the muppet’s guide to: the structure and dynamics of solids 4. phase transitions & crystal...
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The Muppet’s Guide to:The Structure and Dynamics of Solids
4. Phase Transitions & Crystal Growth
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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
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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
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Phase Transitions in BaTiO3
Tcf
[001][011][111]PHONONS
http://www.camsoft.co.kr , Introduction to Solid State Physics, 6th Ed., Kittel
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KxNa1-xNbO3
monoclinic
tetragonal
cubic
Dan Baker, PhD Thesis 2009, University of Warwick
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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
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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
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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
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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).
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Metastable State
E
Potential Energy
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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
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Allotropes - Carbon
GraphiteIn-plane sp2
& van der Waals
Diamond3D - sp3
Fullerenessp2:
Hexagons & pentagons
Carbon structures are stable under different bonding configurations
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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
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Allotropes - Sn
T<13.2°C
Tin Disease – Buttons, Cathedral organ pipes
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HPHT Synthetic Diamonds
http://www.darmann.com/abrasives.htmlhttp://www.diamondlab.org/80-hpht_synthesis.htm
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Polymorphs - ZnS
Zincblende
Cubic, Diamond like
Wurtzite
Hexagonal, hcp like, different number of 2nd nearest neighbours
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Calcite Polymorphs: CaCO3
http://www.u-bourgogne.fr/BIOGEOSCIENCE/images/stories/actu/calcite_polymorphs.jpg
Calcite Aragonite Vaterite
OrthorhombicTrigonal Hexagonal
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Tetragonal RhombohedralAmorphous
Silica Polymorphs: SiO2
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Paracetamol
http://research.jonathanburley.com/files/Paracetamol_for_Web_Page.png
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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
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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
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-adatom
Crystal growth happens at steps
(001)(111)
(101)
(011)
5μm
CdTe in N2
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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)
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-Impurity species
-adatom
Blocking of step flow by impurities
Cohesive energy, Ec
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Growth from SolutionEvaporation of the solvent causes super-saturation and hence the solute comes out of solution