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



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



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



MMAT 305MMAT 305


3.2 Nucleation Rate

MMAT 305MMAT 305

Distribution functions for embryos of different sizes according to Volmer and Becker-Döring theories of nucleation.


3.2 Nucleation Rate



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



Fig. 4.8 The excess free energy of solid clusters for homogeneous and heterogeneous nucleation. Note r* is independent of the nucleation site.



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*.




MMAT 305MMAT 305



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)



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)



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).


3.5 Growth Rate

MMAT 305MMAT 305


3.6 KJMA Model

MMAT 305MMAT 305


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)


Other Modes of Phase Transformations


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.


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)



Fig. 4.14 The influence of interface undercooling (ΔTi ) on growth rate for atomically rough and smooth interfaces.




MMAT 305MMAT 305



MMAT 305MMAT 305

Fig. 4.16 As Fig. 4.15, but for heat conduction into the liquid.

(a) (b) (c)



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)



MMAT 305MMAT 305

Fig. 4.18 Temperature distribution at the tip of a growing thermal dendrite.


3.2 Nucleation Rate

VW

BD

I

ΔT ΔT =0, T=Tm

I: Nucleation Rate

Nuc

leat

ion

Rat

e

chapter 3 nucleation and growth

Documents