Atomic Transport Phase Transformations · (Solidification) Nucleation –Formation of stable...

Atomic Transport

&

Phase Transformations

Lecture III-2

PD Dr. Nikolay Zotov

[email protected]

2

Part III Solid State Reactions

Lectures Short Description

1 Introduction, Interfaces; Interface Thermodynamics

2 Nucleation

3 Growth

4 Transformation kinetics, Coarsening

5 Eutectic decomposition, Spinodal decomposition

6 Summary/Overview

Atomic Transport & Phase Transformations

3

Lecture III-2 Outline

Fluctuations and Correlations in the Melt

Clusters in the melt

Homogeneous nucleation

Heterogeneous nucleation

4

Crystallisation

l ↔ a + l

How is formed the a-phase ?

Melt – Disordered ↓

Crystall - Ordered

5


Diffusion of particles

in the melt

D ~ kT/h(T) ↓

Fluctutaions of density,

variation of velocities

Instantaneous density

r = r(r,t)

r = r(ro,t)

ro

Thermodynamic view ‚Atomistic‘ view

r(r,t)<r>

6


CA(tcorr) = <A(ro,t)A(ro,t+tcorr)>| t Time-correlation function

tcorr – Delay, Measuring Time

7


C(tcorr)

CA(tcorr → ∞) = 0 (Complete Loss of ‚memory‘)

CA(tcorr → ∞) = <A> 2

0

8

G(r,t) = <r(r‘+r,t)r(r‘,0)>|V Van-Hoff Correlation function

G(r,0) = <r(r‘+r,0)r(r‘,0)>| V Space-Correlation function

G(r) = d(r) + rog(r)

ro Average number density (at/Å3)

G( r → ∞) = ro ; g(r) → 1 g(r) Pair correlation function

For isotropic systems:

g(r) = N(r)/4pr2Dr

∫g(r) 4prr2dr = N


9

Fluctuations and Correlations

Holland-Moritz (2002)

The pair correlation functions

shows maxima < 10 Å.

↓

Short-range order (SRO)

in the melt

↓

Clusters of atoms in the melt

10

Clusters

Gerlach et al. (2006)

Dynamic Processes

# ‚Caging‘ (a)

# Diffusion (‚Long-range‘ diffusion)

# Bonding of atoms (b), (c)

# Atom attachement/Detachment (d), (e)

# Formation of dynamically stabe

clusters (g), (h)

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fcc

pentagonale

Dipyramide

Icosahedron

Clusters

6

13

13 Aguado & Jarrold (2010)

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Nucleation Theories

• Classical nucleation theories

Gibbs (Helmholtz) energy of small clusters;

Continuous description (length scale much larger than atomic sizes).

• Kinetic nucleation theories

Collisions between clusters of different sizes;

Rate equations based on attachment/detachment fluxes.

• Atomistic approaches

Potentials between atoms (molecules);

Computer simulations (MD, Monte Carlo, Phase field);

Density functional theory:

13

gL = hL – TsL Gibbs-Energy of the melt per unit volume

gS = hS – TsS Gibbs-Energie of the solid per unit volume

DgV= gS – gL = Dh – TDs

DgV > 0 T > Tm

DgV < 0 T < Tm - Thermodynamic driving

force

Nucleation(Solidification)

Nucleation – Formation of stable clusters

14

Nucleation

L

ß

Homogeneous nucleation

Heterogeneous nucleation

Humphreys and Hatherly (2004)

L-S

L-S

S-S

15

Homogeneous Nucleation(Solidification)

SS

LDG = GS – GL = VSDgV + ASg; L-S Interface!!!

Approximations : spherical solid particle; No strain

AS = 4pR2, VS = (4/3)pR3

DG = (4/3)pR3 DgV + 4pR2 g; T > Tm

Only unstable clusters -

Energy minimization by

remelting of the clusters

T < Tm Stable or

Unstable clusters,

depending on the

relative contribution of

DgV and g;

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T < Tm, DgV < 0

DG = - (4/3)pR3 |DgV | + 4pR2 g;

Condition for extremum: ∂DG/∂R = 0

Assumption: g independent of R

∂DG/∂R = -4pR2 |DgV | + 8pR g = 0; R* = 2g/ |DgV |

∂2DG/∂R2 = -8pR |DgV | + 8pg;

∂2DG/∂R2 |R=R* = -8pg < 0; maximum

DG* = 16 p g3/ 3 |DgV |2 ; Nucleation barrier


R < R* Unstable clusters (embryos)

Growing embryos increase their Gibbs energy (more unstable)

and remelt (dissolve)

R > R* Growing clusters decrease their Gibbs energy and become more stable

R**

17

Homogeneous Nucleation

R > R*

G curves of the Cluster

R < R*

18


DG = - (4/3)pR3 |DgV | + 4pR2 g;

Size-dependent interfacial (surface) energy g(R);

Model of Tolman and Buff (1949)

g(R) = g∞ (1 – 2d/R); d measure of ‚diffusivness‘ of the

interface

0 5 10 15 20

25

30

35

40

45

50

g (m

J/m

2)

R

∂DG/∂R = -4pR2 |DgV | + 8pR g(R) + 8pdg∞ = 0;

R2 – R∞R + d R∞ = 0; R∞ = 2g∞/|DgV|

R1*= R∞ /2 [ 1 + (1 – 4d/ R∞) ½ ] ~ R∞(1 – d/ R∞)

R2* ~ d

DG* ~ DG∞* - 16pd2g∞2/ |DgV |

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Rough (Diffuse) Interfaces

S

LThe critical radius and the

Nucleation barrier both

decrease with increasing d.


20

R* = 2g/ |DgV |

DG* = 16 p g3/ 3 |DgV |2


DgV ~ L DT/Tm ; DT Undercooling

R* = 2g Tm/ L |DT| [L] = J/m3

DG* = 16 p g3 Tm2/ 3 L2 DT2 ; DT ↑ R * ↓ and DG* ↓

21


R* = 2g Tm/ L |DT|

|DT| =2g Tm/ R* L;

T*/Tm = (1 - 2g /LR*)

Lin et al. (2010)

MD Simulations

22

Homogeneous NucleationSolid-Solid Transformation (Precipitation)

Vß > Va

Dilatation strain (uniform expansion/contraction),

both a and ß isotropic deformable solids, having the same elastic constants.

Volume starin DV/V ~ 3e;

Linear strain: e ~ (R‘ – R)/R = dR‘/R (dR‘ < dR)

Elastic energy per unit volume: ~ (1/2)Ee2 E Young‘s modulus

Total elastic energy: ~ +4/3pR3(1/2)Ee2

23

Homogeneous NucleationSolid-Solid Transformations

DG = - (4/3)pR3 |DgV | + (4/3)pR3(1/2)Ee2 + 4pR2 g; Effect of strain

∂DG/∂R = -4pR2 (|DgV | - (1/2)Ee2) + 8pR g = 0;

R* = 2g/ (|DgV | - 1/2Ee2); Nucleation of ß-phase possible only if |DgV | > 1/2Ee2

Strain increases the critical radius

|DgV | ~ La/ß DT/Tc La/ß DT/Tc > 1/2Ee2 or DT > 1/2Ee2 Tc / La/ß

Higher undercooling necessary

DG* = (16 p/3) g3/ [|DgV | - 1/2Ee2]2; Strain increases the nucleation barrier DG*

24

Homogeneous NucleationSolid-Solid Transformation (Precipitation)

a

ß

Surface energy

Strain energy

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Ag precipitates in Al-4 at% Ag alloy

Nicholson et al. (1958)

Ni3Nb precipitates in Ni-based superalloy

E = ½ cijkl eij ekl General elastic energy per

unit volume (anisotropic)

26


Matrix (a) and precipitate (b) have different elastic constants,

the precipitate has in general an ellipsoid shape with axes c and a.

DGstrain ~ Vß X e2

X Shape, El. Constants

3Eß/2(1-2nß) a=c (sphere), incompressible a, soft ß

3Ea/(1+na) a=c (sphere), soft a, incompressible ß

3Ea/(1+na)f(c/a) a≠c (ellipsoid), soft a, incompressible ß

ca

Nabaro (1940)

Precipites tend to take the shape,

which minimizes the Gibbs energy

→ needles or plates.

27

Homogeneous NucleationNucleation rate

+ ● →

Sub-critical

Embryo

Critical Embryo

Ṅ ~ Nn exp(-DĜ/RT)

Ṅ Nucleation rate [nuclei/s.m3]

N Number of species per unit volume

n frequency of atachment/detachment

- • →

Sub-critical

Embryo

28


Ṅ = Nn exp(-DG*/kBT); Volmer & Weber (1925)

DG* = 16 p g3 Tm2/ 3 L2 DT2 ;

Ṅ = Nn exp (- A/(DT)2 ); A = 16 p gSL3 Tm

2/3L2kBT

DTN~ 0.2 Tm (Turnbull‘s Rule)

Ṅ

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Ṅ = Nn exp (- A/(DT)2 );

n ~ no exp (-DGmig/kBT) = no exp [-DGmig/(Tm – DT)], DT = Tm – T

Ṅ = N no exp [-DGmig/(Tm – DT)] exp [- A/(DT)2 ];

Li et al. (2009)

T

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Heterogeneous Nucleation(forein surfaces)

Levitation melting

Casting in moulds

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2Q

A

B

CEquilibrium of the surface tensions along the wall

No diffusion between mould and solid

The angle (ABC) = 2Q

gML = gSM + gSL cos(Q)

DGhet = -VS DgV + ASLgSL + ASMgSM – ASMgML =

= -VS DgV + ASLgSL – ASMgSLcos(Q)

VS = (p/3)[2 – cos(Q) + cos3(Q)]

ASL = 2p(1-cos(Q)]R2;

ASM = p[Rsin(Q)]2;

32

DGhet = {-4/3 p R3 Dgv + 4pR2 gSL } S(Q) =

= DGhom S(Q)

S(Q) = (2 + cosQ)(1 – cosQ)2/4

Q S(Q)

10 0.00017

20 0.0026


0 20 40 60 80 100

0,0

0,1

0,2

0,3

0,4

0,5

S

Contact Angle

33

DGhet = DGhom S(Q)

∂DGhet /∂R = S(Q) [∂DGhom /∂R] = 0

R* = 2g/ |DGV | independent of Q

DG*het = DGhom* S(Q)

Heterogeneous NucleationHeterogeneous Nucleation(forein surfaces)

Q = arccos[ (gML - gSM )/gSL ]

Reduction of Nucleation barrier

# gSL increases

# gML ~ gSM

Potter (2009)

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Heterogeneous NucleationInterfacial Energy

Digilov (2004)

gSL ~ DHm/Wsm2/3 Turnbull‘s Rule

gSL ~ (kBTm/Wsm2/3) exp (DHm/3kBTm)

Wsm Atomic volume of the solid at Tm.

Metal Tm (K) DHm (kJ/mol) g(mJ/m2)

____________________________________

Ni 1726 17,5 255

Cd 594 6,2 58

In 429 3,3 36

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Heterogeneous Nucleation(Solid-Solid Transformations, grain boundaries)

Equilibrium of the surface tensions

gaß sin(Q) = gaßsin(Q*) Y direction

Q = Q*

gaa = 2gaßcos(Q) X direction

--------------------------------------------------------

DG = - Vß|DgV | + Vß (1/2)Ee2 + Aß gaß - Aaagaa; Effect of strain

Vß = 2(p/3)[2 – cos(Q) + cos3(Q)]

Aß = 4p(1-cos(Q)]R2;

Aaa= p[Rsin(Q)]2; (removed area: p[Rsin(Q)]2 )

DG*het = DGhom * 2S(Q) Q = arccos (gaa/2gaß)

Reduction of nucleation barrier

by formation of incoherent gaß interfaces

X

Y

*

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Heterogeneous Nucleation(grain boundaries)

Precipitation of AL6Mn in

Al-1%Mn solid solution

Beavan et al. (1982)

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Heterogeneous Nucleation(Nucleation rate)

Potter et al. (2009)

Ṅhet ~ Nhetn exp(-DG*homS(Q)/kBT)

Nhet – Number of species per unit volume,

which contribute to the nucleation

Heterogeneous nucleation starts at

lower undercoolings

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