The Structure and Dynamics of Solids The Muppet’s Guide to: The Structure and Dynamics of Solids 4. Phase Transitions & Crystal Growth

Phase Transitions The change from one state (or phase) or another is associated with a phase transition and a critical point. In this example it is a structural phase transition that occurs abruptly at a critical temperature, Tc.

Phase Transition ORDERED DISORDERED At the phase transition the Gibbs free energy of the two states is identical

Describing Phase Transitions Ordering Parameter, h: This is the parameter which shows a change at the transition temperature or pressure. Order parameter is a derivative of the Gibbs free energy with respect to a thermodynamic variable Chemical potential Applied Field

1st Order Phase Transitions Ehrenfest classification: Discontinuity in the 1st derivative of Gibbs free energy Transitions that exhibit LATENT HEAT – Energy must be supplied to change the local environment. This results in no temperature change.

Boiling Water First-order transitions are associated with "mixed-phase regimes" Some parts of the system have completed the transition whilst others have not. Water does not instantly change from liquid to gas. Instead it forms a mixture of water and steam bubbles. Similarly it does not instantly freeze.

First Order h Discontinuity Transitions in liquid crystals

1st order Phase Transitions… BaTiO3: Volume change at Tc Thus expect first order phase change with discontinuity in Ps at Tc LaTaO3 shows second order phase transition

Phase Transitions in BaTiO3 [001] [011] [111] PHONONS Tcf http://www.camsoft.co.kr , Introduction to Solid State Physics, 6th Ed., Kittel

KxNa1-xNbO3 monoclinic cubic tetragonal Dan Baker, PhD Thesis 2009, University of Warwick

2nd Order Phase Transitions Ehrenfest classification: Discontinuity in the 2nd derivative of Gibbs free energy Transitions that exhibit NO latent heat correspond to divergences in the susceptibility, an infinite correlation length, and a power law decay of correlations

2nd Order – FM transition

Magnetic Ordering Critical Point Critical Exponents: h Critical Point Critical Exponents: b=0.326(1) MFA, b=0.5 g=1.2378(6) MFA, g=1

Phase Transitions A phase change can occur for thermodynamic reasons: - Ordering of ground state to reduce H BUT Kinetics may produce a non-thermodynamic (metastable) state during growth. Thermodynamically this should decay to the ground state (eventually).

Metastable State

Final Structures: Thermo vs. Kinetics Energy to remain on surface, Ea Energy to diffuse on surface, Ed Cohesive energy, Ec Strain Energy – modification of U

Carbon structures are stable under different bonding configurations Allotropes - Carbon Graphite In-plane sp2 & van der Waals Diamond 3D - sp3 Fullerenes sp2: Hexagons & pentagons Carbon structures are stable under different bonding configurations

Activation Energy Activation Energy Thermodynamically lowest energy state is the most stable, but must overcome the energy barrier to reach it. Multiple structures of the same material with an energy barrier separating the two.

Tin Disease – Buttons, Cathedral organ pipes Allotropes - Sn T<13.2°C Tin Disease – Buttons, Cathedral organ pipes

HPHT Synthetic Diamonds http://www.darmann.com/abrasives.html http://www.diamondlab.org/80-hpht_synthesis.htm

Hexagonal, hcp like, different number of 2nd nearest neighbours Polymorphs - ZnS Zincblende Cubic, Diamond like Wurtzite Hexagonal, hcp like, different number of 2nd nearest neighbours

Calcite Polymorphs: CaCO3 Aragonite Vaterite Trigonal Orthorhombic Hexagonal http://www.u-bourgogne.fr/BIOGEOSCIENCE/images/stories/actu/calcite_polymorphs.jpg

Silica Polymorphs: SiO2 Amorphous Tetragonal Rhombohedral

Paracetamol http://research.jonathanburley.com/files/Paracetamol_for_Web_Page.png

MnSb – polymorph structures n-MnSb(hexagonal) Niccolite (P63/mmc) c-MnSb (cubic) Zincblende (F-43m) w-MnSb (hexagonal) Wurtzite (P63mc) Mn Sb MnSb(0001) GaAs (111)B MnSb on GaAs(111)B MnSb on GaAs(111)A Cubic MnSb GaAs

Crystal Growth Different methods needed for different materials All growth processes require conditions that promote formation of a crystal such as: Condensing from a supersaturated solution Freezing from a melt Evaporation Different methods needed for different materials

Crystal growth happens at steps (001) (111) (101) (011) 5μm CdTe in N2 -adatom

Ce0.5Zr0.5O2 Figures adapted from W.D. McAllister, Materials Science and Engineering, 7th edition, Wiley. Figures after M. Boudart Kinetics of Heterogeneous Catalytic Reactions, Princeton University Press,1984 and W.J. Stark et al. Chem Comm. 588-589 (2003)

Blocking of step flow by impurities Cohesive energy, Ec -adatom -Impurity species

Growth from Solution Evaporation of the solvent causes super-saturation and hence the solute comes out of solution