Sha W., Malinov S. Titanium Alloys: Modelling of Microstructure, Properties and Applications
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Titanium alloys: modelling of microstructure178
7.5 Calculation of continuous-cooling-
transformation diagrams
Usually, the degree of transformation is equal to the fraction of heat absorbed
or released:
Lamellae width (µ m)
8
7
6
5
4
3
2
1
0 102030405060
Cooling rate (°C/min)
(a)
Vickers hardness
400
380
360
340
320
300
280
010 2030405060
Cooling rate (°C/min)
(b)
7.8
Influence of the cooling rate on (a) lamellar thickness and (b)
hardness of IMI 367.
Johnson–Mehl–Avrami method adapted to continuous cooling 179
ft
h
t
t
h
t
t
Ht
Ht
t
t
t
t
T
t
t
t
() =
d
d
=
d
d
S
S
E
S
S
E
∫
∫
∫
∫
∂
∂
∂
∂
[7.1]
where ∂h/∂t (H) is the temperature difference in the case of DTA or the heat
Vickers hardness
400
390
380
370
360
350
340
330
320
310
300
51020304050
Cooling rate (°C/min)
(a)
Vickers hardness
460
440
420
400
380
360
340
320
300
51020304050
Cooling rate (°C/min)
(b)
7.9
Vickers microhardness of (a) Ti-8Al-1Mo-1V and (b) IMI 834 after
cooling at different cooling rates.
Titanium alloys: modelling of microstructure180
flow in the case of DSC, f (t) is the degree of transformation at any given
time t, and t
S
and t
E
are the transformation start and end temperatures
corresponding, respectively to the transformation start and end temperatures
T
S
and T
E
.
Hence, from the DSC signal (Fig. 7.1) and using Eq. [7.1], the degree of
transformation as a function of the time or the temperature can be calculated
and plotted (Fig. 7.11), which traces the course of the β to α transformation
in the Ti-6Al-4V, the Ti-6Al-2Sn-4Zr-2Mo-0.08Si, the Ti-8Al-1Mo-1V, IMI
834 and IMI 367 alloys and the formation of the γ phase in the Ti-46Al-2Mn-
2Nb alloy.
For all cooling rates, a small amount of residual (or retained) β phase,
about 9 wt.% independent of the cooling rate, remains after complete
transformation in Ti-6Al-4V (see Fig. 7.12). Therefore, for all cooling rates,
the same phase transformation is traced (100%β to 91%α + 9%β). The only
difference between X-ray diffraction patterns after different cooling rates is
in the relative intensities of the different {hkl} planes of the α phase (Fig.
7.12). This is probably due to preferred orientations of the α phase under
different cooling rates, as mentioned in Section 7.3 for Ti-6Al-2Sn-4Zr-
2Mo-0.08Si.
A
B
C
(a)
(b)
7.10
Scanning electron micrographs of fully lamellar microstructure
in Ti-8Al-1Mo-1V after continuous cooling from 1100 °C at (a) 5 and
(b) 50 °C/min.
Table 7.2
The local chemical composition of the points A–C shown in Fig. 7.10a for
the cooling rate of 5 °C/min
Ti Al V Mo
Location wt.% at.% wt.% at.% wt.% at.% wt.% at.%
A (α) 91.6 85.9 8.4 14.1 – – – –
B (β) 85.9 83.3 6.5 11.3 4.4 4.0 3.2 1.6
C (α) 91.6 86.0 8.4 14.0 – – – –
Johnson–Mehl–Avrami method adapted to continuous cooling 181
Degree of transformation
1
0.8
0.6
0.4
0.2
0
Degree of transformation
1
0.8
0.6
0.4
0.2
0
820 840 860 880 900 920 940 960 980
T
(°C)
0 100 200 300 400 500 600
Time (sec.)
50 °C/min
40 °C/min
30 °C/min
20 °C/min
10 °C/min
(a)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
800 820 840 860 880 900 920 940 960 9801000
T
(°C)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700
Time (sec.)
50 °C/min
40 °C/min
30 °C/min
20 °C/min
10 °C/min
(b)
7.11
Calculated degree of transformation as a function of the
temperature (left hand side) and the time (right hand side), for
different cooling rates for (a) Ti-6Al-4V, (b) Ti-6Al-2Sn-4Zr-2Mo-0.08Si,
(c) Ti-8Al-1Mo-1V, (d) IMI 834, (e) IMI 367, and (f) Ti-46Al-2Mn-2Nb.
Hence, in Ti-6Al-4V, hereafter in the calculations, 100% completed
transformation corresponds to an actual phase composition of 91%α + 9%β.
All calculations can very easily be converted from degree of transformation
(from 0 to 100%) to amount of α phase (from 0 to 91%).
Degree of transformation
1
0.8
0.6
0.4
0.2
0
(c)
900 950 1000 1050
T
(°C)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
20 °C/min
30 °C/min
40 °C/min
50 °C/min
0 100 200 300 400 500 600
Time (sec.)
Titanium alloys: modelling of microstructure182
Applying Eq. [7.1] again for all the cooling rates used, iso-transformation
contour plots are produced (Fig. 7.13). Such diagrams are a useful tool, since
they can be used to trace the course of transformation in real product, where
the cooling rates in a (usually quite large) piece vary significantly from the
surface to the core.
(f)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
1200 1220 1240 1260 1280 1300 1320
T
(°C)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200 250 300 350 400 450
Time (sec.)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
Degree of transformation
1
0.8
0.6
0.4
0.2
0
10 °C/min
20 °C/min
30 °C/min
40 °C/min
50 °C/min
(e)
850 900 950 1000
T
(°C)
0 200 400 600
Time (sec.)
Degree of transformation
1
0.8
0.6
0.4
0.2
0
Degree of transformation
1
0.8
0.6
0.4
0.2
0
850 900 950 1000
T
(°C)
0 200 400 600
Time (sec.)
20 °C/min
30 °C/min
40 °C/min
50 °C/min
(d)
7.11
Continued
20 °C/min
15 °C/min
10 °C/min
Johnson–Mehl–Avrami method adapted to continuous cooling 183
Finally, the transformation kinetics can be presented in the form of classical
CCT diagrams (Fig. 7.14). In these diagrams, iso-lines of equal degree of
transformation for the different cooling rates are included. The CCT diagrams
are shown for two cases: (i) cooling from 1100 °C, and (ii) cooling from the
β-transus of the alloys.
7.6 Calculation of transformation kinetics
The kinetics of isothermal transformations is usually expressed by the theory
developed by Johnson, Mehl and Avrami (JMA), Eq. [6.1]. However, it must
be emphasised that Eq. [6.1] expresses the JMA theory under the following
assumptions:
40 50 60 70
2θ (°)
Intensity (c.p.s.)
7000
6000
5000
4000
3000
2000
1000
0
{100}α
{002}α
{101}α
{102}α
{103}α
Initial
After cooling
with 5 °C/min
After cooling
with 10 °C/min
After cooling
with 50 °C/min
7.12
X-ray diffraction patterns of Ti-6Al-4V after cooling at different
rates. For clarity, the diffraction patterns are shifted along the vertical
axis with respect to each other.
{110}β
Titanium alloys: modelling of microstructure184
T
(°C)
970
940
910
880
850
820
10 20 30 40 50
Cooling rate (°C/min)
(a)
10 20 30 40 50
Cooling rate (°C/min)
(b)
T
(°C)
950
900
850
800
5
5
5
10
10
10
15
15
15
20
25
30
35
40
50
45
55
65
60
70
75
80
85
90
95
95
95
90
90
85
85
80
80
75
75
70
70
65
65
60
60
55
55
50
50
45
45
40
40
35
35
30
30
25
25
20
20
5
5
5
10
10
10
15
15
15
20
25
30
35
40
50
45
55
65
60
70
75
85
90
95
95
95
90
90
85
85
80
80
75
75
70
70
65
65
60
60
55
55
50
50
45
45
40
40
35
35
30
30
25
25
20
20
80
7.13
Calculated thermo-kinetics diagram – temperature of
transformation versus cooling rate of (a) Ti-6Al-4V, (b) Ti-6Al-2Sn-4Zr-
2Mo-0.08Si and (c) IMI 367. Iso-lines show the iso-degree of
transformation.
Johnson–Mehl–Avrami method adapted to continuous cooling 185
10 20 30 40 50
Cooling rate (°C/min)
(c)
T
(°C)
940
920
900
880
860
5.0
15
25
35
45
55
65
75
85
95
7.13
Continued
• isothermal transformation;
• nucleation frequency is either constant, or else is a maximum at the
beginning of transformation and decreases during the course of
transformation;
• spatially random nucleation.
In the past, this theory has been adapted for non-isothermal transformations
and used as the theoretical basis for interpretation of DTA and DSC data.
Different methods based on the JMA theory have been developed and used
for calculating kinetic parameters. Among them, the most frequently used
are the following:
(i) Direct integration of the generalised form of the JMA equation,
fgIut
t
V
t
m
= 1 – exp – d d
00
∫∫
′
′
τ
[7.2]
where f is transformed volume fraction at time t, I
V
is the nucleation
frequency, u is the growth rate, g is a geometrical factor,
τ
is integrated
variable and m = n − 1.
(ii) Introduction of new state variable,
f = 1 – exp(–
θ
n
) [7.3a]
where f is transformed volume fraction, n is Avrami exponent, and
θ
= ( d ; ( ) = exp
–
R
0
0
t
kT) T kT k
Q
T
∫
[7.3b]
Titanium alloys: modelling of microstructure186
5%
10%
25%
50%
75%
90%
95%
10 100 1.10
3
1.10
4
Time (sec.)
(a)
T
(°C)
1100
1050
1000
950
900
850
800
7.14
Calculated continuous-cooling-transformation (CCT) diagrams of
the β to α (or β to α + β) transformation in (a) and (b) Ti-6Al-4V; (c)
and (d) Ti-6Al-2Sn-4Zr-2Mo-0.08Si; (e) and (f) IMI 367, for cooling
from (a,c,e) 1100 °C and (b,d,f) β-transus.
10 100 1.10
3
1.10
4
Time (sec.)
(b)
T
(°C)
1000
950
900
850
800
Johnson–Mehl–Avrami method adapted to continuous cooling 187
10 100 1000 10000
Time (sec.)
(c)
T
(°C)
1100
1050
1000
950
900
850
800
40 °C/min
50 °C/min
10 °C/min
20 °C/min
30 °C/min
10 100 1000 10000
Time (sec.)
(d)
T
(°C)
1000
980
960
940
920
900
880
860
840
820
800
5%
10%
25%
50%
75%
90%
95%
7.14
Continued