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: 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). AluminumCopper Alloy b (lighter phase) a (darker phase) Figure adapted from Callister, Materials science and engineering, 7th Ed. 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: Answer: 65 wt% sugar. If Co < 65 wt% sugar: syrup If Co > 65 wt% sugar: syrup + sugar. L (liquid) 60 L 40 (liquid solution i.e., syrup) 20 0 + S (solid sugar) 20 40 6065 80 100 Co =Composition (wt% sugar) Pure Sugar solubility limit at 20°C? Solubility Limit 80 Pure Water Question: What is the 100 Temperature (°C) Max concentration for which only a single phase solution occurs. Sucrose/Water Phase Diagram Salt-Water(ice) http://webserver.dmt.upm.es/~isidoro/bk3/c07sol/Solution%20properties_archivos/image001.gif 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). T(°C) • 2 phases: 1600 • Phase Diagram for Cu-Ni at P=1 atm. L (liquid) a (FCC solid solution) L (liquid) 1500 1400 1300 a (FCC solid solution) 1200 1100 1000 0 20 40 60 Figure adapted from Callister, Materials science and engineering, 7th Ed. 80 • 3 phase fields: L L+a a 100 wt% Ni 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). T(°C) 1600 • Phase Diagram for Cu-Ni at P=1 atm. Liquidus: Separates the liquid from the mixed L+a phase L (liquid) 1500 1400 Solidus: Separates the mixed L+a phase from the solid solution 1300 a (FCC solid solution) 1200 1100 1000 0 20 40 60 Figure adapted from Callister, Materials science and engineering, 7th Ed. 80 100 wt% Ni Number and types of phases • Rule 1: If we know T and Co, then we know: - the number and types of phases present. • Examples: T(°C) 1600 A(1100°C, 60): 1 phase: a B (1250°C,35) L (liquid) 1500 1400 B(1250°C, 35): 2 phases: L + a 1300 a (FCC solid solution) 1200 A(1100°C,60) 1100 1000 Cu-Ni phase diagram 0 20 Figure adapted from Callister, Materials science and engineering, 7th Ed. 40 60 80 100 wt% Ni Composition of phases • Rule 2: If we know T and Co, then we know: --the composition of each phase. • Examples: T(°C) Cu-Ni system A TA Co = 35 wt% Ni 1300 L (liquid) At T A = 1320°C: Only Liquid (L) B TB CL = Co ( = 35 wt% Ni) At T D = 1190°C: 1200 D Only Solid ( a) TD Ca = Co ( = 35 wt% Ni) 20 3032 35 At T B = 1250°C: CLCo Both a and L CL = C liquidus ( = 32 wt% Ni here) Ca = C solidus ( = 43 wt% Ni here) Figure adapted from Callister, Materials science and engineering, 7th Ed. tie line a (solid) 4043 50 Ca wt% Ni Cooling a Cu-Ni Binary - Composition • Phase diagram: Cu-Ni system. • System is: --binary i.e., 2 components: Cu and Ni. T(°C) L (liquid) 130 0 L: 35 wt% Ni a: 46 wt% Ni • Consider Co = 35 wt%Ni. Cu-Ni system A 35 32 --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni. L: 35wt%Ni B C 46 43 D 24 L: 32 wt% Ni 36 120 0 a: 43 wt% Ni E L: 24 wt% Ni a: 36 wt% Ni a (solid) 110 0 20 Figure adapted from Callister, Materials science and engineering, 7th Ed. 30 35 Co 40 50 wt% Ni USE LEVER RULE The Lever Rule – Weight % • Tie line – connects the phases in equilibrium with each other - essentially an isotherm T(°C) How much of each phase? Think of it as a lever tie line 1300 L (liquid) B TB a (solid) 1200 R 20 30C S 40 C a L Co R 50 wt% Ni WL Ma ML C C0 ML S a ML M a R S Ca CL Figure adapted from Callister, Materials science and engineering, 7th Ed. S M a S M L R Wa C CL R 0 R S Ca CL 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%). • Examples: Cu-Ni system T(°C) 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 a = 100 wt% At T B : Both a and L 43 35 73 wt % 43 32 WL S R +S Wa R R +S = 27 wt% Figure adapted from Callister, Materials science and engineering, 7th Ed. A TA 1300 TB 1200 TD 20 tie line L (liquid) B S R D 3032 35 C LC o a (solid) 40 4 3 50 C a wt% Ni Cooling a Cu-Ni Binary – wt. % • Phase diagram: Cu-Ni system. • System is: --binary i.e., 2 components: Cu and Ni. T(°C) L (liquid) 130 0 L: 92 wt% a: 8 wt% i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni. • Consider Co = 35 wt%Ni. Cu-Ni system A 32 --isomorphous L: 35wt%Ni 34 B C 46 43 D 24 L: 73 wt% 36 120 0 a: 27 wt% E L: 8 wt% a: 92 wt% a (solid) 110 0 20 Figure adapted from Callister, Materials science and engineering, 7th Ed. 30 35 Co 40 50 wt% Ni 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 dC J D T dx Diffusion coefficient which we expect is a function of the temperature, T Diffusion cont…. • 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: C J 0 t x • BUT dC J D T dx giving Fick’s second law (with D being constant): C C 2C D D T 2 t x x x Solution of Ficks’ Laws t=0 Co For a semi-infinite sample the solution to Ficks’ Law gives an error function distribution whose width increases with time C C t=t x 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 T(°C) L (liquid) L: 35wt%Ni Co = 35 wt%Ni. Enough time is allowed at each temperature change for atomic diffusion to occur. – Thermodynamic ground state Each phase is homogeneous 130 0 L: 35 wt% Ni a: 46 wt% Ni Cu-Ni system A B C D 120 0 L: 32 wt% Ni a: 43 wt% Ni E L: 24 wt% Ni a: 36 wt% Ni a (solid) 110 0 20 Figure adapted from Callister, Materials science and engineering, 7th Ed. 30 35 Co 40 50 wt% Ni 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. α+L α 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 Last a to solidify: < 35 wt% Ni Figure adapted from Callister, Materials science and engineering, 7th Ed. Uniform C a: 35 wt% Ni Binary-Eutectic Systems – Cu/Ag 2 components has a special composition with a min. melting temperature a phase: b phase: Mostly copper Solvus line – the solubility limit Mostly Silver • Limited solubility – mixed phases • 3 phases regions, L, a and b and 6 phase fields - L, a and b, L+a, L+b, ab Figure adapted from Callister, Materials science and engineering, 7th Ed. Binary-Eutectic Systems Cu-Ag system T(°C) The Eutectic point 1200 L (liquid) TE, Eutectic temperature, 779°C 1000 CE, eutectic composition, 71.9wt.% a TE • TE : No liquid below TE Min. melting TE 800 E 779°C CaE=8.0 600 CE=71.9 L +b b CbE=91.2 ab 400 200 • Eutectic transition L(CE) L+a 0 a(CaE) + b(CbE) Figure adapted from Callister, Materials science and engineering, 7th Ed. Co 20 40 60 CE 80 wt% Ag in Cu/Ag alloy 100 Any other composition, Liquid transforms to a mixed L+solid phase Pb-Sn (Solder) Eutectic System (1) • For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find... --the phases present: a + b --compositions of phases: CO = 40 wt% Sn Ca = 11 wt% Sn Cb = 99 wt% Sn --the relative amount of each phase: Wa = S R+S C -C = b O Cb - Ca 99 - 40 59 = = 67 wt% 99 - 11 88 CO - Ca Wb = R = Cb - Ca R+S 40 - 11 29 = = 33 wt% = 99 - 11 88 Pb-Sn system T(°C) 300 200 L (liquid) a L+a 18.3 150 L +b b 183°C 61.9 R 97.8 S 100 a+ b = 0 11 20 Ca 40 Co 60 80 C, wt% Sn 99100 Cb Figure adapted from Callister, Materials science and engineering, 7th Ed. Microstructures in Eutectic Systems: II L: Co wt% Sn T(°C) • 2 wt% Sn < Co < 18.3 wt% Sn • Result: 400 Initially liquid → liquid + a then a alone finally two phases a poly-crystal fine b-phase inclusions L 300 L +a a 200 TE a: Co wt% Sn a b 100 a+ b 0 10 20 Co Co , 2 (sol. limit at T room ) 18.3 (sol. limit at TE) Figure adapted from Callister, Materials science and engineering, 7th Ed. L a Pb-Sn system 30 wt% Sn Microstructures in Eutectic Systems: Co=CE • Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals. T(°C) Micrograph of Pb-Sn eutectic microstructure L: Co wt% Sn 300 Pb-Sn system a 200 L+a L TE 100 ab 0 Lb b 183°C 20 18.3 40 b: 97.8 wt% Sn a: 18.3 wt%Sn 60 CE 61.9 80 100 97.8 C, wt% Sn Figures adapted from Callister, Materials science and engineering, 7th Ed. 160m Microstructures in Eutectic Systems: Co=CE • Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals. T(°C) L: Co wt% Sn 300 Pb-Sn system a 200 L+a Wa Lb b 183°C TE Pb rich L Wb 100 ab 0 20 18.3 40 b: 97.8 wt% Sn a: 18.3 wt%Sn 60 CE 61.9 80 100 97.8 C, wt% Sn Figures adapted from Callister, Materials science and engineering, 7th Ed. 97.8 61.9 45.2% 97.8 18.3 61.9 18.3 54.8% 97.8 18.3 Sn Rich 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. Figure adapted from Callister, Materials science and engineering, 7th Ed. Sn Pb Microstructures IV • 18.3 wt% Sn < Co < 61.9 wt% Sn • Result: a crystals and a eutectic microstructure L: Co wt% Sn T(°C) 300 Pb-Sn system a 200 L+a L a L R a L L+b b S TE a+ b 100 0 20 18.3 40 60 61.9 80 Co, wt% Sn Figure adapted from Callister, Materials science and engineering, 7th Ed. 100 • Just above TE : C a = 18.3 wt% Sn CL = 61.9 wt% Sn Wa= S = 50 wt% R +S WL = (1- W a) = 50 wt% Microstructures IV • 18.3 wt% Sn < Co < 61.9 wt% Sn • Result: a crystals and a eutectic microstructure L: Co wt% Sn T(°C) 300 Pb-Sn system a 200 L+a TE R 100 a+ b L a L R a L L+b b S • Just below TE : C a = 18.3 wt% Sn C b = 97.8 wt% Sn Wa = S = 73 wt% R +S W b = 27 wt% S Primary, a 0 20 18.3 40 60 61.9 80 Co, wt% Sn Figure adapted from Callister, Materials science and engineering, 7th Ed. 100 97.8 Eutectic, a Eutectic, b 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 Eutectoid transition + Figure adapted from Callister, Materials science and engineering, 7th Ed. mixed liquid and solid to single solid transition Solid to solid ‘eutectic’ type transition Iron-Carbon (Fe-C) Phase Diagram L + Fe3C -Eutectoid (B): a + Fe3C T(°C) 1600 L 1400 1200 +L (austenite) 1000 800 a 600 Result: Pearlite = alternating layers of a and Fe3C phases S +Fe3C 727°C = T eutectoid R S 1 0.76 C eutectoid 120 m 0 (Fe) Figure adapted from Callister, Materials science and engineering, 7th Ed. L+Fe3C R B 400 A 1148°C Fe3C (cementite) • 2 important points -Eutectic (A): 2 3 a+Fe3C 4 5 6 6.7 4.30 Co, wt% C Fe3C (cementite-hard) a (ferrite-soft) Iron-Carbon http://www.azom.com/work/pAkmxBcSVBfns037Q0LN_files/image003.gif