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

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Titanium alloys: modelling of microstructure218

and different vanadium concentrations. Subsequently, the difference in the

energies of the two phases, ∆g, and the activation barrier for nucleation,

∆G

, are calculated using Eq. [8.7] (see Fig. 8.4a). In order to limit the

number of unknown parameters in the model, we make the assumption that

the interface mobility is high, so that the atoms are able to move freely

4 wt.% V

6 wt.% V

8 wt.% V

10 wt.% V

12 wt.% V

14 wt.% V

4 wt.% V

6 wt.% V

8 wt.% V

10 wt.% V

12 wt.% V

14 wt.% V

∆

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)

0.01

0.008

0.006

0.004

0.002

8.4

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

the β to α transformation in Ti-6Al-4V alloy as functions of the

vanadium content and the temperature. In calculating ∆

Kelvin is used.



Finite element method: morphology 219

across the interface. The diffusion controls the growth and the lamella grows

just as fast as the diffusion allows. In this case, the vanadium concentration

in β phase at the interface is equal to equilibrium concentration (see Fig.

5.1b) and the free parameters in the model are the number of nucleation sites

and the surface energy γ. The next step is to fit the calculated thickness of

the α plates to the experimental data by setting the N

and γ values free but

constant for all the temperatures, concentrations and cooling rates used.

From the fitting of the free parameters, the values of

= 1.3 10

nucleus/(s · m) and

γ ()

= 0.012 J/m

for the β to α

phase transformation in Ti-6Al-4V alloy are obtained. The plots of the calculated

barrier for nucleation (Fig. 8.4a) and the rate of nucleation (Fig. 8.4b) are for

these values of

and

γ ()

and applying Eqs. [8.7] and [8.8],

respectively.

The simulated distribution of the α lamellae thickness for the above values

of N

and γ is plotted in Fig. 8.5b. This simulation from the model is compared

with the experimentally observed microstructure of the Ti-6Al-4V alloy after

cooling with the same cooling rate (Fig. 8.5a). A good agreement between

the experimental mean lamellae size and the prediction from the FEM is

obtained. The 1-D model predicts the existence of a notable fraction of

lamellae with small thickness, which nucleate at the end of the process of

transformation and do not have enough driving force and time to grow.

Perhaps such lamellae really exist, but they cannot be observed in the optical

microstructure because of their small thickness. Similar good agreement

between the FEM simulations for the α lamellae thickness and experimental

observations is obtained for other cooling rates. The model predictions for

the average thickness of the α lamellae for cooling rates of 20 and 10 °C/min

are 1.7 and 2.1 µm, respectively.

The 1-D model is further used to simulate the kinetics of the β to α phase

transformation in the Ti-6Al-4V alloy. Figure 8.6 shows the model simulations

for the amount of the α phase as a function of the temperature for different

constant cooling rates. For each cooling rate a number of runs were performed,

and the results are reproducible. The shapes of the curves tracing the kinetics

of the β to α phase transformation are classical. The simulated influence of

the cooling rate on the kinetics of the transformation is in agreement with the

experimental data from the differential scanning calorimetry study of the

same transformation (Chapter 7). There are some differences between the

experimental calorimetry kinetics curves in Chapter 7 and the FEM simulations

here. A possible reason for this difference is that the processes of the 2-D and

3-D impingement cannot be taken into account in the 1-D model.

The 1-D model is also used for analysis of the nucleation rate of the β to

α phase transformation in Ti-6Al-4V alloys. Figure 8.7 presents the FEM



Titanium alloys: modelling of microstructure220

simulations for the real nucleation rate for different cooling rates. The nucleation

rates presented in Figs. 8.4b and 8.7 should be well distinguished. Figs. 8.4b

shows the theoretical nucleation rate for different temperatures and vanadium

concentrations and is obtained by the plotting of Eq. [8.8]. Figure 8.7 shows

the model simulations for the real nucleation rate for the current temperature

50 µm

(a)

Cooling rate 40 °C/min

Mean lamellae

size = 1.47 µm

0 0.5 1 1.5 2 2.5 3 3.5 4

Size (m)

(b)

× 10

–6

Frequency

8.5 (a)

Microstructure of Ti-6Al-4V alloy after cooling at a cooling rate

of 40 °C/min and (b) distribution of the lamellae thickness of the α

phase for the same cooling rate calculated from the FEM.



Finite element method: morphology 221

and the vanadium concentration profile in the residual β phase at the

corresponding moment in the β phase. The simulated rate of nucleation first

increases with the decrease in the temperature (increase in the degree of

undercooling below β-transus temperature). The reason for this is that the

barrier for nucleation of α phase significantly decreases with the undercooling.

The maximum of the nucleation rate is around 900 °C. This observation is in

Amount of α phase

0.8

0.6

0.4

0.2

750 800 850 900 950 1000

(°C)

10 °C/min

20 °C/min

40 °C/min

8.6

Kinetics of the β to α transformation in Ti-6Al-4V alloy for

different cooling rates calculated from the FEM.

750 800 850 900 950 1000

(°C)

Cooling rate 10 °C/min

Cooling rate 20 °C/min

Cooling rate 40 °C/min

Rate of nucleation (nuclei/mm.s)

8.7

FEM simulation for the real rate of nucleation at different cooling

rates.



Titanium alloys: modelling of microstructure222

agreement with the TTT diagram for the Ti-6Al-4V alloys (Chapter 14)

according to which the nose point is in the range of 850–900 °C, depending

on the alloy composition. Naturally, higher cooling rates shift the maximum

of the nucleation rate to lower temperatures. Further decrease in the temperature

leads to decrease in the rate of nucleation. There are three concurrent reasons

for this: (i) the diffusion and the migration of the atoms across the interface

are slower, (ii) the amount of the β phase is decreased, and (iii) the vanadium

concentration in the residual β phase is increased.

8.6 The 2-D model

In order to describe more realistically the β to α phase transformation, we

will show 2-D simulation of the nucleation and growth of the α phase lamellae

within a former β grain. We assume that the nucleus of α phase is formed at

the β grain boundary and then grows to the grain interior. The mechanism of

nucleation of the α phase at the β grain boundary, used in 2-D model, is the

same as the mechanism used in the 1-D problem.

After the α/β interface is created during the nucleation stage, it starts

migrating into the surrounding β phase, by the transfer of vanadium atoms

across the interface. Since the α plates grow mainly parallel to {110}β plane,

the mobility M

of vanadium across the interface in the direction of the

longitudinal dimension is much higher than the mobility M

across the interface

in the transverse direction. Hence, the growth oriented parallel to the {110}β

plane can be expected to be mainly diffusion controlled, whilst the growth in

the transverse direction is mixed or even interface controlled. The mobility

determines the corresponding flux of vanadium atoms across the interface,

Eq. [8.12], which in its turn defines the boundary conditions at the α/β

interface for the vanadium diffusion in the β phase.

The growth rate can be evaluated by solving the diffusion equation [8.15]

in the domain occupied by the β phase. The value of vanadium concentration

at any point of the interface C

will be that which satisfies the boundary

condition Eq. [8.17]. The mobility of vanadium atoms M

and M

in the two

directions are free parameters in the model and are determined by fitting the

calculated shape of the α plates to the metallographic observations (Chapters

6 and 7). The following values of

–5

= 6.4 10×

µm · wt.%/(K · s) and

–7

= 3.5 10×

µm · wt.%/(K · s) for the β to α phase transformation in

Ti-6Al-4V are obtained.

Using the computer package, we simulate the process of nucleation and

growth of a colony of α phase lamellae on a square with length 40 µm for

constant cooling rates of 10 °C/min (Fig. 8.8a–d) and 30 °C/min (Fig. 8.8e–h).

Two observations are apparent from the comparison of the simulations:

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Finite element method: morphology 223

(a) Cooling rate 10 °C/min at 910 °C

Temp 910 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

(b) Cooling rate 10 °C/min at 850 °C

Temp 850 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

8.8

FEM simulations for the microstructure evolution of the β to α

phase transformation in Ti-6Al-4V alloy at continuous cooling with

constant cooling rate. The scales to the right of each figure represent

vanadium concentration. (a)-(d) 10 °C/min; (e)-(h) 30 °C/min; (a) and

(e)

= 910 °C; (b) and (f)

= 850 °C; (c) and (g)

= 790 °C; and

(d) and (h)

= 730 °C.

(i) The degree of the β to α phase transformation at the same temperatures

is at much advanced stage when the lower cooling rate is applied

(compare Fig. 8.8a with 8.8e, 8.8b with 8.8f, etc.). This is in agreement

with the experimental study of the kinetics of the same phase

transformation at the same cooling rates using differential scanning

calorimetry (Chapter 7).



Titanium alloys: modelling of microstructure224

Temp 790 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

(d) Cooling rate 10 °C/min at 730 °C

Temp 730 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

(e) Cooling rate 30 °C/min at 910 °C

Temp 910 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

8.8

Continued



Finite element method: morphology 225

(f) Cooling rate 30 °C/min at 850 °C

Temp 850 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

(g) Cooling rate 30 °C/min at 790 °C

Temp 790 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

(h) Cooling rate 30 °C/min at 730 °C

Temp 730 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

8.8

Continued



Titanium alloys: modelling of microstructure226

(ii) The thickness of the grain boundary α phase lamellae of the

Widmanstätten morphology is larger and their number is smaller when

the slower cooling rate is applied (compare Fig. 8.8a–d with Fig. 8.8e–

h). This is in agreement with the metallographic observation of the

features of the phase transformation and a general knowledge of the

influence of the cooling rate on the α phase lamellae morphology and

the size and distribution of the fine microstructure in the Ti-6Al-4V

alloys.

A more realistic numerical simulation of the β to α phase transformation,

when the grain boundary α phase colonies grow in different directions, is

presented in Fig. 8.9. This case takes into account the 2-D impingement of

the colonies growing in the different directions. We consider the nucleation

and growth of α plates on a sector of a β phase grain with a long dimension

of 150 µm. The evolution of the morphology of the microstructure at different

stages (temperatures) simulated from the model is traced in Fig. 8.9a–e. The

transformation starts with the first nucleus at around 960 °C. With further

decrease of the temperature, α phase colonies grow in the different directions

until the transformation is nearly completed at about 800 °C. The calculated

average thickness of the α plates is about 6 µm, which is in good agreement

with the metallographic observations (Chapters 6 and 7).

As the next step, we simulate the evolution of the β to α phase transformation

under isothermal conditions. The starting microstructure is β phase domain

and the temperature is instantly set at any constant temperature in the α + β

equilibrium range. Figure 8.10a–d traces the morphology and the kinetics of

growth for colony of α phase lamellae on a square with length 40 µm. After

long periods of time, the concentration gradients in the β phase decrease, the

interface vanadium concentration converges to the equilibrium concentration

and the chemical potential difference ∆

driving the vanadium atoms across

the interface converges to zero. At this stage, it can be said that the equilibrium

in respect to the amounts of the α and the β phases, as well as their chemical

compositions at this temperature, is reached, and the transformation process

is completed. Note here that the β to α phase transformation at isothermal

conditions predicted from the FEM kinetics does not fit well to the experimental

data. The model presented here shows that, at 900 °C, the transformation is

completed between the 20th

and the 30th minute (Fig. 8.10d). However, the

experimental data in Chapter 6 and the TTT diagram of this alloy conclude

that, at the same temperature, the transformation should complete between

the first and the sixth minute, depending on the real composition of the alloy.

There are at least two reasons for this difference: (i) in this model we consider

nucleation and growth of only one α colony, which is far from the realistic

case of simultaneous nucleation and growth from real starting microstructure

with many grains in both direction of each grain boundary; and (ii) we



Finite element method: morphology 227

consider the 2-D case, which does not take into account the diffusional

redistribution and the growth in the third direction.

It this respect it is worth mentioning that it was possible to fit the kinetics

under isothermal conditions predicted from the FEM to the experimental one

when the diffusion coefficient of vanadium in the β phase was increased by

from 10 to 50 times.

(a) at 930 °C

Temp 930 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

(b) at 880 °C

Temp 880 °C

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

8.9

FEM simulations for the microstructure evolution of the β to α

phase transformation in Ti-6Al-4V alloy at continuous cooling with

constant cooling rate of 5 °C/min, assuming α colony growth in three

different directions. The scales to the right of each figure represent

vanadium concentration. (a) 930 °C; (b) 880 °C; (c) 850 °C; (d) 800 °C;

(e) 730 °C.

