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

2. 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.

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

4. 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

5. 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.

6. 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.

8. First Order h
Discontinuity
Transitions in liquid crystals

9. 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

10. Phase Transitions in BaTiO3
[001]
[011]
[111]
PHONONS
Tcf
, Introduction to Solid State Physics, 6th Ed., Kittel

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

12. 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

13. 2nd Order – FM transition

14. 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

15. 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).

16. Metastable State

17. 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

18. 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

19. 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.

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

21. HPHT Synthetic Diamonds

22. 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

23. Calcite Polymorphs: CaCO3
Aragonite
Vaterite
Trigonal
Orthorhombic
Hexagonal

24. Silica Polymorphs: SiO2
Amorphous
Tetragonal
Rhombohedral

25. Paracetamol

26. 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

27. 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

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

29. 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 (2003)

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

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