The Muppet’s Guide to:

1. The Muppet’s Guide to: The Structure and Dynamics of Solids 7. Phase Diagrams

2. Solid Solutions Solute atoms create strain fields which can inhibit dislocations propagating in a material changing its properties

3. Hume-Rothery Rules – Substitutional Solutions Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics) • The solute and solvent should be of a similar size. (<15% difference) • The crystal structures must match. • Both solute and solvent should have similar electronegativity • The valence of the solvent and solute metals should be similar. Metals – Ni/Cu, Pd/Sn, Ag/Au, Mo/W

4. Phase Equilibria – Example K-Na • Both have the same crystal structure (BCC) and have similar electronegativities but different atomic radii. • Rules suggest that NO solid solution will form. • K and Na sodium are not miscible.

5. Phase Equilibria – Example Simple solution system (e.g., Ni-Cu solution) • 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.

6. Hume-Rothery Rules –Interstitial Solution • The solute must be smaller than the interstitial sites in the solvent lattice • Solute and Solvent should have similar electro-negativities Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics) Light elements – H,C, N and O.

7. Modify Material Properties • Smaller substitutional impurity • Larger substitutional impurity A C D B Impurity generates local stress at A and B that opposes dislocation motion to the right. Impurity generates local stress at C and D that opposes dislocation motion to the right. Increase material strength through substitution • • Impurity atoms distort the lattice & generates stress. • • Stress can produce a barrier to dislocation motion. (Callister: Materials Science and Engineering)

8. Material Properties Dislocations & plastic deformation • Cubic & hexagonal metals - plastic deformation by plastic shear or slip where one plane of atoms slides over adjacent plane by defect motion (dislocations). • If dislocations don't move, deformation doesn't occur! Adapted from Fig. 7.1, Callister 7e.

9. Edge Defect Motion

10. Dislocations & Materials Classes • Metals: Disl. motion easier. -non-directional bonding -close-packed directions for slip. + + + + + + + + + + + + + + + + + + + + + + + + electron cloud ion cores • Covalent Ceramics (Si, diamond): Motion hard. -directional (angular) bonding • Ionic Ceramics (NaCl): Motion hard. -need to avoid ++ and - - neighbours. - - - + + + + - - - - + + + - - - + + + + (Callister: Materials Science and Engineering)

11. Dislocation Motion • Dislocation moves along slip plane in slip direction perpendicular to dislocation line • Slip direction same direction as Burgers vector Edge dislocation Adapted from Fig. 7.2, Callister 7e. Screw dislocation (Callister: Materials Science and Engineering)

12. atom trap Pinning dislocations • dislocations make metals easier to deform • to improve strength of metals, need to stop dislocation motion trap with:- impurity atoms;- other dislocations (work hardening; - grain boundaries. (Callister: Materials Science and Engineering)

13. Modify Material Properties Increase material strength through reducing Grain size • Grain boundaries are barriers to slip. • Barrier "strength" increases with Increasing angle of miss-orientation. • Smaller grain size: more barriers to slip. (Callister: Materials Science and Engineering)

14. Solid Solutions Solid state mixture of one or more solutes in a solvent Crystal structure remains unchanged on addition of the solute to the solvent Mixture remains in a homogenousphase Generally composed on metals close in the periodic table Ni/Cu, Pb/Sn etc. Otherwise compounds tend to form NaCl, Fe2O3 etc.

15. 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 b (lighter phase) a (darker phase) Figure adapted from Callister, Materials science and engineering, 7th Ed.

16. Phase Diagrams • A phase diagram is a graphical representation of the different phases present in a material. • Commonly presented as a function of composition and temperature or pressure and temperature Applies to elements, molecules etc. and can also be used to show magnetic, and ferroelectric behaviour (field vs. temperature) as well as structural information.

17. Unary Phase Diagrams A pressure-temperature plot showing the different phases present in H2O. Phase Boundaries Crossing any line results in a structural phase transition Upon crossing one of these boundaries the phase abruptly changes from one state to another. Latent heat not shown

18. Reading Unary Phase Diagrams Melting Point (solid → liquid) Triple Point (solid + liquid + gas) Boiling Point (liquid→ gas) Sublimation (solid → gas) As the pressure falls, the boiling point reduces, but the melting/freezing point remains reasonably constant.

19. Reading Unary Phase Diagrams P=1atm Melting Point: 0°C Boiling Point: 100°C P=0.1atm Melting Point: 2°C Boiling Point: 68°C

20. Water Ice http://images.jupiterimages.com/common/detail/13/41/23044113.jpg, http://www.homepages.ucl.ac.uk/~ucfbanf/ice_phase_diagram.jpg

21. Binary Phase Diagrams Phase B Phase A Nickel atom Copper atom • 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?

22. Sucrose/Water Phase Diagram 10 0 Solubility L Limit 8 0 (liquid) 6 0 + L Temperature (°C) S (liquid solution 4 0 i.e., syrup) (solid 20 sugar) 0 20 40 60 80 100 65 Co =Composition (wt% sugar) Sugar Water Pure Pure Phase Equilibria: Solubility Limit • Solutions – solid solutions, single phase • Mixtures – more than one phase • Solubility Limit: Max concentration for which only a single phase solution occurs. Question: What is the solubility limit at 20°C? Answer: 65 wt% sugar. If Co < 65 wt% sugar: syrup If Co > 65 wt% sugar: syrup + sugar.

24. Salt-Water(ice) http://webserver.dmt.upm.es/~isidoro/bk3/c07sol/Solution%20properties_archivos/image001.gif

25. Ferroelectric Materials Na½Ba½TiO3 T. Takenaka, K. Maruyama, and K. Sakata, Jpn. J. Appl. Phys. 30, 2236 (1991)

26. B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics (Academic Press, London, 1971). J. Rodel, W. Jo, K. T. P. Seifert, E. M. Anton, T. Granzow, and D. Damjanovic, J. Am. Ceram. Soc. 92, 1153 (2009).

27. T(°C) • 2 phases: 1600 L (liquid) 1500 L (liquid) a (FCC solid solution) • 3 phase fields: 1400 L a liquidus + 1300 a L + L solidus a a 1200 (FCC solid 1100 solution) 1000 wt% Ni 0 20 40 60 80 100 Phase Diagrams • Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components. -independent variables: T and Co (P = 1 atm is almost always used). • Phase Diagram for Cu-Ni at P=1 atm. Figure adapted from Callister, Materials science and engineering, 7th Ed.

28. T(°C) 1600 1500 L (liquid) 1400 a liquidus + 1300 L solidus a 1200 (FCC solid 1100 solution) 1000 wt% Ni 0 20 40 60 80 100 Phase Diagrams • Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components. -independent variables: T and Co (P = 1 atm is almost always used). Liquidus: Separates the liquid from the mixed L+aphase • Phase Diagram for Cu-Ni at P=1 atm. Solidus: Separates the mixed L+aphase from the solid solution Figure adapted from Callister, Materials science and engineering, 7th Ed.

29. T(°C) 1600 L (liquid) 1500 a 1 phase: Cu-Ni phase diagram liquidus B (1250°C, 35): (1250°C,35) 1400 solidus a 2 phases: L + a a + 1300 L B (FCC solid 1200 solution) 1100 A(1100°C,60) 1000 wt% Ni 0 20 40 60 80 100 Number and types of phases • Rule 1: If we know T and Co, then we know: - the number and types of phases present. • Examples: A(1100°C, 60): Figure adapted from Callister, Materials science and engineering, 7th Ed.

30. Cu-Ni system T(°C) A T C = 35 wt% Ni A o tie line liquidus L (liquid) At T = 1320°C: 1300 A a + L Only Liquid (L) B T solidus B C = C ( = 35 wt% Ni) L o a a At T = 1190°C: + D L (solid) 1200 D a Only Solid ( ) T D C = C ( = 35 wt% Ni ) a o 32 35 4 3 20 30 40 50 At T = 1250°C: C C C a B L o wt% Ni a Both and L C = C ( = 32 wt% Ni here) L liquidus C = C ( = 43 wt% Ni here) a solidus Composition of phases • Rule 2: If we know T and Co, then we know: --the composition of each phase. • Examples: Figure adapted from Callister, Materials science and engineering, 7th Ed.

31. Cooling a Cu-Ni Binary - Composition T(°C) L: 35wt%Ni L (liquid) Cu-Ni system a 130 0 A + L L: 35 wt% Ni B a: 46 wt% Ni 35 46 C 32 43 D L: 32 wt% Ni 24 36 a a : 43 wt% Ni + 120 0 E L L: 24 wt% Ni a : 36 wt% Ni a (solid) 110 0 35 20 3 0 4 0 5 0 wt% Ni C o • 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 Co = 35 wt%Ni. USE LEVER RULE Figure adapted from Callister, Materials science and engineering, 7th Ed.

32. T(°C) tie line liquidus L (liquid) 1300 a + M ML L B solidus T B a a + L (solid) 1200 R S S R 20 3 0 4 0 5 0 C C C a L o wt% Ni The Lever Rule – Weight % • Tie line – connects the phases in equilibrium with each other - essentially an isotherm • How much of each phase? Think of it as a lever Figure adapted from Callister, Materials science and engineering, 7th Ed.

33. Weight fractions of phases – ‘lever rule’ Cu-Ni system T(°C) A T C = 35 wt% Ni A o tie line liquidus L (liquid) At T : Only Liquid (L) 1300 a A + L B W = 100 wt%, W = 0 a L T solidus B S R a At T : Only Solid ( ) D a a + W = 0, W = 100 wt% L a L (solid) 1200 D T a D At T : Both and L B 32 35 4 3 20 3 0 4 0 5 0 S = WL C C C a L o wt% Ni R + S R = = 27 wt% Wa R + S • Rule 3: If we know T and Co, then we know: --the amount of each phase (given in wt%). • Examples: Figure adapted from Callister, Materials science and engineering, 7th Ed.

34. Cooling a Cu-Ni Binary – wt. % T(°C) L: 35wt%Ni L (liquid) Cu-Ni system a 130 0 A + L L: 92 wt% B a: 8 wt% 34 46 C 32 43 D L: 73 wt% 24 36 a a : 27 wt% + 120 0 E L L: 8 wt% a : 92 wt% a (solid) 110 0 35 20 3 0 4 0 5 0 wt% Ni C o • 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 Co = 35 wt%Ni. Figure adapted from Callister, Materials science and engineering, 7th Ed.

35. Equilibrium cooling • Multiple freezing sites • Polycrystalline materials • Not the same as a single crystal • 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

36. Diffusion is not a flow Concept behind mean free path in scattering phenomena - conductivity Our models of diffusion are based on a random walk approach and not a net flow http://mathworld.wolfram.com/images/eps-gif/RandomWalk2D_1200.gif