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![Page 1: Equilibrium chemical order and segregation at alloy surfaces and nanoclusters computed using tight-binding derived coordination-dependent bond energies.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649f525503460f94c75858/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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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
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-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 □
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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
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-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
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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
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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
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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
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The surface desegregation process
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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
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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
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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)
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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
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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
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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
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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
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Thank you!
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)(,,,
,,
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)
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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
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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
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Typical shapes of free clusters numbers and colors mark distinct “surface” shells (sites)
Introduction
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- 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
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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)
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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
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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
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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)
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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
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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
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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
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SRO in an alloy with LRO:
- 100% probability
- smallest probability
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Relevant bulk phase diagrams
Rh-Cu
Rh-Pd
Pd-Cu
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The Statistical-Mechanical Theory
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The segregation processBackground:
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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
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Rh inclusion in Rh78Cu845 923-COh
Rh
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Rh core inclusion in Rh19Pd128 147-COh
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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