Transcript פרופ' מיכה פולק

Equilibrium chemical order and segregation at alloy surfaces
and 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
“magic-number” cuboctahedrons (COh)
13
Site
NN coordination #
Core
12
Face (111)
9
Face (100)
8
Edge
7
Vertex
5
55
147
309
36 concentric shells (sites)
around center atom:
0-23 constitute the core
24-28,31 - (100)
29,30,32 - (111)
surface
33-35 - edge
13 inequivalent
36 - vertex
sites (362 atoms)
561
923
The computational approach:
Energetics
Surface/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)
wbII ,
wbJJ ,
VbIJ 
1 II
( wb  wbJJ  2wbIJ )
2
5750
Rh
Cohesive energy
(related to wbRhRh )
V<0, endothermic alloying
(“demixing” tendency)
based on experimental heat of mixing
-23
-35
Pd
3890
+33
V>0 , exothermic alloying
(“mixing” tendency)
Cu
3490
Energetics
Elemental surface-subsurface NN bond energy variation model
wb  w11
outmost layer (l=1)
wb  w12
subsurface layer (l=2)
wb
“bulk”
Surface intra-layer and inter-layer elemental bonds are typically stronger than the bulk value
TB computed variations of elemental NN bond energies vs the number of missing bonds
after Michael I. Haftel et al, Phys. Rev. B 70, 125419 (2004)
0
0
Rh
w (eV)
(111)
-0.02
(100)
0
surf
-0.04
-0.03
(110)
(110)
sub
-0.12
-0.05
-0.06
-0.06
4
5
6
surf
-0.08
-0.04
-0.04
3
sub
-0.02
surf
sub
Cu
Pd
-0.01
(100)
(111)
0.04
7
8
9
10
# missing bonds of NN pairs, DZp
-0.16
3
4
5
6
7
8
9
10
DZp
3
4
5
6
7
9
10
D Zp
0
0
Rh surf
-0.5
w (eV)
8
Pd surf
-0.2
Parabolical fit to □
-1
-0.4
-1.5
-0.6
-2
-0.8
-2.5
-1
6
10
14
18
D22
Zp
Parabolical fit to □
6
10
14
- TB computations for elemental clusters: after C. Barreteau et al. Surf. Sci. 433/435, 751 (1999).
18
DZp
Statistical mechanics
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:


F  kT   N p  c Ip ln c Ip  

p
I



 2V IJ
1


I
I
IJ I J
J I
I J I J
 pq
  N pq   w II
pq c p  cq   V pq c p cq  c p cq  kTc p c p cq cq ln  cosh
kT

2 I
IJ 
pq







   

   

c Ip -concentration of constituent I in shell p
N p - number of atoms belonging to shell p
N pq - number of nearest-neighbor pairs of atoms belonging to shells p,q (related to coordination numbers)
w II
pq - elemental pair interaction energy for constituent I
IJ
w IJ
pq , V pq - heteroatomic interaction and effective interaction energies between constituents I and J
 IJ 1 II
JJ
IJ 
Vpq  ( w pq  w pq  2w pq ) 
2


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
Pt25Rh75(111) as a test case
wpq  wpq  wb
0
Rh
MEIS
1300
K
Medium Energy
Ion
Scattering
(MEIS)
0.8
surf
D. Brown et al, Surf. Sci. 497 (2002) 1
w (eV)
0.6
1300 K
0.5
sub
strengthening
-0.04
0.4
-0.06
0.3
bulk
3
4
5
6
7
8
9
10
DZp
0.2
0.04
0.1
Pt
sub
0
1
2
3
0
wpq  0
-0.04
weakening
4
Layer #
Very small V ~ 4 meV, high temperature
Are surface-subsurface bond strength variations
responsible for the subsurface oscillation?
w (eV)
Pt concentration
wpq  0
-0.02
0.7
surf
-0.08
-0.12
3
4
5
6
7
8
9
10
DZp
Computational Results Part I. Surface segregation profiles for Pt25Rh75(111)
FCEM (no adjustable parameters):
Single layer tension model (SL)
 '1   tot ,  '2  0
0.8
Bond energy variations and corresponding layer tension differences
Two layer tension model (TL)
 1,  2  0
Pt concentration
0.6
(meV)
MEIS: D. Brown et al, Surf. Sci. 497 (2002) 1
1300 K
0.4
0.2
1
1
Temperature evolution
of layer compositions
0
1
2
3
4
Layer
0.8
0.6
LEED: E. Platzgummer et al, Surf. Sci. 419 (1999) 236
0.6
0.4
3
0.2 4
0
0
2
500
1000
1500
Temperature (K)
2000
2500
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
Pt concentration
Pt concentration
0.8
1373 K
0.4
0.2
0
1
2
3
Layer
4
Computation procedure for clusters
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
Cluster thermodynamic functions
111 concentrations for ternary COh-923)
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)
Configurational entropy (J/mol/K)
4
2000 K
3
1500 K
923-COh
1000 K
2
1
0
0
Cu
Rh
500 K
0.2
0.4
0.6
0.8
Overall Rh concentration
2
3
Cusurf → Cucore
2
1
C, J/mol/K
C, J/mol/K
4
The corresponding
heat capacity curves:
E
S 

T
C 


T

T


1
Cuedge → Cu(100)
1
Cuedge/vex → Cucore
Rh791Cu 132
Rh561Cu 362
0
0
2000
T, K
4000
0
0
2000
T, K
4000
The surface desegregation process
Configurational heat-Capacity Schottky anomaly in alloy nanoclusters
Rh561Cu362 923-COh
qCu=1
(111)
4
Site-specific Cu concentrations
(100)
1
0.8
0.6
vertex
(100)
edge
(111)
0.4
0.2
0
0
Heat capacity (J/mol/K)
(Tmax , Cmax)
3
E
2
vert deseg
edge deseg
(100) deseg
1
(111) deseg
core
2000
4000
Temperature (K)
surf segregated
0
0
T0 1000
2000
3000
Temperature (K)
4000
5000
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.
Cmax & number of deseg. excitations per atom vs. cluster size
0.26
Rh-Cu COh 3.5
qCu=1
309
561
923
0.21
3.6
55
2.5
2
3.1
nsnc
3
Cmax, J/mol/K
Cmax, J/mol/K
147
0.16
2.6
2.1
1.6
0.05
13
1.5
0
0.11
0.1
0.15
0.2
0.25
nsnc
ns – fraction of surface sites, nc – fraction of core sites
200
400
600
Number of cluster atoms
800
0.06
1000
Order-disorder transitions and desegregation in “magic number” Pd618/923Cu305/923 COh clusters
2. The case of Pd-Cu (V>0)
Surface “mixed” order
Overall and sublattice concentrations
FCEM computations
based on NRL-TB
energetics ( w  0 )
FCEM computations
based on simple
bond breaking
energetics (uniform
bond-strength, w  0 )
L12-like ordered core
(cross-section)
Schottky type configurational heat capacity
3. Ternary clusters
Surface order-disorder transitions and desegregation
“Magic number” Rh561/923Pd150/923Cu212/923 vs. Pd618/923Cu305/923 923-COh
Pd
Cu
Core
Rh
Heat capacity (J/mol K)
4
(100)
disordering
3
Cu and Pd
desegregation
2
Cu desegregation
1
0
Pd618/923Cu305/923 (“substrate”
effect)
1000
2000
3000
Temperature (K)
4000
Part III. Inter-cluster “phase” separation: The case of Rh-Pd (V<0)
Mixing free-energies computed for 147-COh clusters
Fmix=F-(cRhFRh+cPdFPd)
Rh inclusion
Rh
Pd
Free energy of mixing (kJ/mol)
0
10 K
-1
500 K
-2
-3
0
1000 K
19/147
55/147
79/147
Overall Rh concentration
Convexity between “magic-number” compositional structures (demixed order)
135/147 1
inter-cluster separation
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) & orderdisorder 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)
NRL
M. 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 atom
icosahedron
Thank you!
Bond energy between atoms i and j from states a and b
is estimated as the corresponding contribution to
E  Tr H    ia , jb H jb ,ia (  - the density-operator)
ia jb
(the summation is over atom labels i , j and state labels a, b, and implicitly includes an integral over k)
a , b denote orbitals and angular momenta, s,p,d : pair contributions to the total bond energy ss, sp, pp, sd, pd, dd
(M. I. Haftel, N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 70, 125419 (2004).)
The system is described by an ensemble of quantum states  n
The density-operator  
 n
with probabilities pn , n  i, a , k
p n n
n
 H   p n n H n 
n
 pn
n i i H j
j n
n, i , j


  j   n p n n  i H ij   j  i H ij    ji H ij  Tr ( H )


i, j
i, j
i, j
 n

Effects of elemental bond-energy variations on Pd-Cu cluster surface segregation
Schematics of two models ((100) face)
Simple bond breaking (w  0,  bb )
Two layer tension model (  1 ,  2  0 )
Esite
Pd
surf
subsurf
D 1Cu  Pd   1Cu   1Pd  0
bulk
Cu segregation
&
Cu  Pd
D 1Cu  Pd  D bb
Pd
 bb
  1Pd
 2Pd
D 2Cu  Pd   2Cu   2Pd  0
 0
Cu
surf
subsurf
Cu depletion
Extra Cu enrichment in surface layer
& subsurface oscillation (core depletion)
bulk
Cu
 bb
  1Cu
 2Cu  0
Extra Cu segregation
depth
Effects of cluster size on Cmax
Estimation of the number of surface-core desegregation excitations
cCu 
N surf
N total
Rh
Cu
Initial fully segregated state
The number of excitations (core-surface atomic exchanges):
The number of excitations per atom:

Final randomized state
N surf
Cu
N core  N core  cCu  N core 
N total
N core N surf

 ncore  nsurf  Cmax
N total N total
ncore – fraction of core sites, nsurf – fraction of surface sites
As Ntotal (size)    , Cmax  0
Introduction
Typical shapes of free clusters
numbers and colors mark distinct “surface” shells (sites)
DFT formalism
DFT key variable is the electron density,




 

 

n(r )  N  dr2  dr3 ... drN  * (r , r2 ,...,rN )(r , r2 ,...,rN )
The Kohn-Sham equation is solved in a self-consistent (iterative) way:

- An initial guess for n(r )

- Calculation of the corresponding Kohn-Sham potential VKS [n(r )]
- Solution of the Kohn-Sham equation



  2 n (r )  VKS n (r )   n n (r )

- The eigenvalue spectrum  n and the orbitals  n (r )
(of an auxiliary non-interacting system (1-electron Hamiltonian), which reproduce the density of the original many-body system)
- Calculation of a new density
N

 2
n( r )    i ( r )
i 1
The procedure is repeated until convergence is reached
DFT formalism (continue)
Solution of Kohn-Sham equations by augmented plane-wave method (APW)
Plane waves (PW’s) - inefficient basis set for describing the rapidly varying wave function
(around the nuclei)
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,

Notes: - Only the density n(r ) has strict physical meaning in the Kohn-Sham equations.
- Eigenvalues  n of an auxiliary single-body Schrodinger equation are artificial objects
- Total energy is not simply the sum of all  n :
E
d 3k

 k   F [n(r )]
3 n
2 
n
the integral is over the first Brillouin zone, the first sum is over occupied states

F[n(r )] - a functional of the density
(includes the repulsion of the ionic cores, correlation effects, and part of the Coulomb interaction)
The Naval Research Laboratory tight-binding (NRL-TB) method
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:
1) Construction of a first-principles database of eigenvalues εn(k) and total energies E.

2) Shift of the Kohn-Sham potential VKS by V0  F[n(r )] / N e
( N e - the number of electrons in the unit cell)
3) Definition of shifted the eigenvalues,  n k    n k   V0
 1


The total energy is simply the sum, E  
 n k     ij Dqi Dq j 

 2

2 3 n
ij


4) In the two-center TB approximation,  n k  are dependent on the terms,
d 3k
corresponds to
the self-consistent charge
(SCC-TB) correction
H jb , ia (u)   i*a (r  u) H 2c jb (r )d 3 r ( H 2c - the two-center part of the Hamiltonian)
The integral depends on quantum numbers a , b denoting orbitals and angular momenta, s,p,d,
and on the component of the angular momentum relative to the direction u (specified by  ,  ,  )
and, for non-orthogonal orbitals, are dependent on the terms,
S jb , ia (u)   i*a (r  u) jb (r )d 3 r
H jb ,ia and S jb , ia are parameterized in order to reproduce the eigenvalues  n k 
(for ss, sp, pp, sd, pd, dd at a large number of k-points for fcc and bcc structures for several volumes each).
3D representation of thermodynamic functions of ternary clusters
elucidating composition-dependent properties (Rh-Pd-Cu 147-COh)
Pd
-5
Rh
-10
-15
-20
500 K
4
3
2
1
0
Pd
Cu
Rh
Free energy of mixiing, kJ/mol
0
10 K
Cu
0
1000 K
Cu
Pd
-5
Rh
-10
-15
-20
Configurational entropy (J/mol/K)
Configurational entropy (J/mol/K)
Free energy of mixiing, kJ/mol
Mixing free-energy and configurational entropy plotted with respect to the concentration Gibbs triangle
1000 K
6
4
2
Pd
0
Cu
Rh
- Convexity in Fmix indicates inter-cluster separation. Minima in S indicate intra-cluster separation or ordering
- Note: hundreds computed data points constitute each plot
Pt & Rh (111) surface & subsurface layer tensions
l 
oscillatory profile
1  ml
  nn  m wmn  DZ l wb 

2
(meV)
Schematics of two models
 1,  2  0
Single layer tension model (SL)  '1   tot ,  ' 2  0
Two layer tension model (TL)
El
Rh
 1   tot
Eb
2  0
Pt
Pt enrichment
 tot   1
Eb
2  0
1
2
Pt depletion
3
Layer
C vs. T curves for different overall compositions
Rh-Cu 923-COh
4
Intrasurface
exchange processes
E
(111)
3
Heat capacity (J/mol/K)
Surface-core
processes with increasing
desegregation excitation
energies and Tmax
Cu (111)→Cu core
Rh561Cu 362
(qCu=1)
Cu (100)→Cu core
(100)
Rh641Cu 282
edge
2
Cu edge→Cu(100)
Cu(100)→Cu(111)
Rh791Cu 132
1
0
0
Cu edge/vex→Cu core
1000
2000
3000
Temperature (K)
4000
5000
The Naval Research Laboratory tight-binding (NRL-TB) method
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:
 n k    n k   V0
E
d 3k
  k 
3 n
2  n
- 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.
Estimation of bond energy:
Effective bond energy between nearest neighbor (NN) atoms i and j from states a and b is defined as the
corresponding contribution to E:
E  Tr H    ia , jb H jb ,ia
ia jb
 - the density-operator, H - the Hamiltonian operator
(M. I. Haftel, N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 70, 125419 (2004).)
The total energy of the system in the TB method:
E  Tr H    ia , jb H jb ,ia
ia jb
(the summation is over atom labels i , j and state labels a, b, and implicitly includes an integral over k as well)
 - the density-operator
H - the Hamiltonian operator,
H jb ,ia (k )   exp(ik  R n )  i*a (r  R n  bi ) H jb (r  b j )d 3r
n
 ia - wave-functions associated with atomic orbitals
R n - lattice vector
bi - atom position
k
- Bloch vector
SRO in an alloy with LRO:
- 100% probability
- smallest probability
Relevant bulk phase diagrams
Rh-Pd
Rh-Cu
Pd-Cu
The Statistical-Mechanical Theory
Background:
The segregation process
“Magic number” Rh561Pd302Cu60 923-COh clusters
Surface order-disorder transition and desegregation
1
0.8
Pd
0.6
Cu
34
0.4
Core Rh
33
35
0.2
0
0
1000
2000
3000
Temperature (K)
4000
1
Vertex concentrations
Edge-vex order (40 K)
Pd-Cu site competition and
co-desegregation at vertexes
0.8
Pd
0.6
0.4
Cu
Pd-Cu
desegregation
3
Heat capacity (J/mol/K)
Cu edge site concentrations
Pd-Cu edge disordering
1.5
2
1
1
0
0.2
Rh
0
0
1000
2000
3000
Temperature (K)
4000
Pd-Cu edge
disordering
0.5
0
0
0
200
400
600
Pd863 Cu60 (“substrate” effect)
1000
2000
3000
Temperature (K)
4000
Rh inclusion in Rh78Cu845 923-COh
Rh
Rh core inclusion in Rh19Pd128 147-COh
The attraction of a solute atom to local compositional
fluctuations (SRO) in a binary alloy
segregation suppression
due to higher atomic
bulk coordination