20 °C/min.

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