Thermodynamics
• Is a policeman
• Eliminates the impossible
• Identifies the improbable
• Simplifies your life
– Links what can be measured to what you really want to know
– Limits the number of independent variables
• Is a macroscopic formalism that links to microscopic
insights
– Quantum mechanics  statistical mechanics  thermodynamics
– Links directly to structure and dynamics in solids
Fundamental Concepts of
Thermodynamics
•
•
•
•
•
•
•
First, second, and third law
Entropy
Heat capacity, enthalpy
Reaction enthalpies and thermochemical cycles
Phase transitions
Calorimetry
Pressure as a variable
Why I Count Calories
for a Living
• They are fascinating
– Energetics whisper secrets of the strength of chemical bonds
– Entropies sing of vibrating atoms, moving electrons, and
structural disorder
– Systematics have predictive power
• They pay
– Thermodynamic data are essential to good materials
processing
– Environmental science needs thermodynamics, both for issues
of stability and as a starting point for kinetics
– Mineralogy, petrology, and deep Earth geophysics need
thermodynamic data.
From thermochemical data
one calculates
• Entropies and free energies
• Solubilities
• Phase diagrams
• Petrologic and geochemical
processes
• Materials synthesis and
compatibility
What is a Phase Transition?
• A change in structure usually accompanied by
a change in symmetry
• May occur at a single temperature or spread
over a temperature range
• May be reversible and non-quenchable,
reversible and quenchable, reversible with
hysteresis, or irreversible
• May involve positional, magnetic, or
electronic disordering
Phase Transitions
Example – Heat Capacity and Entropy of Fe2SiO4 Olivine and
Spinel Polymorphs
Navrotsky et al. Proc. Natl. Acad. Sci., 104, 9187-9191 (2007)
Ideal Cubic Perovskite ABO3
Distortions in Perovskites:
octahedral tilting
Why Distortions?
• Ideal cubic structure defines bond distance in
12-coordinated A-site and 6-coordinated B-site
simultaneously
• Many A-site ions are “too small” and prefer
lower coordination
• Octahedra tilt, also sometimes B-cations offcenter ===> a whole family of related structures
of varying symmetry, often with only very small
distortions
• Increasing temperature leads to phase
transitions to higher symmetry structures
Structural Studies: Xrays vs. Neutrons
Xrays
• Lab Xrays easiest but of limited resolution
• Synchrotron Xrays - higher intensity, resolution
Neutrons
• Detect light elements (e.g. H,D), distinguish elements with similar
atomic number
• Accurate crystallographic parameters, see small changes in
symmetry
• See magnetic transitions and ordering
• Obtain vibrational density of states and use to calculate entropy
Whole pattern fitting - Rietveld methods
In situ studies
CaTiO3 Perovskite Transitions
• In situ neutron diffraction and Rietveld
refinement at 296-1729 K and heat capacity
measurements
• Orthorhombic (Pbnm) – tetragonal
• (I4/mcm) reversible transition at 1513 +13 K
and tetragonal (I4/mcm) – cubic (Fm3m)
reversible transition at 1635 +2 K
CaTiO3: Axial Ratios by in situ Neutron Diffraction
M. Yashima and Y. Ali, Solid State Ionics 180, 120-126 (2009)
CaTiO3: Heat Capacity
P. Guyot et al. Phys. Chem. Min. 20, 141-146 (1993)
Thermal Analysis and
Scanning Calorimetry
• Measure a signal (mass, heat, evolved
gas, lemgth, X-ray pattern) at a variable
heating (cooing) rate
• Systems
– Room temp to 600 oC, common
– 600-1500 oC, less common but we have
– 1500-2400 oC, uncommon but we have
Calorimetry above 1500 oC
Example: HfO2
Unit cell volume, Å
3
Mon (P21/c)
144
Cub (Fm3m)
Tetr (P42/nmc)
ZrO2
HfO2
140
J. Wang et al. J. Mat. Sci. 92, 27(20) 5397
400
800
1200
1600
2000
Temperature, °C
Experiment: HfO2_calibr_probe_3rd_heating
Displacement/µm
SETSYS Evolution - 2400
Probe:
Graphite
Carrier gas:
Length (mm): 2.502 mm
Procedure: 1500>2300 (Zone 1)
0.0
TMA on HfO2
Heating
-5.0
-10.0
Delta : -0.71 %
Delta : 1.01 %
-15.0
-20.0
Cooling
-25.0
1800
1850
1900
Furnace temperature /°C
TMA traces of HfO2 (1.5 % Zr) in Ar flow. (HfO2
pellet L 2.5 mm Ø 5 mm sintered at 1700 °C for 2
hours). Heating rate 10 °C/min. 5 gram load. \
SETSYS Evolution - 2400
Experiment: 3rd_HfO2_W_10_1950
Crucible:W 85 µl
Carrier gas:
Procedure: 20>2000 (Zone 3)
Molar mass: 210.49
Mass (mg): 205.71
DTA signal, µV
-bl_cooling - 4 - 3r d_HfO2_W_10_1950/µV
-bl 2/µV
Exo
2
2.5
Exo
1
1.5
Peak :1762.13 °C
Onset Point :1771.45 °C
Enthalpy /µV.s : -126.0198 (Exothermic effect)
Cooling
0
-1
0.5
Heating
-0.5
Peak :1813.08 °C
Onset Point :1802.64 °C
Enthalpy /µV.s : 127.6076 (Endothermic effect)
1700
1750
1800
Sample temperature/°C
DTA traces of HfO2 (99.95% Aldrich) in Ar flow.
Heating rate 10 °/min, cooling rate 20 °/min.
Baseline correction applied.
IMPROVEMENTS: Laser-melted La2O3 sphere, welded tungsten crucible with La2O3
sample, and transparent solidified La2O3 sample after melting in thermal analyzer.
The crucibles and samples are positioned on the tungsten sensor plate.
Temperature ,°C
H
X
cubic
CN6
cubic CN6
C
B
A
2000
cubic CN6
hex CN7
1600
M. Zinkevich, Progress in Mat Sci 52, 597 (2007)
L
2400
mon CN7
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Lanthanum oxide –
phase transitions and fusion
Reversible
DG = DH - T DS
DH, DS > 0 for
low  high
Reversible
with hysteresis
Thermodynamically
reversible, but
kinetically hindered
Thermodynamically
irreversible (initial low T
phase metastable)
Enthalpy of formation
convert material chemically to a state with known enthalpy usually a
solution)
• Thermochemical Cycle
• A => solution DH1
• B => same solution DH2
• A => B DH3= DH1 - DH2
• The task is to find a reaction scheme and solvent that lets
you do this accurately
–
–
–
–
A can be elements, B compound
A can be binary oxides, B ternary compound
A can be end-members, B solid solution or alloy
A and B can be polymorphs, materials with different orderidisorder, different surface areas
Solution Calorimetry: Solvents
• Near room temperature
– Water, aqueous acid of base
– Hydrofluoric acid
– Organic solvents
• At high temperature
– Molten metals, e.g. Sn
– Molten salts, e.g. nitrate or chloride eutectics
– Molten oxides
High Temperature Oxide Melt
Solution Calorimetry
• Dissolve oxide samples (5-15 mg) in a
molten oxide solvent (20 g) at to form a
dilute solution
• Difference in heat of solution of reactants
and products gives heat of reaction
• Oxidative reactions for nitrides, sulfides,
selenides, carbides
• Needed for ceramic materials which do
not dissolve in aqueous solvents
Solvents and Systems
• Lead borate (2PbO-4B2O3, sodium
molybdate (3Na2O-4MoO3), alkali borate
• Oxides dissolve
• H2O and CO2 evolve as gases
• Nitride oxidized to evolved N2
• Sulfide oxidized to dissolved sulfate
Background: High-temperatureoxide melt solution calorimetry
Custom-built Calvet twin microcalorimeter
www.mindat.org
drop tube
silica glass
liner
Drop solution setup
sample
25 °C
alumina plug
platinum
bubbling tube
platinum
crucible
Inconel block
Thermopiles
solvent
702 °C
Heaters
Insulation
µV
Time
Nanovoltmeter
Commercial Setaram AlexSYS Calorimeter
High-Temperature Calorimetry
TTD
SOL
DS
25oC
25oC
25oC
DHDS
DHSOL
DHTTD
700oC
DHTTD  
700C
25C
700oC
CP dT
700oC
DHDS  DHTTD  DHSOL
Thermochemical Cycles Perovskites
• 1. AO(xl, 298K) = AO(dissolved, 973K)
• 2. BO2(xl, 298K) =BO2(dissolved, 973K)
• 3. ABO3(xl, 298K) =
ABO3(dissolved, 973K)
______________________________
• 4. AO(xl, 298K) + BO2(xl, 298K) =
ABO3(xl, 298K) , H4 = H1 + H2 – H3
• Tolerance factor t = dAO/1.414dBO
Gas Adsorption Calorimetry
• Combine sensitive microcalorimeter with
automated gas dosing system
• Measure heat of adsorption and
adsorption isotherm simultaneously
• Apply to high surface area and
microporous materials
Volumetric dosing system
Calvet-type twin
microcalorimeter
Micromeritics ASAP2020
Setaram DSC 111
sample
thermopiles
reference
H2O
to voltmeter
and amplifier
Differential (a) and integral (b) heats of H2O
adsorption for anatase with surface area of 90, 200
and 240 m2/g and rutile of 61 and 103 m2/g
(Levchenko et al. 2006).
The Peter A. Rock
Thermochemistry Laboratory
• A unique suite of equipment and expertise
• Can design a calorimetric experiment to suit
almost any material and problem
High Pressure Phase Transitions
• G = E + PV –TS
• At a given P,T dG = dE + PdV – TdS
• Increasing T, DS has to be positive but DV
can have either sign
• Increasing P, DV has to be negative but DS
can have either sign
• Clausius-Clapeyron Equation
• (dP/dT)equil = DS/DV
Clausius-Clapeyron Equation
(dP/dT)equil = DS/DV
• For solid-solid reactions, to a good approximation,
P-T slope is constant
• For melting, the liquid is often more compressible
than the solid, so phase boundary is curved,
sometimes even showing a maximum
• For dehydration or vaporization, strong curvature is
seen because gas is much more compressible than
condensed phase
H2O Phase Diagram
Al2SiO5 phase diagram
Depth (km)
Upper mantle
Lower mantle
Pressure (GPa)
Outer core
Inner core
Concentric shells of different
phase assemblages with sharp
discontinuities between them
Olivine-spinelloid-spinel at 400
km
Spinel- perovskite + periclase at
670 km
Core-mantle boundary
Phase diagram of Iron
40
(hcp)
Pressure, GPa
30
(fcc)
20
10
(bcc)
(bcc)
liquid
0
0
500
1000
1500
Temperature, K
2000
SILICATE PEROVSKITE IN THE
EARTH’S MANTLE
• Mantle composition between MgSiO3 and
Mg2SiO4 projected on the MgO - SiO2
binary system
• Contains a few percent Fe, Al, Ca, other
minor elements, +/- H2O
• An MgSiO3 - rich perovskite phase
predominates at P > 25 GPa (depth > 670
km)
Mg2SiO4-Fe2SiO4 phase diagram
M2SiO4 transitions
olivine – spinelloid - spinel
30
Pressure (GPa)
PV
20
IL
BP+ST
BETA+ST
GT
PX
10
1000
1500
LIQ
2000
2500
3000
3500
Temperature, K
Phase relations in MgSiO3 composition (PX – pyroxene,
BETA -wadsleyite, LIQ –liquid, SP –spinel, ST –stishovite,
IL – ilmenite, PV -perovskite (After Fei Saxena, Alexandra
Navrotsky, 1990)