Atomic Transport & Phase Transformations Lecture III-1

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1 Atomic Transport & Phase Transformations Lecture III-1 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 2

3 Lecture III-1 Outline 1st Order 2nd Order Phase Transitions (short repetition) Types of Interfaces Thermodynamics of Interfaces 3

4 Phase Transitions Ehrenfest Classification 1. Order 2. Order Latent heat L = T C (S 1 S 2 ) (Heat is released/absorbed at T C = const) Co-existance of the two phase Overheating/Undercooling possible Discontinuous phase transition No latent heat No coexistance of the two phase Overheating/Undercooling not possible Continuous phase transition Solid-Solid phase transitons No Group-Subgroup relations Group Subgroup relations mandatory 4

5 Solidification L Potter et al.(2009) Two-phase region a + l Nucleation of the a phase Growth of the a phase Interface: liquid/solid How the second phase forms? How it grows? Pb Sn What is the effect of the interfaces? 5

6 Nucleation and Growth Importance Govern the kinetics of phase transitions during processing. Control the crystallization, which plays key role in many mineral, climate and industrial processes. Control the average grain size of many materials, which is directly related to the strength of the materials. 6

7 Interfaces Classification Homophase Heterophase Twin Boundaries Grain Boundaries Solid Vapour Solid-Liquid Solid-Solid Liquid-Vapour 7

8 Solidification Eutectic Decomposition l a + ß Solid-Solid Interfaces Lecture III-5 8

9 Precipitation Poter et al.(2009) a a + ß Precipitation of a new solid phase (ß) in a phase diagram with an Eutectic point ß has different structure, composition, and long-range order Lecture III-4 9

10 Precipitation Potter et al.(2009) Precipitation of a new phase in a phase diagram with a peritectic point 10

11 Eutectoid Phase Diagram Precipitation Precipitation ( Proeutectoid reaction) g g + a Eutectoid Decomposition g a + ß 11

12 Solid Liquid Interfacial Energy L S S g SL ~ 0.32 DH m (semi-metals) g SL ~ 0.45 DH m (metals) J. Howe (1997) 12

13 Solid Liquid Interfaces Rough (Diffuse) Interfaces L Flat Interfaces L S S Continuous growth Lateral Growth 13

14 Surfaces of Crystalline Solids Y Atoms like squares with length a (hkl) X G.H. Meier (2014) g(q hkl ) ~ E NN /r a2 cos(q hkl p/4) 14

15 Solid Liquid Interfaces Jackson Parameter (a) [K. A. Jackson, Liquid Metal and Silidification, ASM, Kleveland, OH, 1958] a a ~ DS T m (N A /R) ~ (L/T m ) N A / k B N A = L/k B T m DS T m ~ L/T m ; Jackson Rule : a < 2 rough S-L Interfaces a > 2 flat (faceting) S-L interfaces D.P. Woodruff, The Solid-Liquid Interface, Cambridge Uni Press,

16 Structure Solid Solid Interfaces? a ß Every interface is a crystallographic (surface) plane 16

17 Crystalline Surfaces Fcc structure, C = 1 Different surface densities Different 2D lattices Different 2D Lattices 17

18 2 Dimensional lattices with different symmetry 18

19 Solid Solid Interfaces Classification Types of Solid-Solid Interfaces Coherent The lattice planes of the two phases (grains, crystallites) are continuous across the interface Degree of coherence f Coh = 1 Incoherent The lattice planes of the two phases (grains, crystallites) are not continuous across the interface Degree of coherence f Coh = 0 Semicoherent Regions with coherence and regions without coherence 0 < f Coh < 1 19

20 Coherent Interfaces without Lattice Strain: d a d b d b ~ d a e = (d a d b )/d a ~ 0 Similar interplanar spacing Specific orientation relationships between the two phase (crystallites, grains) 20

21 Coherent Interfaces Cu-Si k-phase (hcp) Cu-Ge z-phase (hcp) Polatidis & Zotov (2017) (111) Cu //(0001) z [-110] Cu //[11-20] z 21

22 Coherent Interfaces 22

23 Coherent Interfaces Ag precipitates in Al-4 at% Ag alloy Nicholson et al. (1958) 23

24 Coherent Interfaces with Lattice Strain: e = (d a d b )/d a 0 24

25 Incoherent Interfaces Large-angle grain boundaries 25

26 Semi-coherent Interfaces 26

27 g = ( G a/b G a G b )/A G a/b = G a + G b + ga Interfacial Energy Solid-Solid interfaces s a b thin interface (Gibbs treatement) V s ~ 0 G = G a + G b + G s + ga extended interface (Guggenheim treatement) (G = G a + G b + G s *) V s > 0 g ~ g Chem + g Strain ; [g] = mj/m 2 g = ( G/ A) T,p,n Interface Type Typical g values (mj/m 2 ) Coherent < 200 Semicoherent Incoherent > 500 Potter et al. (2009) 27

28 G = G a + G b + G s Equilibrium between a, ß and s dg a = dg ß = dg s = 0 Effects of the Interfacial Energy (g) dg s = V s dp S s dt + n As dµ As + n Bs dµ Bs + dg = 0 At T = const and p = const dg = - [n As dµ As + n Bs dµ Bs ]; G A = n As / ; G B = n Bs / ; Surface-excess factors Gibbs-Duhem Equation: X as dµ As + X Bs dµ Bs = 0 G A X Bs /X As G B = dg/dµ B s for diffuse interfaces, the position of the interface is not strictly defined (G A = 0); Equilibrium µ Bs = µ B a = µ Bß = µ B G B = - dg/dµ B 28

29 Effects of the Interfacial Energy (g) G B = - dg/dµ B = - (dg/dx B ) /(dµ B /dx B ) µ B = µ o + RTln(a B ) Dilute solution of B in A (Henry s law) + regular solution model a B = X B exp (W/RT) µ B = µ o + W + RTln(X B ); dµ B /dx B = RT/X B ; dg = - RT G B dx B /X B ; g ~ g o - RT G B ln(x B ) T T m Howe 29

30 Surface Excess X Sn 26 wt% (L) 9 wt% Sn (S) l Langmuir-McLean relation X Bs /X B ~ exp(-dg s /RT) DG s < 0 Excess (Enrichment) of B at the interface DG s > 0 Depletion of B at the interface Pb Sn 30

31 Dependence of Interfacial Energy on Composition The a and b phases are regular solutions with the same structure, respectively the same interaction parameter W, but different compositions. The temperature is low enough DH mix >> TDS mix. Consider a thin coherent interface also as a regular solution model!: DG mix a ~ DH mix a = WX Ba (1- X Ba ); DG mix b ~ DH mix b = WX Bb (1- X Bb ); DG mix a/b ~ DH mix a/b = W [X Ba (1- X Bb ) + X Bb (1- X Ba )] g = (W/A) [ X B a2 2X Ba X Bb + X B b2 ] = (W/A) (X Ba - X Bb ) 2 ; ½ (N/A)zDe (X Ba - X Bb ) 2 (Becker s model) (N/A) ~ 1.5x10 19 at/m 2, z = 8, De = 5x10-20 J, (X Ba - X Bb ) 2 =0.02 g ~ 60 mj/m 2 31

32 Thermodynamics in the presence of Interfaces G = G a + G b + G s ~ ~ G a + G b + ga for each phase G = Sµ i n i ; C = 2 (A and B) Exchange of A and B atoms between a and ß due to diffusion ß a G = (µ Aa n Aa + µ Ba n Ba ) + (µ Aß n Aß + µ Bß n Bß )+ SA i g i ; dg = µ Aa dn Aa + µ Ba dn Ba + µ Aß dn Aß + µ Bß dn Bß + Sg i [( A i / n A ß )dn A ß + ( A i / n Bß )dn Bß ] From mass balance: dn Aß = - dn Aa ; dn Bß = - dn B a dg = (µ Ab - µ Aa )dn Aß + (µ Bb - µ Ba ) dn Bß + Sg i [( A i / n A ß )dn A ß + ( A i / n Bß )dn Bß ] = = (DG Aß DG Aa ) dn Aß + (DG Bß DG Ba ) dn Bß + Sg i [( A i / n A ß )dn A ß + ( A i / n Bß )dn Bß ] 32

33 Thermodynamics in the presence of Interfaces DG k = RTln(a k ) dg = (DG Aß DG Aa ) dn Aß + (DG Bß DG Ba ) dn Bß + Sg i [( A i / n A ß )dn A ß + ( A i / n Bß )dn Bß ] = = -RTln(a Aa /a Aß ) dn Aß RTln(a Ba /a Bß ) dn Bß + Sg i [( A i / n A ß )dn A ß + ( A i / n Bß )dn Bß ] = A/ n = ( A/ V) ( V/ n) = ( A/ R) ( R/ V) ( V/ n) Spherical precipitate A = 4pR 2, V = (4/3)pR 3 ; A/ V = 2/R; 1 Interface V/ n k = V k A/ n k = (2/R) V k ; dg = -RTln(a Aa /a Aß ) dn Aß RTln(a Ba /a Bß ) dn Bß + 2g/R ß [V Aß dn Aß + V Bß dn Bß ] 33

34 dg = -RTln(a Aa /a Aß ) dn Aß RTln(a Ba /a Bß ) dn Bß + 2g/R ß [V Aß dn Aß + V Bß dn Bß ] = = [ -RTln(a Aa /a Aß ) + (2g/R ß )V Aß ] dn Aß + [ -RTln(a Ba /a Bß ) + (2g/R ß )V Bß ] dn Bß ; At equilibrium dg = 0 RTln(a Aa /a Aß ) = (2g/R ß )V A ß ; RTln(a Ba /a Bß ) = (2g/R ß )V B ß ; Thermodynamics in the presence of Interfaces (2g/R ß )V ß = RT[X Bß ln(a Ba /a Bß ) + (1 X Bß ) ln(a Aa /a Aß ) ] The precipitate is ideal solution (a Aß = a Bß ~1) and contains predominantly B atoms (X B ß ~ 1) a Ba = X Ba g Ba = exp(2gv ß /RTR ß ) ~ 1 + 2gV ß /RTR ß ; Setting 1/ g Ba = X Ba ( ) when R ß (planar interface) X Ba (R ß ) = X Ba ( ) (1 + 2gV ß /RTR ß ) Gibbs-Thompson Equation 34

35 Thermodynamics in the presence of Interfaces 0,98 0,96 0,94 0,92 X B a 0,90 0,88 0,86 ß 0,84 0,82 0,80 a 0, R(ß) A B 35

36 Thermodynamics in the presence of Interfaces a a + Cu 4 Ti X Tia (ini) ~ 0.04 X Tia ( ) ~ 0.02 X Tiß ( ) ~

37 Thermodynamics in the presence of Interfaces ß = Cu 4 Ti X Tiß = 0.2 X Tia (ini) = 0.02 Dilute solutions a Tia ~ X Tia exp(w/rt) a Tiß ~ X Tiß exp(w/rt) Qian & Lim (1998) f(x Ba ) = X Bß ln(x Ba ) + (1 X Bß ) ln[(1 - X Ba ) ~ k/r ß + k* (2g/R ß )V ß = RT[X Bß ln(a Ba /a Bß ) + (1 X Bß ) ln(a Aa /a Aß ) ] g ~ 63 mj/m K 53 mj/m K 37

38 Recommended Literature (Part I & III) # E.J. Mittemeijer Fundamentals of Materials Science # H.-G. Lee Materials Thermodynamics # D.A. Porter and K.E. Easterling Phase Transformations in Metals and Alloys # D. R. Gaskell Introduction to the Thermodynamics of Materials # A. Prince Alloy phase equilibria # R. DeHoff Thermodynamics in Material Science 38

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