The Muppet’s Guide to: The Structure and Dynamics of Solids Phase Diagrams 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 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 The Muppet’s Guide to: The Structure and Dynamics of Solids The Final Countdown Characterisation Over the course so far we have seen how thermodynamics plays an important role in defining the basic minimum energy structure of a solid. Small changes in the structure (such as the perovskites) can produce changes in the physical properties of materials Kinetics and diffusion also play a role and give rise to different metastable structures of the same materials – allotropes / polymorphs Alloys and mixtures undergo multiple phase changes as a function of temperature and composition BUT how do we characterise samples? Probes Resolution better than the inter-atomic spacings • Electromagnetic Radiation • Neutrons • Electrons Probes Treat all probes as if they were waves: Wave-number, k: Momentum, p: k k 2 p k; Photons p mv h ‘Massive’ objects Xavier the X-ray Speed of Light E hc Planck’s constant Wavelength Ex(keV)=1.2398/(nm) Elastic scattering as Ex>>kBT Norbert the Neutron De Broglie equation: h mv mass velocity 2 2 1 h Kinetic Energy: En 2 mv 2mn 2 En(meV)=0.8178/2(nm) Strong inelastic scattering as En~kBT Eric the Electron • Eric’s rest mass: 9.11 × 10−31 kg. • electric charge: −1.602 × 10−19 C • No substructure – point particle De Broglie equation: h mv mass velocity Ee depends on accelerating voltage :– Range of Energies from 0 to MeV Probes Resolution better than the interatomic spacings Absorption low – we want a ‘bulk’ probe • Electrons - Eric • quite surface sensitive • Electromagnetic Radiation - Xavier • Optical – spectroscopy • X-rays : • VUV and soft (spectroscopic and surfaces) • Hard (bulk like) • Neutrons - Norbert Interactions Englebert Xavier Norbert 1. Absorption 2. Refraction/Reflection 3. Scattering Diffraction a Crystals are 2D with planes separated by dhkl. There will only be constructive interference when a== - i.e. the reflection condition. Basic Scattering Theory The number of scattered particles per second is defined using the standard expression Unit solid angle d I s I 0 d Differential cross-section Defined using Fermi’s Golden Rule d Final INTERACTION POTENTIAL Initial d d d Spherical Scattered Wavefield exp i k r V r r dr Scattering Potential 2 Incident Wavefield Different for X-rays, Neutrons and Electrons d d exp i k r V r r dr 2 BORN approximation: • Assumes initial wave is also spherical • Scattering potential gives weak interactions d d q k0 k exp i k r V r exp i k r dr 2 0 d d V r exp i q r dr 2 Scattered intensity is proportional to the Fourier Transform of the scattering potential Scattering from Crystal As a crystal is a periodic repetition of atoms in 3D we can formulate the scattering amplitude from a crystal by expanding the scattering from a single atom in a Fourier series over the entire crystal f (r ) exp iq r dV V A(q) f T Atomic Structure Factor j j q exp i q T r j Real Lattice Vector: T=ha+kb+lc The Structure Factor Describes the Intensity of the diffracted beams in reciprocal space exp i q r exp i hu kv lw 2 j j j hkl are the diffraction planes, uvw are fractional co-ordinates within the unit cell If the basis is the same, and has a scattering factor, (f=1), the structure factors for the hkl reflections can be found Weight phase hkl The Form Factor Describes the distribution of the diffracted beams in reciprocal space The summation is over the entire crystal which is a parallelepiped of sides: L q N1a N2b N3c N1 exp i q T exp n i q a 1 T n1 1 N3 N2 exp n i q b exp n i q c 2 n2 1 3 n3 1 The Form Factor Measures the translational symmetry of the lattice q d s Redefine q: L q Deviation from reciprocal lattice point located at d* Ni exp i n s i i i 1,2,3 ni i 1,2,3 sin Ni si sin si sin Ni si i 1,2,3 sj 5 2.5x10 5 2 2.0x10 [L(s1)] The Form Factor has low intensity unless q is a reciprocal lattice vector associated with a reciprocal lattice point 1.5x10 5 1.0x10 5 0.5x10 0 -0.02 N=2,500; FWHM-1.3” N=500 5 -0.01 0 0.01 0.02 Deviation parameter, s1 (radians) The Form Factor L q sin Ni si sj 5 100 2.5x10 80 2.0x10 2 5 60 [L(s1)] [L(s1)] 2 i 1,2,3 40 20 0 5 1.5x10 5 1.0x10 5 0.5x10 -0.6 -0.3 0 0.3 0.6 0 -0.02 -0.01 0 0.01 0.02 Deviation parameter, s1 (radians) Deviation parameter, s1 (radians) N=10 N=500 The square of the Form Factor in one dimension Scattering in Reciprocal Space exp i q T A q f j q exp i q r j j T Peak positions and intensity tell us about the structure: POSITION OF PEAK PERIODICITY WITHIN SAMPLE WIDTH OF PEAK EXTENT OF PERIODICITY INTENSITY OF PEAK POSITION OF ATOMS IN BASIS Powder Diffraction It is impossible to grow some materials in a single crystal form or we wish to study materials in a dynamic process. Powder Techniques Allows a wider range of materials to be studied under different sample conditions 1. Inductance Furnace 290 – 1500K 2. Closed Cycle Cryostat 10 – 290K 3. High Pressure Up-to 5 million Atmospheres • Phase changes as a function of Temp and Pressure • Phase identification Search and Match Powder Diffraction often used to identify phases Cheap, rapid, non-destructive and only small quantity of sample Monochromatic xrays Intensity Diffractometer High Dynamic range detector 2 Angle JCPDS Powder Diffraction File lists materials (>50,000) in order of their dspacings and 6 strongest reflections OK for mixtures of up-to 4 components and 1% accuracy Single Crystal Diffraction 2d hkl sin hkl Monochromatic radiation so sample needs to moved to the Bragg condition…. Angular resolution is the Darwin width of analyser crystal (Typically 10-20”) Detailed Lateral Information obtained XMaS Beamline - ESRF Strain Peak positions defined by the lattice parameters: L q N1 exp i q n i i a, b, c n1 1 Strain is an extension or compression of the lattice, dhkl dhkl Results in a systematic shift of all the peaks Ho Thin Films Intensity (arb. units) XRD measured as a function of temperature 10 4 10 2 10 0 10 -2 10 -4 20 T=294K T=244K T=194K T=94K T=144K T=42K T=10K T=300K 40 60 Scattering Angle () 80 100 Ho Thin Films Substrate and Ho film follow have different behaviour Intensity (arb. units) 1000 T=294K T=244K T=194K T=144K T=94K T=42K T=10K 100 10 1 30 32 Scattering Angle () 34 36 Whole film refinement Peak Broadening Diffraction peaks can also be broadened in qz by: 1. Grain Size 2. Micro-Strains OR Both The crystal is made up of particulates which all act as perfect but small crystals Number of planes sampled is finite Recall form factor: L q i a,b,c sin Ni si si Scherrer Equation Size 2 D cos B NixMn3-xO4+ (400 Peak) As Grown at 200ºC AFM images (1200 x 1200 nm) 1.0 Annealed at 800ºC Intensity 0.8 800C 850C 900C 650C 700C 750C 0.6 0.4 0.2 0 -1.0 400 -0.5 0 0.5 2 450nm thick films 1.0 Peak Broadening Diffraction peaks can also be broadened in qz by: 1. Grain Size 2. Micro-Strains OR Both The crystal has a distribution of inter-planar spacings dhkl ±dhkl. Diffraction over a range, , of angles Differentiate Bragg’s Law: d d Strain 2 2 tan B Width in radians Strain Bragg angle Peak Broadening Diffraction peaks can also be broadened in qz by: 1. Grain Size 2. Micro-Strains OR Both Total Broadening in 2 is sum of Strain and Size: Rearrange Total 2 Williamson-Hall plot D cos B 2 tan B hkl 2 cos Bhkl 2 sin Bhkl y mx c D Powder Diffraction 0.25 2 0.20 400 0.10 222 311 400 0 220 0 10 20 2 2 30 2 (h +k +l ) 200 0.008 100 0 30 40 50 60 Width * cos(B) Intensity (arb. units) 0.05 300 333 422 0.15 2 sin (B) Powder of Nickel Manganite CUBIC Structure 2 y=(1.541 /(4*a ))x a=8.348 ± 0.0036 0.007 y=((1.541/d))+(2s)x Grain ± 19.5 Grain sizeSize=299 = 30±2nm Strain = 0.005±0.001 a/aDispersion = 0.005 ± 0.001 0.006 Detector Angle (°) 0.005 0.05 0.10 0.15 sin(B) 0.20 0.25 Cubic-Tetragonal Distortions CUBIC ac TETRAGONAL ac High Temperature Powder XRD Intensity (counts) Intensity (counts) 0.4BiSCO3 - 0.6PbTiO3 (K. Datta) Tetragonal → Cubic phase transition 30000 30000 20000 10000 20000 30.8 30.9 31.0 31.1 31.2 31.3 31.4 31.5 31.6 31.7 2Theta (°) 10000 0 25 30 35 Courtesy, D. Walker and K. Datta University of Warwick 40 45 50 55 60 65 70 2Theta (°) CsCoPO4 Variable temperature powder X-ray diffraction data show a marked change in the pattern at 170 °C. Dr. Mark T. Weller, Department of Chemistry, University of Southampton, www.rsc.org/ej/dt/2000/b003800h/ Sn in a Silica Matrix 1. What form of tin 2. Particle size 3. Strain 4. Melting Temperature Eutectic’s Following Structural Changes • Consider Cu/Ni with 35 wt.% Ni T(°C) L (liquid) 130 0 L: 35 wt% Ni a: 46 wt% Ni 32 43 L: 32 wt% Ni 36 120 0 a: 43 wt% Ni E L: 24 wt% Ni a: 36 wt% Ni a (solid) D. E. Solid B C D 24 C. Cu-Ni system A A. Liquid B. Mixed Phase L: 35wt%Ni 110 0 20 Figure adapted from Callister, Materials science and engineering, 7th Ed. 30 35 Co 40 50 wt% Ni USE LEVER RULE Cored Samples L α+L α Issues: Lattice Parameter Particle Size Strain Dispersion hkl 2 cos Bhkl 2 sin Bhkl D NiCr Structural Changes? Fcc: hkl are either all odd or all even. Bcc: sum of hkl must be even.