Equilibrium chemical order and segregation at alloy surfacesand nanoclusters computed using tight-binding derived

coordination-dependent bond energies

Micha Polak

Department of Chemistry, Ben-Gurion University

Beer-Sheva, ISRAEL

ACS meeting San Francisco – September 12, 2006

Motivation

Various applications of alloy nanoclusters in heterogeneous catalysis, magnetic media, etc

Full atomic scale chemical-structural information for alloy nanoclusters is inaccessible by current experimental techniques 

55

309

“magic-number” cuboctahedrons (COh)

36 concentric shells (sites)around center atom: 0-23 constitute the core24-28,31 - (100)29,30,32 - (111)33-35 - edge36 - vertex

147

561

923

SiteNN coordination#

Core12

Face (111)9

Face (100)8

Edge7

Vertex5

13 inequivalent sites (362 atoms)

surface

13

The computational approach:

EnergeticsSurface/subsurface bond energy variation (2-layer) model with data computed by DFT-based Tight-Binding method (NRL-TB)

Statistical Mechanics The “Free energy Concentration Expansion Method” (FCEM)

adapted to a system of atom-exchanging equilibrated nanoclusters

Computational Results: I. Surface segregation profiles for Pt25Rh75(111) – a test case

II. Binary & ternary Rh-Pd-Cu 923 atom cuboctahedral clusters 1. Site specific concentrations, surface segregation, core depletion and order-disorder transitions (highlighting bond energy variation effects)

Cluster thermodynamic properties:2. Entropy, Internal-Energy: configurational heat capacity III. Mixing Free-Energy: inter-cluster separation

The alloy systems: basic empirical interatomic energetics (meV)

V>0 , exothermic alloying(“mixing” tendency)

V<0, endothermic alloying (“demixing” tendency)

based on experimental heat of mixing

Cohesive energy

(related to )

5750Rh

Pd3890

Cu3490

-23-35

+33

)2(2

1 , , IJ

bJJb

IIb

IJb

JJb

IIb wwwVww

RhRhbw

Elemental surface-subsurface NN bond energy variation model

Surface intra-layer and inter-layer elemental bonds are typically stronger than the bulk value

11 wwb

21 wwb

bw

outmost layer (l=1)

subsurface layer (l=2)

“bulk”

Energetics

-0.16

-0.12

-0.08

-0.04

0

0.04

3 4 5 6 7 8 9 10

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

3 4 5 6 7 8 9 10

-0.06

-0.04

-0.02

0

3 4 5 6 7 8 9 10

-1

-0.8

-0.6

-0.4

-0.2

0

6 10 14 18

TB computed variations of elemental NN bond energies vs the number of missing bondsafter Michael I. Haftel et al, Phys. Rev. B 70, 125419 (2004)

w (

eV)

sub surf

Rh

sub

surfPd Cu

sub

surf

-2.5

-2

-1.5

-1

-0.5

0

6 10 14 18 22

w (

eV)

Rh surf

Zp

Parabolical fit to □

Pd surf

- TB computations for elemental clusters: after C. Barreteau et al. Surf. Sci. 433/435, 751 (1999).

Zp

ZpZpZp# missing bonds of NN pairs,

(110)(100)

(111)

(110)

(100)(111)

Parabolical fit to □

FCEM adapted to alloy clusters

The FCEM expressions were obtained using NN pair-interaction model Hamiltonian and expanding the free energy in powers of constituent concentrations. The free energy of a system of multi-component alloy clusters capable of atomic exchange:

qp IJ

IJpqJ

qIq

Jp

Ip

Iq

Jp

Jq

Ip

IJpq

I

Iq

Ip

IIpqpq

p I

Ip

Ipp

kT

VccckTcccccVccwN

ccNkTF

2coshln

2

1

ln

advantages:

This analytical formula (that takes into account inter-atomic correlations) makes FCEM much more efficient than computer simulations. It can yield large amounts of data: site-specific concentrations and corresponding thermodynamic properties vs. cluster size (up to ~1000 atoms), multi-component composition and temperature

-concentration of constituent I in shell p

- number of atoms belonging to shell p

- number of nearest-neighbor pairs of atoms belonging to shells p,q (related to coordination numbers)

- elemental pair interaction energy for constituent I

- heteroatomic interaction and effective interaction energies between constituents I and J

Ipc

pN

pqN

IIpqw

IJpqV,IJ

pqw

)( IJ

pqJJpq

IIpq

IJpq wwwV 2

2

1

Statistical mechanics

-0.06

-0.04

-0.02

0

3 4 5 6 7 8 9 10

-0.12

-0.08

-0.04

0

0.04

3 4 5 6 7 8 9 10

1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Layer #

Pt co

nce

ntr

atio

n

MEIS 1300 K

bulk

Pt25Rh75(111) as a test case

w (

eV)

sub

surfRh

w (

eV)

sub

surf

Pt

D. Brown et al, Surf. Sci. 497 (2002) 1

Very small V ~ 4 meV, high temperature

Are surface-subsurface bond strength variations responsible for the subsurface oscillation?

bpqpq www

Zp

Zp

0pqw

0pqw

strengthening

weakening

Medium Energy Ion Scattering (MEIS)

1300 K

1 2 3 4

0

0.2

0.4

0.6

0.8

Layer

Pt c

on

cen

tra

tion

1 2 3 40

0.2

0.4

0.6

0.8

Layer

Pt c

on

cen

tra

tion

Part I. Surface segregation profiles for Pt25Rh75(111)

MEIS: D. Brown et al, Surf. Sci. 497 (2002) 1

LEED: E. Platzgummer et al, Surf. Sci. 419 (1999) 236

Computational Results

0 500 1000 1500 2000 25000

0.2

0.4

0.6

0.8

1

Temperature (K)

Pt c

on

cen

tra

tion

1

4

3

2

Single layer tension model (SL)

Two layer tension model (TL)

Temperature evolution of layer compositions

FCEM (no adjustable parameters):

1300 K

1373 K

(meV)

Bond energy variations and corresponding layer tension differences0',' 21 tot

0, 21

In the SL model ignoring surface-subsurface bond variations, the subsurface oscillation due to V only is much weaker than in the TL model at all temperatures

Input: - Cluster geometrical parameters- Energetic parameters

Free energy numerical minimization (MATLAB - including Genetic Algorithm confirmation, under the constraint of conservation of the system overall concentration)

Output: - set of all site/shell concentrations (e.g., 37 inequivalent sites, 72 independent variables 111 concentrations for ternary COh-923)

Cluster thermodynamic functions

Computation procedure for clusters

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Overall Rh concentration

Co

nfig

ura

tion

al e

ntr

op

y (

J/m

ol/K

) 2000 K

1000 K

500 K

1500 K

0 2000 40000

1

2

T, K

C,

J/m

ol/K

Rh791Cu 132

0 2000 40000

1

2

3

4

T, K

C,

J/m

ol/K

Rh561Cu 362

The correspondingheat capacity curves:

Cusurf → Cucore

Cuedge → Cu(100)

Cuedge/vex → Cucore

T

ST

T

EC

Rh

Cu

Part II. Cluster site specific concentrations, ordering and configurational heat capacity

1. The case of Rh-Cu (V<0)Surface/core segregation/separation and surface “demixed order” at compositional “magic numbers”

Rh inclusion (Rh78Cu845)

923-COh

The surface desegregation process

Configurational heat-Capacity Schottky anomaly in alloy nanoclusters

Desegregation contribution to the cluster heat capacity. The lowest level in the energy scheme corresponds to completely Cu surface segregated cluster. Desegregation excitations of single Cu atom to the Rh core are indicated by vertical arrows. T0 signifies the onset of the desegregation effect involving the lowest (111) excitation.

0 1000 2000 3000 4000 50000

1

2

3

4

Temperature (K)

He

at c

ap

aci

ty (

J/m

ol/K

)

(Tmax , Cmax)

T0

Evert deseg

edge deseg

(100 )deseg

(111 )deseg

surf segregated

(111)

(100)

0 2000 40000

0.2

0.4

0.6

0.8

1

Temperature (K)

Site

-spe

cific

Cu

conc

entr

atio

ns

(100)

vertex

(111)

edge

core

Rh561Cu362 923-COh

Cu=1

1.5

2

2.5

3

3.5

0 200 400 600 800 1000

0.06

0.11

0.16

0.21

0.26

1.6

2.1

2.6

3.1

3.6

0.05 0.1 0.15 0.2 0.25

Number of cluster atoms

Cm

ax, J

/mol

/K

n snc

nsnc

Cm

ax, J

/mol

/K

ns – fraction of surface sites, nc – fraction of core sites13

55

147309 561

923

Rh-Cu COh Cu=1

Cmax & number of deseg. excitations per atom vs. cluster size

Order-disorder transitions and desegregation in “magic number” Pd618/923Cu305/923 COh clusters

Overall and sublattice concentrations

FCEM computationsbased on NRL-TB energetics ( )

FCEM computations based on simple bond breaking energetics (uniformbond-strength, )

Schottky type configurational heat capacity

Surface “mixed” order L12-like ordered core

(cross-section)

0w

0w

2. The case of Pd-Cu (V>0)

1000 2000 3000 4000

0

1

2

3

4

Temperature (K)

He

at c

ap

aci

ty (

J/m

ol K

)

(100)disordering Cu and Pd

desegregation

Cu desegregation

Pd

Cu

Pd618/923Cu305/923 (“substrate” effect)

Surface order-disorder transitions and desegregation“Magic number” Rh561/923Pd150/923Cu212/923 vs. Pd618/923Cu305/923 923-COh

Core Rh

3. Ternary clusters

0 19/147 55/147 79/147 135/147 1

-3

-2

-1

0

Overall Rh concentration

Fre

e e

ne

rgy o

f m

ixin

g (

kJ/m

ol)

10 K

500 K

1000 K

Mixing free-energies computed for 147-COh clusters

Convexity between “magic-number” compositional structures (demixed order) inter-cluster separation

Fmix=F-(cRhFRh+cPdFPd)Rh

Pd

Part III. Inter-cluster “phase” separation: The case of Rh-Pd (V<0)

Rh inclusion

Concluding Remarks

The test case for the FCEM/TB approach: good agreement between the two-layer oscillatory profile computed for Pt25Rh75(111) surface and reported experimental data,

highlighting the role of subsurface tensions

The relatively high efficiency of FCEM in computing binary & ternary alloy nanocluster compositional structures and related thermodynamic properties enables to predict a variety of phenomena:

• Cluster ordering involving “magic-number” low-temperature structures that exhibit - core & segregated surface order-disorder transitions, - enhanced elemental segregation due to preferential surface bond strengthening (Pd-Cu),

• Configurational heat capacity Schottky-type anomaly: reflect distinctly the various atomic exchange excitation processes: C vs. T experimental measurements are expected to elucidate the energetics of alloy cluster surface segregation (via desegregation peaks) & order- disorder transitions

• Surface-Segregation related intra & inter-cluster separation (Rh-Pd)

• Ternary alloying effects on surface transitions and segregation

Relevant publications:

M. Polak and L. Rubinovich, Surface Science Reports 38, 127 (2000)L. Rubinovich and M. Polak, Phys. Rev. B 69, 155405 (2004)M. Polak and L. Rubinovich, Surf. Sci. 584, 41 (2005)M. Polak and L. Rubinovich, Phys. Rev. B 71, 125426 (2005)L. Rubinovich, M.I. Haftel, N. Bernstein, and M. Polak, Phys. Rev. B 74, 035405 (2006)

NRLM. Polak and L. Rubinovich, (submitted to Phys. Rev. B, 2006)

Future Plans: 

1. Refinement of FCEM energetics:

TB-computed bonding in clusters, including also:

Hetero-atomic interactions; NNN pairs; deeper subsurface layers;

Higher accuracy by inclusion of on-site contributions

2. Comparative computations for icosahedrons

3. Effects of chemisorption (O,S)

This research is supported by THE ISRAEL SCIENCE FOUNDATION

923 atomicosahedron

Thank you!

)(,,,

,,

HTrHHijHipj

jjHiipHpH

jiijji

jiij

jiij

nnnn

njin

nnnn

nn

n

nnn p

Bond energy between atoms i and j from states and is estimated as the corresponding contribution to

The density-operator

i j

ijji HHTrE ,,

(M. I. Haftel, N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 70, 125419 (2004).)

(the summation is over atom labels i , j and state labels , and implicitly includes an integral over k)

k,,in The system is described by an ensemble of quantum states with probabilities , n np

denote orbitals and angular momenta, s,p,d : , pair contributions to the total bond energy ss, sp, pp, sd, pd, dd

( - the density-operator)

Two layer tension model ( )

Effects of elemental bond-energy variations on Pd-Cu cluster surface segregation

0111 PdCuPdCu

0222 PdCuPdCu

PdPdbb 1

CuCubb 1

Extra Cu enrichment in surface layer& subsurface oscillation (core depletion)

Cu depletion

Pd

Cu

Simple bond breaking ( ),0w

02 Pd

02 Cu

surf subsurf bulk

surf subsurf bulk

Esite

Schematics of two models ((100) face)

bb

Cu segregation

depth

Extra Cu segregationPdCubb

PdCu 1

&

0, 21

maxCnnN

N

N

Nsurfcore

total

surf

total

core

Effects of cluster size on Cmax

Estimation of the number of surface-core desegregation excitations

The number of excitations (core-surface atomic exchanges) :total

surfcoreCucore

Cucore N

NNcNN

The number of excitations per atom:

Rh

Cu

Initial fully segregated state Final randomized state

total

surfCu N

Nc

ncore – fraction of core sites, nsurf – fraction of surface sites

0, max CN total (size) As

Typical shapes of free clusters numbers and colors mark distinct “surface” shells (sites)

Introduction

- The eigenvalue spectrum and the orbitals

DFT formalism

),...,,(),...,,(...)( 22*

32 NNN rrrrrrrdrdrdNrn

DFT key variable is the electron density,

(of an auxiliary non-interacting system (1-electron Hamiltonian), which reproduce the density of the original many-body system)

)()()(2 rrVr nnnKSn

N

ii rrn

1

2)()(

The Kohn-Sham equation is solved in a self-consistent (iterative) way:

- An initial guess for

- Calculation of the corresponding Kohn-Sham potential

)(rn

)]([ rnVKS

The procedure is repeated until convergence is reached

n )(rn

- Calculation of a new density

- Solution of the Kohn-Sham equation

Solution of Kohn-Sham equations by augmented plane-wave method (APW)

In the APW scheme the unit cell is divided into two regions (mixed basis set):(i) The muffin-tin (MT) region which consists of spheres centered at the nuclear position, inside which the APW’s satisfy the atomic Schrodinger equation(ii) The interstitial region I, where the APW’s consist of PW’s,

Plane waves (PW’s) - inefficient basis set for describing the rapidly varying wave function (around the nuclei)

DFT formalism (continue)

- Eigenvalues of an auxiliary single-body Schrodinger equation are artificial objectsn

- Total energy is not simply the sum of all :

- Only the density has strict physical meaning in the Kohn-Sham equations. )(rn

Notes:

n

)]([

2 3

3rnF

kdE

nn

k

)]([ rnF

the integral is over the first Brillouin zone, the first sum is over occupied states

- a functional of the density (includes the repulsion of the ionic cores, correlation effects, and part of the Coulomb interaction)

NRL-TB maps results of a limited set of first-principles calculations to a two-center non-orthogonal Slater-Koster TB Hamiltonian:

( - the two-center part of the Hamiltonian)

4) In the two-center TB approximation, are dependent on the terms,

and, for non-orthogonal orbitals, are dependent on the terms,

The Naval Research Laboratory tight-binding (NRL-TB) methodR. E.Cohen, M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 50 14694 (1994)M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54 4519 (1996)

2) Shift of the Kohn-Sham potential by ( - the number of electrons in the unit cell)

0Vnn kk

1) Construction of a first-principles database of eigenvalues εn(k) and total energies E.

KSV

3) Definition of shifted the eigenvalues,

eNrnFV /)]([0

eN

and are parameterized in order to reproduce the eigenvalues (for ss, sp, pp, sd, pd, dd at a large number of k-points for fcc and bcc structures for several volumes each).

nn

kdE k

3

3

2The total energy is simply the sum,

kn

rdHH jciij3

2*

, )()()( ruru

kn

rdS jiij3*

, )()()( ruru ijH , ijS ,

The integral depends on quantum numbers denoting orbitals and angular momenta, s,p,d, and on the component of the angular momentum relative to the direction (specified by )

cH 2

, ,,u

ijjiij qq

2

1 corresponds to the self-consistent charge (SCC-TB) correction

0

2

4

6

Co

nfig

ura

tion

al e

ntr

op

y (J

/mo

l/K)

Rh

Cu

Pd

-20

-15

-10

-5

0

Fre

e e

ne

rgy

of m

ixiin

g, k

J/m

ol

Rh

CuPd

0

1

2

3

4

Co

nfig

ura

tion

al e

ntr

op

y (J

/mo

l/K)

Rh

Cu

Pd

Mixing free-energy and configurational entropy plotted with respect to the concentration Gibbs triangle

-20

-15

-10

-5

0

Fre

e e

ne

rgy

of m

ixiin

g, k

J/m

ol

Rh

CuPd

1000 K

- Note: hundreds computed data points constitute each plot- Convexity in Fmix indicates inter-cluster separation. Minima in S indicate intra-cluster separation or ordering

3D representation of thermodynamic functions of ternary clusterselucidating composition-dependent properties (Rh-Pd-Cu 147-COh)

500 K

1000 K

10 K

Pt & Rh (111) surface & subsurface layer tensions oscillatory profile

bllm

mnn mnl wZw2

1

tot 1

1 tot

Rh

Pt

02

02

El

321Layer

Eb

Eb

0',' 21 tot

0, 21 Single layer tension model (SL)

Two layer tension model (TL)

Pt enrichment

Pt depletion

Schematics of two models (meV)

0 1000 2000 3000 4000 50000

1

2

3

4

Temperature (K)

Hea

t cap

acity

(J/

mol

/K)

C vs. T curves for different overall compositions Rh-Cu 923-COh

Rh561Cu 362

(Cu=1)

Rh641Cu 282

Rh791Cu 132

Cu edge→Cu(100)Cu(100)→Cu(111)

Surface-coreprocesses with increasing desegregation excitation

energies and Tmax

(111)

(100)

edge

E

Cu edge/vex→Cu core

Cu (100)→Cu core

Cu (111)→Cu core

Intrasurfaceexchange processes

The Naval Research Laboratory tight-binding (NRL-TB) method

Estimation of bond energy:Effective bond energy between nearest neighbor (NN) atoms i and j from states and is defined as the corresponding contribution to E:

R. E.Cohen, M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 50 14694 (1994)M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54 4519 (1996)

NRL-TB maps results of a limited set of first-principles calculations to a two-center non-orthogonal Slater-Koster TB Hamiltonian:- Construction of a first-principles database of eigenvalues εn(k) and total energies E.- Finding “shift potential” V0 and shift the eigenvalues in order to get total energy:

- Finding a set of parameters which generate non-orthogonal, two-centre Slater-Koster Hamiltonians H which will reproduce the energies and eigenvalues in the database.

nn

nn

kdE

V

k

kk

3

30

2

i j

ijji HHTrE ,,

(M. I. Haftel, N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 70, 125419 (2004).)

H- the density-operator, - the Hamiltonian operator

The total energy of the system in the TB method:

i j

ijji HHTrE ,,

H

- the density-operator

- the Hamiltonian operator,

(the summation is over atom labels i , j and state labels , , and implicitly includes an integral over k as well)

rdHiH jjinin

nij3*

, )()()exp()( brbRrRkk

nR - lattice vector

ib - atom position

k - Bloch vector

i - wave-functions associated with atomic orbitals

SRO in an alloy with LRO:

- 100% probability

- smallest probability

Relevant bulk phase diagrams

Rh-Cu

Rh-Pd

Pd-Cu

The Statistical-Mechanical Theory

The segregation processBackground:

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Temperature (K)

Cu

ed

ge

site

co

nce

ntr

atio

ns

35

33

34

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Temperature (K)

Ve

rte

x co

nce

ntr

atio

ns

Cu

Rh

Pd

0 1000 2000 3000 40000

1

2

3

Temperature (K)

He

at c

ap

aci

ty (

J/m

ol/K

)

0 200 400 6000

0.5

1

1.5

Pd

Cu

Pd-Cu site competition andco-desegregation at vertexes

Pd-Cu edge disordering

Pd-Cu edgedisordering

Pd-Cu desegregation

Pd863 Cu60 (“substrate” effect)

“Magic number” Rh561Pd302Cu60 923-COh clusters Surface order-disorder transition and desegregation

Edge-vex order (40 K)

Core Rh

Rh inclusion in Rh78Cu845 923-COh

Rh

Rh core inclusion in Rh19Pd128 147-COh

solute atom

solvent atom

repulsive interactions

(demixing tendency)

attractive interactions

(mixing tendency)

surface

bulk

segregation suppression

due to higher atomic

bulk coordination

The attraction of a solute atom to local compositional fluctuations (SRO) in a binary alloy

segregation suppressiondue to higher atomic

bulk coordination