Thermal Behavior – II
Free Energy & Phase Diagrams
[partly based on Chapter 7, Sholl & Steckel]
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Free energy changes
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excluding vibrational entropy of solids
including just the harmonic vibrational part
including the quasi-harmonic/anharmonic contribution
Case studies
Also see:
The Free Energy
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At non-zero temperatures, we need to consider the free energy
Gibbs free energy: G = E – TS + PV
Helmholtz free energy: F = E – TS
E is the ground state (zero temperature) energy which can be
computed using DFT
The entropy: S = Selec + Slattice (+ Sconfig)
Selec is the electronic contribution to entropy; easy to calculate but is
generally small (depends on the situation and our goal)
Slattice is very difficult to compute as it involves treatment of phonons
(more later)
Sconfig is the configurational entropy which can be used when
appropriate
Oxidation
• When is a metal unstable to oxidation?
• In other words, under what conditions is the following reaction
favorable: M + (x/2)O2  MOx
• … when DG is negative
 DG = GMOx – [GM + (x/2)GO2]
 DG = EMOx – EM – (x/2)GO2 – T(SMOx – SM) + P(VMOx – VM)
• If P = 0 and SMOx = SM
 DG = EMOx – EM – (x/2)GO2
 DG = EMOx – EM – x mO = [EMOx– (x/2)mO2] – EM
DFT energies
O chemical potential
(depends on T & P)
• Note that the TS and PV terms are retained for oxygen, and hence
the T & P dependence of DG is preserved
Stability of Oxides
 DG = [EMOx– (x/2)mO2] – EM
• [EMOx– (x/2)mO2] and EM may be plotted versus mO2 for
various choices of x (i.e., various types of oxides)
M3O2 not stable
M2O stable above this
chemical potential
(But what is this?)
The O Chemical Potential
[See Hill, Statistical Thermodynamics, Chapter 8 for details]
Zero K DFT energy for
isolated O2 molecule
Chemical potential at temperature T
and some reference pressure p0 (this
includes translational, rotational,
vibrational and electronic contributions,
and can be computed using statistical
mechanics without any fitting to
experimental data)
• Thus the (T, P) combination that corresponds to the
critical O chemical potential of the previous slide may be
determined
• This will provide us with a phase diagram for the
equilibrium between the metal and its oxide
Phase (Ellingham) Diagram
• Qualitatively correct,
but large quantitative
errors to be
expected
• Sources of errors
– DFT
approximations
– Neglect of the
vibrational entropy
of the solids
involved
Surface Phase Diagrams
• Ag is a good catalyst
for many reactions
involving O
• It was later realized
that it was a thin
layer of surface oxide
that was catalytically
active, rather than
pure Ag itself
• DFT work shows that
under the catalytically
active conditions a
surface oxide is
thermodynamically
favored
The Free Energy Revisited
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For the solid, G = E – T(Selec + Slattice) + Ezero-pt + PV
E is the energy we have been talking about so far. This is the total (internal)
energy of the electron-ion system
The electronic entropy Selec(T,V) can also be easily calculated
Well known statistical mechanics formulae available for Sel(T,V), which is
written in terms of the electronic “density of states”
hcpbcc transition
The Free Energy: Lattice part
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G = E – T(Selec + Slattice) + Ezero-pt + PV
Slattice is the entropy due to lattice vibrations
Before we get to lattice vibrations, let us consider “normal modes”
A simple illustration … going back to simple harmonic oscillators
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Just one mode or
frequency of vibration
Two modes or
frequencies of vibration
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r(w) is called the vibrational
density of states
What is a “mode”?
If the system is perturbed so that one of
the modes is excited, then the system
will display periodic vibration at the
frequency corresponding to that mode
If the system is arbitrarily perturbed, the
vibration that results will not be periodic
with any particular frequency, but will be
a linear combination of the modes
For N masses, we will have N modes or
frequencies
In general, as N is large (like in a solid),
we will have a continuum of
modes/frequencies, resulting in the
phonon band structure or spectrum (very
similar to electronic band structure)
Phonons
• Just like electronic states in a periodic lattice, the lattice vibrational
modes may be classified using a k-point within the 1st BZ
• Thus, if there are N atoms within a periodically repeating unit cell (in
3-d), there will be a total of 3N vibrational modes per k-point, each
with a distinct frequency wik (i = 1-3N, k = a point in the 1st BZ)
• Out of the 3N modes, 3 will be acoustic and the others will be optical
“branches”
• The low frequency (or long wavelength) portion of the acoustic
branches are essentially elastic acoustic waves which travel at the
velocity of sound in that material
• The modes of the optical branch may result in time-varying dipoles,
and hence can be excited by electromagnetic (or optical) waves
• Phonons: Quantized lattice waves
• For an interesting applet, see:
http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html
Phonons in Bulk Silicon
The Free Energy: Lattice part
(contd.)

E zero  pt  TS lattice  harmonic 

0
1

r (w )  w  kT ln (1  e
2
w / kT

d w
Vibrational density
of states

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G = E – T(Selec + Slattice) + Ezero-pt + PV
Computation of the vibrational contribution to the free energy boils
down to the determination of the phonon band structure of the
system, which is extremely computationally demanding
Note: For a single harmonic oscillator, the density of states is just
the Dirac delta function
The above picture is still not complete, as it includes only the
harmonic contribution to vibrations
The anharmonic contribution is necessary to implicitly include
thermal expansion, which will alter the vibrational contribution for
increasing temperatures
The Quasi-harmonic Approximation
• Within the harmonic approximation
– G(T) = E – T[Selec(T) + Slattice(T)] + Ezero-pt
– That is, there is no volume dependence!
– Phonon frequencies are determined at the equilibrium volume
• In the quasi-harmonic approximation, the phonon
problem is solved at several choices of the unit cell
volume, and the following free energy is evaluated
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G(V,T) = E(V) – T[Selec(V,T) + Slattice(V,T)] + Ezero-pt(V) + PV
Note the explicit dependence of G on V
For each T, the V corresponding to the minimum G is determined
This will give the T dependence of V, and hence the thermal
expansion
The Quasi-Harmonic Approximation
A Schematic
• For each choice of unit cell volume, compute G as a function of T
• Then, at each T, the V corresponding to the minimum G is
determined
Separate phonon calculation for each column
V1
Minimization
of G for each
choice of T
T1
V2
…
VN
G(V1,T1)
T2
TN
G(VN,TN)
Once the correct volume for a
given T is known, the phonon
results for that volume may be
used to determine the free energy
for that phase, and can be
compared to the free energy of
other competing phases to
determine the phase diagram
Phase transformations involving solids
Experiments
Boron Nitride
Kern et al, PRB 59, 8551 (1999)
Tin
Pavone et al, PRB 57, 10421 (1998)
The Einstein crystal
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To understand the main
aspects/reasons of solid-solid
transformations such as fccbcc,
we can make some simplifying
assumptions
We assume that the entire fcc
lattice has only one characteristic
phonon frequency (or mode)
Lets assume that the bcc lattice too
has just one phonon frequency, but
smaller than that of the fcc lattice
Lets further assume that the
energy Efcc < Ebcc
Then:
Gfcc = Efcc + 0.5ħwfcc +
kTln[1-exp(-ħwfcc/kT)]
At low T, fcc will be more stable,
however, at high T, as wbcc is
smaller than wfcc, the phonons in
bcc are easier to excite, resulting in
a larger free energy drop in bcc for
entropic reasons
DFT
expt.
DFT
expt.
Phase transformations involving melting
Phase diagram of elemental Mg
Experimental results
Experimental result
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Note: easy to consider high pressure using computational methods;
sometimes computation may be the only option
Calculation of alloy phase diagrams is considerably more complex:
composition variable(s) need to be included (makes calculation laborious) &
configurational entropy needs to be added (easy)
Extreme pressures
• Extreme geophysical pressures may be difficult to create in the lab,
but can be simulated easily
Extreme pressures – contd.
Liquid
Solid
Extreme pressures – contd.
Thermal expansion
• Can a material contract when heated?
Term Paper Guidelines
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Presentations on December 9 (Friday)
Choose topic preferably close to your research
DFT has to be a necessary major component of term paper
Actual computations are optional
Without actually doing any elaborate calculations, come up with a “case” or
a “proposal” for doing calculations that could go well with your own primary
research topic, which ideally may be pursued after this course
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Identify a specific problem, and explain why this is important
Do a thorough literature search to find out what is already known (experimentally &
computationally) – in other words, educate me and the class!
Explain the specific DFT strategies undertaken in the past, and the results and
insights that have already emerged (this should be the main component of the term
paper)
Identify a couple of open issues that may be worth pursuing in the future
Finalize topic by November 16
Suggested journals for literature search: Phys. Rev. B, Phys. Rev. Lett.,
Appl. Phys. Lett., J. Appl. Phys., J. Phys. Chem., Nano Letters, etc., within
the last 10 years.
Term Paper Topic Suggestions
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Modeling of amorphous materials (metals, insulators, polymers)
Point defects in semiconductors and insulators
Metal-oxide interfaces (electronic properties, phase equilibria, adhesion, etc.)
Materials for phase change memory applications
Materials for hydrogen storage
Ferroelectricity/multiferroicity in thin films, multi-layers, etc.
Catalytic chemical reactions
Materials under extreme conditions (pressure, temperature, radiation, etc.)
Magnetism in bulk and nanostructures
Modeling of crystal growth (metals, insulators, semiconductors)
Methods for computing the dielectric constant of materials
Methods for handling the “band gap” and “defect state” problems within traditional
DFT
Methods for computing energy offsets (e.g., work function, electron affinity, ionization
potential, Schottky barrier heights, etc.)
Note: The above suggestions are quite broad. You
need to choose a specific subtopic within these