Crystal GrowthGeneral Formalism

Phase growing into with velocity v : v f k

f ( “site factor” ) : fraction of sites where a new atom

can be incorporated 0 < f < 1

: interatomic distance k : jump frequency

*at equilibrium temperature and ( T ) the net k must be zero.

k k +

G*

G = G

*

exp Gk kRT

below T: G = G - G

k k +

G*

G

G

G

*Gk expRT

*G Gk exp

RT

net jump rate =

RTGkkkk exp1

1 Gv f k expRT

for small undercooling RT ;ΔG xex 1~

Gv f kRT

fS Tv f k

RT

Same as for nucleationCollision limited : s / diff limited : 6D / 2

For metal solidification:

1

RS f (Richards’ rule)

8ff

m

LS

T

+ Tv f kT

Small undercooling

Crystal growth velocity normal to area A is v.

Consider growth to occur by nucleation of monolayer patches .

Growth governed by two – dimensional nucleation

vl

v

area A

area concentration of critical nuclei.

# at. in cluster **sss iknI Nuc.rate :

h

growth of patches occurs by lateral spreading at a velocity vl (ledge velocity)

l lv f kfraction of sites along ledge that can

incorporate new atoms.

k : net jump rate to ledge

vapor solid; k is collision rate, surface migration

The time for 1 layer of area A to form : t =v

RTGkk exp1for condensed phase

Number of nuclei formed in that amount of time

AtIN ss

Average area of each patch grown in that amount of time -

2v tA ls

In that time the whole area must be covered:

AtAtI ls 2v 3 2 1s lI t v

# of nuclei

so 1/3 2/3 s lv I vt

AAN ss , or

or

Large undercooling required

1

0

f

T

v

T

f(G)

Since nucleation of ledges is difficult at low undercoolings in many systems

crystal growth is governed by intrinsic ledge structure.

Plug in eqns. for Is and vl

Probability that an atom is in a critical nucleus - very T dependent

Growth on Surface Defects -

Observation : many crystals grow at small undercooling supersaturations

F. C. Frank : Screw dislocation mechanism

Spiral growth around screw dislocation

t0 t1 t2 t3 t4

~ r*

After some time tn a spiral forms.

R

Top View time evolution

Ledge Size* l

v

R Kr KG

~ critical nucleus size

K ~ 4; Archimedes spiral

Area fraction of growth sites : (Assume attachment at all step sites; fl = 1 )

vl

GKR

f

: is inter-atomic spacing.

and Tv f kT

1

0

f

T T

parabolic

v

Fraction transformed in isothermal process – Avrami analysis

Consider transformation

How do you deal with the overlap?

Mathematical device : extended volume fraction Xex volume fraction

transformed disregarding overlap.

The actual volume fraction grows in a relative amount to the unconsumed

fraction, at the same rate the extended volume fraction does.:

//

1 ex

dx dtdX dt

x

Unconsumed fraction

exdXx

dx

1

Integrate

exXx exp1 Avrami equation

Expand : 32

!31

21

exexex XXXx

dilute overlap of two overlap of three

or

Application to nucleation & growth : ( Johnson - Mehl)

Case (1) constant number of heterogeneous nuclei present from the

beginning.

concentration: N

growth rate of crystals : v

33

4 vtNX ex

33

34exp1 tNvx

x

t

Plot of ln t vs ln[-ln(1-x)]

should have slope of 3.

Case (2) Assume a constant nucleation rate I, # of nuclei formed between t’

and t’ + dt’ ; concentration, N = I dt’ and at some later time ( t >

t’ ) the “radius” of transformed phase is v (t – t’)

so 4333

0 3'

34' tIvttvIdtX

t

ex

3 413

x exp Iv t

Plot of ln t vs ln[-ln(1-x)] slope of 4

These plots are called Johnson- Mehl –Arami plots

(JMA plots)

Calorimetry results

pow

er

DSCisothermals

Time (min) 20 40 60 80 100

329K 328K

327K

326K 325K

324K

Time

1/2

1

X

329K 328K 327K 326K 325K 324K

0

Fraction transformed

Case study : Devitrification of Au65Cu12Si9Ge14 glass

C. Thompson et. al., Acta Met., 31, 1883 (1983)

-16 -8 00 08 16

(b)

ln (1-) 02 06 10 14 18

-100-80-60-40-200204060

(a)

ln (t)

ln [-

ln(1

-x)]

JMA plot (327K)

must be introducedN = Iss(t -)Slope = 4.0

ln [-

ln(1

-x)]

slope = 4

Transient time

304 306 308 31038

40

42

44

46

48

50

ln [

(s)]

103 / T (K-1)

activation energy : 2.0 eV

Interface Stability during Growth

(i) Consider solidification in a one component system:

The solidification process is controlled by the rate at which the latent heat of

solidification can be conducted away from the solid / liquid interface.

Solid growth into a liquid @ T > Tm

T

Solid LiquidTm

heat flux

v - interface velocity

(a)

The heat flux away from the interface through the solid must balance that

from the liquid plus the latent heat generated at the interface.:

s L fs ,i L,i

dT dTK K vLdx dx

Here K is the thermal conductivity, and Lf the latent heat of fusionper unit volume.

heat

Solid

Liquid

The dotted lines are isotherms.

Consider the stability of a perturbation that may develop at the interface:

Since the perturbation is an the high temperature side of the interface, the thermal gradient in the protuberance is less than it is in the planar portion of the interface and the protuberance will disappear. Rate limiting step in the growing phase.

T

Solid Liquid

heat flux

v

x

(b) Solidification in a super-cooled liquid:

The perturbation in the super-cooled liquid sees a higher temperature gradient than the planar portion of the interface perturbation is stable.Rate limiting step in the shrinking phase.

Solid Liquid

heat flux

Sieradzki’s rule of interface stability:

If the rate limiting step is in the phase that is growing a planar interface will be stable to a geometrical perturbation. If the rate limiting step is in the phase that is shrinking the planar interface will be unstable.

Alloy Solidification

Def. Partition coefficient k

L

s

xxk

xs and xL are the mole fraction of solute in the solid and liquid respectively.

T3

T1

T2

kx0 x0 x0 / k

We will consider 3 limiting cases of the solidification process. :

(a) Infinitely slow (equilibrium) solidification

(b) Solidification with no diffusion in the solid and perfect mixing in the liquid.

(c) Solidification with no diffusion in the solid and only diffusional mixing in

the liquid.

(a) Equilibrium solidification.

@ composition x0

0kxxs will be the composition of the 1st amount of solid to solidify.

* Note that k is constant for straight liquidus and solidus.

As the temperature is lowered more solid forms.

For slow enough cooling mixing in liquid and solid is perfect and xs and xL will

follow the solidus and liquidus lines respectively

At T3 the last liquid to freeze out has a composition xo