Atomic Transport & Phase Transformations. PD Dr. Nikolay Zotov

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1 Atomic Transport & Phase Transformations PD Dr. Nikolay Zotov

2 Atomic Transport & Phase Transformations Part II Lecture Diffusion Short Description 1 Introduction; Non-equilibrium thermodynamics 2 Diffusion, Diffusion Equations 3 Solutions of the Diffusion Equations 4 Atomic Mechansims of Diffusion 5 Statistical Interpretation 6 Measuremens of Diffusion Coefficents 7 8 Diffusion in Ordered Alloys 9 High Diffusion Pathways 2

3 Atomic Transport & Phase Transformations Lecture II-8 Outline Ordered Intermetallic Alloys Long-range Order Parameters Atomic Diffusion Mechanisms Reaction Diffusion Effect of Order Parameters on the Diffusion 3

4 4

Ordered Alloys Intermetallic Phases Stoichiometric Ordered Structures Different atoms occupying different positions Sub-Lattices Occupied by only 1 type of

5 Ordered Alloys Intermetallic Phases Stoichiometric Ordered Structures Different atoms occupying different positions Sub-Lattices Occupied by only 1 type of atoms (molecules) Fractional coordinates in the unit cell + Translations {L} = {SLa} + {SLß} + a Bcc Fe ß Sub-Lattice a: {Fe, 0,0,0} Sub-Lattice ß: {Fe, ½ ½ ½ } 5

6 Ordered Alloys Intermetallic Phases Al B2 type NiAl Ni Sub-Lattice a: {Al, 0,0,0} Sub-Lattice b: {Ni, ½ ½ ½ } 6

7 Ordered Alloys Intermetallic Phases CuAu, L1 o type structure Tetragonal ; P 4/m mm Sub-Lattice a: {Au, 0 ½ ½ } Sub-Lattice b: {Cu, 0 0 0} Sub-Lattice g: {Cu, ½ ½ 0} 7

8 Ordered Alloys Intermetallic Phases Crystal structure Strukturbericht symbol Pearson symbol fcc A1 cf4 bcc A2 ci2 hcp A3 hp2 Diamond (C) A4 cf8 White Tin (Sn) A5 ti4 aas A7 hr2 Graphite (C) A9 hp4 a-mn A12 ci58 b-w (WO 3 ) A15 cp8 NaCl B1 cf8 Strukturbericht Symbol A Elements B XY Structures C - XY 2 Structures D - X m Y n Structures E - > 2 Elements 8

9 B2 Ordered Alloys Intermetallic Phases Cu 3 Au L1 2 CuAu L1 o Fe 3 Al DO 3 Mg 3 Cd DO 19 R.E. Smallman (1970) 9

10 Ordered Alloys Defects Mono-vacancies; Di-vacancies (clusters of vacancies); Substitutional Vacancies (Ni Al or Al Ni ); Anti-Site Defects (Ni Al + Al Ni ). Minimization of the Gibbs Energy with respect to the number of (different) defects Fu et al. (1993) 10

11 Non-Stoichiometric Ordered Structures B2 NiAl (AB) Sub-Lattice a: {A, 0,0,0} Sub-Lattice b: {B, ½ ½ ½ } Ordered Alloys Intermetallic Phases A-rich alloys A 1+x B 1-x ; Substituion of A on ß sub-lattice (A b ) Vacancies on the ß sub-lattice (V b ) B-rich alloys A 1-x B 1+x ; Substituion of B on a sub-lattice (B a ) Vacancies on the a sub-lattice (V a ) Experimental Observations: for A=Co,Ni,Fe, Pd and B = Al, Ga, In A-rich alloys: A b mechanism B-rich alloys: V a mechanism Bocquet et al. (1996) 11

12 Long-range Order Parameters Definitions: r a fraction of a sites occupied by A atoms r b fraction of b sites occupied by B atoms (Lecture I 9) w a fraction of wrong (B) atoms occupying a sites w a = 1 r a ; w b fraction of wrong (A) atoms occupying b sites w b = 1 r b ; h LRO = (r a X A )/(1 X A ) h LRO = (r b X B )/(1 X B ) 12

13 Long-range Order Parameters Non-Stoichiometric CuAu h LRO = r a w b or h LRO = r b w a ; CuAu X Au < 0.5 Maximum Order: Cu site fully occupied, some Cu on the Au site r a = 1; w b = (X Cu -0.5)/0.5 = [(1-X Au ) 0.5]/ 0.5 = 1 2X Au ; h = 2X Au X Au > 0.5 Maximum order: Au site fully occupied; some Au on the Cu site r b = 1; w a = (X Au - 0.5)/ 0.5 = 2X Au - 1; h = 2-2X Au LRO Parameter 1,0 0,8 0,6 0,4 0,2 Maximum LRO 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 X Au

Cu ß CuZn T < T c = 727 K B2 structure ordered ß CuZn T > T c = 727 K A2

14 Ordered Alloys Atomic Mechanisms of Diffusion Ring Mechanism (Cu and Zn exchange positions) This would lead to disorderd (random) alloy. Cu ß CuZn T < T c = 727 K B2 structure ordered ß CuZn T > T c = 727 K A2 structure Disordered Zn What atomic jump mechanisms could preserve the average LRO? 14

15 Ordered Alloys Atomic Mechanisms of Diffusion Type Proposed 6-jump cycle (6JC) Elcock and McCombie (1958), Hunigton (1961) Triple Defect (TD) Stolwijk et al. (1980) Anti-site bridge (ASB) Kao & Chang (1993) Path Probability Mechanism Kikuchi & Sato (1969) Preservation of (local) stoichiometry Only nearest-neighbour jumps! 15

16 Ordered Alloys Atomic Mechanisms of Diffusion 6JC 16

17 Ordered Alloys Atomic Mechanisms of Diffusion 6JC Wynblatt (1967) Limited applicability, because the individual energy barriers relatively high. Fig Paul et al. (2014) 17

18 Ordered Alloys Atomic Mechanisms of Diffusion Triple Defect (TD) For Al-rich alloys 2 Vacancies on the Ni sub-lattice + Ni Al ; Only Ni diffuses 18

19 Ordered Alloys Atomic Mechanisms of Diffusion TD For Al-rich alloys 2 Vacancies on the Ni sub-lattice + Ni Al ; Both Ni and Al diffuse 19

20 Ordered Alloys Atomic Mechanisms of Diffusion Anti-site Bridge (ABS) For Ni-rich alloys Ni on Al sites + V Al ; Only Ni diffuses 20

21 Diffusion in Ordered Alloys Reaction Diffusion, Thermodynamic Discussion 2 Peritectic Reactions L + a ß L + ß g 21

22 Diffusion in Ordered Alloys Thermodynamic Discussion DG 22

23 Diffusion in Ordered Alloys Formation of Intermetallic Compounds Diffusion Couple a + g; ß a J g g J a xa xg 23

24 Diffusion in Ordered Alloys Formation of Intermetallic Phases Diffusion Couple Al/Cu G. Schmitz 24

25 J g = - Ď c Bg / x xg ; J a = - Ď c Ba / x xa ; Diffusion in Ordered Alloys Kinetic c/ x = (c/rt) µ/ x Ideal Solution J g = - Ď (c Bg /RT) µ Bg / x ; J a = - Ď (c Ba /RT) µ Ba / x ; J g v g c B g = dx g /dt c Bg ; J a v a c a = dx a /dt c Ba ; w = x g - x a w - Width of the diffusion zone of the (ß) phase 25

26 Diffusion in Ordered Alloys Kinetic dw/dt = d/dt (x g - x a ) = d x g /dt d x a /dt = J g / c Bg - J a / c a ; dw/dt = Ď/RT (µ Ba - µ Bg )/ x ~ Ď/RT Dµ/ x ~ Ď/RT DG/w ; µ Ba - µ Bg = G a G g + RT ln(x Ba /X Bg ) wdw = Ď/RT DG dt w 2 = kt; k = 2DG Ď/RT ; w(0) = 0, [k] = m 2 /s Parabolic growth 26

27 Diffusion in Ordered Alloys Effect of Order Parameter A* D A * = G A <r 2 > f/6 G = vexp(-q o /k B T) (Lecture II-5) AB G A = <n A > exp[ - Q o (1 + a A h 2 )/ (T/T c ) ] Girifalco (1964) D A * = D oa exp[- Q o (1 + a A h 2 )/ (T/T c ) ] ln(d A * ) = ln (D oa ) - Q o (1 + a A h 2 )/(T c /T) D Zn Girifalco (1964) 27

28 Diffusion in Ordered Alloys Effect of Order Parameter Experimenatl Observations for B2 Compounds (Bocquet et al. 1996) # The Diffusion coefficient varies with composition and exhibits a minimum at the stoichiometric composition h has a maximum for the stoichiometric composition Q has a maximum at the stoichiometric composition Variation of activation energy in NiAl with composition LRO Parameter 1,0 0,8 0,6 0,4 0,2 0,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 X Au Bakker (1984) Q = Q o (1 + a A h 2 )/ (T/T c ) 28

29 Diffusion in Ordered Alloys Effect of Order Parameter Experimenatl Observations for B2 Compounds (Bocquet et al. 1996) # The Activation energy for tracer diffusion is higher in the ordered state than in the disordered state (typical for B2 alloys with substitutional disorder) D A * = D oa exp[- Q o (1 + a A h 2 )/ (T/T c ) ] Change of slope at T > T c. Cu ( ) Zn (o) D Bakker (1984) O (Roman numbers denote different Zb concentrations) 29

30 Diffusion in Ordered Alloys Effect of Order Parameter D A *= G A <r 2 > f/6 f is a function of temperature and u = e BB e AA /<e> [<e> = e AB (e AA + e BB )/2] u Bakker (1984) T c /T The correlation factor also depends indirctly on the order parameter 30

Atomic Transport & Phase Transformations Lecture 7

Atomic Transport & Phase Transformations Lecture 7 PD Dr. Nikolay Zotov zotov@imw.uni-stuttgart.de Lecture I-7 Outline Limited solubility in the solid state Effect of the maximum temperature of the miscibility