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

4. Phase Transitions & Crystal Growth

1st Order Phase TransitionsEhrenfest 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.

Discontinuity in

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

Tcf

[001][011][111]PHONONS

http://www.camsoft.co.kr , Introduction to Solid State Physics, 6th Ed., Kittel

KxNa1-xNbO3

monoclinic

tetragonal

cubic

Dan Baker, PhD Thesis 2009, University of Warwick

2nd Order Phase TransitionsEhrenfest 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

,Discontinuity inT P

2nd Order – FM transition

G

Magnetisation mH

0

0.5

1.0

0 50 100 150 200 250

Temperature (K)

Ma

gn

etis

atio

n

0

0.5

1.0

1.5

2.0

0 100 200 300 400 500

Temperature (K)

Su

sce

ptib

ility

,

1

CT T 0.5

CM T T

2

2

dm GSusceptibility

dH H

250CT K

Magnetic Ordering

CT T

CM T T

Critical Exponents:

b=0.326(1) MFA, b=0.5

g=1.2378(6) MFA, g=1

Critical Point

h

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

E

Potential Energy

Final Structures:Thermo vs. Kinetics

phase phase

Tc

Temperature

Fre

e E

nerg

y

Energy to remain on surface, Ea

Energy to diffuse on surface, Ed

Cohesive energy, Ec

Strain Energy – modification of U

, ,G U P V PV TS

Allotropes - Carbon

GraphiteIn-plane sp2

& van der Waals

Diamond3D - sp3

Fullerenessp2:

Hexagons & pentagons

Carbon structures are stable under different bonding configurations

Activation Energy

exp A

B

ED A k T

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.

Activation Energy

Allotropes - Sn

T<13.2°C

Tin Disease – Buttons, Cathedral organ pipes

HPHT Synthetic Diamonds

http://www.darmann.com/abrasives.htmlhttp://www.diamondlab.org/80-hpht_synthesis.htm

Polymorphs - ZnS

Zincblende

Cubic, Diamond like

Wurtzite

Hexagonal, hcp like, different number of 2nd nearest neighbours

Calcite Polymorphs: CaCO3

http://www.u-bourgogne.fr/BIOGEOSCIENCE/images/stories/actu/calcite_polymorphs.jpg

Calcite Aragonite Vaterite

OrthorhombicTrigonal Hexagonal

Tetragonal RhombohedralAmorphous

Silica Polymorphs: SiO2

Paracetamol

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

MnSb – polymorph structuresn-MnSb(hexagonal)

Niccolite (P63/mmc)

c-MnSb (cubic)

Zincblende (F-43m)

w-MnSb (hexagonal)

Wurtzite (P63mc)

MnSb

MnSb on GaAs(111)A

Cubic MnSb

GaAs

MnSb(0001)

GaAs (111)B

MnSb on GaAs(111)B

Crystal Growth

• 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

-adatom

Crystal growth happens at steps

(001)(111)

(101)

(011)

5μm

CdTe in N2

Ce0.5Zr0.5O2

Figures adapted from W.D. McAllister, Materials Science and Engineering, 7 th 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)

-Impurity species

-adatom

Blocking of step flow by impurities

Cohesive energy, Ec

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