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

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

Hume-Rothery Rules – Substitutional Solutions 1.The solute and solvent should be of a similar size. (<15% difference) 2.The crystal structures must match. 3.Both solute and solvent should have similar electronegativity 4.The valence of the solvent and solute metals should be similar. Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics) Metals – Ni/Cu, Pd/Sn, Ag/Au, Mo/W

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

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

Hume-Rothery Rules – Interstitial Solution 1.The solute must be smaller than the interstitial sites in the solvent lattice 2.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.

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

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.

Edge Defect Motion

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

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

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. atom trap (Callister: Materials Science and Engineering)

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

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 homogenous phase Generally composed on metals close in the periodic table Ni/Cu, Pb/Sn etc. Otherwise compounds tend to form NaCl, Fe 2 O 3 etc.

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.,  and  ). Aluminum- Copper Alloy Components and Phases  (darker phase)  (lighter phase) Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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.

Unary Phase Diagrams A pressure-temperature plot showing the different phases present in H 2 O. Phase Boundaries Upon crossing one of these boundaries the phase abruptly changes from one state to another. Latent heat not shown Crossing any line results in a structural phase transition

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

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

Water Ice

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? Binary Phase Diagrams Phase B Phase A Nickel atom Copper atom

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 C o < 65 wt% sugar: syrup If C o > 65 wt% sugar: syrup + sugar. 65 Sucrose/Water Phase Diagram Pure Sugar Temperature (°C) CoCo =Composition (wt% sugar) L (liquid solution i.e., syrup) Solubility Limit L (liquid) + S (solid sugar) PureWater

Salt-Water(ice)

Ferroelectric Materials Na ½ Ba ½ TiO 3 T. Takenaka, K. Maruyama, and K. Sakata, Jpn. J. Appl. Phys. 30, 2236 (1991)

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).

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 at P=1 atm. 2 phases: L (liquid)  (FCC solid solution) 3 phase fields: L L +   wt% Ni T(°C) L (liquid)  (FCC solid solution) L +  liquidus solidus Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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 at P=1 atm. wt% Ni T(°C) L (liquid)  (FCC solid solution) L +  liquidus solidus Figure adapted from Callister, Materials science and engineering, 7 th Ed. Liquidus: Separates the liquid from the mixed L+  phase Solidus: Separates the mixed L+  phase from the solid solution

wt% Ni T(°C) L (liquid)  (FCC solid solution) L +  liquidus solidus Cu-Ni phase diagram 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:  B(1250°C, 35): 2 phases: L +  B (1250°C,35) A(1100°C,60) Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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

wt% Ni L (liquid)  (solid) L +  L +  T(°C) A 35 C o L: 35wt%Ni Cu-Ni system 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;  phase field extends from 0 to 100 wt% Ni. Consider C o = 35 wt%Ni. Cooling a Cu-Ni Binary - Composition  :43 wt% Ni L: 32 wt% Ni L: 24 wt% Ni  :36 wt% Ni B  : 46 wt% Ni L: 35 wt% Ni C D E Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE

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

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

wt% Ni L (liquid)  (solid) L +  L +  T(°C) A 35 C o L: 35wt%Ni Cu-Ni system 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;  phase field extends from 0 to 100 wt% Ni. Consider C o = 35 wt%Ni. Cooling a Cu-Ni Binary – wt. %  :27 wt% L: 73 wt% L: 8 wt%  :92 wt% B  : 8 wt% L: 92 wt% C D E Figure adapted from Callister, Materials science and engineering, 7 th Ed.

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

Diffusion is not a flow Our models of diffusion are based on a random walk approach and not a net flow Concept behind mean free path in scattering phenomena - conductivity