Post on 15-Sep-2018
Atomic Transport
&
Phase Transformations
Lecture III-2
PD Dr. Nikolay Zotov
zotov@imw.uni-stuttgart.de
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
5
Fluctuations and Correlations in the Melt
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
Fluctuations and Correlations in the Melt
CA(tcorr) = <A(ro,t)A(ro,t+tcorr)>| t Time-correlation function
tcorr – Delay, Measuring Time
7
Fluctuations and Correlations in the Melt
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
Fluctuations and Correlations in the Melt
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)
12
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;
16
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
Homogeneous Nucleation(Solidification)
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**
18
Homogeneous Nucleation
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.
Homogeneous Nucleation
20
R* = 2g/ |DgV |
DG* = 16 p g3/ 3 |DgV |2
Homogeneous Nucleation(Solidification)
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
Homogeneous Nucleation(Solidification)
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
25
Homogeneous NucleationSolid-Solid Transformations
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
Homogeneous NucleationSolid-Solid Transformations
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
Homogeneous NucleationNucleation rate
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
29
Homogeneous NucleationNucleation rate
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
31
Heterogeneous Nucleation(forein surfaces)
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;
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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
Heterogeneous Nucleation(forein surfaces)
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)
34
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
35
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
*
36
Heterogeneous Nucleation(grain boundaries)
Precipitation of AL6Mn in
Al-1%Mn solid solution
Beavan et al. (1982)