Order-Disorder Phase Transitions in Metallic Alloys

Ezio Bruno and Francesco Mammano, Messina, Italy

collaborations:

Antonio Milici, Leon Zingales (Messina)

Yang Wang (Pittsburg)

Bronze Age little statue from Olimpia (Greece)

Bronze Age helmet probably from Magna Grecia (Souther Italy).

The dawn of metallurgy coincides with the beginning of human history. In the earl Bronze Age, metallic alloys technologies brought new efficient tools thus permitting the development of agriculture and towns. The availability of sufficient food and the henceforth born complex social organization freed the people from the necessity of hunting for surviving and allowed the discovery of writing. Since then, almost any human handcraft is made of metallic alloys.

The development of new technologies in high-tech areas, such as medical prosthesis or jet engines, requires a careful design of alloys mechanical properties. The determination of the phase diagrams is crucial to this purpose since the performances of alloys are strongly influenced by the various crystalline phases of which they are made.

fcc, disordered ’ fcc, ordered bcc, ordered

A new experimental method for alloy phase diagrams measurements [Zhao, J.-C. Adv. Eng. Mater. 3, 143 (2001)]

A

B

Metallic alloys

Fixed “geometrical” lattice

Different chemical species

SRO

Warren-Cowley Short-Range Order Parameter

random alloys

ordered compoundssegregation

-1 1

ordereddisorderedsegregated

0

TE

MP

ER

AT

UR

Ehigh

low

need for a theory

• ab initio

• finite T

• able to deal with metallic alloys (regardless of the ordering status)

• able to make quantitative predictions about ordering or segregation

• should contain the electronic structure

SRO vs. T

phase diagrams

Existing schemes strenghts problemsPerturbative theories

based on CPA

(Concentration waves, GPM)

Spectral properties electrostatics

perturbative

Theories based on effective Ising Hamiltonians

(Connolly-Williams, CVM)

convergence

(how many clusters?)

iiii kVqa -0.2

-0.1

0

0.1

0.2

-0.2 -0.1 0 0.1 0.2

ZnCu

V

q

Cu0.50Zn 0.50 bcc

1024 atoms sample simulating a random alloy

‘qV’ laws

ai , ki aA , kA if the site i is occupied by a A atom

aB, kB if the site i is occupied by a B atom

ieli Zrrdqi

j

jiji qMV 2

Faulkner, Wang &Stocks, 1995LSMS Density Functional theory calculations

The distribution of chargesin random alloys is

continuous

bcc random Cu0.50Zn0.50

•To date ‘qV’ laws are only a numerical evidence, i.e. a proof within DFT is still missing.

•Deviations from ‘qV’ laws (if any) are not larger than numerical errors in LSMS or LSGS, at least for the systems already investigated.

•Not clear wether or not ‘qV’ laws are due to the approximations made (spherical potentials, LDA)

•Arbitrariety in the choice of the crystal partition in ‘atomic volumes’,

However : Different partitions (e.g. like in Singh &

Gonis, Phys. Rev. B 49, 1642 (1994) ) always lead to linear ‘qV’ laws (actual values of the coefficients are a function of the chosen partition, see Ruban & Skriver, Phys. Rev. B 66, 024201 (2002) )

ieli Zrrdqi

i

It is possible to obtain the qV laws using a Coherent Potential Approximation that

includes Local Fields (CPA+LF)

[Bruno, Zingales & Milici, Phys. Rev. B. 66, 245107, (2002)]

CPA+LF analysis of ‘qV’ laws

CPA+LF simulates the Madelung field V by an external field that is non zero only within the impurity site

aA, aB are related to the response of the impurity sites to

kA, kB are not independent:

at =0 a q0 =k

global electroneutrality implies cA q0A +cB q0

B =0

iiii kVqa

kA- kB is related to some

electronegativity difference

Charge Excess Functional (CEF) theory

Linear ‘qV’ laws

iiii kVqa

j

jiji qMV 2

Ground state charge excesses satisfy the linear eqs.

ij jijii kqMqa 2

Can be derived from a functional quadratic in the qi

E. Bruno, III International Alloy Conference, Estoril (2002)E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

q , 1

2ai qi bi 2

i Mij qi q j

ij qi

i

Charge Excess Functional (CEF) theory

E. Bruno, III International Alloy Conference, Estoril (2002)E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

Linear ‘qV’ laws

iiii kVqa

j

jiji qMV 2

Ground state charge excesses satisfy the linear eqs.

ij jijii kqMqa 2

Charge electroneutrality

qii 0

q , 1

2ai qi bi 2

i Mij qi q j

ij qi

i

Charge Excess Functional (CEF) theory

Linear ‘qV’ laws

iiii kVqa

j

jiji qMV 2

Ground state charge excesses satisfy the linear eqs.

ij jijii kqMqa 2

E. Bruno, III International Alloy Conference, Estoril (2002)E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

q , 1

2ai qi bi 2

i Mij qi q j

ij qi

i

Charge Excess Functional (CEF) theory

Linear ‘qV’ laws

iiii kVqa

j

jiji qMV 2

Ground state charge excesses satisfy the linear eqs.

ij jijii kqMqa 2

Madelung Energy

E. Bruno, III International Alloy Conference, Estoril (2002)E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

q , 1

2ai qi bi 2

i Mij qi q j

ij qi

i

Charge Excess Functional (CEF) theory

Linear ‘qV’ laws

iiii kVqa

j

jiji qMV 2

Ground state charge excesses satisfy the linear eqs.

ij jijii kqMqa 2

“Elastic” local charge relaxation energy

E. Bruno, III International Alloy Conference, Estoril (2002)E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

i

iij

jiiji

iii qqqMbqaq 2

21

,

ai qi bi 2 Mij q jj

qii 0

The Charge Excess Functional (CEF):

Euler-Lagrange equations:

‘qV’ laws: ai qi + Vi = ki

Charge electroneutrality

j

jiji qMV 2

iii kba E. Bruno, III International Alloy Conference, Estoril (2002)E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

bcc random Cu0.50Zn0.50

Ab initio calculations

CEF calculations

CEF-CPA calculations

• The CEF provides the distribution of local charge excesses and the electrostatics of metallic alloys regardless of the amount of order that is present. Differences with respect to LSMS are comparable with numerical errors.

• CEF parameters (3 for a binary alloy) depend on the mean alloy concentration only. Hence they can be calculated for one supercell and used for any other supercell at the same mean conc.

• The CEF is based on a coarse graining of the electronic density, (r), i.e.: electronic degrees of freedom are reduced to one for each

atom, the local charge excess qi.

• The CEF theory is founded on the ‘qV’ laws. Need for understanding the limits of their validity.

• What is the relationship between the ‘true’ energy (e.g. from Kohn-Sham calculations) and the value of the CEF functional at its minimum ?

j

jijiiMAD qMq,

CEF predictions:

By eliminating the Madelung terms via the Euler-Lagrange equations:

qqba

ii ,, 2

i

jijiiii

i qMqbqa 2

2

ij

jiiji

iiiTOT qqMbqaqE 2

21

,

“local energies” linear vs. local charge excesses

-3.665

-3.655

-0.2 -0.1 0charge shifts q

i (a.u.)

Eig

enva

lues

(R

yd.) bcc

fcc

Pd 4p3/2 core statesin Cu0.5Pd0.5 alloysbcc (2 samples) and fcc (3 samples)

PCPA calculations

[Ujfalussy et al.,Phys. Rev. B 61, 12005

(2000)] 

-0.02

0

0.02

0.04

-0.1 0 0.1

Ei-E

i,Mad

Ei,Mad

Ei

qi (a.u.)

r=0.99998

PCPA calculations for Cu0.50 Pd0.50

Within LDA and CPA-based theories for alloys of specified mean at. concentration all the site-diagonal electronic properties are unique functions of

the site chemical occupation, Z, and of the Madelung potential Vi.

It follows that once the functional forms are determined (e.g. from PCPA

calculs.) the knowledge of the distribution of the Vi (e.g. from CEF calculs.) is

sufficient to determine any site-diagonal electronic property.

PCPA on sample A

CEF coefficients

CEF on sample B

Oi= O(Vi) p(V)

properties of sample B

rrr

rr i

LDAi

i

ii

Vde

Vr

ZerW

i

,,,2

2

,

Ei,t

EEE CCF Gt

ri,

j

iijiii qMeVq 2, r

Kohn-Sham eff. potential

s.s. scattering matrices

Fermi level,

CPA coherent medium

ITE

RA

TIO

NS

0 1

ordereddisordered

A

B

SROWarren-Cowley Short-Range Order Parameter

Random alloys

Intermetallic compounds

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

0 1

ordereddisordered

Order-Disorder phase transition

SRO

0

50 CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=10 KSRO=1

0 1

ordereddisordered

0

50

CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=300 KSRO=0.95

0 1

ordereddisordered

0

CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=400 KSRO=0.67

0 1

ordereddisordered

0

10 CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=430 KSRO=0.48

0 1

ordereddisordered

0

10 CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=450 KSRO=0.34

0 1

ordereddisordered

0

10

CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=800 KSRO=0.15

0 1

ordereddisordered

0

10

CuZn

Cou

nts

q0-0.2 0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

T=3200 KSRO=0

0 1

ordereddisordered

0

0.05

0.09

0.1

0.11

0.12

0.13

0.14

0

0.25

0 0.5 1(1/2,1/2,1/2)

<q>

<2q>

<q>2

<3q>

<q>3

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

SRO(1/2,1/2,1/2)

T

Moments of the charge distributions vs. SRO

0

50 CuZn

Cou

nts

q0-0.2 0.2

0 1

ordereddisordered

0

10

CuZn

Cou

nts

q0-0.2 0.2

Charge polydisperse Charge monodisperse

Summary

1. The CEF theory constitutes a simple, very realistic model for the energetics of metallic alloys

2. The CEF can be regarded as a coarse grained density functional ((r) qi). If the CEF parameters are extracted from PCPA

calculations, then the energy from CEF coincide with total electronic energy from the PCPA theory.

3. A MonteCarlo-CEF algorithm allows for the study of order-disorder phase transitions.

0

0.1

Cu-Cu

0

0.1

Zn-Zn

0

0.1

0.2

0 2 4 6

Cu-Zn

Nor

mal

ised

fre

quen

cyDistribution of interaction energies for nearest neighbours

ji

jiijM qqME,

R ijij RR

M

1

Madelung energy

2q

qq jiij

Interaction strength for the (i,j) pair

0

0.1

Cu-Cu

0

0.1

Zn-Zn

0

0.1

0.2

0 2 4 6

Cu-Zn

Nor

mal

ised

fre

quen

cyDistribution of interaction energies for nearest neighbours

ji

jiijM qqME,

R ijij RR

M

1

2q

qq jiij

The variation of has effects similar to the variation of R in

ionic glasses

Charge correlations in random alloys

-1

-0.5

0

0 0.5 1

ij=<q

iq

j>/<q>2

(1/2,1/2,1/2)

(1/2,1/2,1/2)

bcc random Cu0.50Zn0.50

Zn n1=4

bcc random Cu0.50Zn0.50

Zn n1=4n2=3

Zn n1=4

bcc random Cu0.50Zn0.50

Zn n1=4n2=3

Zn n1=4