Download - Chapter 3 Nucleation and Growth
Institute of Materials Science
Chapter 3: Nucleation and Growth
3.1 Homogeneous Nucleation – Driving Force
3.2 Nucleation Rate
3.3 Heterogeneous Nucleation
3.4 Nucleation in the Solid State
3.5 Growth Rate
3.6 KJMA Model
3.7 Heat Flow, Interface Stability and Dendritic Growth
MMAT 305MMAT 305
Institute of Materials Science
3.1 Homogeneous Nucleation – Driving Force
Table 8.1 Major Types of Phase Transformations
Type of Transformation Example
1. Vapor liquid Condensation of moisture
2. Vapor solid Formation of frost on a window
3. Liquid crystal Formation of ice on a lake
4. Crystal 1 crystal 2
(a) Precipitation Formulation of Fe3C on cooling austenite
(b) Allotropic α-Fe γ-Fe at 910 ºC
(c) RecrystallizationCold-worked Cu new grains at high
temperatures
From J.D. Verhoeven, “Fundamentals of Physical Metallurgy,” Wiley, 1974
MMAT 305MMAT 305
Institute of Materials Science
3.1 Homogeneous Nucleation – Driving Force
MMAT 305MMAT 305
Table 8.2 Degree of Complexity Involved in Phase Transformations
(a) Structure change
(b) Structure change + composition change
(c) Structure change + strain formation
(d) Structure change + strain formation + composition change
From J.D. Verhoeven, “Fundamentals of Physical Metallurgy,” Wiley, 1974
Institute of Materials Science
3.1 Homogeneous Nucleation – Driving Force
MMAT 305MMAT 305
Institute of Materials Science
3.2 Nucleation Rate
MMAT 305MMAT 305
Distribution functions for embryos of different sizes according to Volmer and Becker-Döring theories of nucleation.
Institute of Materials Science
3.2 Nucleation Rate
Institute of Materials Science
3.3 Heterogeneous Nucleation
MMAT 305MMAT 305
Wall
S
SL
SW X
YL
WL Wall
S
SL
SW X
YL
WL
Institute of Materials ScienceMMAT 305MMAT 305
ΔT =0, T=Tm
HeterogenousI: Nucleation Rate
ΔT
IN
ucle
atio
n R
ate
Homogenous
3.3 Heterogeneous Nucleation
Institute of Materials ScienceMMAT 305MMAT 305
Fig. 4.8 The excess free energy of solid clusters for homogeneous and heterogeneous nucleation. Note r* is independent of the nucleation site.
3.3 Heterogeneous Nucleation
Institute of Materials ScienceMMAT 305MMAT 305
Fig 4.9 (a) Variation of ΔG* with undercooling (ΔT ) for homogeneous and heterogeneous nucleation. (b) The corresponding nucleation rates assuming the same critical value of ΔG*.
3.3 Heterogeneous Nucleation
Institute of Materials Science
3.4 Nucleation in the Solid State
MMAT 305MMAT 305
Institute of Materials Science
3.4 Nucleation in the Solid State
MMAT 305MMAT 305
Institute of Materials Science
3.4 Nucleation in the Solid State
MMAT 305MMAT 305
Fig. 3.48 For a coherent thin disc there is little misfit parallel to the plane of the disc. Maximum misfit is perpendicular to the disc.
Fig. 3.47 The origin of coherency strains. The number of lattice points in the hole is conserved.
(a) (b) (c)
Institute of Materials Science
3.4 Nucleation in the Solid State
MMAT 305MMAT 305
Fig. 3.51 Coherency strains caused by the coherent broad faces of precipitates.
Fig. 1. Coherent plate and plate with incoherent edge.
(a)
(b)
Institute of Materials Science
3.4 Nucleation in the Solid State
MMAT 305MMAT 305
Fig. 3.50 The variation of misfit strain energy with ellipsoid shape, f(c/a). (After F.R.N. Nabarro, Proceedings of the Royal Society A, 175 (1940) 519.)
(a) (b)
Fig. 3.49 The origin of misfit strain for an incoherent inclusion (no lattice matching).
Institute of Materials Science
3.5 Growth Rate
MMAT 305MMAT 305
Institute of Materials Science
3.6 KJMA Model
MMAT 305MMAT 305
Institute of Materials Science
3.6 KJMA Model
MMAT 305MMAT 305
Institute of Materials Science
3.6 KJMA Model
MMAT 305MMAT 305
Institute of Materials Science
3.6 KJMA Model
MMAT 305MMAT 305
Institute of Materials Science
Nucleation Rate – Limitations to KJMA Model
MMAT 305MMAT 305
Fig. 5.24 (a) Nucleation at a constant rate during the whole transformation. (b) Site saturation – all nucleation occurs a the beginning of transformation. (c) A cellular transformation.
(a)
(b)
(c)
Institute of Materials ScienceMMAT 305MMAT 305
Other Modes of Phase Transformations
Institute of Materials Science
Other Modes of Solidification
MMAT 305MMAT 305
(a)
(b)
Fig. 4.11 Atomically smooth solid/liquid interfaces with atoms represented by cubes. (a) Addition of a single atom onto a flat interface increases the number of ‘broken bonds’ by four. (b) Addition to a ledge (L) only increases the number of broken bonds by two, whereas at a jog in a ledge (J) there is no increase.
Fig. 4.12 Ledge creation by surface nucleation.
Institute of Materials ScienceMMAT 305MMAT 305
Fig. 4.13 Spiral growth. (a) A screw dislocation terminating in the solid/liquid interface showing the associated ledge. (After W.T. Read Jr., Dislocations in Crystals, © 1953 McGraw-Hill. Used with the permission of McGraw-Hill Book Company.) Addition of atoms at the ledge causes it to rotate with an angular velocity decreasing away from the dislocation core so that a growth spiral develops as shown in (b). (After J.W. Christian, The Theory of Phase Transformations in Metals and Alloys, Pergamon Press, Oxford, 1965.)
(a)
(b)
Other Modes of Solidification
Institute of Materials ScienceMMAT 305MMAT 305
Fig. 4.14 The influence of interface undercooling (ΔTi ) on growth rate for atomically rough and smooth interfaces.
Other Modes of Solidification
Institute of Materials Science
3.7 Heat Flow, Interface Stability and Dendritic Growth
MMAT 305MMAT 305
Institute of Materials Science
3.7 Heat Flow, Interface Stability and Dendritic Growth
MMAT 305MMAT 305
Fig. 4.16 As Fig. 4.15, but for heat conduction into the liquid.
(a) (b) (c)
Institute of Materials Science
3.7 Heat Flow, Interface Stability and Dendritic Growth
MMAT 305MMAT 305
Fig. 4.17 The development of thermal dendrites: (a) a spherical nucleus; (b) the interface becomes unstable; (c) primary arms develop in crystallographic directions (<100> in cubic crystals); (d) secondary and tertiary arms develop (after R.E. Reed-Hill, Physical Metallurgy Principles, 2nd. Edn., Van Nostrand, New York, 1973.)
(a)(b)
(c)
(d)
Institute of Materials Science
3.7 Heat Flow, Interface Stability and Dendritic Growth
MMAT 305MMAT 305
Fig. 4.18 Temperature distribution at the tip of a growing thermal dendrite.
Institute of Materials Science
3.2 Nucleation Rate
VW
BD
I
ΔT ΔT =0, T=Tm
I: Nucleation Rate
Nuc
leat
ion
Rat
e