Nucleation

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Nucleation and growth kinetics

Homogeneous nucleation Critical radius, nucleation rate Heterogeneous nucleationNucleation in melting and boilingGrowth mechanismsRate of a phase transformation

Reading: Chapters 4.1and 4.2 of Porter and Easterling,


Nucleation and growth - the main mechanism of phase transformations in materials

αB

1X

αB

2X 0

BX

coordinate spatial

αB

1X

αB

2X 0

BX

αB

1X

αB

2X 0

BX

atomsB

atomsB

T

P

solid

liquid


Phase transformations involve change in structure and (for multi-phase systems) composition ⇒ rearrangement and redistribution of atoms via diffusion is required.

The process of phase transformation usually involves:

Kinetics of phase transformations

Nucleation of the new phase(s) - formation of stable small particles (nuclei) of the new phase(s). Nuclei are often formed at grain boundaries and other defects.

Growth of the new phase(s) at the expense of the original phase(s).

S-shape curve: percent of material transformed vs. the logarithm of time.


NucleationNucleation can be Heterogeneous – the new phase appears on the walls of the container, at impurity particles, etc.Homogeneous – solid nuclei spontaneously appear within the undercooled phase.

Let’s consider solidification of a liquid phase undercooled below the melting temperature as a simple example of a phase transformation.

solid solid

liquid liquid

homogeneousnucleation

heterogeneousnucleation

supercooledliquid


Homogeneous nucleation

solid

liquid

Is the transition from undercooled liquid to a solid spherical particle in the liquid a spontaneous one?

That is, does the Gibbs free energy decreases?

supercooledliquid

The formation of a solid nucleus leads to a Gibbs free energy change of ΔG = G2 - G1 = -VS (Gv

L – GvS) + ASLγSL

negative below Tm

always positive

1 2

VS – volume of the solid sphereASL – solid/liquid interfacial areaγSL – solid/liquid interfacial energyΔGv = Gv

L – GvS is the difference between free energies per

unit volume of solid and liquid

at T < Tm, GvS < Gv

L – solid is the equilibrium phase


When a liquid is cooled below the melting temperature, there is a driving force for solidification, ΔGv = Gv

L - GvS

G

T*

ΔGv

GvS

GvL

Tm

ΔT

Reminder: Driving force for solidification (ΔGv)

At any temperature below Tm there is a driving force for solidification. The liquid solidify at T < Tm. If energy is added/removed quickly, the system can be significantly undercooled or (supercooled).

As we will see, the contribution of interfacial energy (γSL) results in a kinetic barrier for the phase transformation.

At temperature T*

Lv

*Lv

Lv ST-HG =

Sv

*Sv

Sv ST-HG =

v*

vv ST-HG ΔΔ=Δ

At temperature Tm

0ST-HG mvm

mvv =ΔΔ=Δ

m

mvm

v THS Δ

=Δ

m

mv

m

mv*m

vv TΔTΔH

TΔHTΔHΔG =−≈

For small undercooling ΔT we can assume that ΔHv and ΔSv are independent of temperature (neglect the difference in Cp between liquid and solid)


Origin of the interfacial energy (γSL)Consider a solid-liquid interface. Depending on the type of material and crystallographic orientation of the interface, the interface can be atomically flat (smooth, faceted) or rough (diffuse).

liquid

solid

liquid

solid

HSvH

LvH

TS-SvmST-

G

LvmST-

spatial coordinate

SvG L

vGγSL

inte

rfac

e



ΔG = G2 - G1 = -VS Δ Gv + ASLγSL

For a spherical nucleus with radius r: 3S r π

34V =

SL2v

3r γr 4πΔGr π

34-ΔG += 2SL r 4πA =

rΔG

*ΔG

*r

ΔGinterfacial energy ~ r2

volume energy ~ r3

For nucleus with a radius r > r*, the Gibbs free energy will decrease if the nucleus grows. r* is the critical nucleus size, ΔG* is the nucleation barrier.



At r = r* 0r γ 8πΔGr -4πdr

G dΔ SLv

2 =+=

v

SL*

ΔG γ2r = ( )

( )2v

3SL*

ΔG3γ 16πΔG =

G

T*

GvS

GvL

Tm

ΔT

*rr =*

SL

v r γ2ΔG =

Temperature of unstable equilibrium of a solid cluster of radius r* with undercooled liquid.



v

SL*

ΔG γ2r = ( )

( )2v

3SL*

ΔG3γ 16πΔG =

m

mv T

ΔTΔHΔG =

ΔT1

ΔHT γ2rm

mSL

*⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )( ) ( )22

m

2m

3SL*

ΔT1

ΔH3Tγ 16ΔG ⎟

⎟⎠

⎞⎜⎜⎝

⎛ π=

Both r* and G* decrease with increasing undercooling

The difference between the Gibbs free energy of liquid and solid (also called “driving force” for the phase transformation) is proportional to the undercooling below the melting temperature, ΔT = Tm – T:

where Hm is the latent heat of melting (or fusion)

Therefore:



ΔT1

ΔHT γ2rm

mSL

*⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )( ) ( )22

m

2m

3SL*

ΔT1

ΔH3Tγ 16ΔG ⎟

⎟⎠

⎞⎜⎜⎝

⎛ π=

Both r* and G* decrease with increasing undercooling

rΔG

*ΔG 1

*1r

ΔG

*2r

*ΔG 2

m12 TTT <<


MD simulation of laser melting of nanocrystalline AuLin, Leveugle, Bringa, Zhigilei, J. Phys. Chem. C 114, 5686, 2010

20 nm Au films irradiated with 200 fs laser pulse

Model:~500,000 atoms, nanocrystalline sample has 31 grains, average grain diameter is 8 nm

laser pulse laser pulse


Laser melting of single crystal Au film

20 ps

500 psTime (ps)

Tem

pera

ture

(K)

0 100 200 300 400 500

1000

2000

3000

4000

5000

6000

Tl

Fabs = 45 J/m2

Te

Time (ps)

T/T m

0 100 200 300 400 5000.95

1

1.05

1.1

1.15

1.2

Tm = 963 K

45 J/m2

200 ps

Melting starts at free surfaces of the free-standing film, two melting fronts propagate from the surfaces, temperature drops (energy goes into ΔHm×Vl). Melting stops when T approaches the equilibrium melting temperature Tm.


Laser melting of nanocrystalline Au film

20 ps

100 ps

45 J/m2

50 ps

Melting starts at grain boundaries, temperature drops (energy goes into ΔHm×Vl). Melting continues even after T drops below the equilibrium melting temperature Tm at ~30 ps and the last crystalline region disappears at ~250 ps.

Time (ps)

T/T m

0 100 200 300 400 5000.9

0.95

1

1.05

1.1

Tm = 963 K


The continuation of the melting process below Tm can be explained based on the nucleation theory

SLvr rGrG γπ+Δπ−=Δ 23 434

THT

Gr

m

mSL

v

SL

Δ⎟⎟⎠

⎞⎜⎜⎝

⎛Δγ

=Δγ

=122

*

⎥⎦

⎤⎢⎣

⎡Δγ

−=rH

TTm

SLm

121*

100 ps50 ps

Melting of nanocrystalline film (melted regions are blanked)

temperature of theequilibrium between the cluster of size r and the surrounding liquid

critical radius at ΔT


Unstable equilibrium between a crystalline cluster of radius r and undercooled liquid

Time (ps)

Num

bero

fato

ms

inth

ecr

ysta

lline

clus

ter

0 50 100 150 200 250 3001000

2000

3000

4000

5000

6000

T* ≈ 845 K

820 K

840 K

835 K

850 K

841 K

844 K

843 K

845 K

847 K

⎥⎦

⎤⎢⎣

⎡Δγ

−=rH

TTm

SLm

121*


Critical undercooling temperature of a crystalline cluster surrounded by undercooled liquid

⎥⎦

⎤⎢⎣

⎡Δγ

−=rH

TTm

SLm

121*



There is an energy barrier of ΔG* for formation of a solid nucleus of critical size r*. The probability of energy fluctuation of size ΔG* is given by the Arrhenius equation and the rate of homogeneous nucleation is

⎟⎟⎠

⎞⎜⎜⎝

⎛−ν

kTΔGexpN

*

d~& nuclei per m3 per s

where νd is the frequency with which atoms from liquid attach to the solid nucleus. The rearrangement of atoms needed for joining the solid nucleus typically follows the same temperature dependence as the diffusion coefficient:

⎟⎠⎞

⎜⎝⎛ −ν

kTexp d

dQ~

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −

kTΔGexp

kTQexpN

*d~&Therefore:


Rate of homogeneous nucleation

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −

kTΔGexp

kTQexpN

*d~&

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTΔGexp

*

⎟⎠⎞

⎜⎝⎛ −

kTQexp d

N&

mTTemperature

ΔG* is too high - nucleation is suppressed

( ) ( )kTQexpkTΔGexp d* −<<−d

* QΔG >

d* QΔG ≤ ( ) ( )kTQexpkTΔGexp d

* −>−

ΔG*~ 1/ΔT2 – decreases with T – sharp rise of homogeneous nucleation (diffusion is still active)

( )kTQexp d− – too small – low atomic mobility suppresses the nucleation rate


Rate of Homogeneous Nucleation

In many phase transformations, it is difficult to achieve the level of undercooling that would suppress nucleation due to the drop in the atomic mobility (regime 3 in the previous slide). The nucleation typically happens in regime 2 and is defined by the probability of energy fluctuation sufficient to overcome the activation barrier ΔG*r:

ΔTΔTcr

( )( ) ( )22m

v

2m

3SL*r ΔT

1ΔH3

Tγ 16ΔG ⎟⎟⎠

⎞⎜⎜⎝

⎛=

πUsing

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ−= 20 T

AexpIN&where A has a relatively weak dependence on temperature (as compared to ΔT2)

very strong temperature dependence!

There is critical undercooling for homogeneous nucleation ΔTcr ⇒there are virtually no nuclei until ΔTcr is reached, and there is an “explosive” nucleation at ΔTcr.

0

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTΔGexp~N

*&

N&


Heterogeneous nucleation

Let’s consider a simple example of heterogeneous nucleation of a nucleus of the shape of a spherical cap on a wall of a container. Three interfacial energies:

γLC – liquid container interface,γLS – liquid-solid interface, γSC – solid-container interface.

Balancing the interfacial tensions in the plane of the container wall gives γLC = γSC + γLS cos(θ) and the wetting angle θ is defined by cos(θ) = (γLC - γSC)/ γLS

solidnucleus

liquid

θ

γSC

γLSγLC

solid

liquidsupercooled

liquid

the new phase appears on the walls of the container, at impurity particles, grain boundaries, etc.


How about the out-of-plane component of the liquid-vapor surface tension?

γLS = γSV + γLV cos(θ) θ = 90º

↓γLS = γSV

The out-of-plane component of the liquid-vapor surface tension is expected to be balanced by the elastic response of the solid, but theoretical analysis is not straightforward due to an apparent divergence of stress at the contact line.

See [Physical Review Letters 106, 186103, 2011] for extracurricular reading


Heterogeneous Nucleation

solidnucleus

liquid

θ

γSC

γLSγLC

The formation of the nucleus leads to a Gibbs free energy change of ΔGr

het = -VS ΔGv + ASLγSL + ASCγSC - ASCγLC

VS = π r3 (2 + cos(θ)) (1 – cos(θ))2/3ASL = 2π r2 (1 – cos(θ)) and ASC = π r2 sin2(θ)

( ) ( )θSΔGθSγr 4πΔGr π34-ΔG hom

rSL2

v3het

r =⎭⎬⎫

⎩⎨⎧ +=

( ) ( )( ) /4cos θ1cos θ2θS 2−+=

One can show that


Heterogeneous nucleation

( ) ( )θSΔGθSγr 4πΔGr π34-ΔG hom

rSL2

v3het

r =⎭⎬⎫

⎩⎨⎧ +=

where

At r = r* ( ) ( ) 0θS r γ 8πΔGr 4π-dr

G dΔ SLv

2r =+=

v

SL*

ΔG γ2r =

ΔG*

ΔT

( ) ( )( )

( ) *hom2

v

3SL*het ΔGθS

ΔG3γ 16 πθSΔG ==

( ) ( )( ) 1/4cos θ1cos θ2θS 2 ≤−+=

*hetΔG

*homΔG

Active nucleation starts

crhetΔT cr

homΔT


Heterogeneous Nucleation

v

SL*

ΔG γ2r = ( ) *

hom*het ΔGθSΔG =

( ) ( )( ) 4-2 10/4cos θ1cos θ2θS 10θ if ≈−+== o

*hetΔG

*r

ΔG

*homΔG

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTΔGexp~N

*homhom&

ΔT

⎟⎟⎠

⎞⎜⎜⎝

⎛−

kTΔGexp~N

*hethet&

homhet NN && >>

heterogeneous nucleation starts at a lower undercooling

hetN& homN&N&


Nucleation of melting

For solid/liquid/vapor interfaces, often γSolid-Vapor > γSolid-Liquid + γLiquid-Vapor

In this case, no superheating is required for nucleation of the liquid and surface melting can take place below Tm. (remember our discussion of the paper on ice skating)

Melting starts from free surfaces.

Melting of small atomic clusters, a cross-section through the center of the cluster is shown

(simulations by J. Sethna, Cornell University)


Growth mechanisms

The next step after the nucleation is growth. Atomically rough (diffuse) interfaces migrate by continuous growth, whereas atomically flat interfaces migrate by ledge formation and lateral growth.

liquid

solid

liquid

solid

The rate of the continuous growth (typical for metals) is typically controlled by heat transfer to the interfacial region for pure materials and by solute diffusion for alloys.

Growth in the case of atomically flat interfaces can proceed from existing interfacial steps (e.g. due to the screw dislocations or twin boundaries) or by surface nucleation and lateral growth of 2D islands.


Growth mechanisms

Smooth solid-liquid interfaces typically advance by the lateral growth of ledges. Ledges can result from surface nucleation or from dislocations that is intersecting the interface.

Spiral growth on dislocationsAFM images of growing crystal of KDP (potassium dihydrogenphosphate) by De Yoreo and Land, LLNL and Malkin and Kuznetsov, University of California


Growth: Temperature dependenceIn multi-component systems, non-congruent phase transformations typically involve long-range diffusion of components necessary for achieving the equilibrium phase composition.

The atomic rearrangements necessary for growth of a one-component phase or growth in a congruent phase transformation also involve thermally-activated elementary processes (diffusion). When growth is diffusion controlled, it slow down with decreasing temperature.

⎟⎠⎞

⎜⎝⎛ −=

RTEexpDD d

0


Rate of phase transformations

mTtemperature

nucleation rate

Total rate of a phase transformation induced by cooling is a product of the nucleation rate (driving force increases with undercooling but diffusion needed for atomic rearrangement slows down with T decrease) and growth rate (diffusion controlled - slows down with T decrease).

overall transformation rate

growth rate

high T (close to Tm): low nucleation and high growth rates coarse microstructure with large grains

low T (strong undercooling): high nucleation and low growth rates fine structure with small grains


Growth mechanisms – directional dependence

The shape of a growing crystal can be affected by the fact that different crystal faces have different growth rates. Close-packed low-energy faces tend to grow slower and, as a result, they are the ones that are mostly present in a growing crystallite.

For example, water ice I(h) has hexagonal crystal symmetry that is reflected in the symmetry of snow crystals. The growth rate is fast parallel to the basal {0001} and prism {1010} faces. As a result, very small snow crystals have shape of hexagonal prisms. As they grow, growth instabilities result in more complex shapes of larger snow crystals.

P. V. Hobbs, Ice Physics, Oxford Univ. Press, Oxford, 1974.


Growth instabilities, dendrites

Material and heat diffusion limits the rate at which a crystal can grow, often greatly affecting the shape of the growing crystals.

As a result the flat growth is unstable, and a crystal tend to grow into more complex shapes, e.g. snowflakes

An example is the Mullins-Sekerkainstability. Consider a flat solid surface growing into a supersaturated vapor. If a small bump appears on the surface, then the bump sticks out farther into the supersaturated medium, and hence tends to grow faster than the surrounding surface.

supersaturated vapor

crystal

http://www.its.caltech.edu/~atomic/snowcrystals/


Patterns of snow crystals: “A letter from the sky”

http://www.lowtem.hokudai.ac.jp/ptdice/english/aletter.html

Nakaya diagram: The shape of snow crystals depends on the temperature and humidity of the atmosphere in which they have grown.Vertical axis shows the density of water vapor in excess of saturation with respect to ice.The black curve shows the saturation with respect to liquid water as a function of temperature.Physics Today, Dec. 2007, pp. 70-71, 2007


Two-dimensional ice dendrites on windows

by Nadezhda Bulgakova

by Harry Bhadeshia


Water dendrites in ice

http://www.msm.cam.ac.uk/


Crystal growth and heat flow during solidificationThermal dendrites

http://www.msm.cam.ac.uk/phase-trans/

Bragard et al., Interfacial Science 10, 121, 2002

microstructure of a metal ingot


Crystal growth and heat flow during solidificationThermal dendrites

What are the processes leading to the formation of the microstructure schematically shown in this figure?

cold mould

heat flow

liquid metal


SummaryMake sure you understand language and concepts:

Homogeneous nucleation Interfacial energyCritical radius, nucleation rate Heterogeneous nucleationTemperature dependence homogeneous and heterogeneous nucleation rates Nucleation in melting and boilingNucleation in solidificationGrowth mechanismsRate of a phase transformation Growth instabilities, dendrites

H. Imai and Y. Oaki, MRS Bulletin 35, February issue, 138-144 (2010)

Nucleation

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