898
Int. J. Nanotechnol., Vol. 8, Nos. 10/11/12, 2011
Coordination-dependent bond energies derived from
DFT surface-energy data for use in computations of
surface segregation phenomena in nanoclusters
Micha Polak* and Leonid Rubinovich
Department of Chemistry,
Ben-Gurion University,
Beer-Sheva 84105, Israel
Fax: +972-8-6472943
E-mail: [email protected]
*Corresponding author
E-mail: [email protected]
Abstract: Theoretical computations of alloy surface phenomena, such as
elemental segregation, within atomic pair-interaction models, necessitate the
use of reliable bond energies as input. This work introduces the idea to extract
the coordination dependence of bond energies from density-functional theory
(DFT) computed surface energy anisotropy. Polynomial functions are fitted to
DFT data reported recently for surface energies of pure Pt, Rh and Pd.
Compared to other approaches, the proposed method is highly transparent, and
is expected to yield better insight into the origin of alloy segregation
phenomena at surfaces of bulk and nanoclusters.
Keywords: coordination-dependent bond energies; surface energy anisotropy;
surface segregation; alloy nanoclusters; Pt; Rh; Pd.
Reference to this paper should be made as follows: Polak, M. and
Rubinovich, L. (2011) ‘Coordination-dependent bond energies derived from
DFT surface-energy data for use in computations of surface segregation
phenomena in nanoclusters’, Int. J. Nanotechnol., Vol. 8, Nos. 10/11/12,
pp.898–906.
Biographical notes: Micha Polak received BSc Degree in Chemistry (1968),
MSc in Physical Chemistry (1971) and PhD (NMR of Solids, 1975), all from
Tel-Aviv University, Israel. He did Post-doctoral NMR research in the
University of Utah, Salt-Lake-City (1975–1977) and in Caltech, Pasadena
(1977–1979). In 1979 he received a Senior Lecturer position at Ben-Gurion
University for the development of a Surface Science laboratory (XPS, AES),
mainly used for alloy segregation experiments. Currently his research activities
include theories of (i) alloy nanoclusters and (ii) chemical equilibrium
involving a small number of molecules.
Leonid Rubinovich (1955) received his MSc Degree in Physics from the
University of Tomsk (Russia) in 1976 and PhD (Alloy Physics, 1990) at
Institute of Strength Physics and Materials Science (Tomsk). He joined the
group of Professor Micha Polak in Beer-Sheva, Israel, as a researcher in 1993,
where he, among other things, gained experience in the theory of
surface segregation. His current research interests include applications of
statistical-mechanics to alloy clusters and to chemical reactions involving a
small number of molecules.
Copyright © 2011 Inderscience Enterprises Ltd.
Coordination-dependent bond energies
899
The nature of bond-energy variations at solid surfaces still constitutes an important issue
in surface science, including its relatively unexplored role in surface segregation
phenomena, which are manifested as deviations of the surface composition from bulk
values [1]. This enrichment in alloy constituent is not limited to the outmost surface
layer, but typically extends either monotonously or oscillatorically to subsurface layers.
Such a composition gradient can affect physical and chemical properties relevant to
heterogeneous catalysis, oxidation, magnetism, adhesion, etc. The term “surface
segregation” usually refers to the free (clean) surface, whereas under gaseous
environment chemisorption usually modifies the segregation characteristics [2]. Since the
intrinsic and chemical driving forces operate concomitantly, study of free surface
segregation forms a basis for the other case.
Experimental measurement of layer-by-layer compositions is quite tedious and is
limited to just a few dedicated techniques. This has motivated numerous theoretical
computational studies in this field [1]. Modelling alloy surfaces ranged from simplified
empirical bond energies [3] or site energies (bond order simulation model, BOS) [4], to
somewhat more realistic models, such as the Tight-Binding (TB) method in the Second
Moment Approximation (SMA) [5–8], the Embedded Atom Method (EAM) [9], the
Equivalent Crystal Theory (ECT) [10], or the Equivalent-Medium Approximation with
bond-strength modifications at surfaces [11], some of which are based on semi-empirical
many-body potentials. More reliable first-principles Cluster Expansion (CE) involves
fitting of cluster interactions to energies of configurations created by different
substitutions of alloy atoms and computed by the density-functional theory (DFT)
in conjunction with the mean-field approximation or Monte Carlo (MC) simulations
[12–16]. However, such simulations are time-consuming, and typically provide results
for quite limited sets of temperatures and alloy compositions.
In the case of alloy clusters, the segregation behaviour can differ considerably from
that of bulk systems, due to more complicated geometries having a diversity of
nonequivalent surface sites [17] and the limited supply of atoms for segregation [18].
Furthermore, simulations based on DFT energetics are limited by the particle size [17],
and to our best knowledge, hitherto CE has not been applied to alloy clusters, unlike
other methods mentioned above. Employing analytical expressions for the alloy free
energy based on certain statistical-mechanical approximations has certain advantages.
In particular, the Free energy Concentration Expansion Method (FCEM), which takes
into account Short-Range Order (SRO) [1], has proven to be highly efficient in
computations of site-specific concentrations vs. overall composition, temperature and
cluster size. It revealed diverse phenomena and properties specific to alloy binary and
ternary clusters, such as mixed and demixed type compositional order and order-disorder
transitions [19–21], as well as inter-cluster separation and Schottky–type peaks in the
nanocluster configurational heat capacity [21,22]. However, rather simplistic energetics
were used in these FCEM computations involving elemental pair interactions estimated
from the BOS model [4], with the assumption of equal distribution of the site energy
among its nearest neighbour (NN) bonds [23–26]. More recently, we studied [27,28]
alloy nanoclusters using the FCEM with improved energetics, which incorporates
elemental bond energies and their surface-induced variations, obtained by means of the
NRL Tight-Binding method [29]. Furthermore, as a test case we studied the role of such
variations in the emergence of oscillatory segregation profiles in alloys with weak mixing
or even demixing tendency [28]. In particular, using FCEM with TB data, while
neglecting alloying effects on elemental bond energies, a distinct two-layer oscillatory
900
M. Polak and L. Rubinovich
profile in Pt25Rh75(111) was obtained in reasonable agreement with previously reported
experimental data [30]. While the above-mentioned DFT/CE/MC combination, applied to
the same surface [13–15], also yielded good agreement with experimental data, no
straightforward explanation of segregation profiles by means of elemental bond energies
is possible. On the other hand, the TB/FCEM pair-bond approach (as well as the one
introduced below), provide some extra physical insight into the phenomena. For example,
as explained in [28], the subsurface will be enriched by the element that exhibits a higher
interlayer bond strengthening.
A comprehensive evaluation of the role of elemental bond energy variations in
surface segregation in bulk alloys and alloy clusters necessitates even more reliable
energetics data (than TB) as input to the FCEM. In the present work this is achieved by a
novel method we introduced briefly elsewhere [40], which is based on the extraction of
coordination dependent bond energies from DFT computed surface energies. Such
valuable data concerning pure element bond energy variations are valid especially in
studies of alloys with small atomic size mismatch and relatively small effective
heteroatomic interactions (e.g., Pt-Rh and Pt-Pd), as well as of pure metal surfaces,
surface defects etc. In this approach the bond energies are treated as parameterised and
fitted coordination-dependent functions, rather than numerical values. Such approach
helps to circumvent a transferability problem existing in CE, for example, namely
the need to repeat fitting of interactions in every specific surface geometry.
As a first step, coordination variables are introduced and the functional dependence of the
Coordination-induced Bond Energy Variations (CBEV) is approximated by a polynomial
with coefficients fitted to DFT data reported recently for energies of six surfaces (111),
(100), (110), (311), (331) and (210) in Pt, Rh and Pd [31]. Thus, for an m–n pair-bond
with ∆Zm and ∆Zn NN broken bonds at the m and n sites, the variation, δwmn, is
considered as a polynomial function of two bond coordination variables: one
‘symmetric’, xmn = ∆Zm+∆Zn, taking into account the total number of broken bonds, and
the second ‘anti-symmetric’, ymn = ∆Zm – ∆Zn, taking into account a possible
non-equivalency of the two sites. Then, the array of the DFT computed surface energies,
(E111, E100, E110, E311, E331, E211), is mapped into an array of six coefficients,
(a1,0, a2,0, a0,2, a3,0, a1,2, a4,0), of a polynomial (P6) of sequentially ascending powers of the
coordination variables,
2
2
3
2
4
δ wmn = a1,0 xmn + a2,0 xmn
+ a0,2 ymn
+ a3,0 xmn
+ a1,2 xmn ymn
+ a4,0 xmn
.
(1)
It can be noted that in the present approach, any preliminary assumptions concerning
bond variations are absent and the polynomial function is chosen due to its generality and
simplicity. Furthermore, the odd powers of ymn are omitted because of symmetry
considerations (δwmn = δwnm).
When incorporated in the FCEM (and in equation (2) below), the required pair-bond
energy, wb, can be derived straightforwardly from the bulk energy. It should be noted that
experimental cohesive energies have been quite often used as bulk energies in semiempirical or empirical modelling of transition metal and alloy properties, thus neglecting
the free-atom spin-polarisation and promotion energy contributions [32]. However, wb
should be obtained either by direct DFT computations [33] or, as we demonstrated
recently [34], by the use of DFT corrected cohesive energies [32,35]. This work uses the
latter approach [34] for deriving the bulk energies (see bottom of Table 1).
Coordination-dependent bond energies
Table 1
901
Coefficients of the polynomial expansion of pair-bond energy variations (eV) in terms
of coordination variables: extraction from computed surface energy anisotropies of Pt,
Rh and Pd
Metal
Polynomial terms
Pt
Rh
–2
–5.58·10
x2
3.59·10–2
1.69·10–2
8.41·10–3
w
y2
4.28·10–3
–3.04·10–3
–9.05·10–5
w/s
–2.11·10
–3
–2.46·10–3
s
–2.53·10
–4
–4
x
xy
x4
2
–9.05·10
–3
–4.89·10
–4
5.61·10–4
–2.84·10
–2
x
3
–7.57·10
Tendency
Pd
–2
–3.60·10
7.88·10–5
1.48·10–4
s
s (small)
w
Bulk energies (eV)
–7.28
–8.46
–6.65
Notation: s – bond strengthening contribution (minus sign), w – bond weakening
contribution (plus sign).
The surface energy mapping is accomplished by solving a system of six linear equations
of the form given in [28]:
Es ≡ Ehkl = ∑
m

1
 ∑ δ wmn − ∆Z m wb 
2  n( n ≠ m)

(2)
with respect to the set of the polynomial coefficients given in equation (1). (The second
summation goes over the n atomic NN of surface site m.) It can be noted that the
polynomial terms for all three metals reflect almost systematic strengthening or
weakening of bond energy contributions (Table 1). As can be seen in Figure 1 for the
case of Pd, both interlayer and intralayer bond energy variations exhibit the expected
tendency to increase for lower coordinations. These polynomial functions can be used
straightforwardly in the derivation of the energy variation for any other surface bond
coordinations, such as most of the bonds in nanoclusters.
While P6 exactly reproduces the six DFT computed input surface energies, its
transferability, or the capability to predict variations of other bonds, can be estimated by a
cross-validation score that is often used for evaluation of accuracy of the cluster variation
(CV) method [36,37]. For example, for a 5-coefficient polynomial (P5),
2
2
3
4
δ wmn = a1,0 xmn + a2,0 xmn
+ a0,2 ymn
+ a3,0 xmn
+ a4,0 xmn
,
the relative cross-validation score, ξ, is given by,
2

1 6  EsCBEV(5)
ξ=
 DFT − 1 ,
∑
6 s =1  Es

(3)
902
M. Polak and L. Rubinovich
where EsCBEV(5) denotes the energy of surface s predicted by P5 without using the
corresponding EsDFT . The reasonably low relative cross-validation scores of 0.026,
0.012 and 0.023 obtained for Pt, Rh and Pd, respectively, indicate good accuracy
(transferability) of P5. Yet, with the increase of the number of input surface orientations
and correspondingly of the polynomial coefficients, the accuracy of P6 is expected to be
even better than of P5.
Figure 1
Energy variations of palladium bonds at the six surfaces involved in the mapping. Bond
energy variations within atomic layers (‘intra’, y = 0) and between adjacent layers
(‘inter’, y ≠ 0) at the fcc(311) surface (inset) are indicated by black circles, and
variations related to the other five surfaces are indicated by open circles. The
corresponding coordination variables are shown in the table
For nanocluster computations, energy variations of lower coordinated bonds are often
needed. For this goal, the coefficients of the polynomial were fitted also to DFT
computed energies [38,39] of small Pt, Rh and Pd clusters (dimer, triangle, tetrahedron
and octahedron). The results shown in Figures 1 and 2 have been used as input energetics
for statistical-mechanical FCEM computations of Pt-Pd 923-atom cuboctahedron cluster
compositional structures (and Pt25Rh75(111) as a test case) [40]. These CBEV/FCEM
computations reveal unique effects of the variations in the energies of the cluster
intra-surface bonds on its segregation characteristics (particularly, a reversal from Pd to
Pt at certain surface sites). On the other hand, strong Pd segregation at the subsurface
shell is predicted due to surface-subsurface interlayer bond energy variations [40].
To summarise, formulation of a new physically transparent approach to surface bond
energetics, incorporating Coordination-induced Bond Energy Variations extracted from
DFT-based data, is presented. The use of polynomial coordination functions in the CBEV
provides transferability of energetic parameters to different surface geometries, including
low-coordination sites in nanoclusters having different structures, as verified by
reasonable cross-validation scores. The combination of the straightforward and highly
Coordination-dependent bond energies
903
efficient analytical statistical-mechanical FCEM and energetics based on the CBEV can
facilitate studying effects of bond energy variations on segregation in medium to large
size (up to at least 1000 atoms) cluster surfaces of alloys having weak mixing tendency.
In addition to surface segregation computations for binary and ternary alloy nanoclusters,
the coordination functions obtained for Pt, Rh and Pd are expected to be useful also in
studies of surface defects in pure metals and alloys (e.g., steps, kinks etc.). The present
version of the CBEV method uses a limited DFT database of surface energies. However,
it can be improved by including:
•
higher order polynomials, based on more DFT computed elemental surface energies
in order to take into account possible surface-induced variations of pair-interactions
beyond NN
•
effective hetero-atomic interactions, enabling the applicability of the new approach
to alloys with relatively strong mixing/demixing tendencies, and possibly
•
the extension of the pair-based approach to multi-site interactions.
Figure 2
Least square values (circles) fitted to the Figure 1 variations and to DFT computed
energies of small clusters [38,39] (squares); Diamond – interpolated value of the energy
variation for the low coordinated (x = 12) vertex-edge bond in cuboctahedron (shown in
the inset for the 923-atom cluster)
Acknowledgement
This work was supported by the Israel Science Foundation (grant No.1204/04).
904
M. Polak and L. Rubinovich
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