MAE 715, Professor Zabaras Phase diagram calculation based on cluster expansion and Monte Carlo methods Wei LI 05/07/2007

MAE 715, Professor ZabarasMAE 715, Professor Zabaras

CCOORRNNEELLLL U N I V E R S I T Y


Phase diagram calculation based on cluster Phase diagram calculation based on cluster expansion and Monte Carlo methodsexpansion and Monte Carlo methods

Wei LI

05/07/2007




The Alloy Theoretic Automated Toolkit

Lattice geometry Ab initio code parameters

Effective cluster interactions Ground states

Thermodynamic properties Phase diagrams

MAPS (MIT Ab initioPhase Stability Code)

Cluster expansion construction

Ab initio code(e.g. VASP, Abinit)

Emc2 (Easy Monte Carlo Code)




In an A−B alloy, only the composition x of element A and the difference in chemical potential between the two species μ =μA−μB needs to be specified. For conciseness, we simply refer to the quantity μ as the “chemical potential” and x as the concentration. The natural thermodynamic potential (expressed per atom) associated with the semi-grand-canonical ensemble can be defined as

Semi-Grand-Canonical Monte Carlo

1, ln exp i i

i

N E xN

The potential can also be defined by the following total differential:

,d E x d xd




1 1

0 0

,

1 1 0 0

,

1, , , ,E x x d

This enables the calculation of the thermodynamic function through thermodynamic integration

, ,

1, expg g s g s g

s

E xN

The potential at the initial point

The starting point is in the limit of low temperature at a chemical potential stabilizing a given ground state g.





Free energy function (temperature fixed, potential fixed):

Ts x F x

Bx

F

FF

F F

x x

x x

( )F F x x

, ,const N V T

Phase equilibrium:

, ,

( )F F x x





log( )

exp ( ) exp ( )log( )

exp ( ) ( ) log

exp ( ) exp ( ) log

B n nall states

n n n n n nBall states

n n n n n nBall states

n n nB n n n Ball states

S k P P

N E x N E xS k

N E x N E xk

N E xk N E x k

( )

log

( ) log

n n n

all states

B n n n B

B B

N E x

k N E x k

k N E x k

Entropy:

1 1( ) ( ( )) ( )

( )

B B

d d Ts x d x s x d d x dsk k

x d xd d dx Tds





Using the LTE for each ground state of the system as a starting point, one can map out the whole potential surface associated with any given ordered phaseSimilarly, the high temperature limit can be used as a starting point to obtain for the disordered phase. Once has been determined for all phases, the boundary between two given phases can be located by

,

,

, : , ,

,x

,

,x





Phase boundary tracing

Along the phase boundary:

( ) ( ) ( ) ( )

( )

( )

d d d d

d E E

d x x

Perform two MC simulations simultaneously for two different phases.

Need to deal with the meta-stable phase.




Phonon calculation

Steps:

(1) Fully relax the structure of interest.

The unrelaxed geometry will be used to determine the neighbor shells and measure distances between atoms. Typically the user would specify the str.out file, then obtain the str_relax.out file by runing an ab initio code with a command of the form runstruct_vasp.

(2) Generation of the pertubations.

Each volume subdirectory now contains a str.out file which is stretched version of the main str_relax.out file provided. You then need to run the ab initio code to rerelax the geometry at the various levels of imposed strain and obtain the energy as a function of strain.




(3) We then need to use the ab initio code to calculate reaction forces for each perturbation.

(4) Fitting the force constants and phonon calculations. In addition, we need to specify the range of the springs included in the fit.

(5) Do cluster expansion of high-T limit of vibrational entropy.

(6) Do cluster expansion of vibrational free energy.

(7) Do electronic excitations and cluster expansion.

(8) Calculate the new ECI related with temperature.

Phonon calculation




Phonon calculation




Phonon calculation




Cluster expansion method




Cluster expansion method

1 1

1 1

1

1 1

N n n N

T Tn N N n n n N N

T Tn n N Nn n

A x b

A A x A b

x A A A b




Cross Validation

While the statistical noise vanishes in the large sample limit, it may still affect finite-sample performances. As the number of alternative cluster choices increases, the likelihood that one suboptimal choice happens to give a smaller cross-validation score than the true optimal choice increases.




Cluster Hierarchy

Now we define what a “physically meaningful” cluster choice is. Define the diameter of a cluster as the maximum distance between two sites in the cluster.

1. A cluster can be included only if all its subclusters have already been included.

2. An m-point cluster can be included only if all m-point clusters of a smaller diameter have already been included.




Ground state prediction

The accuracy of a cluster expansion is not solely measured by the error in the predicted energies. It is particularly important that it is able to predict the correct ground states.

Since the mean squared error focuses on optimizing the absolute energy values, while the ground states are determined by the ranking of energies, the lowest mean squared error does not necessarily lead to the most accurate prediction of the ground state line.

For this reason, the search for the best cluster expansion in our algorithm gives an absolute preference to choices of clusters which yield the same ground states as the calculated energies. More specifically, a candidate cluster expansion which has a higher crossvalidation score but predicts the correct ground states will be preferred to a cluster expansion which has a lower cross-validation score but predicts incorrect ground states.




For a given set of structures, it is possible that no candidate cluster expansion predicts the right ground states. In these cases, a simple way to ensure that the ground states are correctly predicted is to give extra “weight” to specifically chosen structures in order to obtain the correct ground states.


For the calculation, it is rewritten as








My result

Results




My result

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Results

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MAE 715, Professor Zabaras Phase diagram calculation based on cluster expansion and Monte Carlo methods Wei LI 05/07/2007