Lecture 7: Solid State Reactions Phase Diagrams and Mixing

Transcription

1 Lecture 7: Solid State Reactions Phase Diagrams and Mixing Prof Ken Durose, Univ of Liverpool Text book for this lecture: Callister Materials Science and Engineering

2 Learning objectives 1.Solid state reactions Recap: how Gibbs Free Energy can be used to predict reactions 2 Phase diagrams Be able to read a phase diagram Be able to identify the phases present at a given point Be able to calculate how much of those phases there are Be able to predict the microstructures from the phase diagram 3 Mixing Understand the thermodynamic basis of mixing Understand the origin of the miscibility gap Be able to use the phase diagram to predict miscibility phenomena There will be worked examples on this bit for you to practice.

3 1. Prediction of reactions e.g. solar cell contact metal M to semiconductor AB M AB stable or unstable M AB M MB + A M + AB MB + A k eqm = [MB]*[A] /[M]*[AB] AB Will it happen at equilibrium? If k eqm is large, the reaction goes from l to r Find the Gibb s free energy for the reaction. G RT ln( k eqm ) If G is large and negative, k eqm is large and the reaction goes from l to r

4 1. Prediction of reactions G G H S reaction reaction reaction reaction H G H S final final final reaction G S T S H initial initial initial reaction G etc M + AB MB + A ( G G ) ( G G reaction MB A M AB) If G is large and negative, k eqm is large and the reaction goes from l to r In this case the contact would be unstable to reaction with the solar cell materials. The contact could become non-ohmic, the series resistance could increase, and the performance could become unstable.

5 2. Equilibrium phase diagrams ISSUES TO ADDRESS... When we combine two elements... what equilibrium state do we get? In particular, if we specify... --a composition (e.g., wt% Cu - wt% Ni), and --a temperature (T) then... How many phases do we get? What is the composition of each phase? How much of each phase do we get? Phase A Phase B Nickel atom Copper atom

6 Components and Phases Components: The elements or compounds which are present in the mixture (e.g., Al and Cu) Phases: The physically and chemically distinct material regions that result (e.g., a and b). Aluminum- Copper Alloy Adapted from chapter-opening photograph, Chapter 9, Callister 3e. b (lighter phase) a (darker phase)

7 Phase Equilibria Simple solution system (e.g., Ni-Cu solution) Crystal Structure electroneg r (nm) Ni FCC Cu FCC Both have the same crystal structure (FCC) and have similar electronegativities and atomic radii (W. Hume Rothery rules) suggesting high mutual solubility. Ni and Cu are totally miscible in all proportions.

8 Phase Diagrams Indicate phases as function of T, C o, and P. For this course: -binary systems: just 2 components. -independent variables: T and C o (P = 1 atm is almost always used). Phase Diagram for Cu-Ni system T( C) L (liquid) a (FCC solid solution) phases: L (liquid) a (FCC solid solution) 3 phase fields: L L + a a Adapted from Fig. 9.3(a), Callister 7e. (Fig. 9.3(a) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH (1991). wt% Ni

9 Phase Diagrams: number and types of phases Rule 1: If we know T and C o, then we know: --the number and types of phases present. Examples: A(1100 C, 60): 1 phase: a B(1250 C, 35): 2 phases: L + a T( C) L (liquid) B (1250 C,35) a (FCC solid solution) Cu-Ni phase diagram Adapted from Fig. 9.3(a), Callister 7e. (Fig. 9.3(a) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991) A(1100 C,60) wt% Ni

10 Phase Diagrams: composition of phases Rule 2: If we know T and C o, then we know: --the composition of each phase. Examples: Co = 35 wt% Ni At TA = 1320 C: Only Liquid (L) CL = Co ( = 35 wt% Ni) At TD = 1190 C: Only Solid ( a) Ca = Co ( = 35 wt% Ni) At TB = 1250 C: T( C) TA 1300 TB 1200 TD 20 Both a and L CL = Cliquidus ( = 32 wt% Ni here) Ca = Csolidus ( = 43 wt% Ni here) L (liquid) Cu-Ni system A B D 3235 C L C o tie line a (solid) 4 3 C a wt% Ni Adapted from Fig. 9.3(b), Callister 7e. (Fig. 9.3(b) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991.)

11 Rule 3: If we know T and C o, then we know: --the amount of each phase (given in wt%). Examples: Co = 35 wt% Ni At TA: Only Liquid (L) Phase Diagrams: weight fractions of phases lever rule WL = 100 wt%, Wa = 0 At TD: Only Solid ( a) At TB: W L W a WL = 0, Wa = 100 wt% Both a and L S R +S R R +S = 27 wt% 73wt% T( C) TA 1300 TB 1200 TD 20 L (liquid) Cu-Ni system A B R S D 3235 C L C o tie line a (solid) 4 3 C a wt% Ni Adapted from Fig. 9.3(b), Callister 7e. (Fig. 9.3(b) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991.)

12 e.g.: Cooling in a Cu-Ni Binary Phase diagram: Cu-Ni system. System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni. Consider C o = 35 wt%ni. T( C) 1300 L: 35 wt% Ni a: 46 wt% Ni L (liquid) a (solid) Adapted from Fig. 9.4, Callister 7e A B C D E 35 Co L: 35wt%Ni wt% Ni Cu-Ni system L: 32 wt% Ni a: 43 wt% Ni L: 24 wt% Ni a: 36 wt% Ni

13 Equilibrium cooling The compositions that freeze are a function of the temperature At equilibrium, the first to freeze composition must adjust on further cooling by solid state diffusion We now examine diffusion processes how fast does diffusion happen in the solid state?

14 Diffusion Fick s law for steady state diffusion t = 0 t = 0 C o C J = flux amount of material per unit area per unit time C = conc dc t = t C o /4 C x 3C o /4 C o /2 J D dx 2 x Dt 2 x x

15 Diffusion cont. Diffusion coefficient D (cm 2 s -1) Non-steady state diffusion Fick s second law C 2 C t D x 2 Diffusion is thermally activated D D 0 exp Q d RT

16 L Non equilibrium cooling α + L α

17 Cored vs Equilibrium Phases C a changes as we solidify. Cu-Ni case: Fast rate of cooling: Cored structure First a to solidify has Ca = 46 wt% Ni. Last a to solidify has Ca = 35 wt% Ni. Slow rate of cooling: Equilibrium structure First a to solidify: 46 wt% Ni Last a to solidify: < 35 wt% Ni Uniform Ca: 35 wt% Ni

18 Tensile Strength (MPa) Elongation (%EL) Mechanical Properties: Cu-Ni System Effect of solid solution strengthening on: --Tensile strength (TS) --Ductility (%EL,%AR) TS for pure Cu Cu Ni TS for pure Ni Composition, wt% Ni Cu Ni Composition, wt% Ni Adapted from Fig. 9.6(a), Callister 7e. Adapted from Fig. 9.6(b), Callister 7e %EL for pure Cu %EL for pure Ni --Peak as a function of Co --Min. as a function of Co

19 Binary-Eutectic Systems 2 components Ex.: Cu-Ag system 3 single phase regions (L, a, b) Limited solubility: a: mostly Cu b: mostly Ag T E : No liquid below T E C E : Min. melting T E composition Eutectic transition T( C) L(C E ) a(c ae ) + b(c be ) has a special composition with a min. melting T. a C o L + a L (liquid) a b Cu-Ag system L+b b T C E C E wt% Ag in Cu/Ag alloy Adapted from Fig. 9.7, Callister 7e.

20 e.g. Pb-Sn Eutectic System (1) For a 40 wt% Sn-60 wt% Pb alloy at 150 C, find... --the phases present: a + b --compositions of phases: C O = 40 wt% Sn C a = 11 wt% Sn C b = 99 wt% Sn --the relative amount of each phase: S C W b - C O a = R+S = C b - C a = W b = = R R+S = = C O - C a C b - C a = = 67 wt% = 33 wt% T( C) 0 L+a a 183 C 18.3 R L (liquid) a + b Pb-Sn system L+b S Ca C o C b Adapted from Fig. 9.8, Callister 7e. C, wt% Sn b

21 Microstructures in Eutectic Systems: II 2 wt% Sn < C o < 18.3 wt% Sn Result: Initially liquid + a then a alone finally two phases a poly-crystal fine b-phase inclusions T( C) T E L L + a a L: C o wt% Sn L a a: C o wt% Sn a b 100 a+ b Pb-Sn system Adapted from Fig. 9.12, Callister 7e C o (sol. limit at Troom) 18.3 (sol. limit at T E ) 30 C o, wt% Sn

22 Microstructures in Eutectic Systems: III C o = C E Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals. 300 Pb-Sn system 200 T E T( C) a L+a L 183 C L: C o wt% Sn L b b Micrograph of Pb-Sn eutectic microstructure 100 a b b: 97.8 wt% Sn a: 18.3 wt%sn 160 m Adapted from Fig. 9.14, Callister 7e. 0 Adapted from Fig. 9.13, Callister 7e C E C, wt% Sn

23 Lamellar Eutectic Structure Adapted from Figs & 9.15, Callister 7e.

24 Intermetallic Compounds Adapted from Fig. 9.20, Callister 7e. Mg 2 Pb Note: intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact.

25 Eutectoid & Peritectic Peritectic transition + L Cu-Zn Phase diagram Eutectoid transition + Adapted from Fig. 9.21, Callister 7e.

26 Summary Phase diagrams are useful tools to determine: --the number and types of phases, --the wt% of each phase, --and the composition of each phase for a given T and composition of the system. Alloying to produce a solid solution usually --increases the tensile strength (TS) --decreases the ductility. Binary eutectics and binary eutectoids allow for a range of microstructures.

27 3)Thermodynamics of mixing Why do some things mix and others not? Why does heating promote mixing? Is an alloy of two semiconductors a random solid solution, or is it just a mixture of two phases? Solid solution OR Mixture of phases?

28 Free energy of mixing G mix H mix T S mix Enthalpy term bond energies Entropy term stats of mixing behaviour of mixtures consider the case: Pure A + Pure B mixture A x B 1-x

29 Enthalpy term: Compare the energy in bonds (V AB etc) before and after Heat of mixing Binding energy in mixture compared to average in components mixing Binding energy change on mixing ΔH mix Exothermic V AB > 1/2(V AA + V BB ) Stronger -ve Ideal V AB = 1/2(V AA + V BB ) Same 0 endothermic V AB < 1/2(V AA + V BB ) Weaker +ve

30 For reference only ΔH mix derived ΔH mix = (H for mixture AB) (H for pure A + H for pure B) ΔH mix = znx A x B {1/2(V AA + V BB ) V AB } N AA etc = number of bonds of type AA, AB etc N A, N B, number of atoms of A, B. N A + N B = N V AA, V BB, V AB, = binding energies of pairs A-A, x A etc = N A /N = mole fraction of A x B = N B /N = mole fraction of B. x A = (1-x B ) z = co-ordination number

31 For reference only ΔS mix derived ΔS = (S final S initial ) S final from the statistical definition S = k B lnw ΔS mix = k B ln N! ( Nx A )!( Nx B )! Using Stirling s approximation lnn! = nlnn-n gives ΔS mix = -Nk B [ x A lnx A + x B lnx B ]

32 For reference only ΔG mix - the formula G mix H mix T S mix ΔG mix = znx A x B {1/2(V AA + V BB ) V AB } + Nk B T[ x A lnx A + x B lnx B ] Binding energy term {...} may be +ve or ve Entropy term [...] is negative

33 Examples changing ΔH mix e.g. #1 i) Negative ΔH mix The free energy curve is concave downwards at all points at all x. A mixes with B in all proportions. 2 x G 2 Energy TΔS ΔH ΔG molar fraction x 0 for all x

34 changing ΔH mix e.g. #2 ii) ΔH mix = 0 ideal solution mixing is entropy driven. miscibility over the whole composition range. Energy TΔS ΔH ΔG molar fraction x

35 Energy changing ΔH mix e.g. #3 iv) ΔH mix large and positive Enthalpy dominates miscibility gap exists In gap, solid segregates into x A and x A x x Miscibility gap x T ΔS ΔH ΔG 1

36 Low T T ΔS ΔH ΔG Energy molar fraction x

37 Moderate T TΔS ΔH ΔG Energy molar fraction x

38 High T mixing is promoted TΔS ΔH ΔG Energy molar fraction x

39 Binary isomorphous phase diagram T L L + solid solid Mole fraction x

40 Phase diagram with miscibility gap T L L + solid B solid locus 2 G 0 2 x or spinode x x A Mole fraction x

41 Example phase diagram CdTe-CdS

42 Worked example of earlier concept Pb-Sn Eutectic System (2) For a 40 wt% Sn-60 wt% Pb alloy at 200 C, find... --the phases present: a + L --compositions of phases: C O = 40 wt% Sn C a = 17 wt% Sn C L = 46 wt% Sn --the relative amount of each phase: Wa = C L - C O C L - C a = = 6 29 = 21 wt% W L = C O - C a = 23 C L - C a = 79 wt% T( C) 0 a 17 Ca L+a R L (liquid) 183 C a + b Pb-Sn system L+b C o Adapted from Fig. 9.8, Callister 7e. S C L C, wt% Sn b