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

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…. 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 C x C CoCo t = 0 t = t For a semi-infinite sample the solution to Ficks’ Law gives an error function distribution whose width increases with time

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

wt% Ni L (liquid)  (solid) L +  L +  T(°C) A 35 C o L: 35wt%Ni Cu-Ni system C o = 35 wt%Ni. Slow Cooling in a Cu-Ni Binary  :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. Enough time is allowed at each temperature change for atomic diffusion to occur. – Thermodynamic ground state Each phase is homogeneous

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

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

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

Min. melting T E Binary-Eutectic Systems Eutectic transition L(C E )  (C  E ) +  (C  E ) T E : No liquid below T E T E, Eutectic temperature, 779°C C E, eutectic composition, 71.9wt.% The Eutectic point Cu-Ag system L (liquid)  L +  L+    CoCo wt% Ag in Cu/Ag alloy T(°C) CECE TETE C  E =8.0 C E =71.9C  E = °C Figure adapted from Callister, Materials science and engineering, 7 th Ed. Any other composition, Liquid transforms to a mixed L+solid phase E

L+  L+   +  200 T(°C) 18.3 C, wt% Sn L (liquid)  183°C  For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find... --the phases present: Pb-Sn system Pb-Sn (Solder) Eutectic System (1)  +  --compositions of phases: C O = 40 wt% Sn --the relative amount of each phase: CoCo 11 CC 99 CC S R C  = 11 wt% Sn C  = 99 wt% Sn W  = C  - C O C  - C  = = = 67 wt% S R+SR+S = W  = C O - C  C  - C  = R R+SR+S = = 33 wt% = Figure adapted from Callister, Materials science and engineering, 7 th Ed.

2 wt% Sn < C o < 18.3 wt% Sn Result:  Initially liquid → liquid +   then  alone   finally two phases   poly-crystal  fine  -phase inclusions Microstructures in Eutectic Systems: II Pb-Sn system L +  200 T(°C) CoCo,wt% Sn CoCo L  30  +  400 (sol. limit at T E ) TETE 2 (sol. limit at T room ) L  L: C o wt% Sn    : C o wt% Sn Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of  and  crystals. Microstructures in Eutectic Systems: C o =C E 160  m Micrograph of Pb-Sn eutectic microstructure Pb-Sn system LL  200 T(°C) C, wt% Sn L   L+  183°C 40 TETE 18.3  : 18.3 wt%Sn 97.8  : 97.8 wt% Sn CECE 61.9 L: C o wt% Sn Figures adapted from Callister, Materials science and engineering, 7 th Ed.

Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of  and  crystals. Microstructures in Eutectic Systems: C o =C E Pb-Sn system LL  200 T(°C) C, wt% Sn L   L+  183°C 40 TETE 18.3  : 18.3 wt%Sn 97.8  : 97.8 wt% Sn CECE 61.9 L: C o wt% Sn Figures adapted from Callister, Materials science and engineering, 7 th Ed. Pb rich Sn Rich

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

18.3 wt% Sn < C o < 61.9 wt% Sn Result:  crystals and a eutectic microstructure Microstructures IV SR WLWL = (1-W  ) = 50 wt% C  = 18.3 wt% Sn CLCL = 61.9 wt% Sn S R +S W  = = 50 wt% Just above T E : Pb-Sn system L+  200 T(°C) C o, wt% Sn L   L+  40  +  TETE L: C o wt% Sn L  L  Figure adapted from Callister, Materials science and engineering, 7 th Ed.

18.3 wt% Sn < C o < 61.9 wt% Sn Result:  crystals and a eutectic microstructure Microstructures IV SR 97.8 S R Primary,  Eutectic,  Eutectic,  Just below T E : C  = 18.3 wt% Sn C  = 97.8 wt% Sn S R +S W  = = 73 wt% W  = 27 wt% Pb-Sn system L+  200 T(°C) C o, wt% Sn L   L+  40  +  TETE L: C o wt% Sn L  L  Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Intermetallic Compounds Mg 2 Pb Note: intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact. Figure adapted from Callister, Materials science and engineering, 7 th Ed.  phase: Mostly Mg  phase: Mostly Lead

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

Iron-Carbon (Fe-C) Phase Diagram 2 important points -Eutectoid (B):  + Fe 3 C -Eutectic (A): L  + Fe 3 C Fe 3 C (cementite) L  (austenite)  +L+L  +Fe 3 C   +  L+Fe 3 C  (Fe) C o, wt% C 1148°C T(°C)  727°C = T eutectoid A SR 4.30 Result: Pearlite = alternating layers of  and Fe 3 C phases 120  m    RS 0.76 C eutectoid B Fe 3 C (cementite-hard)  (ferrite-soft) Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Iron-Carbon

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 meta- stable 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: Photons‘Massive’ objects

Xavier the X-ray E x (keV)=1.2398/ (nm) Speed of Light Planck’s constant Wavelength Elastic scattering as E x >>k B T

Norbert the Neutron E n (meV)=0.8178/ 2 (nm) De Broglie equation: mass velocity Kinetic Energy: Strong inelastic scattering as E n ~k B T

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: mass velocity E e 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 1. Absorption 2. Refraction/Reflection 3. Scattering Diffraction Englebert Xavier Norbert

Crystals are 2D with planes separated by d hkl. There will only be constructive interference when  =  =  - i.e. the reflection condition. 

Basic Scattering Theory The number of scattered particles per second is defined using the standard expression Unit solid angle Differential cross-section Defined using Fermi’s Golden Rule

Spherical Scattered Wavefield Scattering Potential Incident Wavefield Different for X-rays, Neutrons and Electrons

BORN approximation: Assumes initial wave is also spherical Scattering potential gives weak interactions 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 Atomic Structure Factor Real Lattice Vector: T=ha+kb+lc

The Structure Factor Describes the Intensity of the diffracted beams in reciprocal space 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

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:

The Form Factor Measures the translational symmetry of the lattice The Form Factor has low intensity unless q is a reciprocal lattice vector associated with a reciprocal lattice point N=2,500; FWHM-1.3” N=500 Deviation from reciprocal lattice point located at d* Redefine q:

The Form Factor The square of the Form Factor in one dimension N=10N=500

Scattering in Reciprocal Space 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 JCPDS Powder Diffraction File lists materials (>50,000) in order of their d- spacings and 6 strongest reflections OK for mixtures of up-to 4 components and 1% accuracy  Monochromatic x- rays  Diffractometer  High Dynamic range detector

Single Crystal Diffraction 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: Strain is an extension or compression of the lattice, Results in a systematic shift of all the peaks

Ho Thin Films XRD measured as a function of temperature

Ho Thin Films Substrate and Ho film follow have different behaviour

Whole film refinement

Peak Broadening Diffraction peaks can also be broadened in q z 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: Scherrer Equation

Ni x Mn 3-x O 4+  (400 Peak) As Grown at 200ºC AFM images (1200 x 1200 nm) 400  450nm thick films Annealed at 800ºC

Peak Broadening Diffraction peaks can also be broadened in q z by: 1.Grain Size 2. Micro-Strains OR Both The crystal has a distribution of inter-planar spacings d hkl ±  d hkl. Diffraction over a range,  of angles Differentiate Bragg’s Law: Width in radians Strain Bragg angle

Peak Broadening Diffraction peaks can also be broadened in q z by: 1.Grain Size 2. Micro-Strains OR Both Total Broadening in 2  is sum of Strain and Size: Rearrange Williamson-Hall plot

Powder Diffraction Powder of Nickel Manganite CUBIC Structure Grain size = 30±2nm Strain Dispersion = 0.005±0.001

Cubic-Tetragonal Distortions CUBIC TETRAGONAL

High Temperature Powder XRD 0.4BiSCO PbTiO 3 (K. Datta) Tetragonal → Cubic phase transition Courtesy, D. Walker and K. Datta University of Warwick

CsCoPO 4 Dr. Mark T. Weller, Department of Chemistry, University of Southampton, Variable temperature powder X-ray diffraction data show a marked change in the pattern at 170 °C.

Sn in a Silica Matrix 1.What form of tin 2.Particle size 3.Strain 4.Melting Temperature

Eutectic’s

wt% Ni L (liquid)  (solid) L +  L +  T(°C) A 35 C o L: 35wt%Ni Cu-Ni system Consider Cu/Ni with 35 wt.% Ni Following Structural Changes  :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 A.Liquid B.Mixed Phase C. D. E. Solid

Cored Samples α L α + L Issues: Lattice Parameter Particle Size Strain Dispersion

NiCr Structural Changes? Fcc: hkl are either all odd or all even. Bcc: sum of hkl must be even.