Automating first-principles phase diagram calculations€¦ · · 2013-01-01Automating...
Transcript of Automating first-principles phase diagram calculations€¦ · · 2013-01-01Automating...
Automating first-principles phase diagram calculations
Axel van de Walle (Northwestern University)
Mark D. Asta
(Northwestern University)
Gerbrand Ceder (MIT)
This work was supported by: DOE under contract no. DE-F502-96ER 45571. NSF under program DMR-0080766 and DMR-0076097.
Composition-temperature phase diagrams
Thermodynamic properties of stable and metastable phases,
Short-range order in solid solutions,
Thermodynamic properties of planar defects
Morphology of precipitate microstructures
Previous successes of
first-principles thermodynamic
calculations
(Ducastelle (1991), Fontaine (1994), Zunger (1994,1997),
Ozolins et al. (1998), Wolverton et al. (2000),
Ceder et al. (2000), Asta et al. (2001).)
Automate the process in order to make
this tool available to a wider community
First-principles
Phase Diagram Calculation
Theory is well established:
(Ducastelle (1991), Fontaine (1994), Zunger (1994))
Our goal:
But: the task is tedious in practice.
Year
Tim
e
1980 2000
Computer
Human
E(σ1,...,σn)
The cluster expansion
Alloy system Lattice model
+1+1
+1
+1+1
+1+1+1
+1+1
+1
+1 +1
E(r1,...,rn)
ri
Parametrize configurational-dependence of the energy E
with a polynomial in the occupation variables σi:
E(σ1,...,σn) = Σ Jij σiσj + Σ Jijk σiσjσk + ...{i,j} {i,j,k}
= Σ Jα σααwhere
and
, ,... ..., , ,α =
i
∋
αΠ σiσ
α
αcluster+1
+1 +1
+1-1
-1 -1
-1
+1
+1= −1
-1
+1=
Structures Clusters
, ,... ..., , ,...,, , ,
Least Squares Fit
E(1) = Σα
Jα σα(1)
E(2) = Σα
Jα σα(2)
Jα
First-principles
Calculations
(2)(1) (3) (4)α1 α10α2 α11
Cluster expansion fit
Questions:
1) Which clusters α1, α2, ... to include?
2) Which structures (1), (2), ..., to include?
Strange ground state?
Decaying ECI?
Mean Square Error small?
Error on ground states small?
Have supercomputer time to burn?
Deadline coming up?
Tired already?
Look for the magic clusterFind bad structure and kill it
Force this structure not to be a ground state
Can I remove one structure
and predict its energy?
Add not-too-complicated cluster
Add a structure that makes things work better
Start with simple clusters and structures
Compute many many structural energies
Add simple structure and/or clusters
least-squares fit
Stop doing cluster expansions
Cluster expansions construction:
The Old Way
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
YesNo
Yes
No
Done!
Cluster expansion construction
MAPS
Monte Carlo Simulations
Emc2
(MIT Ab-initio
Phase Stability Code)
(Easy Monte Carlo Code)
Lattice geometry Ab-initio code parameters
Ab-initio code
Thermodynamic quantities Phase diagrams
Effective cluster interactions Ground states
"Predictors"
Elastic or
Electrostatic
Energy
ex:ex: VASP
Alloy Theoretic Automated Toolkit
12
3
Find best
cluster expansion
Filter
Filter
Job manager
geometry
energy
Find best
structureinput files
output files
"ready"
"error"
cluster
expansionground
states
Array of
processors
MAPS
Lattice geometry Ab-initio code parameters
Alloy theory
"Engine"
Ab-initio code
Interface
Processes
InterfaceHardware
Cluster expansion construction module
(MAPS)
Cross-Validation
score (CV)
Cross-Validation
score (CV)
Number of regressor
Cross-Validation
CV = Σ ( Ei − E-i )2
E E E
Example for polynomial fit
i=1
n
Energy of structure i
predicted from a fit to structures
1, 2, ... , i−1, ,i+1, ... , n
i removed
True energy of structure i
Squar
e er
ror
Mean Square
Error (MSE)
Mean Square
Error (MSE)
CV prevents under- or over- fitting
"Leave-one-out estimator"
Cluster Hierarchy
Example: square lattice
Rules:1) Include cluster only if all its subclusters are also included.
2) Include cluster only if all smaller clusters are also included.Note: α smaller than β iff
max ||ri − rj|| ≤ max ||ri − rj||i,j∈α i,j∈β
rirj
Include these clusters only if those clusters included.
0 20 40 60 80 100
En
erg
y
Concentration (%)
Incorrect ground state
predicted energypredicted energy ab-initio energy
Choose ECI with
cross-validation
Flag incorrectly
ordered structures
none ?
done
Increase their
weight in the fit
Ensuring ground state accuracy
(from LS fit)
Structure generation
0 20 40 60 80 100
Ener
gy
Concentration (%)
Criteria:
1) look for predicted ground states
2) Minimize variance
(look for noncoplanar structure in correlation space)
ρ1ρ2
ρ3
3 3 7.35 90 90 120
1 2 1.33
-2 -1 1.33
1 -1 1.33
0 0 1.0000 O
2 1 0.6666 Co
1 2 0.3333 O
0 0 0.0000 Li,Vac
Axes
Cell
Atoms
a b c α β γ
[INCAR]
PREC = high
ISMEAR = -1
SIGMA = 0.1
NSW=41
IBRION = 2
ISIF = 3
KPPRA = 1000
DOSTATIC
do static run
k-point density
(k point per reciprocal atom)
Standard VASP tokens
1) Lattice geometry
2) First-principles code parameters
Input Files
-450-400-350-300-250-200-150-100-50
0
0 20 40 60 80 100
Ener
gy (m
eV)
Concentration(at % Ti)
generated calculated
1) Formation Energies and Ground States
-40
-20
0
20
40
60
1 1.5 2 2.5 1 1.5
ECI (
meV
)
r/rnn:
pairs triplets
2) Effective Cluster Interactions
Example output (hcp Ti-Al system)
Energy Predictors
Presence of long range interactions:
pai
r E
CI
distance
Elastic interactions
Electrostatic interactions
Hard to fit so many ECI...
1) Look for analytic solution for long-range ECI behavior:
True ECITrue ECI
Analytic prediction
2) Subtract out long-range analytic contribution
Energy predictors easy to add to MAPS:
No change needed in the code,
just link with user-specified routines
3) Fit short-range contribution to ab initio calculations
Solution:
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.2 0.4 0.6 0.8 1
1010 0
110110
111111
210210
-0.08
0
0.08
-0.08
0
0.08
-0.08
0
0.08
-0.08
0
0.08
-0.08
0
0.08
-0.08
0
0.08
-0.040
0.04
-0.04
0
0.04
-0.04
0
0.04
Const
ituen
t st
rain
ener
gy (
eV/a
tom
)Angular-dependence of the
k-space expansion coefficients
Concentration (at % Au)Cu Au
Constituent strain formalism implementation
Concentration-dependence of the
constituent strain energy
Input:
Output:
100
110
111
210
Superlattice
Directions
ab-initio
code
parameters
(Laks, Ferreira, Froyen, and Zunger, 1992)
+
Monte Carlo Simulations
E = Σα Jα σα
From cluster expansions:
to thermodynamic quantities:
Phase diagrams:
Free energies:
F
concentration
Tasks to perform:
How to determine when target accuracy reached?
How to detect phase transitions?
How to efficiently calculate phase boundaries?
Equilibration and Averaging times
Equilibration time Averaging time
Equilibrium not reached at t1
Equilibrium reached at t2
Given tolerance: ε
> ε
< ε
Choose averaging time such that
(E, x,E, x, SRO) SRO)on some thermodynamic quantity Q =
Detecting phase transitions
2) Use cross-validation to find optimal order of polynomial
(Accounting for noise in MC averages)
1) Fit polynomial through (Qi, Ci)
4) Perform statistical test of equality between Qn+1 and Qn+1
3) Extrapolate Qn+1 using fitted polynomial
Q(C)
Thermodynamic
quantity
Control
variable
Goal: find discontinuity in
(E, x,E, x, SRO) SRO) (µ, µ, T)
µ
φ
β
β + dβ
dµ E − E µ
dβ β(x − x ) β=
γ
αγ
α
γ
α
Phase boundary tracing
µ(β)µ(β + dβ)
as temperature β changes,
need to update µ to preserve φγ = φαPrinciple:
µ(β) is known find xγ(µ(β)) and xα(µ(β))
x
γ α
-gs=3
-mu0=3.5
-mu1=4.5
-dmu=0.05
Unit cell
Simulation Cell
enclosed
sphere
radius
0 0.2 0.4 0.6 0.8 1
Concentration
0
1
2
3
4
5
µ=1
µ=2
µ=3µ=4
µ=5
µ = 3.5µ = 3.5µ = 4.5µ = 4.5
-er=35
-dx=1e-3
-T0=100
-T1=1000
-dT=20
-k=8.617e-5
Temperature range
Boltzman's kB
-dx=1e-3
-er=35
-dT=20
-k=8.617e-5
Target precision
Thermodynamic
Integration
Phase
Boundaries
Determination
Input parameters
(Monte Carlo Code)
-gs1=3
-gs2=4
E
Free energy surfaces
CaO-MgO system
Ti-Al system
Emc2 output:
Emc2 ouput:Phase diagrams
Short-range order calculations
Calculated diffuse X-ray scattering in Ti-Al hcp solid-solution
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
05
101520253035404550
0 1 2 3 4 5 6
γk (
mJ/
m2)
k
T/Tc =0.60xAl =6%xAl =8%xAl =10%xAl =12%
Energy cost of creating a diffuse anti-phase boundary
in a Ti-Al short-range ordered alloy by sliding k dislocations