Atomic Transport & Phase Transformations Lecture III-2

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1 Atomic Transport & Phase Transformations Lecture III-2 PD Dr. Nikolay Zotov

2 Atomic Transport & Phase Transformations Part III Lectures Solid State Reactions Short Description 1 Introduction, Interfaces; Interface Thermodynamics 2 Nucleation 3 Growth 4 Transformation kinetics, Coarsening 5 Eutectic decomposition, Spinodal decomposition 6 Summary/Overview 2

3 Lecture III-2 Outline Fluctuations and Correlations Clusters Homogeneous nucleation Heterogeneous nucleation 3

4 Crystallisation l a + l How is formed the a-phase? 4

5 Fluctuations and Correlations Thermodynamic view Atomistic view r o Diffusion of particles D ~ kt/h Fluctutaions of density, variation of velocities Instantaneous density r = r(r,t) Dr = r(r,t) - <r> r = r(r o,t) 5

6 Fluctuations and Correlations C(t corr ) = <r(r o,t)r(r o,t+t corr )> t Time-correlation function 6

7 Fluctuations and Correlations C(t corr ) 0 C(t corr ) = 0 (Loss of memory ) 7

Fluctuations and Correlations G(r,t) = <r(r +r,t)r(r,0)> V Van-Hoff Correlation function G(r,t) = G s (r,t) + G d (r,t) G s Self-Correlation function G d Distict correlation function G(r,0) = <r(r

8 Fluctuations and Correlations G(r,t) = <r(r +r,t)r(r,0)> V Van-Hoff Correlation function G(r,t) = G s (r,t) + G d (r,t) G s Self-Correlation function G d Distict correlation function G(r,0) = <r(r +r,0)r(r,0)> V G(r) = d(r) + r o g(r) Space-Correlation function r o Average number density (at/å 3 ) G( r ) = r o ; g(r) 1 g(r) Pair correlation function For isotropic systems: g(r) = N(r)/4pr 2 Dr g(r) 4prr 2 dr = N 8

9 Fluctuations and Correlations Clusters Short-range order (SRO) in the melt Holland-Moritz (2002) 9

10 Clusters Dynamic Processes Diffusion ( Long-range diffusion) Caging (a) Bonding of atoms (b), (c) Atom attachement/detachment (d), (e) Formation of clusters (g), (h) Gerlach et al. (2006) 10

11 pentagonale Dipyramide Clusters 6 fcc 13 Icosahedron 13 Aguado & Jarrold (2010) 11

12 Nucleation Theories Classical nucleation theories Gibbs (Helmholz) 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 (Monte Carlo, Phase field); Density functional theory: 12

13 Nucleation (Solidification) g L = h L Ts L g S = h S Ts S Gibbs-Energy of the melt per unit volume Gibbs-Energie of the solid per unit volume Dg V = g S g L = Dh TDs Dg V > 0 T > T m Dg V < 0 T < T m Nucleation Formation of stable (condensed-matter) clusters 13

14 Nucleation Homogeneous nucleation L-S Heterogeneous nucleation L L-S S-S ß Humphreys and Hatherly (2004) 14

15 Homogeneous Nucleation (Solidification) DG = G S G L = V S Dg V + A S g; L Approximations : spherical solid particle; No strain S S A S = 4pR 2, V S = (4/3)pR 3 DG = (4/3)pR 3 Dg V + 4pR 2 g; T > T m Unstable clusters - Energy minimization by remelting of clusters T < T m Stable Unstable clusters 15

16 T < T m, Dg V < 0 Homogeneous Nucleation (Solidification) DG = - (4/3)pR 3 Dg V + 4pR 2 g; Condition for extremum: DG/ R = 0 Assumption: g independent of R DG/ R = -4pR 2 Dg V + 8pR g = 0; R* = 2g/ Dg V R** 2 DG/ R 2 = -8pR Dg V + 8pg; 2 DG/ R 2 R=R* = -8pg < 0; maximum DG* = 16 p g 3 / 3 Dg V 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 > R** = 3g/ Dg V Stable clusters 16

17 Homogeneous Nucleation 17

18 Homogeneous Nucleation (Solidification) DG = - (4/3)pR 3 Dg V + 4pR 2 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 DG/ R = -4pR 2 Dg V + 8pR g(r) + 8pdg = 0; R 2 R R + d R = 0; R = 2g / Dg V g (mj/m 2 ) R R 1 *= R /2 [ 1 + (1 4d/ R ) ½ ] ~ R (1 d/ R ) R 2 * ~ d The critical radius decreases with increasing d. DG* ~ DG * - 16pd 2 g 2 / Dg V 18

19 R* = 2g/ Dg V Homogeneous Nucleation (Solidification) DG* = 16 p g 3 / 3 Dg V 2 DG V ~ L DT/T m ; DT Undercooling R* = 2g T m / L DT [L] = J/m 3 DG* = 16 p g 3 T m2 / 3 L 2 DT 2 ; DT R * and DG* 19

20 Homogeneous Nucleation (Solidification) R* = 2g T m / L DT DT =2g T m / R* L; T * /T m = (1-2g /LR*) Lin et al. (2010) MD Simulations Lin et al. (2010) 20

21 Homogeneous Nucleation Solid-Solid Transformation (Precipitation) V ß > V a 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)Ee 2 E Young s modulus Total elastic energy: ~ 4/3pR 3 (1/2)Ee 2 21

22 Homogeneous Nucleation Solid-Solid Transformations DG = - (4/3)pR 3 Dg V + (4/3)pR 3 (1/2)Ee 2 + 4pR 2 g; Effect of strain DG/ R = -4pR 2 ( Dg V - (1/2)Ee 2 ) + 8pR g = 0; R* = 2g/ ( Dg V - 1/2Ee 2 ); Nucleation of ß-phase possible only if Dg V > 1/2Ee 2 Strain increases the critical radius Dg V ~ L a/ß DT/T c L a/ß DT/T c > Ce 2 or DT > 1/2Ee 2 T c / L a/ß Higher undercooling necessary DG* = (16 p/3) g 3 / [ Dg V - 1/2Ee 2 ] 2 ; Strain increases the nucleation barrier DG* R** = 3g/ ( Dg V - 1/2Ee 2 ); 22

23 Homogeneous Nucleation Solid-Solid Transformation (Precipitation) a ß Strain energy Surface energy 23

24 Homogeneous Nucleation Solid-Solid Transformations Ag precipitates in Al-4 at% Ag alloy Ni 3 Nb precipitates in Ni-based superalloy Nicholson et al. (1958) E = ½ c ijkl e ij e kl General elastic energy per unit volume 24

25 Homogeneous Nucleation Solid-Solid Transformations Matrix (a) and precipitate (b) have different elastic constants, the precipitate has in general an ellipsoid shape with axes c and a. DG strain ~ V ß X e 2 X 3E ß /2(1-2n ß ) Shape, El. Constants a=c (sphere), incompressible a, soft ß a c 3E a /(1+n a ) a=c (sphere), soft a, incompressible ß 3E a /(1+n a )f(c/a) a c (ellipsoid), soft a, incompressible ß Precipites tend to take the shape, which minimizes the Gibbs energy. Nabaro (1940) 25

26 Homogeneous Nucleation Nucleation rate + - Sub-critical Embryo Critical Embryo Sub-critical Embryo Ṅ ~ (J + - J - )/l ~ N sub n exp(-dg*/rt) ~ Nn exp(-dg*/rt) Ṅ Nucleation rate [nuclei/s.m 3 ] N sub Number of sub-critical embryos N Number of species per unit volume n frequency of atachment/detachment J flux [species/s.m 2 ] l Free diffusion path of the species [m] 26

27 Homogeneous Nucleation Nucleation rate Ṅ = Nn exp(-dg*/k B T); Volmer & Weber (1925) DG* = 16 p g 3 T m2 / 3 L 2 DT 2 ; Ṅ = Nn exp (- A/(DT) 2 ); A = 16 p g SL3 T m2 /3L 2 k B T DT N ~ 0.2 T m (Turnbull s Rule) 27

28 Homogeneous Nucleation Nucleation rate Ṅ = Nn exp (- A/(DT) 2 ); n ~ n o exp (-DG mig /k B T) = n o exp [-DG mig /(T m DT)], DT = T m T Ṅ = N n o exp [-DG mig /(T m DT)] exp [- A/(DT) 2 ]; T Li et al. (2009) 28

29 Heterogeneous Nucleation (forein surfaces) Casting in moulds Levitation melting 29

30 Heterogeneous Nucleation (forein surfaces) Equilibrium of the surface tensions along the wall No diffusion between mould and solid The angle (ABC) = 2Q g ML = g SM + g SL cos(q) A 2Q B C DG het = -V S Dg V + A SL g SL + A SM g SM A SM g ML = = -V S Dg V + A SL g SL A SM g SL cos(q) V S = (p/3)[2 cos(q) + cos 3 (Q)] A SL = 2p(1-cos(Q)]R 2 ; A SM = p[rsin(q)] 2 ; 30

31 DG het = {-4/3 p R 3 Dg v + 4pR 2 g SL } S(Q) = = DG hom S(Q) Heterogeneous Nucleation (forein surfaces) S(Q) = (2 + cosq)(1 cosq) 2 /4 0,5 0,4 Q S(Q) S 0,3 0,2 0,1 0, Contact Angle 31

32 Heterogeneous Nucleation (forein surfaces) DG het = DG hom S(Q) DG het / R = S(Q) [ DG hom / R] = 0 R* = 2g/ DG V independent of Q DG* het = DG hom * S(Q) Q = arccos[ (g ML - g SM )/g SL ] Reduction of Nucleation barrier # g SL increases # g ML ~ g SM Potter (2009) 32

33 Heterogeneous Nucleation Interfacial Energy g SL ~ DH m /W sm 2/3 Turnbull s Rule g SL ~ (k B T m /W sm 2/3 ) exp (DH m /3k B T m ) W sm Atomic volume of the solid at T m. Metal T m (K) DH m (kj/mol) g(mj/m 2 ) Ni ,5 255 Digilov (2004) Cd 594 6,2 58 In 429 3,

34 Heterogeneous Nucleation (Solid-Solid Transformations, grain boundaries) Equilibrium of the surface tensions g aß sin(q) = g aß sin(q*) Y direction Y Q = Q* g aa = 2g aß cos(q) X direction * X DG = - V ß Dg V + V ß (1/2)Ee 2 + A ß g aß - A aa g aa ; Effect of strain V ß = 2(p/3)[2 cos(q) + cos 3 (Q)] A ß = 4p(1-cos(Q)]R 2 ; A aa = p[rsin(q)] 2 ; (removed area; p[rsin(q)] 2 ) DG* het = DG hom * 2S(Q) Q = arccos (g aa /2g aß ) Reduction of nucleation barrier by formation of incoherent g aß interfaces 34

35 Heterogeneous Nucleation (grain boundaries) Precipitation of AL 6 Mn in Al-1%Mn solid solution Beavan et al. (1982) 35

36 Heterogeneous Nucleation (Nucleation rate) Ṅ het ~ N het n exp(-dg* hom S(Q)/k B T) N het Number of species per unit volume, which contribute to the nucleation N het < N Potter et al. (2009) Heterogeneous nucleation starts at lower undercoolings 36

Atomic Transport & Phase Transformations Lecture III-1

Atomic Transport & Phase Transformations Lecture III-1 PD Dr. Nikolay Zotov zotov@imw.uni-stuttgart.de Atomic Transport & Phase Transformations Part III Lectures Solid State Reactions Short Description