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

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.

Components and Phases • Components: • Phases: b (lighter phase) a 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.

Crossing any line results in a structural phase transition 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

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.

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

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

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?

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. Sucrose/Water Phase Diagram Pure Sugar Temperature (°C) 20 40 60 80 100 Co =Composition (wt% sugar) L (liquid solution i.e., syrup) Solubility Limit (liquid) + S (solid sugar) 4 6 8 10 Water 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. 65

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

Phase Diagrams • Phase Diagram for Cu-Ni at P=1 atm. • 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). • 2 phases: L (liquid) a (FCC solid solution) • 3 phase fields: L + wt% Ni 20 40 60 80 100 1000 1100 1200 1300 1400 1500 1600 T(°C) L (liquid) (FCC solid solution) + liquidus solidus • Phase Diagram for Cu-Ni at P=1 atm. Figure adapted from Callister, Materials science and engineering, 7th Ed.

Phase Diagrams • Phase Diagram for Cu-Ni at P=1 atm. • 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). wt% Ni 20 40 60 80 100 1000 1100 1200 1300 1400 1500 1600 T(°C) L (liquid) a (FCC solid solution) L + liquidus solidus Liquidus: Separates the liquid from the mixed L+a phase • Phase Diagram for Cu-Ni at P=1 atm. Solidus: Separates the mixed L+a phase from the solid solution Figure adapted from Callister, Materials science and engineering, 7th Ed.

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

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

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

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

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

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

Diffusion in 1 Dimension Fick’s First Law J = flux – amount of material per unit area per unit time C = concentration Diffusion coefficient which we expect is a function of the temperature, T

Diffusion cont…. BUT Requires the solution of the continuity equation: The change in concentration as a function of time in a volume is balanced by how much material flows in per time unit minus how much flows out – the change in flux, J: giving Fick’s second law (with D being constant): BUT

Solution of Ficks’ Laws For a semi-infinite sample the solution to Ficks’ Law gives an error function distribution whose width increases with time t = 0 C Co C x t = t

Consider slabs of Cu and Ni. Interface region will be a mixed alloy (solid solution) Interface region will grow as a function of time

Slow Cooling in a Cu-Ni Binary wt% Ni 20 120 130 3 4 5 110 L (liquid) a (solid) L + T(°C) A 35 C o L: 35wt%Ni Cu-Ni system Co = 35 wt%Ni. a: 46 wt% Ni L: 35 wt% Ni Enough time is allowed at each temperature change for atomic diffusion to occur. – Thermodynamic ground state B C a : 43 wt% Ni L: 32 wt% Ni D L: 24 wt% Ni a : 36 wt% Ni E Each phase is homogeneous Figure adapted from Callister, Materials science and engineering, 7th Ed.

Non – equilibrium cooling α L α + L Non – equilibrium cooling No-longer in the thermodynamic ground state Reduces the melting temperature Figure adapted from Callister, Materials science and engineering, 7th Ed.

Cored vs Equilibrium Phases • Ca changes as we solidify. • Cu-Ni case: First a to solidify has Ca = 46 wt% Ni. Last a to solidify has Ca = 35 wt% Ni. • Fast rate of cooling: Cored structure • Slow rate of cooling: Equilibrium structure First a to solidify: 46 wt% Ni Uniform C : 35 wt% Ni Last < 35 wt% Ni Figure adapted from Callister, Materials science and engineering, 7th Ed.

Binary-Eutectic Systems – Cu/Ag has a special composition with a min. melting temperature 2 components a phase: Mostly copper b phase: Mostly Silver Solvus line – the solubility limit • Limited solubility – mixed phases • 3 phases regions, L, a and b and 6 phase fields - L, a and b, L+a, L+b, a+b Figure adapted from Callister, Materials science and engineering, 7th Ed.

Binary-Eutectic Systems Cu-Ag system L (liquid) a L + b Co wt% Ag in Cu/Ag alloy 20 40 60 80 100 200 1200 T(°C) 400 600 800 1000 CE TE CaE=8.0 CE=71.9 CbE=91.2 779°C The Eutectic point TE, Eutectic temperature, 779°C CE, eutectic composition, 71.9wt.% E • TE : No liquid below TE Min. melting TE • Eutectic transition L(CE) (CE) + (CE) Any other composition, Liquid transforms to a mixed L+solid phase Figure adapted from Callister, Materials science and engineering, 7th Ed.

Pb-Sn (Solder) Eutectic System (1) • For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find... --the phases present: a + b Pb-Sn system --compositions of phases: L + a b 200 T(°C) 18.3 C, wt% Sn 20 60 80 100 300 L (liquid) 183°C 61.9 97.8 CO = 40 wt% Sn Ca = 11 wt% Sn Cb = 99 wt% Sn --the relative amount of each phase: W a = C - CO C - C 99 - 40 99 - 11 59 88 = 67 wt% S R+S 150 R 11 C 40 Co S 99 C W  = CO - C C - C R R+S 29 88 = 33 wt% 40 - 11 99 - 11 Figure adapted from Callister, Materials science and engineering, 7th Ed.

Microstructures in Eutectic Systems: II L: Co wt% Sn • 2 wt% Sn < Co < 18.3 wt% Sn • Result: Initially liquid → liquid +  then  alone finally two phases a poly-crystal fine -phase inclusions Pb-Sn system L + a 200 T(°C) Co , wt% Sn 10 18.3 20 300 100 30 b 400 (sol. limit at TE) TE 2 (sol. limit at T room ) L a a: Co wt% Sn a b Figure adapted from Callister, Materials science and engineering, 7th Ed.

Microstructures in Eutectic Systems: Co=CE • Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals. Pb-Sn system L a 200 T(°C) C, wt% Sn 20 60 80 100 300 L a b + 183°C 40 TE 160 m Micrograph of Pb-Sn eutectic microstructure L: Co wt% Sn CE 61.9 18.3 : 18.3 wt%Sn 97.8 : 97.8 wt% Sn Figures adapted from Callister, Materials science and engineering, 7th Ed.

Microstructures in Eutectic Systems: Co=CE • Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals. Pb-Sn system L a 200 T(°C) C, wt% Sn 20 60 80 100 300 L a b + 183°C 40 TE L: Co wt% Sn CE 61.9 18.3 : 18.3 wt%Sn 97.8 : 97.8 wt% Sn Pb rich Sn Rich Figures adapted from Callister, Materials science and engineering, 7th Ed.

Lamellar Eutectic Structure At interface, Pb moves to a-phase and Sn migrates to b- phase Lamellar form to minimise diffusion distance – expect spatial extent to depend on D and cooling rates. Sn Pb Figure adapted from Callister, Materials science and engineering, 7th Ed.

Microstructures IV • 18.3 wt% Sn < Co < 61.9 wt% Sn • Result: a crystals and a eutectic microstructure WL = (1- W a ) = 50 wt% C = 18.3 wt% Sn CL = 61.9 wt% Sn S R + = • Just above TE : T(°C) L a L: Co wt% Sn L a 300 L Pb-Sn system L + a a 200 18.3 61.9 S R L + b b TE 100 a + b 20 40 60 80 100 Co, wt% Sn Figure adapted from Callister, Materials science and engineering, 7th Ed.

Microstructures IV • 18.3 wt% Sn < Co < 61.9 wt% Sn • Result: a crystals and a eutectic microstructure • Just below TE : C a = 18.3 wt% Sn b = 97.8 wt% Sn S R + W = = 73 wt% = 27 wt% T(°C) L a L: Co wt% Sn L a 300 L Pb-Sn system L + a a b 200 18.3 61.9 S R L + b TE 97.8 S R 100 a + b Primary, a 20 40 60 80 100 Eutectic, a Eutectic, b Co, wt% Sn Figure adapted from Callister, Materials science and engineering, 7th Ed.

Intermetallic Compounds a phase: Mostly Mg Mg2Pb b phase: Mostly Lead Note: intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact. Figure adapted from Callister, Materials science and engineering, 7th Ed.

Eutectoid & Peritectic Peritectic transition  + L  Cu-Zn Phase diagram mixed liquid and solid to single solid transition Eutectoid transition   +  Solid to solid ‘eutectic’ type transition Figure adapted from Callister, Materials science and engineering, 7th Ed.

Iron-Carbon (Fe-C) Phase Diagram • 2 important T(°C) points 1600 d -Eutectic (A): L g + Fe3C L 1400 g +L g A S R 4.30 1200 L+Fe3C -Eutectoid (B): g a + Fe3C 1148°C (austenite) 1000 g g +Fe3C 120 mm a 0.76 C eutectoid B + Fe3C (cementite) 800 a g 727°C = T eutectoid R S Fe3C (cementite-hard) a (ferrite-soft) 600 a +Fe3C 400 1 2 3 4 5 6 6.7 (Fe) Co, wt% C Result: Pearlite = alternating layers of a and Fe3C phases Figure adapted from Callister, Materials science and engineering, 7th Ed.

Iron-Carbon http://www.azom.com/work/pAkmxBcSVBfns037Q0LN_files/image003.gif