introduction to materials science and engineering · 2021. 2. 17. · introduction to materials...
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Introduction to Materials Science and Engineering
Eun Soo Park
Office: 33-313 Telephone: 880-7221Email: [email protected] hours: by appointment
2020 Fall
11. 03. 2020
1
2
• Phase diagrams are useful tools to determine:-- the number and types of phases present,-- the composition of each phase,-- and the weight fraction of each phase given the temperature and composition of the system.
• The microstructure of an alloy depends on-- its composition, and-- whether or not cooling rate allows for maintenance of
equilibrium.
• In phase diagram, important phase transformations include eutectic, eutectoid, and peritectic.
SummaryContents for previous class
Chapter 9 Phase Diagrams
Metatectic reaction: β ↔ L + α Ex. Co-Os, Co-Re, Co-Ru 3
Syntectic reaction
Liquid1+Liquid2 ↔ α
L1+L2
α
K-Zn, Na-Zn,K-Pb, Pb-U, Ca-Cd
4
5
L+αL+β
α +β
200
C, wt% Sn20 60 80 1000
300
100
L
α βTE
40
(Pb-Sn System)
Hypoeutectic & Hypereutectic
Fig. 11.7, Callister & Rethwisch 9e. [Adapted from Binary Alloy Phase Diagrams, 2nd edition, Vol. 3, T. B. Massalski (Editor-in-Chief), 1990. Reprinted by permission of ASM International, Materials Park, OH.]
160 μmeutectic micro-constituent
Fig. 11.13, Callister & Rethwisch 9e.
hypereutectic: (illustration only)
β
βββ
β
β
Adapted from Fig. 11.16, Callister & Rethwisch 9e.(Illustration only)
(Figs. 11.13 and 11.16 from Metals Handbook, 9th ed., Vol. 9, Metallography and Microstructures, 1985.Reproduced by permission of ASM International,Materials Park, OH.)
175 μm
α
α
α
ααα
hypoeutectic: C0 = 50 wt% Sn
Fig. 11.16, Callister & Rethwisch 9e.
T(°C)
61.9eutectic
eutectic: C0 =61.9wt% Sn
6
• For alloys for which 18.3 wt% Sn < C0 < 61.9 wt% Sn• Result: α phase particles and a eutectic microconstituent
Microstructural Developments in Eutectic Systems
18.3 61.9
SR
97.8
SR
primary αeutectic α
eutectic β
WL = (1-Wα) = 0.50
Cα = 18.3 wt% SnCL = 61.9 wt% Sn
SR + S
Wα = = 0.50
• Just above TE :
• Just below TE :Cα = 18.3 wt% SnCβ = 97.8 wt% Sn
SR + S
Wα = = 0.73
Wβ = 0.27Fig. 11.15, Callister & Rethwisch 9e.
Pb-Snsystem
L+β200
T(°C)
C, wt% Sn
20 60 80 1000
300
100
L
α βL+α
40
α+ β
TE
L: C0 wt% Sn LαLα
α ironFerrite(BCC)
soft & ductile
γ ironaustenite
(FCC)
Cementite (Fe3C)hard & brittle
A; eutecticB; eutectoid
B
A
c. Fe-C phase diagram
C concentration 0.008w% 2.14w% 6.7w%iron steel cast iron
7
Hypoeutectoid Steel
Proeutectoid
ferrite
Pearlitedark – ferritelight - cementite
8
Hypereutectoid Steel
Proeutectoid cementite
Pearlite
9
10
Alloying with Other Elements
• Teutectoid changes:
Fig. 11.33, Callister & Rethwisch 9e. (From Edgar C. Bain, Functions of the Alloying Elementsin Steel, 1939. Reproduced by permission of ASMInternational, Materials Park, OH.)
T Eut
ecto
id(º
C)
wt. % of alloying elements
Ti
Ni
Mo SiW
Cr
Mn
• Ceutectoid changes:
Fig. 11.34,Callister & Rethwisch 9e. (From Edgar C. Bain, Functions of the Alloying Elements in Steel, 1939. Reproduced by permission of ASM International, Materials Park, OH.)
wt. % of alloying elementsC
eute
ctoi
d(w
t% C
)
Ni
Ti
Cr
SiMnWMo
Ternary Eutectic System(with Solid Solubility)
11
Ternary Eutectic System(with Solid Solubility)
12
13
Ternary Eutectic System(with Solid Solubility)
13
Ternary Eutectic System(with Solid Solubility)
14
Ternary Eutectic System(with Solid Solubility)
15
16
Ternary Eutectic System(with Solid Solubility)
16
17
Ternary Eutectic System(with Solid Solubility)
17
Ternary Eutectic System(with Solid Solubility)
18
Ternary Eutectic System(with Solid Solubility)
T= ternary eutectic temp.
L+β+γ
L+α+γL+α+β
19
Ternary Eutectic System(with Solid Solubility)
http://www.youtube.com/watch?v=yzhVomAdetM20
• Isothermal section (TA > T > TB)
THE EUTECTIC EQUILIBRIUM (l = α + β + γ)
(a) TA > T >TB (b) T = e1 (c) e1 > T > e3 (d) T = e3
(g) TA = E(e) T = e2 (f) e2 > T > E (h) E = T 21
Vertical section Location of vertical section
THE EUTECTIC EQUILIBRIUM (l = α + β + γ)
22
Vertical section Location of vertical section
THE EUTECTIC EQUILIBRIUM (l = α + β + γ)
23
Vertical section Location of vertical section
THE EUTECTIC EQUILIBRIUM (l = α + β + γ)
25
Vertical section Location of vertical section
THE EUTECTIC EQUILIBRIUM (l = α + β + γ)
< Quaternary phase Diagrams >
Four components: A, B, C, D
A difficulty of four-dimensional geometry→ further restriction on the system
Most common figure: “ equilateral tetrahedron “
Assuming isobaric conditions,Four variables: XA, XB, XC and T
4 pure components6 binary systems4 ternary systemsA quarternary system
ISSUES TO ADDRESS...• Transforming one phase into another takes time.
• How does the rate of transformation depend ontime and T?
• How can we slow down the transformation so thatwe can engineering non-equilibrium structures?
• Are the mechanical properties of non-equilibriumstructures better?
Feγ
(Austenite)
Eutectoid transformation
C FCC
Fe3C(cementite)
α(ferrite)
+(BCC)
Chapter 10. Phase Transformations: Development of Microstructure and Alteration of Mechanical Properties
Contents for today’s class
27
2 1 0G G G∆ = − <
Microstructure-Properties Relationships
Microstructure Properties
Alloy design & Processing
Performance
“ Phase Transformation ”
“Tailor-made Materials Design”down to atomic scale
28
(Ch1) Thermodynamics and Phase Diagrams
(Ch2) Diffusion: Kinetics
(Ch3) Crystal Interface and Microstructure
(Ch4) Solidification: Liquid → Solid
(Ch5) Diffusional Transformations in Solid: Solid → Solid
(Ch6) Diffusionless Transformations: Solid → Solid
Backgroundto understand
phase transformation
RepresentativePhase
transformation
Contents_Phase transformation course
29
4 Fold Anisotropic Surface Energy/2 Fold Kinetics, Many Seeds
Solidification: Liquid SolidPhase Transformations
30
- casting & welding- single crystal growth- directional solidification- rapid solidification
4.1. Nucleation in Pure MetalsTm : GL = GS
- Undercooling (supercooling) for nucleation: 250 K ~ 1 K
<Types of nucleation>- Homogeneous nucleation - Heterogeneous nucleation
Solidification: Liquid Solid
31
Driving force for solidification
4.1.1. Homogeneous Nucleation
mTTLG ∆
=∆
L : ΔH = HL – HS
(Latent heat)T = Tm - ΔT
GL = HL – TSL
GS = HS – TSS
ΔG = Δ H -T ΔS ΔG =0= Δ H-TmΔS
ΔS=Δ H/Tm=L/Tm
ΔG =L-T(L/Tm)≈(LΔT)/Tm
V Variation of free energy per unit volume obtained from undercooling (ΔT)
32
33 33
4.1.1. Homogeneous Nucleation
LVLS GVVG )(1 += SLSL
LVL
SVS AGVGVG γ++=2
LV
SV GG ,
SLSLS
VL
VS AGGVGGG γ+−−=−=∆ )(12
SLVr rGrG γππ 23 434
+∆−=∆
: free energies per unit volume
for spherical nuclei (isotropic) of radius : r
Surface Free Energy- destabilizes the nuclei (it takes energy to make an interface)
Volume (Bulk) Free Energy –stabilizes the nuclei (releases energy)
r < r* : unstable (lower free E by reduce size)r > r* : stable (lower free E by increase size)r* : critical nucleus size
Why r* is not defined by ∆Gr = 0?
r* dG=0Unstable equilibrium
SLVr rGrG γππ 23 434
+∆−=∆
Calculation of critical radius, r*
embryo nucleus
Gibbs-Thompson Equation
V
SL
Gr
∆=∗ γ2
SLVr rGrG γππ 23 434
+∆−=∆
22
23
2
3
)(1
316
)(316*
TLT
GG
V
mSL
V
SL
∆
=
∆=∆
πγγπ
mTTLG ∆
=∆V
Critical ΔG of nucleation at r*
nuclei < r* shrink; nuclei>r* grow (to reduce energy)
35
* SL SL m
V V
2 2 T 1rG L Tγ γ
= = ∆ ∆ 22
23
2
3
)(1
316
)(316*
TLT
GG
V
mSL
V
SL
∆
=
∆=∆
πγγπ
36
Fig. 4.5 The variation of r* and rmax with undercooling ∆T
∆T↑ → r* ↓→ rmax↑
The creation of a critical nucleus ~ thermally activated process
of atomic cluster
* SL SL m
V V
2 2 T 1rG L Tγ γ
= = ∆ ∆ mTTLG ∆
=∆
∆TN is the critical undercooling for homogeneous nucleation.
The number of clusters with r* at ∆T < ∆TN is negligible. 37
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Os
W
Re
TaMo
NbIr
Ru
Hf
RhZr
Pt
Fe
TiPd
Co
Ni
MnCu
AuAg
Ge
Al
SbTe
Pb
Cd
Bi
Sn
Se
In
Ga
Hg
Melting temperature > 3000K
: Ta, Re, Os, W measured by ESL
Slope : 0.18
JAXA(2007)JAXA(2005)
JAXA(2003)
JAXA(2003)
Δ = 55.9 + 1.83 x 10-1T
Measuring high temperature properties!
Maximum undercooling vs. Melting temperature
38
39
* Copper Homogeneous nucleation
ΔT = 230 K → r* ~ 10-7 cm < 4 * (Diameter of Cu atom)If nucleus is spherical shape,
V = 4.2 * 10-21 cm3 ~ 360 atoms (∵one Cu atom 1.16 *10-23 cm3)
“Typically in case of metal” ΔT * ~ 0.2 TE / σSL ~ 0.4 L
r* (critical nucleus for homogeneous nucleation) of metal ~ 200 atoms
But, if cluster radius ~ (only 4 * atom diameter),
“no spherical shape”(large deviation from spherical shape) →
→ Possible structure for the critical nucleus of Cu
: bounded only by {111} and {100} plane
- σSL may very with the crystallographic nature of the surface.
- The faces of this crystal are close to their critical size for2D nucleation at the critical temp for the nucleus as a whole.
4.1.2. The homogeneous nucleation rate - kinetics
How fast solid nuclei will appear in the liquid at a given undercooling?
C0 : atoms/unit volumeC* : # of clusters with size of C* ( critical size )
)exp(*hom
0 kTGCC ∆
−=∗
clusters / m3
The addition of one more atom to each of these clusters will convert them into stable nuclei.
)exp(*hom
0hom kTGCfN o
∆−=
nuclei / m3∙s
fo ~ 1011 s-1: frequency ∝ vibration frequency energy of diffusion in liquid surface area (const.)
Co ~ typically 1029 atoms/m322
23
)(1
316*
TLTG
V
mSL
∆
=∆
πγ
-hom
*3 11 cm when G ~ 78s kN T− ∆≈
Homogeneous Nucleation rate
임계핵 크기의 cluster 수
한 개 원자 추가로 확산시 핵생성
Reasonable nucleation rate40
kTLTA
V
mSL2
23
316πγ
=})(
exp{ 20hom TACfN o ∆
−≈A = relatively insensitive to Temp.
Fig. 4.6 The homogeneous nucleation rate as a function of undercooling ∆T. ∆TN is the critical undercooling for homogeneous nucleation.
→ critical value for detectable nucleation- critical supersaturation ratio- critical driving force- critical supercooling
How do we define ∆TN?
where
4.1.2. The homogeneous nucleation rate - kinetics
2hom1~T
N∆
→ for most metals, ΔTN~0.2 Tm (i.e. ~200K)
Changes by orders of magnitude from essentially zero to very high values over a very narrow temperature range
41
0* exp( * / )C C G kT= −∆
Homogeneous Nucleation Rate
hom 0*exp expmG GN C
kT kTω ∆ ∆ = − −
Concentration of Critical Size Nuclei per unit volume
C0 : number of atoms per unit volume in the parent phase
hom *N f C= ( )kTGexpf m∆−= ωvibration frequency, area of critical nucleusω ∝
:mG activation energy for atomic migration∆
Homogeneous Nucleation in Solids
If each nucleus can be made supercritical at a rate of f per second,
: f depends on how frequently a critical nucleus can receive an atom from the α matrix.
: This eq. is basically same with eq (4.12) except considering temp. dependence of f.
42
hom 0*exp expmG GN C
kT kTω ∆ ∆ = − −
= Qd
4.1.2. The homogeneous nucleation rate - kinetics
43
Under suitable conditions, liquid nickel can be undercooled (or supercooled) to 250 K below Tm (1453oC) and held there indefinitely without any transformation occurring.
Normally undercooling as large as 250 K are not observed.The nucleation of solid at undercooling of only ~ 1 K is common.
The formation of a nucleus of critical size can be catalyzed by a suitable surface in contact with the liquid. → “Heterogeneous Nucleation”
Which equation should we examine?
)exp(*hom
0hom kTGCfN o
∆−=*
( ) ( )
3 3 2SL SL m
2 2 2V V
16 16 T 1G3 G 3 L T
πγ πγ ∆ = = ∆ ∆
Why this happens? What is the underlying physics?
Real behavior of nucleation: metal ΔTbulk < ΔTsmall drop
Ex) liquid
containeror
liquid
Solid thin film (such as oxide)
44
Introduction to Materials Science and Engineering
Eun Soo Park
Office: 33-313 Telephone: 880-7221Email: [email protected] hours: by appointment
2020 Fall
11. 05. 2020
1
2
Ternary Eutectic System(with Solid Solubility)
2
< Quaternary phase Diagrams >
Four components: A, B, C, D
A difficulty of four-dimensional geometry→ further restriction on the system
Most common figure: “ equilateral tetrahedron “
Assuming isobaric conditions,Four variables: XA, XB, XC and T
4 pure components6 binary systems4 ternary systemsA quarternary system
Microstructure-Properties Relationships
Microstructure Properties
Alloy design & Processing
Performance
“ Phase Transformation ”
“Tailor-made Materials Design”down to atomic scale
4
Contents for previous class
5
(Ch1) Thermodynamics and Phase Diagrams
(Ch2) Diffusion: Kinetics
(Ch3) Crystal Interface and Microstructure
(Ch4) Solidification: Liquid → Solid
(Ch5) Diffusional Transformations in Solid: Solid → Solid
(Ch6) Diffusionless Transformations: Solid → Solid
Backgroundto understand
phase transformation
RepresentativePhase
transformation
Contents_Phase transformation course
2 1 0G G G∆ = − <A B
Time
Chapter 10. Phase Transformations: Development of Microstructure and Alteration of Mechanical Properties
Liquid Undercooled Liquid Solid
<Thermodynamic>
Solidification: Liquid Solid
• Interfacial energy ΔTN
Tm
Melting and Crystallization are Thermodynamic Transitions
6
7 7
Fig. 4.5 The variation of r* and rmax with undercooling ∆T
∆T↑ → r* ↓→ rmax↑
The creation of a critical nucleus ~ thermally activated process
of atomic cluster
* SL SL m
V V
2 2 T 1rG L Tγ γ
= = ∆ ∆ mTTLG ∆
=∆
∆TN is the critical undercooling for homogeneous nucleation.
The number of clusters with r* at ∆T < ∆TN is negligible. 8
0 500 1000 1500 2000 2500 3000 3500 4000
0
200
400
600
800
1000
Os
W
Re
TaMo
NbIr
Ru
Hf
RhZr
Pt
Fe
TiPd
Co
Ni
MnCu
AuAg
Ge
Al
SbTe
Pb
Cd
Bi
Sn
Se
In
Ga
Hg
Melting temperature > 3000K
: Ta, Re, Os, W measured by ESL
Slope : 0.18
JAXA(2007)JAXA(2005)
JAXA(2003)
JAXA(2003)
Δ = 55.9 + 1.83 x 10-1T
Measuring high temperature properties!
Maximum undercooling vs. Melting temperature
9
10
* Copper Homogeneous nucleation
ΔT = 230 K → r* ~ 10-7 cm < 4 * (Diameter of Cu atom)If nucleus is spherical shape,
V = 4.2 * 10-21 cm3 ~ 360 atoms (∵one Cu atom 1.16 *10-23 cm3)
“Typically in case of metal” ΔT * ~ 0.2 TE / σSL ~ 0.4 L
r* (critical nucleus for homogeneous nucleation) of metal ~ 200 atoms
But, if cluster radius ~ (only 4 * atom diameter),
“no spherical shape”(large deviation from spherical shape) →
→ Possible structure for the critical nucleus of Cu
: bounded only by {111} and {100} plane
- σSL may very with the crystallographic nature of the surface.
- The faces of this crystal are close to their critical size for2D nucleation at the critical temp for the nucleus as a whole.
kTLTA
V
mSL2
23
316πγ
=})(
exp{ 20hom TACfN o ∆
−≈A = relatively insensitive to Temp.
Fig. 4.6 The homogeneous nucleation rate as a function of undercooling ∆T. ∆TN is the critical undercooling for homogeneous nucleation.
→ critical value for detectable nucleation- critical supersaturation ratio- critical driving force- critical supercooling
How do we define ∆TN?
where
4.1.2. The homogeneous nucleation rate - kinetics
2hom1~T
N∆
→ for most metals, ΔTN~0.2 Tm (i.e. ~200K)
Changes by orders of magnitude from essentially zero to very high values over a very narrow temperature range
11
hom 0*exp expmG GN C
kT kTω ∆ ∆ = − −
= Qd
4.1.2. The homogeneous nucleation rate - kinetics
12
Under suitable conditions, liquid nickel can be undercooled (or supercooled) to 250 K below Tm (1453oC) and held there indefinitely without any transformation occurring.
Normally undercooling as large as 250 K are not observed.The nucleation of solid at undercooling of only ~ 1 K is common.
The formation of a nucleus of critical size can be catalyzed by a suitable surface in contact with the liquid. → “Heterogeneous Nucleation”
Which equation should we examine?
)exp(*hom
0hom kTGCfN o
∆−=*
( ) ( )
3 3 2SL SL m
2 2 2V V
16 16 T 1G3 G 3 L T
πγ πγ ∆ = = ∆ ∆
Why this happens? What is the underlying physics?
Real behavior of nucleation: metal ΔTbulk < ΔTsmall drop
Ex) liquid
containeror
liquid
Solid thin film (such as oxide)
13
Nucleation becomes easy if ↓ by forming nucleus from mould wall.SLγ
From
Fig. 4.7 Heterogeneous nucleation of spherical cap on a flat mould wall.
4.1.3. Heterogeneous nucleation
22
23
)(1
316*
TLTG
V
mSL
∆
=∆
πγ
SMSLML γθγγ += cos
SLSMML γγγθ /)(cos −=
het S v SL SL SM SM SM MLG V G A A Aγ γ γ∆ = − ∆ + + −
3 24 4 ( )3het V SLG r G r Sπ π γ θ ∆ = − ∆ +
where 2( ) (2 cos )(1 cos ) / 4S θ θ θ= + −
In terms of the wetting angle (θ ) and the cap radius (r) (Exercies 4.6)
14
S(θ) has a numerical value ≤ 1 dependent only on θ (the shape of the nucleus)
* *hom( )hetG S Gθ∆ = ∆ )(
316*2* 2
3
θπγγ SG
GandG
rV
SL
V
SL ⋅∆
=∆∆
=
( )S θ
θ = 10 → S(θ) ~ 10-4
θ = 30 → S(θ) ~ 0.02 θ = 90 → S(θ) ~ 0.5
15
S(θ) has a numerical value ≤ 1 dependent only on θ (the shape of the nucleus)
* *hom( )hetG S Gθ∆ = ∆
Fig. 4.8 The excess free energy ofsolid clusters for homogeneous andheterogeneous nucleation. Note r* isindependent of the nucleation site.
)(3
16*2* 2
3
θπγγ SG
GandG
rV
SL
V
SL ⋅∆
=∆∆
=
16
θ
AV
BV
Barrier of Heterogeneous Nucleation
4)coscos32(
316)(
316*
3
2
3
2
3 θθπγθπγ +−⋅
∆=⋅
∆=∆
V
SL
V
SL
GS
GG
θ θ − +∆ = ∆
3* 2 3cos cos
4*
sub homoG G
32 3cos cos ( )4
A
A B
V SV V
θ θ θ− += =
+
How about the nucleation at the crevice or at the edge?
* *hom( )hetG S Gθ∆ = ∆
17
*homo
34∆G *
homo12∆G *
homo14∆G
Nucleation Barrier at the crevice
contact angle = 90groove angle = 60
What would be the shape of nucleus and the nucleation barrier for the following conditions?
*homo
16∆G
18
How do we treat the non-spherical shape?
∆ = ∆ +
* * Asub homo
A B
VG GV V
Substrate
Good Wetting
AV
BV Substrate
Bad Wetting
AV
BV
Effect of good and bad wetting on substrate
19
Although nucleation during solidification usually requires some undercooling, melting invariably occurs at the equilibrium melting temperature even at relatively high rates of heating.
Because, melting can apparently, start at crystal surfaces without appreciable superheating.
Why?
SVLVSL γγγ <+
In general, wetting angle = 0 No superheating required!
3.7 The Nucleation of Melting
(commonly)In the case of gold,
SLγLVγSVγ
20
Liquid Undercooled Liquid Solid
<Thermodynamic>
Solidification: Liquid Solid
• Interfacial energy ΔTN
Melting: Liquid Solid
• Interfacial energy
SVLVSL γγγ <+
No superheating required!
No ΔTN
Tm
vapor
Melting and Crystallization are Thermodynamic Transitions
21
The Effect of ΔT on ∆G*het & ∆G*hom?
Fig. 4.9 (a) Variation of ∆G* with undercooling (∆T) for homogeneous and heterogeneous nucleation.(b) The corresponding nucleation rates assuming the same critical value of ∆G*
-3 1hom 1 cmN s−≈
Plot ∆G*het & ∆G* hom vs ΔTand N vs ΔT.
*( ) ( )
3 3 2SL SL m
2 2 2V V
16 16 T 1G3 G 3 L T
πγ πγ ∆ = = ∆ ∆
)(3
16* 2
3
θπγ SG
GandV
SL ⋅∆
=∆
n1 atoms in contactwith the mold wall
)exp(*
1*
kT
Gnn het∆
−=
)exp(*
11 kT
GCfN het
het
∆−=
22
입자 성장은 장거리 원자 확산에 의함
성장속도 G의 온도의존성은 확산계수 경우와 동일.
23
5.4 Overall Transformation Kinetics – TTT DiagramThe fraction of Transformation as a function of Time and Temperature
Plot the fraction of
transformation (1%, 99%)
in T-log t coordinate.
→ f (t,T)
Plot f vs log t.
- isothermal transformation
- f ~ volume fraction of β at any time; 0~1
Fig. 5.23 The percentage transformation versus timefor different transformation temperatures.
If isothermal transformation,
24
Transformation Kinetics
Avrami proposed that for a three-dimensional nucleation and growth process kinetic law
( )nktf −−= exp1
specimen of Volumephase new of Volume
=f : volume fraction transformed
Assumption : √ reaction produces by nucleation and growth
√ nucleation occurs randomly throughout specimen
√ reaction product grows rapidly until impingement
Johnson-Mehl-Avrami equation
25
Constant Nucleation Rate Conditions
Nucleation rate (I ) is constant.
Growth rate (v) is constant.
No compositional change
f
tτ
δτ
t-τ
f(t)
specimen of Volume during formed
nuclei ofnumber tat time measured during
nucleated particle one of Vol.
×
=ττ dd
dfe
( )[ ] ( )
0
03
34
V
dIVtvdfe
ττπ ×−=
( )[ ]∫ −⋅=t 3
34)(
oe dtvItf ττπ
( ) 43
0
43
31
41
34 tIvtvI
t
πτπ =
−−⋅=
- do not consider impingement & repeated nucleation
- only true for f ≪ 1
As time passes the β cells will eventually impinge on one another and the rate of transformation will decrease again.
Constant Nucleation Rate Conditions consider impingement + repeated nucleation effects
( ) edffdf −= 1f
dfdfe −=
1( )ffe −−= 1ln
( )
−−=−−= 43
3exp1)(exp1)( tIvtftf e
π
Johnson-Mehl-Avrami Equationf
t
1
4t∝
beforeimpingement
1-exp(z)~Z (z ≪1 )
( )nktf −−= exp1k: T sensitive f(I, v)n: 1 ~ 4 (depend on nucleation mechanism)
Growth controlled. Nucleation-controlled.
i.e. 50% transform Exp (-0.7) = 0.5 7.05.0 =nkt nk
t /15.07.0
= 4/34/15.09.0vN
t =
Rapid transformations are associated with (large values of k), or (rapid nucleation and growth rates)
* Short time:
t→∞, f →1* Long time:
If no change of nucleation mechanism during phase transformation, n is not related to T.
Rate of Phase Transformation
Avrami rate equation → y = 1- exp (-ktn)
– k & n fit for specific sample
All out of material - done
log tFrac
tion
trans
form
ed, y
Fixed T
fraction transformed
time
0.5
By convention rate = 1 / t0.5
Adapted from Fig. 10.10, Callister 7e.
maximum rate reached – now amount unconverted decreases so rate slows
t0.5
rate increases as surface area increases & nuclei grow
28
Rate of Phase Transformations
• In general, rate increases as T ↑
r = 1/t0.5 = A e -Q/RT
– R = gas constant– T = temperature (K)– A = preexponential factor– Q = activation energy
Arrhenius expression
• r often small: equilibrium not possible!
Adapted from Fig. 10.11, Callister 7e.(Fig. 10.11 adapted from B.F. Decker and D. Harker, "Recrystallization in Rolled Copper", Trans AIME, 188, 1950, p. 888.)
135°C 119°C 113°C 102°C 88°C 43°C
1 10 102 104
29
Time–Temperature–Transformation Curves (TTT)
How much time does it take at any one temperature for a given fraction of the liquid to transform (nucleate and grow) into a crystal?
f(t,T) ~πI(T)µ(T)3t4/3where f is the fractional volume of crystals formed, typically taken to be 10-6, a barely observable crystal volume.
∆−
∆
∆=
RTE
TT
RTH
nI Dmcryst exp81
16exp
2πν
∆
∆−
=
m
m
TT
RTH
TaNfRTT exp1
)(3)( 2ηπ
µ
Nucleation rates Growth rates
30
Nucleation and Growth Rates
Nulceation and Growth for Silica
0.E+00
2.E+07
4.E+07
6.E+07
8.E+07
1.E+08
1.E+08
1.E+08
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
Temperature (K)
Nu
clea
tion
Rat
e (s
ec-1
)
0
5E-26
1E-25
1.5E-25
2E-25
2.5E-25
Gro
wth
Rat
e (m
-sec
-1)
Nucleation rate (sec-1) Growth rate (m/sec)
31
Time Transformation Curves for Silica
T-T-T Curve for Silica
200400600800
100012001400160018002000
1 1E+13 1E+26 1E+39 1E+52 1E+65 1E+78
Time (sec)
Tem
per
atu
re (K
)
32
33
* Time-Temperature-Transformation diagrams
TTT diagrams
33
34
* Continuous Cooling Transformation diagrams
CCT diagrams
34