Kinetics. Rate of change in response to thermodynamic forces

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1 Kinetics Rate of change in response to thermodynamic forces Deviation from local equilibrium continuous change T heat flow temperature changes µ atom flow composition changes Deviation from global equilibrium discontinuous change ΔG (ΔF) phase change (or other change of structure)

2 Atom Diffusion J B 12 c c c J B 23 g = g(t,p,c) g c T,P = µ (T,P,c) Mass flux in response to a gradient in chemical potential Assume constant T,P Flux of solute Evolution of composition J B = M d g = M dµ µ = M dc dx c dx c dx J B = nd dc dx c B t = J x = nd 2 c x 2 Fick s First Law Fick s Second Law

3 Diffusion J B 12 J B 23 J B = nd dc dx c c c The diffusivity (D) is the material property that governs diffusion Diffusion in solids requires Atoms jump from position to position Atom jumps result in net flux Diffusion mechanism depends on atom site Interstitial Substitutional

4 Jumps of Interstitial Atoms G ΔG m Atom motion Must overcome barrier ΔG m to move from site to site (~ 1eV) Attempts with vibrational frequency ν e ~ /sec Number of jumps per unit time ω = (# attempts/time)(probability of jump/attempt) = ν e exp ΔG m kt = ν exp Q (of the order of 10 5 /sec at room T) m kt x

5 Jumps of Substitutional Atoms For a substitutional atom to jump There must be a neighboring vacancy to permit the jump The atom must overcome its barrier and jump ω = [P(vacant site)][p(jump given vacant site)] = [c v ]ν exp Q m kt = exp Q v kt ν exp Q m kt = ν exp (Q + Q ) v m kt

6 Concentration-Driven Diffusion a J 12 J 21 c 1 c 2 J = J 12 J 21 = nd dc dx Diffusion by interchange of atoms on adjacent planes J 12 = N 1 ω x = nc 1 aω x J = J 12 J 21 = naω x (c 1 c 2 ) N 1 = atoms of type 1 per unit area n = atom sites per unit volume a = volume per unit area of plane ω x = jumps in the x-direction/time = 1 6 naω(c 1 c 2 ) = 1 6 na2 ω dc dx D = 1 6 a2 ω

7 Concentration-Driven Diffusion a J 12 J 21 c 1 c 2 J = J 12 J 21 = nd dc dx Diffusion by interchange of atoms on adjacent planes Random-walk diffusion Atoms do not jump preferentially in either direction Net flux because there are more atoms on plane 1 than on 2 Diffusivity governs random-walk diffusion D = 1 6 a2 ω = 1 6 a2 ν exp Q D kt D = D 0 exp Q D kt Q Q D = m Q V + Q m interstitial substitutional

8 Random-Walk Diffusion X Let an atom move by random steps of length a Its position at time t is X(t) X(t) = 0 X 2 (t) = X X = na 2 = ωa 2 t (X and -X are equally likely) x 2 = 1 3 X 2 = 1 3 ωa2 t = 2Dt x = x 2 1/ 2 = 2Dt

9 Influence of Microstructure on Diffusivity Interstitial species Usually no effect from microstructure Stress may enhance diffusion Substitutional species Raising vacancy concentration increases D Quenching from high T Solutes Irradiation Defects provide short-circuit paths Grain boundary diffusion Dislocation core diffusion

10 Adding Vacancies Increases D ln(d) t D eq D = c v ν exp Q m kt Quench from high T Rapid cooling freezes in high c v D decreases as c v evolves to equilibrium Add solutes that promote vacancies High-valence solutes in ionic solids Mg ++ increases vacancy content in Na + Cl - Ionic conductivity increases with c Mg Large solutes in metals Interstitials in metals Processes that introduce vacancies directly Irradiation Plastic deformation

11 Grain Boundary Diffusion ln(d) D 0 D 0 B -Q decreasing D grain size 1/kT -Q B d D = D 0 exp Q D kt = D 0 exp (Q m + Q V ) kt δ D GB D 0 exp Q m d kt Grain boundaries have high defect densities Effectively, vacancies are already present Q D ~ Q m Grain boundaries have low cross-section Effective width = δ Areal fraction of cross-section: δ A GB A = Kδd d 2 δ d δ D GB 0 D 0 d

12 Kirkendall Effect A and B both substitutional They exchange with vacancies independently J A = J B + J V Flux of vacancies leads to Decreasing volume of A Kirkendall voids

13 Pores along the Interface of Cu/Cu 3 Au Formed by Kirkendall effect due to fast diffusion of Cu Cu 3 Au Cu Aged joint at 200 C for 10 days (SEM) solder Cu Aged bump at 200 C for 31days (TEM) Cu 3 Au Cu Pores can weaken joint interface and affect joint failure mechanism.

14 Kinetics Rate of change in response to thermodynamic forces Deviation from local equilibrium continuous change T heat flow temperature changes µ atom flow composition changes Deviation from global equilibrium discontinuous change ΔG (ΔF) phase change (or other change of structure)

15 Global Equilibrium: Kinetics of Phase Transformations unstable å G metastable G x stable T å T T å å Phases usually represent distinct free energy minima Requires a finite change to accomplish phase transformation Phase may be preserved in a metastable state Phase transformations ( 1 st order ) Thermodynamics: driving force is free energy difference Kinetics: rate of transformation depends on mechanism (path)

16 Basic Mechanisms of Phase Transformations G instability nucleated β T0 T å α => metastable å => equilibrium å β α Two basic mechanisms of phase transformation Nucleation and growth A discrete particle of β phase forms in the interior of α Grows to consume α phase Instability Parent phase becomes internally unstable; must transform

17 The Thermodynamic Barrier to Nucleation β α A spherical nucleus of β phase forms in the interior of α Free energy change is the sum of volume and surface terms ΔG = ΔG V V β + σa β = 4 3 πr3 ΔG V + 4πr 2 σ ΔG* = W H = 16πσ 3 3(ΔG V ) 2

18 Nucleation Rate ÎG r c ÎG* r = W H Nucleation rate Activation barrier must be surmounted Nucleus must grow by adding atoms (diffusion) N = ν N P(ΔG > W H ) = A 1 exp Q D A kt 2 exp W H kt = Aexp (W + Q ) H D kt

19 Nucleation Rate N = Aexp (W H + Q D ) kt W H = 16πσ 3 3(ΔG V ) 2 Nucleation balances the kinetics of activation and diffusion For small ΔT, W H is large, dominates nucleation rate For large ΔT, W H is small, Q D dominates rate Nucleation rate has c behavior Negligible at small and large ΔT Maximum at intermediate ΔT

20 Kinetics of Nucleated Transformations ÎT ln( ) nucleation growth Time to initiate transformation: τ = ( N ) 1 The kinetics of nucleated transformations follow c -curves For small or large ΔT, transformation is sluggish Transformation initiates most quickly at intermediate ΔT (nose) Nucleated transformations can be controlled Transformation can be suppressed by rapid cooling Transformation can be targeted to nucleation or growth control

21 Heterogeneous Nucleation Nucleation is always easier at defects such as boundaries Get energy back from destruction of boundary area within nucleus At high T, nucleation barrier dominates kinetics Nuclei form preferentially at boundaries or other defects At low T, nucleation barrier is easily overcome Nuclei form in the bulk where material can gather most easily

22 Examples: Intergranular and Transgranular Precipitates Boundary precipitates in 6Ni steel Intragaranular precipitates in Al-Cu-Si Left: illustration of intergranular precipitation in 6Ni steel Left figure is bright-field transmission electron micrograph Right figure is dark-field that lights up precipitate phase Right: example of intragranular precipitates in Al-Cu-Si Precipitates are platelets in this case, seen in three prientations

23 Age Hardening ln( ) T å å+l L å + +L ÎT boundaries heterogeneities homogeneous A x B Many structural alloys are strengthened by age hardening Require dense distribution of precipitates in grain interiors Steps in age hardening: Composition chosen in α field Alloy is solidified, then homogenized (at T below T E ) Alloy is quenched to suppress intergranular precipitation Alloy is reheated to aging T at which precipitates form inside grains

24 Chemical Heterogeneity during Solidification L α 1 T å å+l å+ α 2 α 3 α 4 First α to form (α 1 )is rich in A Second α (α 2 ) has less A For equilibrium, α 1 should adjust to α 2, requiring solid state diffusion If diffusion cannot equilibrate composition, α 1, α 2 are removed When T reaches T E, there is remnant liquid The last liquid solidifies to eutectic, located in interstices of α