BME 303 - Lesson 7 - Phase Transformation.pdf

CHAPTER
12
Phase
Transformations
BME 303 - Biomaterials
Lesson 7

2
 Introduction
 Basic concepts
 The kinetics of phase transformations
 Metastable versus equilibrium states
 Isothermal transformation diagrams
 Continuous cooling transformation diagrams
 Mechanical behavior of iron-carbon alloys
 Tempered martensite
 Review of phase transformations and mechanical
properties for iron-carbon alloys

3
Phase Transformations
 Phase transformations occur, when phase
boundaries (the red curves) on these plots are
crossed as temp. and/or pressure is changed.
 Upon passing across the solid-gas phase
boundary of the CO2 phase diagram, dry ice
(solid CO2) sublimes (gaseous CO2).
an arrow delineates this phase transformation
H2O
CO2

4
ISSUES TO ADDRESS...
• Transforming one phase into another takes time.
• How does the rate of transformation depend on
time and temperature?
• Is it possible to slow down transformations so that
non-equilibrium structures are formed?
• Are the mechanical properties of non-equilibrium
structures more desirable than equilibrium ones?
Fe
g
(Austenite)
Eutectoid
transformation
C FCC
Fe3C
(cementite)
a
(ferrite)
+
(BCC)

Three types of phase transformation
• Simple diffusion-dependent transformations -- no change in
either the number or composition of the phase present.
– solidification of a pure metal, allotropic transformations,
recrystallization, grain growth.
• Diffusion-dependent transformations -- some alteration in
phase compositions and often in the number of phases present;
the final micro-structure ordinarily consists of two phases.
– eutectoid reaction.
• Diffusionless, wherein a metastable phase is produced.
– martensitic transformation.
5

6
 Introduction
 Basic concepts

7
 Phase transformation
- At least one new phase is formed that has different physical/chemical
characteristics and/or a different structure than the parent phase.
- Begin by the formation of numerous small particles of the new phase(s),
which increase in size until the transformation has reached completion.
 Gibbs free energy
- Gibbs free energy is a function of the internal energy of the system (enthalpy,
H) and a measurement of the randomness or disorder of the atoms or molecules
(entropy, S). (G = H-TS )
- A transformation will occur spontaneously only when the change in free energy
G has a negative value.
Kinetics of Phase Transformations I

8
Phase transformation may be broken down into two distinct stages:
 Nucleation (孕核，成核)
- Nucleation involves the appearance of very small particles, or nuclei of the new
phase, which are capable of growing.
- For the homogeneous type, nuclei of the new phase form uniformly throughout the
parent phase, whereas for the heterogeneous type, nuclei form preferentially at
structural inhomogeneities, such as, container surfaces, insoluble impurities, grain
boundaries, dislocations, and so on.
 Growth
During the growth stage these nuclei increase in size, which results in the
disappearance of some (or all) of the parent phase. The transformation reaches
completion if the growth of these new phase particles is allowed to proceed until the
equilibrium fraction is attained.
Kinetics of Phase Transformations II

Nucleation
– nuclei (seeds) act as templates on which crystals grow
– for nucleus to form rate of addition of atoms to nucleus must be
faster than rate of loss
– once nucleated, growth proceeds until equilibrium is attained
9
Driving force to nucleate increases as we increase T
– supercooling (eutectic, eutectoid)
– superheating (peritectic)
Small supercooling  slow nucleation rate - few nuclei - large crystals
Large supercooling  rapid nucleation rate - many nuclei - small crystals

Solidification: Nucleation Types
• Homogeneous nucleation
– nuclei form in the bulk of liquid
metal
– requires considerable
supercooling
(typically 80-300ºC)
10
• Heterogeneous nucleation
– much easier since stable “nucleating surface” is
already present — e.g., mold wall, impurities in
liquid phase
– only very slight supercooling (0.1-10ºC)

Homogeneous Nucleation & Energy Effects
11
r* = critical nucleus: for r < r* nuclei shrink; for r >r* nuclei grow (to reduce energy)
GT = Total Free Energy
= GS + GV
Surface Free Energy - destabilizes
the nuclei (it takes energy to make
an interface)
g


 2
4 r
GS
g = surface tension
Volume (Bulk) Free Energy –
stabilizes the nuclei (releases energy)




 G
r
GV
3
3
4
volume
unit
energy
free
volume

 
G

12
g


g 2
3
4
3
4
r
G
r
A
G
V
G v
v 






0
8
4
)
( 2





g

 r
G
r
dr
G
d
v
When r = r*
v
G
r




g
2
*
V: volume of spherical nucleus,
Gv: volume free energy,
A: surface area of spherical nucleus,
g: surface free energy,
r: radius of spherical nucleus,
r*: critical radius of spherical nucleus,
G*: critical free energy, activation energy
Kinetics of Phase Transformations
(10.1)
(10.2)
(10.3)
2
3
2
3
)
(
3
16
)
2
(
4
)
2
(
3
4
*
v
v
v
v G
G
G
G
G











g
g
g

g
 (10.4)

Solidification
13
T
H
T
r
f
m


g


2
*
Note: Hf and g are weakly dependent on T
 r* decreases as T increases
For typical T r* ~ 10 nm
Hf = latent heat of solidification
Tm = melting temperature
g = surface free energy
T = Tm - T = supercooling
r* = critical radius
m
m
f
v
T
T
T
H
G
)
( 



Gv: volume free energy

14
- The volume free energy change Gv is the driving force for the solidification
transformation, and its magnitude is a function of temperature. At the equilibrium
solidification temperature Tm, the value of Gv is zero, and with diminishing
temperature its value becomes increasingly more negative. It can be shown that Gv
is a function of temperature as
Hf: latent heat of fusion, Tm: equilibrium melting temperature (K),
T: real solidification temperature (K).
m
m
f
v
T
T
T
H
G
)
( 


 (10.5)





























T
T
H
T
T
T
T
H
G
r
m
f
m
m
m
f
v
1
2
)
(
2
2
*
g
g
g
(10.6)
2
2
2
3
2
3
2
3
)
(
1
3
16
)
(
3
16
)
(
3
16
*
T
T
H
T
T
T
T
H
G
G
m
f
m
m
m
f
v 
















 






g
g
g
(10.7)

15
FIGURE 10.3 Schematic free energy-
versus-embryo/nucleus radius curves
for two different temperatures. The
critical free energy change (G*) and
critical nucleus radius (r*) are indicated
for each temperature.





















T
T
H
T
r
m
f
m 1
2
*
g
2
2
2
3
)
(
1
3
16
*
T
T
H
T
G
m
f
m













g
- Both the critical radius r* and the activation free energy G* decrease as temperature
T decreases. With a lowering of temperature at temperatures below the equilibrium
solidification temperature (Tm), nucleation occurs more readily.

16
- Supposed that there is a spherical embryo has a radius of r*. If this embryo catches
an atom, then it will become a stable nucleus, and it will grow up, since its Gibbs free
energy gets smaller. If this embryo loses an atom, then it will disappear later because
of the decrease of Gibbs free energy. Therefore, the computation of nucleation rate
can be treated as two steps. The first is the probability of forming a embryo with a
radius of r*. The second is that the rate of this embryo getting an additional atom.
r*

17
- The number of stable nuclei n* (having radii greater than r*) is a function of
temperature as
K1: total number of nuclei of the solid phase.
- As the temperature is lowered below Tm, the exponential term in Equation 10.8 also
decreases such that the magnitude of n* increases.
)
*
exp(
* 1
kT
G
K
n



2
2
2
3
)
(
1
3
16
*
T
T
H
T
G
m
f
m













g
T   Tm-T   G*   exp(-G*/kT)   n* 
FIGURE 10.4 For solidification, schematic
plots of (a) number of stable nuclei versus
temperature, (b) frequency of atomic
attachment versus temperature.
(10.8)

18
FIGURE 10.4 For solidification, schematic plots of (c) nucleation
rate versus temperature.
- The diffusion effect is related to the frequency at which atoms from the liquid attach
themselves to the solid nucleus, vd.
Qd: temperature-independent activation energy for diffusion,
K2: temperature-independent constant.
- The nucleation rate is simply proportional to the product of n* and vd.
K3: number of atoms on a nucleus surface.
- Note (Fig. 10.4c) that, with a lowering of temperature from
below Tm, the nucleation rate first increases, achieves a
maximum, and subsequently diminishes.
)
/
exp(
)
*
exp(
3
2
1 kT
Q
kT
G
K
K
K
N d





)
exp(
2
kT
Q
K
v d
d 
 (10.9)
(10.10)

19

20
FIGURE 10.5 Heterogeneous nucleation of
a solid from a liquid. The solid–surface (gSI),
solid–liquid (gSL), and liquid–surface (gIL)
interfacial energies are represented by
vectors. The wetting angle () is also shown.
 Heterogeneous nucleation
- It is easier for nucleation to occur at surfaces and interfaces than at other sites. This
type of nucleation is termed heterogeneous.
- Let us consider the nucleation, on a flat surface, of a solid particle from a liquid
phase. It is assumed that both the liquid and solid phases “wet” this flat surface.
Taking a surface tension force balance in the plane of the flat surface (Fig. 10.5)
leads to the following expression:
- It is possible to derive equations for r* and G*,

g
g
g cos
SL
SI
IL 

(10.12)
v
SL G
r 

 /
2
* g (10.13)
  










 2
3
3
16
*
v
SL
G
S
G
g
 (10.14)

21


r
r
r sin r sin
IL
SI
SI
SI
SL
SL
v
S
het A
A
A
G
V
G g
g
g 





v
SL
G
r




g
2
*
)
(
)
(
3
16
2
3
*

g
S
G
G
v
SL
het 




2
)
sin
( 
 r
ASI 
)
cos
1
(
2 2

 
 r
ASL
)
cos
cos
3
2
(
3
3
3






r
VS

g
g
g cos
SL
SI
IL 

)
(
S
G
Ghet 




SL
v r
G
r
G g

 2
3
4
3
4




4
)
cos
1
)(
cos
2
(
)
(
2






S
)
(
*
hom
*

S
G
Ghet 




VS: volume of heterogeneous nucleus,
Gv: volume free energy,
ASL: area of solid-liquid interface,
ASI: area of solid-surface interface,
gSL: surface free energy of solid-liquid interface,
gSI: surface free energy of solid-surface interface,
r: radius of spherical nucleus,
r*: critical radius of spherical nucleus,
Ghet*: critical free energy, activation energy

22
The S() of this last equation is a function only of , which will have a numerical
value between zero and unity.
- The critical radius r* for heterogeneous nucleation is the same as for homogeneous,
inasmuch as gSL is the same surface energy as g in Equation 10.3 (r* = -2g / Gv).
- In Fig. 10.6, the lower G* for heterogeneous means that a smaller energy must be
overcome during the nucleation process (than for homogeneous).
  *
hom
*
G
S
Ghet 


  (10.15)
)
(
*
hom
*

S
G
Ghet 



4
cos
cos
3
2
)
(
3






S
FIGURE 10.6 Schematic free energy-versus-
embryo/nucleus radius plot on which is
presented curves for both homogeneous and
heterogeneous nucleation. Critical free energies
and the critical radius are also shown.

23
2
2
2
3
)
(
1
3
16
*
T
T
H
T
G
m
f
m













g
)
exp(
)
*
exp(
kT
Q
kT
G
N d





FIGURE 10.7 Nucleation rate versus temperature for both
homogeneous and heterogeneous nucleation. Degree of
supercooling (T) for each is also shown.
- In terms of the nucleation rate, the -versus-T curve is shifted to higher temperatures
for heterogeneous. This effect is represented in Fig. 10.7, which also shows that a
much smaller degree of supercooling (T) is required
for heterogeneous nucleation.

N

24
FIGURE 10.8 Schematic plot showing curves for
nucleation rate ( ), growth rate ( ), and overall
transformation versus temperature.

G

N
 Growth
- The growth step in a phase transformation begins once an embryo has exceeded the
critical size, r*, and becomes a stable nucleus.
- Particle growth occurs by long-range atomic diffusion, which normally involves
several steps-for example, diffusion through the parent phase, across a phase
boundary, and then into the nucleus. Consequently, the growth rate is determined
by the rate of diffusion, and its temperature dependence is the same as for the
diffusion coefficient.
Q: activation energy, independent of
temperature;
C: constant, independent of temperature.

G










kT
Q
C
G exp (10.16)

25
FIGURE 10.9 Schematic plots of (a)
transformation rate versus
temperature, and (b) logarithm time
versus temperature. The curves in
both (a) and (b) are generated from
the same set of data-i.e., for horizontal
axes, the time is just the reciprocal of
the rate from plot (a).
- At a specific temperature, the overall transformation rate is equal to some product of
and .
- Whereas the treatment on transformations has been developed for solidification, the
same general principles also apply to solid-solid and solid-gas transformations.
- The rate of transformation and the time required for the transformation to proceed to
some degree of completion are inversely proportional to one another.
- The kinetics of phase transformations are often represented using logarithm time-
versus-temperature plots (Fig. 10.9).

G

N

26
- Transformation rate-versus-temperature curve
1. The size of the product phase particles will depend on transformation temperature.
For example, for transformations that occur at temperature near Tm, corresponding
to low nucleation and high growth rates, few nuclei form that grow rapidly. Thus,
the resulting microstructure will consist of few and relatively large phase particles.
2. When a material is cooled very rapidly through the temperature range
encompassed by the transformation rate curve to a relatively low temperature
where the rate is extremely low, it is possible to produce nonequilibrium phase
structures.

Rate of Phase Transformations
27
Kinetics - study of reaction rates of phase
transformations
• To determine reaction rate – measure degree
of transformation as function of time (while
holding temp constant)
measure propagation of sound waves –
on single specimen
electrical conductivity measurements –
on single specimen
X-ray diffraction – many specimens required
How is degree of transformation measured?

Rate of Phase Transformation
Avrami equation => y = 1- exp (-ktn)
– k & n are transformation specific parameters
28
transformation complete
log t
Fraction
transformed,
y
Fixed T
fraction
transformed
time
0.5
By convention rate = 1 / t0.5
maximum rate reached – now amount
unconverted decreases so rate slows
t0.5
rate increases as surface area increases
& nuclei grow

Temperature Dependence of Transformation Rate
• For the recrystallization of Cu, since
rate = 1/t0.5
rate increases with increasing temperature
29
• Rate often so slow that attainment of equilibrium
state not possible!
135C 119C 113C 102C 88C 43C
1 10 102 104

30
 Introduction
 Basic concepts

Transformations & Undercooling
31
• For transf. to occur, must
cool to below 727ºC
(i.e., must “undercool”)
• Eutectoid transf. (Fe-Fe3C system): g  a + Fe3C
0.76 wt% C
0.022 wt% C
6.7 wt% C
Fe
3
C
(cementite)
1600
1400
1200
1000
800
600
400
0 1 2 3 4 5 6 6.7
L
g
(austenite)
g+L
g +Fe3C
a +Fe3C
L+Fe3C
d
(Fe) C, wt%C
1148ºC
T(ºC)
a
ferrite
727ºC
Eutectoid:
Equil. Cooling: Ttransf. = 727ºC
T
Undercooling by Ttransf. < 727C
0.76
0.022

The Fe-Fe3C Eutectoid Transformation
32
Coarse pearlite  formed at higher temperatures – relatively soft
Fine pearlite  formed at lower temperatures – relatively hard
• Transformation of austenite to pearlite:
g
a
a
a
a
a
a
pearlite
growth
direction
Austenite (g)
grain
boundary
cementite (Fe3C)
Ferrite (a)
g
• For this transformation,
rate increases with
[Teutectoid – T ] (i.e., T).
675ºC
(T smaller)
0
50
y
(%
pearlite) 600ºC
(T larger)
650ºC
100
Diffusion of C
during transformation
a
a
g
g
a
Carbon
diffusion

33
Generation of Isothermal Transformation Diagrams
• The Fe-Fe3C system, for C0 = 0.76 wt% C
• A transformation temperature of 675ºC.
100
50
0
1 102 104
T = 675ºC
y,
%
transformed
time (s)
400
500
600
700
1 10 102 103 104 105
Austenite (stable)
TE (727ºC)
Austenite
(unstable)
Pearlite
T(ºC)
time (s)
isothermal transformation at 675ºC
Consider:

34
Austenite-to-Pearlite Isothermal Transformation
• Eutectoid composition, C0 = 0.76 wt% C
• Begin at T > 727ºC
• Rapidly cool to 625ºC
• Hold T (625ºC) constant (isothermal treatment)
400
500
600
700
Austenite (stable)
TE (727ºC)
Austenite
(unstable)
Pearlite
T(ºC)
1 10 102 103 104 105
time (s)
g g
g
g g
g

Transformations Involving Noneutectoid Compositions
35
Hypereutectoid composition – proeutectoid cementite
Consider C0 = 1.13 wt% C
a
TE (727ºC)
T(ºC)
time (s)
A
A
A
+
C
P
1 10 102 103 104
500
700
900
600
800
A
+
P
Fe
3
C
(cementite)
1600
1400
1200
1000
800
600
400
0 1 2 3 4 5 6 6.7
L
g
(austenite)
g+L
g +Fe3C
a+Fe3C
L+Fe3C
d
(Fe)
C, wt%C
T(ºC)
727ºC
T
0.76
0.022
1.13

Bainite: Another Fe-Fe3C Transformation Product
36
10 103
105
time (s)
10-1
400
600
800
T(ºC)
Austenite (stable)
200
P
B
TE
A
A
• Bainite:
-- elongated Fe3C particles in
a-ferrite matrix
-- diffusion controlled
• Isothermal Transf. Diagram,
C0 = 0.76 wt% C
Fe3C
(cementite)
5 mm
a (ferrite)
100% bainite
100% pearlite

37
Spheroidite: Another Microstructure for the Fe-
Fe3C System
• Spheroidite:
-- Fe3C particles within an a-ferrite matrix
-- formation requires diffusion
-- heat bainite or pearlite at temperature
just below eutectoid for long times
-- driving force – reduction
of a-ferrite/Fe3C interfacial area
60 mm
a
(ferrite)
(cementite)
Fe3C

38
Martensite: A Nonequilibrium Transformation
Product
• Martensite:
-- g(FCC) to Martensite (BCT)
Martensite needles
Austenite
60
mm
x
x x
x
x
x
potential
C atom sites
Fe atom
sites
• Isothermal Transf. Diagram
• g to martensite (M) transformation..
-- is rapid! (diffusionless)
-- % transf. depends only on T to
which rapidly cooled
10 103
105
time (s)
10-1
400
600
800
T(ºC)
Austenite (stable)
200
P
B
TE
A
A
M + A
M + A
M + A
0%
50%
90%

Martensite Formation
g (FCC) a (BCC) + Fe3C
39
slow cooling
tempering
quench
M (BCT)
Martensite (M) – single phase
– has body centered tetragonal (BCT)
crystal structure
Diffusionless transformation BCT if C0 > 0.15 wt% C
BCT  few slip planes  hard, brittle

Phase Transformations of Alloys
4340 Alloy Steel
Effect of adding other elements
Change transition temp.
Cr, Ni, Mo, Si, Mn
retard g  a + Fe3C
reaction (and formation of
pearlite, bainite)
40

41
 Introduction
 Basic concepts

Continuous Cooling
Transformation Diagrams
Conversion of isothermal
transformation diagram to
continuous cooling
transformation diagram
42
Cooling curve

Isothermal Heat Treatment Example Problems
On the isothermal transformation diagram for
a 0.45 wt% C, Fe-C alloy, sketch and label the
time-temperature paths to produce the
following microstructures:
a) 42% proeutectoid ferrite and 58% coarse pearlite
b) 50% fine pearlite and 50% bainite
c) 100% martensite
d) 50% martensite and 50% austenite
43

Solution to Part (a) of Example Problem
a) 42% proeutectoid ferrite and 58% coarse pearlite
44
Isothermally treat at ~ 680ºC
-- all austenite transforms
to proeutectoid a and
coarse pearlite.
A + B
A + P
A + a
A
B
P
A
50%
0
200
400
600
800
0.1 10 103 105
time (s)
M (start)
M (50%)
M (90%)
Fe-Fe3C phase diagram,
for C0 = 0.45 wt% C
Wpearlite 
C0  0.022
0.76  0.022
=
0.45  0.022
0.76  0.022
= 0.58
W 
a = 1  0.58 = 0.42
T (ºC)

Solution to Part (b) of Example Problem
b) 50% fine pearlite and 50% bainite
45
T (ºC)
A + B
A + P
A + a
A
B
P
A
50%
0
200
400
600
800
0.1 10 103 105
time (s)
M (start)
M (50%)
M (90%)
for C0 = 0.45 wt% C
Then isothermally treat
at ~ 470ºC
– all remaining austenite
transforms to bainite.
Isothermally treat at ~ 590ºC
– 50% of austenite transforms
to fine pearlite.

Solutions to Parts (c) & (d) of Example Problem
c) 100% martensite – rapidly quench to room
temperature
46
d) 50% martensite
& 50% austenite
-- rapidly quench to
~ 290ºC, hold at this
temperature
T (ºC)
A + B
A + P
A + a
A
B
P
A
50%
0
200
400
600
800
0.1 10 103 105
time (s)
M (start)
M (50%)
M (90%)
for C0 = 0.45 wt% C
d)
c)

47
 Introduction
 Basic concepts

Mechanical Props: Influence of C Content
48
• Increase C content: TS and YS increase, %EL decreases
C0 < 0.76 wt% C
Hypoeutectoid
Pearlite (med)
ferrite (soft)
C0 > 0.76 wt% C
Hypereutectoid
Pearlite (med)
Cementite
(hard)
300
500
700
900
1100
YS(MPa)
TS(MPa)
wt% C
0 0.5 1
hardness
0.76
Hypo Hyper
wt% C
0 0.5 1
0
50
100
%EL
Impact
energy
(Izod,
ft-lb)
0
40
80
0.76
Hypo Hyper

Mechanical Props: Fine Pearlite vs.
Coarse Pearlite vs. Spheroidite
49
• Hardness:
• %RA:
fine > coarse > spheroidite
fine < coarse < spheroidite
80
160
240
320
wt%C
0 0.5 1
Brinell
hardness
fine
pearlite
coarse
pearlite
spheroidite
Hypo Hyper
0
30
60
90
wt%C
Ductility
(%RA)
fine
pearlite
coarse
pearlite
spheroidite
Hypo Hyper
0 0.5 1

Mechanical Props: Fine Pearlite vs. Martensite
50
• Hardness: fine pearlite << martensite.
0
200
wt% C
0 0.5 1
400
600
Brinell
hardness
martensite
fine pearlite
Hypo Hyper

51
 Introduction
 Basic concepts

Tempered Martensite
52
• tempered martensite less brittle than martensite
• tempering reduces internal stresses caused by quenching
• tempering decreases TS, YS but increases %RA
• tempering produces extremely small Fe3C particles surrounded by a.
9
mm
YS(MPa)
TS(MPa)
800
1000
1200
1400
1600
1800
30
40
50
60
200 400 600
Tempering T(ºC)
%RA
TS
YS
%RA
Heat treat martensite to form tempered martensite

Summary of Possible Transformations
53
Austenite (g)
Pearlite
(a + Fe3C layers + a
proeutectoid phase)
slow
cool
Bainite
(a + elong. Fe3C particles)
moderate
cool
Martensite
(BCT phase
diffusionless
transformation)
rapid
quench
Tempered
Martensite
(a + very fine
Fe3C particles)
reheat
Strength
Ductility
Martensite
T Martensite
bainite
fine pearlite
coarse pearlite
spheroidite
General Trends

55
Shape-Memory Alloys
 Shape-memory alloys (形狀記憶合金)
- One of the shape-memory alloys (SMAs), after being deformed, has the ability to
return to its predeformed size and shape upon being subjected to an appropriate
heat treatment-that is, the material “remembers” its previous size/shape.
- Deformation normally is carried out at a relatively low temperature, whereas shape
memory occurs upon heating.
- A shape-memory alloy is polymorphic-that is, it may have two crystal structures
(or phases), and the shape-memory effect
involves phase transformations between
them.
Time-lapse photograph that demonstrates the
shape-memory effect. A wire of a shape-
memory alloy (Nitinol) has been bent and
treated such that its memory shape spells the
word Nitinol. The wire is then deformed and,
upon heating, springs back to its predeformed
shape; this shape recovery process is recorded
on the photograph.

56
Shape-Memory Alloys
Fig. 10.37 Diagram illustrating the shape-memory effect. This insets are schematic re-
presentations of the crystal structure at the four stages. Ms and Mf denote temperatures
at which the martensitic transformation begins and ends. Likewise for the austenite
transformation, As and Af represent beginning and end transformation temperatures.

BME 303 - Lesson 7 - Phase Transformation.pdf

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