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Order-Disorder Phase Transitions in Metallic Alloys

EzioBruno and Francesco Mammano, Messina, Italy

collaborations:

Antonio Milici, Leon Zingales (Messina)

Yang Wang (Pittsburg)

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.

Bronze Age little statue from

Olimpia (Greece)

Bronze Age helmet probably from

Magna Grecia (Souther Italy).

bbcc, ordered

g fcc, disordered

g’ fcc, ordered

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.

A new experimental method for alloy phase diagrams measurements

[Zhao, J.-C. Adv. Eng. Mater.3, 143 (2001)]

random alloys

segregation

ordered compounds

disordered

ordered

segregated

0

-1

1

high

TEMPERATURE

low

SRO

Warren-Cowley Short-Range Order Parameter

SRO vs. T

phase diagrams

- 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

Cu

Zn

bcc

0.50

0.50

1024 atoms sample simulating a random alloy

0.2

Zn

Cu

0.1

V

0

-0.1

-0.2

-0.2

-0.1

0

0.1

0.2

q

Faulkner, Wang &Stocks, 1995

LSMS Density Functional theory calculations

‘qV’ laws

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))

CPA+LF analysis of ‘qV’ laws

It is possible to obtain the qV laws using aCoherentPotential Approximation that includes Local Fields (CPA+LF)

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

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

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

kA, kB are not independent:

at F=0aaq0a=ka

global electroneutrality implies cAq0A +cBq0B =0

kA- kB is related to some electronegativity difference

Linear ‘qV’ laws

Ground state charge excesses satisfy the linear eqs.

Charge Excess Functional (CEF) theory

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)

Linear ‘qV’ laws

Ground state charge excesses satisfy the linear eqs.

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

Ground state charge excesses satisfy the linear eqs.

Charge electroneutrality

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

Ground state charge excesses satisfy the linear eqs.

Charge Excess Functional (CEF) theory

Madelung Energy

E. Bruno, III International Alloy Conference, Estoril (2002)

E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

Linear ‘qV’ laws

Ground state charge excesses satisfy the linear eqs.

Charge Excess Functional (CEF) theory

“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)

The Charge Excess Functional (CEF):

Euler-Lagrange equations:

‘qV’ laws:ai qi + Vi = ki

Charge electroneutrality

E. Bruno, III International Alloy Conference, Estoril (2002)

E. Bruno, L. Zingales and Y. Wang, Phys. Rev. Lett. 91, 166401 (2003)

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(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 ?

CEF predictions:

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

“local energies” linear vs. local charge excesses

Pd 4p3/2 core states

in Cu0.5Pd0.5 alloys

bcc (2 samples) and fcc (3 samples)

PCPA calculations

[Ujfalussy et al.,Phys. Rev. B 61, 12005 (2000)]

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, Za, 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= Oa(Vi)

pa(V)

properties of sample B

A

B

ordered

disordered

0

1

Intermetallic

compounds

Random

alloys

SRO

Warren-Cowley Short-Range Order Parameter

Summary

- The CEF theory constitutes a simple, very realistic model for the energetics of metallic alloys
- The CEF can be regarded as a coarse grained density functional (r(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.
- A MonteCarlo-CEF algorithm allows for the study of order-disorder phase transitions.

Madelung energy

Interaction strength for the (i,j) pair

Distribution of interaction energies for nearest neighbours

Normalised frequency

The variation of e has effects similar to the variation of R in ionic glasses

Distribution of interaction energies for nearest neighbours

Normalised frequency

bcc random Cu0.50Zn0.50

Zn n1=4

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