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Symmetry breaking and morphological instabilities in core-shell metallic nanoparticles

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 013003 (http://iopscience.iop.org/0953-8984/27/1/013003) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 013003 (35pp)

doi:10.1088/0953-8984/27/1/013003

Topical Review

Symmetry breaking and morphological instabilities in core-shell metallic nanoparticles Riccardo Ferrando Dipartimento di Fisica and CNR/IMEM, Universit`a degli Studi di Genova, Via Dodecaneso 33, 16146, Genova, Italy E-mail: [email protected] Received 17 September 2014, revised 17 November 2014 Accepted for publication 18 November 2014 Published 8 December 2014 Abstract

Nanoalloys are bi- or multi-component metallic particles in the size range between 1 and 100 nm. Nanoalloys present a wide variety of structures and properties, which make them suitable for many applications in catalysis, optics, magnetism and biomedicine. This topical review is devoted to the structural properties of nanoalloys of weakly miscible metals, which are expected to present phase-separated arrangements of their components, such as core-shell and Janus arrangements. The focus is on singling out size- and composition-dependent transitions between these arrangements, showing that several transitions can be rationalized by a unifying concept, that is symmetry breaking, caused by the accumulation of strain at the atomic level and its subsequent release. The driving forces that rule the interplay between core-shell and other structures and determine the actual shapes of core and shell, and the placement of the core inside the shell are analyzed. Several systems, such as Ag–Cu, Ag–Co, Ag–Ni, Au–Co, Au–Pt, and Ir–Pt are treated, comparing computational results to experimental observations and simple analytical models. After treating the lowest-energy structures, which are representative of the equilibrium configurations at sufficiently low temperatures, high-temperature and growth kinetics effects are considered. Keywords: nanoparticles, metals, nanoalloys (Some figures may appear in colour only in the online journal)

structure. When the components of a binary nanoparticle are poorly miscible, the core-shell chemical ordering is often expected in the equilibrium configuration. In core-shell nanoparticles, an external shell of a given element (say element B) covers a core of element A. This is often denoted as A@B structure. In the following, element A will be the element of higher surface energy and higher cohesion, unless otherwise specified, and we will use the notation A@B to indicate any phase-separated chemical ordering in which element A is covered by element B. Nanoparticles made of weakly miscible metals are of interest for several applications in catalysis, optics, magnetism, biomedicine and environmental chemistry [1–3]. For example,

1. Introduction

Nanoalloys are bi- or multi-component metallic particles in the size range between 1 and 100 nm [1]. These nanoparticles are of great interest for both basic science and technological applications, which stem from the very high degree of tunability of the physical and chemical properties of these systems, which is a direct consequence of the great variety of morphologies that nanoalloys can assume. The morphology of a binary nanoparticle is specified when both its geometric structure and its chemical ordering pattern are specified. The chemical ordering pattern is the way in which the two elements are arranged within its geometric 0953-8984/15/013003+35$33.00

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© 2015 IOP Publishing Ltd Printed in the UK

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Topical Review

Ag–Co, Ag–Ni, and Au–Co nanoalloys are composed by a ferromagnetic (Co or Ni) and a non-ferromagnetic metal (Ag or Au), the latter presenting a sharp surface plasmon resonance (SPR). This causes interesting magneto-optical properties [4]. The frequency of the SPR in these nanoalloys depends on composition and ordering pattern, so that it can be tuned to a good extent [5]. Applications to catalysis have been also found. Here we mention just a few examples. Ag–Co nanoalloys are used for the oxygen reduction in alkaline media [6], and Ag–Cu clusters are more efficient catalysts for oxygen reduction reactions [7] than the corresponding pure metallic nanoparticles. Au–Pt nanoalloys have recently been the subject of wide interest due to their possible application in fuel cells. Their efficiency in the oxygen reduction reaction has been demonstrated in several cases [8–11]. Co–Cu nanoalloys have been proposed for the selective conversion of synthesis gas to ethanol and higher alcohols [12]. Pt–Ir nanoparticles were found to be more active in cyclohexene disproportionation reaction than the corresponding monometal catalysts [13], and are also employed in the oxygen reduction reaction [14]. Core-shell Ni@Ag nanoparticles show enhanced CO oxidation activity [15]. Core-shell Au–Co nanoparticles have also been proposed for biomedical applications, such as thermal ablation therapies and drug delivery, for which the magnetic Co core is coated by a shell of Au, which is a biocompatible material that can be easily functionalized [16, 17]. Au–Pt nanoparticles have revealed an anti-bacterial activity [18], and being fully biocompatible, they are foreseen as new type of antibiotic. Ag–Cu nanoparticles find application in the fabrication of Pb-free solder interconnects, because of their low melting temperatures [19]. The specific morphology of both the shell and the core is often of crucial importance in determining the properties of the nanoparticles, and can have a strong influence on applications. A stable and regular coating of the core can be necessary when the core must be protected or separated from the environment the nanoparticle is embedded in, as may happen for biomedical applications. The SPR frequency is known to depend on the chemical ordering of the nanoparticles, in particular on the shape and placement of the core [5, 20]. Besides core-shell (figure 1(a)) and phase-separated Janus arrangements (figure 1(d)), other chemical ordering patterns are possible in weakly miscible systems. These comprise multishell [21] (figure 1(b)), quasi-Janus [22–29] (figure 1(e)) and ball-and-cup patterns [30, 31] (figure 1(c)). In the multishell configuration, a core of B element is covered by an intermediate shell of A atoms which is in turn covered by an outer shell of B atoms. The quasi-Janus configuration may be either considered as a Janus configuration in which the A part is covered by a very thin shell of B atoms, or a coreshell nanoparticle in which the core is asymmetrically placed. In the ball-and-cup configuration, the core is covered by an incomplete shell on one of its sides. As we will see, there is a non-trivial interplay between the geometric structures and the preferential chemical ordering patterns of nanoparticles [28]. This Topical Review is devoted to the structural properties of nanoalloys of weakly miscible metals. Its focus is

on singling out size- and composition-dependent structural transitions, showing that several transitions can be rationalized by a unifying concept, that is symmetry breaking, caused by the accumulation of strain at the atomic level and its subsequent release [28, 32–34]. The analysis of the most stable structures and of their structural transitions allows us to discuss the driving forces that rule the interplay between chemical ordering patterns of great interest in the experiments. Specifically, we discuss the driving forces that (i) Stabilize the core-shell chemical ordering with respect to other possible arrangements, depending on the geometric structure of the nanoparticle. (ii) Determine the actual shapes of core and shell, and the placement of the core inside the shell. After treating the lowest-energy structures, which are representative of the equilibrium structures at sufficiently low temperatures, we deal with temperature effects and with growth kinetics effects. In general, it is rather difficult to disentangle equilibrium and formation kinetics effects for generic nanoparticles in actual experimental procedures. For example, lowtemperature equilibration of nanoalloys may be difficult to obtain in the experiments. For this reason, we focus first on computational results, in which equilibrium and kinetics are easily treated separately, and then we compare the theoretical results with the experiments when possible. In the following we mainly treat isolated nanoparticles, showing results that are valid in the gas phase or for nanoparticles that are weakly interacting with their environment. For nanoparticles that interact strongly with their environment through their surface, the effects are very specific depending on the environment itself. These effects are especially important for small clusters (up to at least ∼2 nm diameter), in which the majority of atoms is indeed at the surface, but may become less important as the nanoparticle size increases. This review article is structured as follows. In section 2 we deal with the driving forces that stabilize the different types of chemical ordering patterns in nanoparticles of weakly miscible elements. We present an idealized model that uses macroscopic concepts, such as surface and interface energies, to show that transitions between A@B, B@A and Janus configurations are possible depending on compositions for what concerns the lowest-energy chemical ordering pattern. Then we discuss the physical factors that appear in real systems and may alter the simple picture of the idealized model. In section 3 we briefly review the computational methods for searching the lowest-energy structures of nanoparticles. These methods are usually known as global optimization methods. Review articles on these methods specifically applied to metal nanoparticles and nanoalloys are found in [35–37]. Here we mostly focus on the specific methods that are employed to optimize chemical ordering within a given geometric motif. In sections 4–6 we review the present knowledge about the most energetically stable structures of nanoalloys for a series 2

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Figure 1. Schematic representation of the different chemical ordering patterns in weakly miscible systems. A and B atoms are shown in orange and grey, respectively. From top to bottom: (a) core-shell A@B; (b) multishell B@A@B; (c) ball-and-cup A@B; (d) Janus; (e) quasi-Janus A@B. Each pattern is shown in three different views. Left column: external structure of the nanoparticle. Middle column: cross-section of the nanoparticle. Right column: B atoms are shown as small spheres.

of poorly miscible element pairs. These structures correspond to the nanoparticle equilibrium configurations at sufficiently low temperatures. In section 4 we consider small clusters, up to sizes of about 50 atoms (less than 2 nm in diameters), with special attention given to those sizes and compositions at which high-symmetry structures are the lowest in energy. Sections 5 and 6 deal with the main subject of this Topical Review. In these sections, we consider larger nanoparticles, in the A-rich and B-rich cases, respectively. For these nanoparticles, the high-symmetry structures encountered at small sizes may

undergo different kinds of symmetry breaking when size increases. Morphological instabilities may develop when changing composition. These instabilities are of completely static character and they are not to be confused with other morphological instabilities that may occur in nanoparticle growth [38]. In section 7 the temperature effects at equilibrium are treated, starting from the previous knowledge of the lowestenergy structures that correspond to the equilibrium in the limit T → 0. 3

J. Phys.: Condens. Matter 27 (2015) 013003

A

B

Topical Review

B

A

A

B

Figure 2. From left to right: A@B, B@A core-shell configurations and Janus configuration in a spherical nanoparticle of radius R.

Section 8 deals with non-equilibrium effects, such as growth and coalescence. In section 9 we summarize the main points of the review comparing the theoretical results with the experimental evidence. Finally, section 10 contains the conclusions.

For the A@B configuration, the radius of the core is Rc = Rx 1/3 and therefore the excess energy is   A@B Eexc (1) = 4π R 2 γB + γAB x 2/3 . The excess energy is the energy difference between the energy of the nanoparticle and the energy of the same number of A and B atoms in their pure bulk crystal [1]. For the B@A configuration, the radius of the core is Rc = R(1 − x)1/3 and therefore the excess energy is   B@A Eexc (2) = 4π R 2 γA + γAB (1 − x)2/3 .

2. Core-shell and Janus nanoparticles

In this chapter we introduce the driving forces that stabilize the different types of chemical ordering in weakly miscible nanoparticles. We focus on the energetics in the limit of T → 0, in order to discuss what the ground state configurations are. As a first step, we discuss a simplified isotropic model showing that composition-dependent transitions between different types of core-shell and Janus arrangements can take place. Then we introduce further effects appearing in real systems, such as anisotropy of surface and interface energies, edge and corner effects, and strain due to lattice mismatch between the elements. This latter effect leads to the introduction of atomic strain and pressure.

The A@B configuration is favourable over the inverted B@A B@A A@B configuration when Eexc > Eexc , giving   2/3 γA − γB > x − (1 − x)2/3 γAB . (3) The left member in equation (3) is always positive, whereas the right member is negative if x < 1/2, and becomes positive for x > 1/2. This means that for x < 1/2 the direct A@B core-shell configuration is always lower in energy. For x > 1/2, the result depends on the actual values of γA , γB , γAB and x. In this case, the inverted B@A becomes lower in energy if γA − γB < γAB and x is sufficiently close to 1. For the Janus configuration, the excess energy is given by

2.1. A simple model for the transition from core-shell to Janus nanoparticles

J Eexc = SA γA + SB γB + γAB SAB ,

Let us consider a binary nanoparticle of composition Ax B1−x and use a simple model to compare the energetics of three possible configurations, two core-shell (A@B and B@A) and a Janus configuration (see figure 2). Multishell configurations are always energetically unfavourable in this model and will not be considered. We assume that our nanoparticle is spherical of radius R. We neglect the lattice mismatch between the elements, assuming that A and B atoms share the same lattice constant. We assume also that surface energies (γA , γB ) and interface energy (γAB ) do not depend on orientation. To fix the ideas, we put γA > γB . All these quantities are per unit area. The volumes occupied by the two elements in the nanoparticle are VA = 43 π R 3 x and VB = 43 πR 3 (1 − x). In the Janus configuration, we consider a flat interface. This is a further simplifying assumption. However, qualitatively analogous results can be found considering convex or concave interfaces. A model of this kind has been used in [39] to compare core-shell and Janus configurations in Ag–Cu nanoparticles.

(4)

where SA and SB are the free surface area of the A and B parts, respectively, and SAB √ is the interface area. If x  √ 1, SA  SAB  4π R 2 x 1/2 / 3 and SB = 4π R 2 (1 − x 1/2 / 3) which gives 4π R 2 J Eexc  √ (γA − γB + γAB ) x 1/2 + 4π R 2 γB .. 3

(5)

From equations (1) and (5) we find that the A@B configuration is favourable when √  γA − γB > 3 x 1/6 − 1 γAB , (6) which is always satisfied for x  1/27, while for x > 1/27 the Janus configuration may in principle become lower in energy. For example, if we consider x = 1/2 we find J Eexc = π R 2 (2γA + 2γB + γAB ) x 1/2

4

(7)

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to the Wulff construction [40, 41] they determine faceted equilibrium shapes of nanocrystals (i.e. of nanoparticles that can be considered as fragments of bulk crystals). Therefore, the equilibrium shapes of nanocrystals are not spherical, but polyhedral (truncated octahedra and rhombic dodecahedra for fcc and bcc nanocrystals, respectively). Segregation properties in nanoparticles may therefore depend on facet orientation [42], originating in complex chemical ordering patterns. This indicates that geometric structure and chemical ordering are closely interrelated. • If the nanoparticles are small (at least up to few hundred atoms, corresponding to diameters of 2–3 nm), there is a non-negligible portion of edge and vertex sites. These edge and vertex sites are both at the external nanoparticle surface and at the interfaces between different species. Segregation at edge and vertex sites can be different than on terraces [43]. The importance of edges and vertices may imply also that the energetics derived from surface and interface energies of bulk crystals may be insufficient to characterize the energetics of these nanoparticles. • Strain can give a volume contribution to the excess energy of nanoparticles. This is true for single-component nanoparticles [44], because of the occurrence of noncrystalline geometric structures, such as icosahedra and decahedra [45]. In these non-crystalline nanoparticles, interatomic distances are non-optimal, being either expanded or compressed with respect to the distances in bulk crystals. Strain effects are even more important in binary and multicomponent systems, because of the size mismatch between different atomic species, which can introduce strong strain effects not only in non-crystalline but also in crystalline structures. Strain is different in different geometries, and this is another indication that preferred geometry and preferred chemical ordering are interrelated. In systems with non-negligible size mismatch, the definition itself of interface energy is problematic. As we will see in the following, these strain effects are extremely important for a series of nanoalloys. A typical strain effect is the size effect, which is a driving force that helps in bringing the element with larger atomic size to the nanoparticle surface [46, 47]. In order to quantify the effects of local distortions in the nanoparticles it is useful to use the local stress tensor σi , which is defined for each atom i in the nanoparticle. σi is a 3 × 3 matrix whose Cartesian components are defined as [48]

and therefore the condition for A@B being lower in energy becomes: γA − γB >

 1  4/3 2 − 1 γAB  0.76 γAB , 2

(8)

which means that if γA − γB < 0.76 γAB a transition from A@B to Janus configurations takes place with increasing x, for x smaller than 1/2. For x close to 1 one finds that the condition for A@B being lower in energy is √ 2/3 3 x − (1 − x)1/2 γ A − γB > √ γAB (9) 3 − (1 − x)1/2 which becomes γA − γB > γAB for x → 1. This last result can be understood by noting that the condition γA − γB < γAB

(10)

is the same for the dewetting of a thin film of B above an infinite A surface. Finally, it can be easily shown that when the condition (10) holds, the B@A configuration becomes favourable over the Janus configuration too, when x → 1. These results show that if γAB > γA − γB , transitions from A@B, to Janus and finally to B@A configurations may take place with x increasing from 0 to 1. On the other hand, if γAB < γA − γB , the A@B configuration prevails for all x. It must be noted that the model gives the same energy to all possible placements of the core (given that its shape is fixed). This fact, for purely statistical reasons, leads to the conclusion that off-centre cores are more likely than centred ones, so that A@B and B@A configurations should appear with somewhat off-centre cores. This is however evident when the cores are small, while for large cores the off-centre preference tends to vanish. In fact, the core can be centred with equal probability in all positions that are at a distance within R − Rc from the centre of the nanoparticle. It is also worth noting that the transitions in this simplified model are size-independent, because all energetic contributions scale as R 2 . In spite of that, these transitions are due to the finite size of the nanoparticle, and would not be recovered in an infinite system without surfaces. This model is however oversimplified, since it does not include important physical effects which may significantly alter the scenario sketched above. The main effects that are present in real systems are qualitatively discussed in the next section. 2.2. Anisotropy, edge, and strain effects

σiab =

The analysis of the simple model in section 2.1 has revealed a rather complex behaviour, with composition-dependent transitions between chemical ordering patterns. However, that model misses a series of physical effects that will be crucial in determining chemical ordering in several weakly-miscible systems.

a b 1  ∂Ei rij rij . Vi j =i ∂rij rij

(11)

In this equation, rija and rijb (with a, b = x, y, z) are the Cartesian components of the vector rij which joins atoms i and j , with rij being its modulus. Ei is the binding energy of atom i and Vi is the atomic volume. The isotropic local pressure Pi is proportional to the trace of σ i

• Surface and interface energies are in general not isotropic, but depend on orientation. Low-index surfaces (such as the [1 1 1], [1 0 0] and [1 1 0] surfaces in fcc crystals) usually present lower surface energies, and, according

1 Pi = − Tr(σ i ). 3 5

(12)

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Topical Review

In the following another invariant of the stress tensor will be useful, i.e. the quantity τi  1 yy yy τi = (σixx − σi )2 + (σi − σizz )2 + (σizz − σixx )2 3 (13)

simulation at a given temperature T in which local relaxation is performed after each move [51]. In the exchange basin hopping, an atom pair of species A and B is chosen at each step. Their chemical identity is swapped and the system is locally relaxed to the closest local minimum after swapping. If Ei and Ef are the relaxed energies of the nanoparticle before and after swapping, the move is always accepted if E = Ef − Ei  0, and accepted with probability exp(−E/(kB T )) if E > 0. The simulation temperature T is chosen to have the best efficiency of the simulations. In the systems treated here, the most efficient searches are obtained at low temperatures T  300 K [28, 32]. One of the most important points in the algorithm is the way in which the atoms A and B in the pair are chosen as candidates for swapping. The standard procedure is to choose both of them randomly. These random exchanges are reasonably efficient for nanoparticle sizes that are not too large, but they become quite inefficient when dealing with nanoparticles of sizes from several hundred atoms on. One possible way to overcome this problem is to use tailored exchanges. In tailored exchanges, atoms are chosen with different probabilities according to their local environment. A form of tailored exchanges was proposed in [52] for simulation of the strongly miscible system Pd–Pt. In [52] the pair of atoms to swap was chosen according to a probability depending on their mutual distance. This form of tailored exchanges has not been used for weakly miscible systems so far. An effective scheme for weighing probabilities in systems with tendency to form A@B core-shell or quasi-Janus structures is the following [28, 34]. In an A@B cluster, to each B atom i is given a weight wB (i)

which measures the anisotropy of the atomic stress. τi = 0 corresponds to the perfectly isotropic case. • As mentioned in the introduction, more complex chemical ordering patterns than pure core-shell or Janus are possible in nanoparticles of weakly miscible elements, such as multishell, quasi-Janus and ball-and-cup configurations. These patterns may energetically compete with A@B, B@A and Janus patterns. 3. Computational optimization of chemical ordering in weakly miscible systems

Searching for the lowest-energy structures of nanoparticles is quite a challenging task. This is even more difficult for nanoalloys, in which one has to optimize both geometric shape and chemical ordering. In general, the latter introduces a further degree of complexity, due to the problem of the huge number of homotops [49]. Homotops are isomers that share the same geometric structure (neglecting local relaxation effects) but a different chemical ordering. In principle, in a nanoalloy of N = NA + NB atoms, an estimate of the number of homotops is given by Nhomotops =

N! . NA ! NB !

(14)

This formula somewhat overestimates Nhomotops because it neglects the fact that some homotops can be symmetryequivalent. However, it is clear that Nhomotops is simply enormous for intermediate compositions even in a nanoalloy of 100 atoms. In a weakly miscible system the situation is somewhat easier than in systems with a tendency to intermixing, because the significant homotops are those that present a high degree of separation between the two elements. However the problem of finding the optimal chemical ordering is quite complex also in the case of weak mixing, so that specific algorithms have to be developed. Since full optimization at the density-functional theory (DFT) level is now limited to sizes of a few atoms [50], most results are obtained by extensive searches within atomistic models to build up a database of structures with different geometries and chemical ordering patterns. Selected structures in the database are subsequently locally reoptimized at the DFT level. In the literature, there are a few review papers that report on the recent developments in the global optimization of nanoalloys [35–37]. Here we focus only on the issue of optimizing chemical ordering within a given geometric structure, and specifically on those algorithms that are more useful when dealing with weakly miscible systems. Most searches of the optimal chemical ordering use a basin-hopping procedure with exchange moves only [36]. The basin-hopping algorithm is a Metropolis Monte Carlo

wB (i) = [nA (i) + 1]m ,

(15)

where nA (i) is the number of A nearest neighbours of atom i, and m is a positive integer (the choice of m = 2 has given good results [28, 34]). The weight given to A atoms depends on whether they stay at the nanoparticle surface or inside the nanoparticle. For A atoms inside the nanoparticle wA (i) = [nB (i) + 1]m ,

(16)

where nB (i) is the number of B nearest neighbours of atom i. For A atoms at the nanoparticle surface wA (i) = [nB (i) + 1 + 12 − nB (i) − nA (i)]m = [13 − nA (i)]m ,

(17)

as if the missing neighbours of the surface atom were B atoms (we assume that maximum coordination is 12). In this algorithm, the atoms inside the nanoparticle are chosen with higher probability if they are surrounded by neighbours of a different species. Surface atoms are chosen for exchange with a higher probability if they are of A species. Surface B atoms surrounded only by B neighbours have small (but not zero) probability of being chosen. A similar form of tailored exchanges was used also in [32]. In order to check whether the tailored exchanges were introducing undesired bias in the search, extensive comparison 6

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arrangements. In these searches, the interactions between the atoms have been modelled by second-moment tight binding potentials (SMTB) [66–68]. Selected structures of the different motifs have been reoptimized locally at the DFT level [32, 46, 56, 69]. The main results of these computational works is that a family of structures of enhanced stability has been singled out. These structures are core-shell polyicosahedra [46, 56, 70], which are made of 13-atom elementary icosahedra sharing some atom. In these clusters, the surface contains only Ag atoms, whereas the internal part is made either of Cu, Ni or Co. The atomistic models single out some magic compositions at sizes 34 and 38. For Ag–M (M = Cu, Ni or Co), the magic compositions are Ag27 M7 for size 34 and Ag30 M8 for size 38. These magic compositions correspond to minima of the excess energy at a fixed size depending on composition [46], a result that is confirmed both at the atomistic and at the DFT level [32, 46, 56]. The Ag27 M7 is highly symmetric, of D5h symmetry group, whereas the Ag30 M8 has only one symmetry plane. The D5h is called fivefold pancake (pc5). Other high symmetry polyicosahedra are found for compositions Ag32 M6 (D5h group) for size 38 and Ag32 M13 (icosahedral group) for size 45. The Ag32 M6 structure, called in the following sixfold pancake (pc6), has a slightly higher excess energy than Ag30 M8 , and at variance with other polyicosahedra, it does not contain a compact core, but M atoms form a hexagon. The Ag32 M13 structure is an anti-Mackay icosahedron [71], i.e. a cluster that has the full icosahedral symmetry, but with the external shell whose atoms are of hcp stacking on the pseudo-(1 1 1) facets of the subsurface shell. The Ih group is the largest finite symmetry group in three-dimensional space that has more than one rotation axis. Mackay and anti-Mackay icosahedra will be treated in more detail in section 5. High-symmetry coreshell polyicosahedra are shown in figure 3. High-symmetry structures are especially interesting because they are likely to be well separated in energy from higher isomers, as we will see in section 7. What are the driving forces that stabilize the core-shell polyicosahedra? Ag, Cu, Ni and Co crystals belong either to the fcc or the hcp lattices, showing that in these metals there is a clear tendency to maximize coordination. This tendency is still present at the nanoscale, and therefore we expect that the geometric structures that maximize the number of nearest-neighbour bonds should be energetically favourable. Polyicosahedra present a high number of nearest neighbour bonds. For example, at size 38, in the regular truncated octahedron (a high-symmetry piece of the fcc lattice), there are 144 nearest-neighbour bonds, while in the D6h polyicosahedral structure of figure 3(b) the bonds are 157. The same considerations apply also for the structures of sizes 34 and 45. A polyicosahedron made of atoms of a single species is not expected to be especially favourable. In fact, interatomic distances are non-optimal in this structure (as happens in icosahedra), with internal nearest neighbour bonds being shorter than surface bonds. The latter are even longer than bonds in the bulk crystal. For example, in the pure Ag pc5

with standard random exchanges have been made for the systems of [28, 34], finding that at small sizes, both random and tailored exchanges lead to the same results, the latter being faster in reaching the lowest-energy structures. For sizes above 500 atoms the efficiency gain of tailored exchanges was notable. Other schemes for optimizing chemical ordering have been proposed recently. Some of them rely on the use of symmetry [53, 54] to restrict the number of alternatives to be explored. Another scheme for selecting atomic pairs to be swapped has been proposed very recently [55], finding that it gives notable efficiency improvements over random exchanges for segregating systems with modest size mismatch. 4. Highly symmetric lowest-energy structures and chemical ordering patterns of small clusters 4.1. Ag–Cu, Ag–Ni and Ag–Co

Here and in the following sections we consider the size range up to ∼50 atoms, corresponding to diameters of less than 2 nm. For such small sizes, there is a notable variety of non-crystalline morphologies and there are several very stable structures of high symmetry. These highly symmetric motifs appear for specific sizes and compositions, which will be denoted as magic. As we will see in sections 5 and 6, these structures may undergo symmetry breaking with increasing size. In Ag–Cu, Ag–Ni and Ag–Co, there is a notable size mismatch between the atoms, more than 10% in favour of Ag. Besides this size effect, Ag is also expected to segregate to the cluster surface because of its lower surface energy and weaker cohesion [46, 47, 56]. In fact, the tendency of Ag to occupy low-coordination sites in Ag–Cu clusters is already evident at very small sizes. This has been shown by DFT calculations for sizes up to 6 atoms [57, 58] and of 11 atoms [59]. Lowestenergy configurations of clusters are planar up to 6 atoms, with rhombic, trapezoidal and triangular shapes for sizes 4, 5 and 6, respectively. For size 7, DFT calculations indicate the three-dimensional pentagonal bipyramid as the most stable structure [57, 60], with Ag occupying sites of lower coordination than Cu. For size 13, the DFT calculations show that a single Cu impurity is sufficient to change the preferred geometry from fcc cuboctahedral (the most stable shape for pure Ag [61]) to icosahedral [62]. The impurity is placed in the central site of the icosahedron. As we will see in the following, this type of transition between fcc and icosahedral structures upon the insertion of a few impurities that can be rationalized in terms of strain release, and it appears in several systems for surprisingly large sizes [63] with consequences also on the melting temperatures [64]. According to atomistic calculations using an embedded atom model, Ag–Co clusters follow an icosahedral growth pattern with Ag outside for sizes between 10 and 20 atoms [65]. Increasing the size to a few ten atoms, extensive global optimization searches have been performed to build up a database of structural motifs and chemical ordering 7

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stability of core-shell polyicosahedra. Electronic shell closure effects can be expected for Ag27 Cu7 , since 34 is a magic number for the spherical jellium model [73]. This is indeed the case. The electronic shell closure enhances the stability of this cluster and it is the origin of the large HOMO-LUMO gap calculated by DFT, of about 0.8 eV [46, 74]. This cluster also shows a quite stiff vibrational mode, which consists in a breathing of the Cu core [75]. On the other hand, the same kind of electronic effects destabilize the D6h polyicosahedral structure in Ag–Cu in favour of incomplete anti-Mackay icosahedra [69]. This is especially evident for size 40, another spherical jellium magic number. In Ag–Co, magnetic effects are stronger. These effects tend to destabilize the Co core at the DFT level compared to atomistic calculations, rendering the core-shell structure still favourable, but in a somewhat weaker way than in Ag–Cu or Ag–Ni [56]. The comparison between the pc5 structures of Ag–Co and Ag–Ni shows that the Ag external cage is practically unaltered in the two clusters with differences in the interatomic distances of the order of 0.3%, whereas the Co atoms at the apexes of the fivefold bipyramid shrink their distance by 5.5%. In parallel, the electron spin is quenched to S = 0, and the gap in the one-electron energy spectrum drops from 0.81/0.46 eV to 0.22 eV. In summary, the notable stability of core-shell polyicosahedra is also confirmed at the DFT level, with some caveats for Ag–Co in which magnetic effects are more important [46, 56, 69, 74]. Core-shell polyicosahedra are possible only for Ag-rich compositions, because for these small sizes the majority of atoms is at the cluster surface. Other composition ranges have been analyzed in a series of articles [30, 76–78]. Several polyicosahedral structures have been singled out [76]. Some of them are highly symmetric. For example, the lowest-energy structures according to the SMTB model for Ag17 Cu17 are shown in figure 5(a). It is a nicely symmetric structure of D5h group. When the number of Ag atoms is insufficient to cover the Cu core, ball-and-cup configurations may occur as lowest energy structures [30]. A nice example is given in figure 5(b). For these systems, the global optimization searches have also singled out a perfect core-shell Mackay icosahedron [79] for composition Ag42 M13 . The stability of this structure has been confirmed at the DFT level at least for Ag–Cu [32]. In figure 6 we compare the anti-Mackay and the Mackay icosahedra. They have the same core of 13 atoms, but they differ because of the shell, which contains 32 and 42 atoms in the anti-Mackay and in the Mackay icosahedron, respectively. The Mackay shell is therefore denser, with smaller intrashell nearest-neighbour distances. The stability of both icosahedra depends on the size mismatch between core and shell atoms. The anti-Mackay icosahedron requires that the atoms of the shell are larger than those of the core. On the other hand, this kind of size mismatch is not necessary in the Mackay icosahedron, and it may become a destabilizing factor when it is too large. This issue will be discussed when dealing with larger icosahedra. Comparison with experiments at these small sizes is complicated by the fact that the majority of the computational

Figure 3. High-symmetry core-shell polyicosahedra: (a) five-fold pancake (pc5) Ag27 M7 , of D5h symmetry; (b) six-fold pancake (pc6) Ag32 M6 , of D6h symmetry; (c) anti-Mackay icosahedron Ag32 M13 , of Ih symmetry. Ag atoms are shown in light grey and M atoms in orange.

structure (figure 3(a)) nearest-neighbour bond lengths on the surface vary from 2.92 to 3.08 Å, being therefore significantly expanded compared to the distance in the Ag bulk crystal (2.89 Å). On the other hand, inner bonds are heavily contracted (between 2.58 and 2.73 Å). The structure is therefore strongly strained. Moreover, one should consider also that in metals there is a bond-order / bond-length correlation, so that optimal surface bonds should be shorter than inner bonds. However, there is a simple way to improve the stability of this structure. If the internal Ag atoms are substituted by smaller atoms, the strain is released to a large extent, because surface bonds can contract. This is evident by the pressure maps (according to equation (12)) of figure 4, in which pure Ag polyicosahedra are compared to those obtained after substituting inner atoms with Cu atoms. In the pure Ag clusters, pressures on inner atoms are strongly positive (these atoms are compressed), and pressures on surface atoms are negative (these atoms suffer tensile strain). In the Cu@Ag clusters, all pressure values are much closer to 0. The discussion reported above does not consider specific quantum effects that may either enhance or decrease the 8

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Figure 4. Pressure maps of pure Ag (left column) and of core-shell Cu@Ag (right column) polyicosahedra, for the three geometric structures of figure 3. The clusters in the right column are obtained from those of the left column after substituting the inner Ag atoms with Cu atoms: (a) pc5 Ag34 → Ag27 Cu7 ; (b) pc6 Ag38 → Ag32 Cu6 ; (c) anti-Mackay icosahedron Ag45 → Ag32 Cu13 . Results are obtained by the SMTB potential of [72]. All structures are locally relaxed. Pressure values are in GPa. A notable strain release is obtained after substitution with Cu, as shown by the decrease of the absolute values of atomic pressures.

results are related to neutral clusters, while the experiments are able to detect gas-phase charged clusters. On the computational side, the search for the lowest-energy structures of charged clusters requires DFT global optimization, which is already very cumbersome for clusters of a few ten atoms [50]. On the experimental side, the characterization of neutral gas phase clusters is extremely challenging. The structure and stability in neutral and charged clusters of the same size may differ significantly when electronic shell closure effects (which depend on the number of free electrons) are more important than geometric effects. We have already seen that this point is quite complex and requires specific analysis for different sizes and compositions. The importance of the shell closure effects has been experimentally demonstrated in the size range up to 50 atoms for Ag cations doped by a single Cu or Ni impurity [80]. Keeping in mind these caveats, the comparison with the experiments allows us to confirm at least some aspects of the scenario emerging from the calculations. First of all, the tendency of Ag to occupy the cluster surface has been confirmed for both Ag–Cu and Ag–Co [65, 81]. In [81] mass-spectrometry analysis of the Ag–Cu clusters produced by an inert-gas aggregation source was performed. The abundances of the different compositions at fixed size were measured for a series of sizes, including 34, 38, and

45. In this kind of experiment, it is difficult to evaluate to what extent the abundances of the different compositions are due to the condensation kinetics in the source or reflect the stability of the different clusters at equilibrium. The condensation kinetics have of course at least one evident effect, because Cu-rich clusters are more abundant than Agrich clusters [81], due to the higher condensation temperature of Cu. Nevertheless, the measured prominent peaks single out many of the magic compositions predicted by the theory [81], including Ag32 Cu13 , Ag32 Cu6 and Ag17 Cu17 . On the other hand, some of the most stable structures predicted by the theory are not abundant, for example Ag27 Cu7 and Ag30 Cu8 . This may reflect the fact that experimental clusters are charged, while calculations refer to neutral clusters. Moreover, the condensation kinetics in the vapour alters the abundances favouring Cu-rich compositions. Finally, as recently noted in [82], equilibration within a given cluster very strongly depends on its size and composition, so that the clusters observed in a limited time scale may not represent the global minimum configurations. 4.2. Au–Co

In Au–Co, size mismatch and cohesive energy are driving forces for the formation of Co@Au structures. This trend 9

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Therefore Au optimal shells allow a smaller space to inner cores than Ag optimal shells, even though Au and Ag crystals have almost the same lattice constant. This effect destabilizes the pc5 and the anti-Mackay structures. In order to stabilize the pc5 and the anti-Mackay structure, the inner atoms should be even smaller than Co atoms. At variance with the results for Ag–Cu and Ag–Ni, these results for Au–Co have been obtained at the atomistic level and are still to be validated by DFT calculations. In fact, DFT calculations have been performed up to now only to locally relax pc5, pc6 and anti-Mackay core-shell structures, without comparison with other isomers [56]. The DFT calculations however hint of a good stability of these structures, without the destabilizing effects related to magnetism that have been found in Ag–Co. 4.3. Au–Pt

At variance with the systems considered so far, in which lattice mismatch is more than 10%, for Au–Pt there is a much smaller mismatch, being (aAu − aPt )/aAu = 3.6% (aAu and aPt are the lattice constants of Au and Pt, respectively). Because of its lower surface energy and cohesion, a surface enrichment in Au is expected. Also the size effect, which is however weaker than in the systems of sections 4.1 and 4.2, should contribute in bringing Au to the cluster surface. Another difference with the systems of sections 4.1 and 4.2 is that bulk Au–Pt presents a miscibility gap at low temperatures, but there is a hightemperature interval in which solid solutions are possible for all compositions [85]. The structures of Au–Pt nanoalloys in the size range up to 50 atoms have been studied by global optimization searches within atomistic models [33, 86, 87] and by DFT calculations comparing selected isomers obtained in the atomistic searches [33, 86–90]. These studies confirm the preference for Pt@Au arrangements. However, at variance with the systems with large atomic size mismatch, Au-rich clusters do not show a predominance of polyicosahedral structures. At intermediate compositions, global optimization searches within the SMTB model have singled out icosahedral fragments (especially for sizes below 50), fcc twins, capped decahedra, and the Leary tetrahedron for size 98 [87]. The Pt@Au pattern has been confirmed up to size 40 atoms by DFT calculations [86, 88]. For size 55, combined SMTB global-optimization searches and DFT calculations have singled out a competition between two main motifs: a Mackay icosahedron (for which size 55 is a geometric magic size) and an asymmetric capped decahedron. These structures are shown in figures 8(a) and (e), respectively. The asymmetric capped decahedron differs from the symmetric capped decahedron (figure 8(d)) because the two vertex atoms are displaced to a position in the mid plane of the cluster. Other significant motifs are the cuboctahedron and the Ino decahedron (figures 8(b) and (c), respectively). The asymmetric capped decahedron is favourable at intermediate compositions, whereas the icosahedron is more stable on both Au-rich and Pt-rich sides. The energetic stability of the asymmetric capped decahedron can be rationalized again in terms of local atomic

Figure 5. (a) Lowest-energy configuration of Ag17 Cu17 according to SMTB model [76], shown from two different perspectives. The structure is a pc5 (D5h symmetry group). (b) Ball-and-cup configuration of Ag21 Co13 . This is the global minimum according to the SMTB model used in [70]. The geometric structure is a polyicosahedron, essentially the same pc5 as in Ag27 M7 (figure 3(a)). However, 6 Co atoms appear at the cluster surface so that the horizontal symmetry plane is lost and the symmetry group becomes C5v . The same kind of structure was found for Ag21 Cu13 in [30].

results from both DFT [47] and atomistic calculations [28, 83]. Here we focus on the results of global optimization searches within the SMTB model of [28, 83]. Also for Au–Co the global optimization runs confirm the preference for core-shell structures on the Au-rich side [83]. However, the geometries are rather different from those of Ag–Cu, Ag–Ni and Ag–Co (section 4.1). In fact, for Au27 Co7 the lowest energy structure is not the pc5, but a low-symmetry structure with only six internal atoms and one Co atom at the surface. This structure becomes a perfect core-shell for composition Au28 Co6 , as shown in figure 7(a). For Au32 Co13 the lowestenergy structure is not the anti-Mackay icosahedron, but an incomplete Mackay icosahedron of ball-and-cup structure. Only for Au32 Co6 do we recover the pc6 as the lowest-energy structure. However the pc6 is only marginally more stable than an incomplete core-shell structure, whose completion is found for composition Au32 Co7 . That structure is shown in figure 7(b). It is a chiral structure, of C5 symmetry group. The tendency of pure Au cluster to present chiral structures is well known [84]. In the bimetallic case, the chiral structures are often of high symmetry, presenting several rotational symmetries. This is related to the effect of size mismatch, as we will see in the following. Compared to Ag–Cu, Ag–Ni and Ag–Co systems, Au–Co perfect core-shell structures have smaller cores. This is related to the stronger bond-order/bond-length correlation in Au, which causes a stronger contraction of Au surface bonds. 10

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Figure 6. (a) Anti-Mackay icosahedron Ag32 M13 . (b) Mackay icosahedron Ag42 M13 .

In summary, Pt@Au structures are the most stable in this size range according to all calculations, a result that is also confirmed by global optimization searches by genetic algorithms [91], and, for larger sizes, by an analytical embedded-atom model [92].

5. Symmetry breaking in nanoparticles of A-rich compositions

In this section and in section 6 we consider larger nanoparticles, containing from a few hundred to several thousand atoms (up to diameters of ∼5 nm). These nanoparticles are more easily produced and characterized in the experiments than smaller clusters. Moreover, it is reasonable to assume that the interaction of the nanoparticles with the environment becomes less important as the nanoparticle size increases, because the proportion of atoms on the nanoparticle surface decreases with size. Therefore the comparison between theoretical results on isolated nanoparticles and experimentally produced nanoparticles becomes more meaningful with increasing sizes. Of course this statement must be taken with care, since there are environments that may alter the segregation patterns even in the limit of large objects (for example, ligands that bind more strongly to A atoms than to B atoms may drive A atoms to the nanoparticle surface). In the following, we define A-rich compositions as those compositions that contain no more B atoms than those that are barely sufficient to cover the A core by a single atomic layer. Therefore the meaning of A-rich is size-dependent. In a Mackay icosahedron of 309 atoms, the outer atomic shell contains 162 atoms (more than half of the total). It follows that the A-rich regime means 162 B atoms or less (i.e. 52% B atoms or less). On the other hand, in a Mackay icosahedron of 6525 atoms, the outer shell contains 1442 atoms, so that the A-rich regime means 22% B atoms or less.

Figure 7. Lowest-energy structures according to the SMTB model [28, 83]. (a) Low-symmetry structure of Au28 Co6 . (b) C5 structure of Au32 Co7 . Both structures are chiral.

pressure, as shown in figure 9 [33]. The central site is heavily compressed in the icosahedron. In the asymmetric capped decahedron all pressures are closer to zero. This compensates for the lower number of nearest neighbour bonds in this structure. On the other hand, in the symmetric capped decahedron (figure 8(d)) the presence of vertex atoms induces a high compressive strain in subvertex atoms that makes the structure unfavourable [33], so that a less symmetric arrangement is lower in energy. This is an example of symmetry breaking that allows strain release, a phenomenon that will be encountered in several cases in the following sections when dealing with larger nanoparticles. 11

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5.1. Symmetry breaking in Mackay icosahedra

The Mackay icosahedron is an onion-like structure, with the atoms arranged in concentric layers of monoatomic thickness1 . The geometric magic numbers of Mackay icosahedra are those at which concentric layers are completed. They are given by the formula NM (n) = (10n3 − 15n2 + 11n − 3)/3

(18)

where n  1 is the number of concentric layers. By using this formula, it turns out that the series of Mackay geometric magic sizes is NM = 1, 13, 55, 147, 309, 561, 923.... In this section we concentrate on shells of monoatomic thickness, i.e. B atoms fully occupy only the outer concentric layer of the Mackay structure. Taking Cu@Ag as an example, we have compositions Ag12 Cu1 , Ag42 Cu13 , Ag92 Cu55 , Ag162 Cu147 , Ag252 Cu309 , Ag372 Cu561 for perfect Ag shells of monoatomic thickness. Most of the results presented here will refer to Cu@Ag, but the same kind of behaviour holds also for Ni@Ag, Co@Ag, Co@Au, and Ni@Au, which share the same segregation trends and values of lattice mismatch above 10%, with the ‘large’ atoms having the tendency to surface segregation. In weakly miscible systems with smaller lattice mismatch, such as Pt@Au and Ni@Cu, this behaviour can be recovered but for larger nanoparticle sizes. In figure 10 the series of Cu@Ag Mackay icosahedra of increasing size is shown. The first icosahedron (figure 10(a)) has composition Ag42 Cu13 . This structure has already been considered in section 4.1. According to global optimization searches within the SMTB model, it is the lowest-energy structure for this composition [32]. This result has also been confirmed at the DFT level by comparing selected isomers [32]. The second icosahedron and the third icosahedron have compositions Ag92 Cu55 and Ag162 Cu147 , respectively. From the pressure maps in figure 10 it turns out that, as the size increases, larger and larger positive pressures appear in Ag atoms, especially for those that are in the middle of the facets. At the same time, negative pressures appear in the Cu atoms of the subsurface layer. These pressures are more negative in subvertex sites. Now, if an atom at a given site feels a negative pressure, it means that it is suffering tensile strain, so that its optimal size at that site would be larger than what it actually is. Therefore, a strain release would be obtained by substituting that atom by a larger atom, i.e. in our case, by substituting a Cu subvertex atom with an Ag atom. On the other hand, if the atom feels a positive pressure, the strain is compressive, and substitution by a smaller atom may be beneficial. This means that the strain on the Ag atoms at the facet centres can be released by substituting them with the smaller Cu atoms. Strain release is associated to more optimal bond lengths, and this can give an energy gain. Therefore, exchanging subvertex Cu atoms and Ag atoms at the facet centres may be beneficial for the structure. However, these exchanges decrease the number of Cu–Cu and of Ag–Ag bonds to the

Figure 8. Structural motifs of Au–Pt for size 55. Representative clusters of each motif are shown for composition Au32 Pt23 . Au and Pt atoms are shown in yellow (light grey) and purple (dark grey), respectively. Each cluster is shown in two views. (a) Mackay icosahedron; (b) cuboctahedron; (c) Ino decahedron; (d) symmetric capped decahedron; (e) asymmetric capped decahedron. The most energetically stable motifs are the Mackay icosahedron and the asymmetric capped decahedron. Reprinted with permission from [33]. Copyright 2013 Elsevier.

1 These layers are usually called shells in the literature, but we choose here not to use this word to avoid confusion, because we use the word shell for the whole complex of B atoms in A@B structures.

advantage of Ag–Cu bonds, and since Ag and Cu are weakly miscible, this has an energetic cost. 12

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P

Au 32 Pt 23

40 30 20 10 0 -10 0

10

20

30

40

50

i

Figure 9. Local pressure for the asymmetric capped decahedron (circles) and the icosahedron (crosses) in Au32 Pt23 . Atoms are ordered by increasing distance from the cluster centre. In the pictures below the graph, cross-sections of the icosahedron (left) and of the asymmetric capped decahedron (right) are shown, where atoms are coloured according to their local pressure. In the icosahedron, the central atom is in red indicating a strong positive pressure. The intermediate shell is in green corresponding to a smaller positive pressure, and the outer shell is in blue indicating negative pressure. In the asymmetric capped decahedron all inner atoms have small positive pressure, so that they are in green. Reprinted with permission from [33]. Copyright 2013 Elsevier.

The balance of these two opposite effects is against the exchanges for small nanoparticle sizes, but becomes favourable for the exchanges as the nanoparticle size increases and, correspondingly, the strain in the perfect core-shell structure becomes larger and larger. In fact, if we optimize chemical ordering by exchange moves, we find that the perfect core-shell structure is the lowest-energy Mackay icosahedron for Ag42 Cu13 and Ag92 Cu55 , while for Ag162 Cu147 the optimal chemical ordering within the Mackay icosahedral motif (shown in figure 11) presents a seven-atom Cu patch at the nanoparticle surface and, correspondingly, seven Ag atoms in subvertex positions. This chemical ordering allows a notable strain release, as results from the pressure map. An energy gain is thus obtained by breaking the perfect icosahedral symmetry of the Cu@Ag icosahedron in such a way that the only remaining symmetry element of the final structure is a mirror plane. We note finally that there is another, more efficient way to improve the energetics of Ag162 Cu147 by symmetry breaking. This symmetry breaking involves the Ag shell only and is reported in figure 12, which shows the lowest-energy structure obtained by full global optimization within the SMTB model. This structure has the same Cu core as the Mackay icosahedron,

but this core is covered by an anti-Mackay concentric layer (containing 30 atoms less than a Mackay arrangement) which is in turn partially covered by an Ag island of 30 atoms. Anti-Mackay structures are treated in detail in the next section. 5.2. Breaking of mirror symmetries in anti-Mackay icosahedra: the chiral icosahedron

As we have seen in the previous section, when B atoms are larger than A atoms, the A shell can become too dense in a Mackay icosahedron, so that its atoms are subjected to compressive strain. This strain can be released as in Ag162 Cu147 (figure 12), by creating a less dense concentric layer in contact with the core and putting the excess atoms in an island above that layer. It is possible to form B shells of monoatomic thickness and lower density that preserve the full symmetry of the icosahedron. One well-known possibility is given by the antiMackay shells, whose structure is explained and compared to Mackay shells in figures 13(a)–(b). The formation of an antiMackay shell corresponds to placing the atoms on hcp-like instead of fcc-like sites of the substrate (see figure 13(b)), i.e. of the core. In this case, for an A@B icosahedron with n 13

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Figure 10. Local atomic pressure in Mackay Cu@Ag icosahedra of increasing size. Pressures are given in GPa. The external Ag shell is of monoatomic thickness. (a) Ag42 Cu13 ; (b) Ag92 Cu55 ; (c) Ag162 Cu147 . Each cluster is shown in four views. From left to right: cluster surface, pressure map of the surface, cluster cross-section (Ag atoms in grey, Cu atoms in orange), pressure map of the cross-section. As size increases, significantly negative pressures appear in the core, especially for the Cu atoms that are in contact with the shell. Moreover, significantly positive pressures appear in the shell, for Ag atoms in the middle of the facets. Note that the pressure on the central atom is always strongly positive and larger than 6 GPa (it is in the range 10–20 GPa). The scale has been limited to the interval [−6, 6] GPa to have a better colour contrast. Only the pressure on the central atom is outside this interval.

concentric layers, the layers from 1 to n − 1 are occupied by A atoms and have Mackay structure, while the n-th layer (the shell) is occupied by B atoms and its geometric structure is different, and less dense. In fact, on the core surface there are fewer hcp-like sites than fcc-like adsorption sites, because of the finite size of the facets. Therefore, a smaller number of atoms can be accommodated, so that the magic numbers for icosahedra with anti-Mackay shells (in the following named anti-Mackay icosahedra for the sake of brevity) are NaM (n) = NM (n) − 10(n − 2)

clusters, incomplete Mackay shells are more stable even for anti-Mackay magic numbers (for example, in 45 atom pure Cu or Ag clusters, incomplete Mackay shells win by 3.52 and 3.06 eV, respectively, according to DFT calculations [32]). But for binary nanoparticles the situation may be different, as has been shown in Cu@Ag, Ni@Ag, Co@Ag, Co@Au, Ni@Au [32]. In these systems, the lattice mismatch in favour of the shell element (more than 10%) is a driving force for an outer shell of lower density. The global optimization and DFT results in [32] confirm that anti-Mackay shells are favourable for sufficiently small icosahedra at magic numbers and compositions, for example for Ag72 Cu55 . Global optimization searches within the SMTB model confirm that defective anti-Mackay shells are possible for non-magic sizes or compositions [31]. However, as size increases, another motif comes into play: the chiral icosahedron (shown in figure 13(c) for Ag132 Cu147 ). The chiral icosahedron shares the same magic sizes and compositions with the anti-Mackay icosahedron. According to SMTB and DFT calculations, for Ag132 Cu147 the chiral icosahedron is not the global minimum, but it is separated from the anti-Mackay achiral structure by a

(19)

with n  3 and NM (n) given by equation (18). This formula corresponds to the sequence NaM = 45, 127, 279, 521, 873, ..., with compositions (referring to Cu@Ag as an example) Ag32 Cu13 , Ag72 Cu55 , Ag132 Cu147 , Ag212 Cu309 , Ag312 Cu561 . In single-component clusters, the anti-Mackay shell is usually energetically unfavourable. Since surface nearestneighbour distances are expanded in single-component Mackay icosahedra, an even denser shell may become favourable when the size of the icosahedron increases, instead of the less dense anti-Mackay shell. In single-component 14

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Figure 11. Mackay Cu@Ag icosahedron Ag162 Cu147 . In the top row we present the perfect core-shell structure, as in figure 10(c). In the bottom row, we present the best structure obtained starting from the perfect core-shell and allowing only exchange optimization moves. Seven Cu atoms (in orange) appear on the nanoparticle surface, and correspondingly seven Ag atoms are now inside the nanoparticle. This causes a notable strain release at the nanoparticle surface, as shown by the pressure map. The seven inner Ag atoms (not visible in the figure) occupy subvertex positions, which are those of lowest negative pressure in the perfect core-shell structure. The transformation from the perfect core-shell structure to the structure of the bottom row implies a considerable lowering of the symmetry, from the Ih group (120 elements) to the Cs group (2 elements, reflection about a single mirror plane). The structure of the bottom row is lower in energy by 0.23 eV according to the SMTB calculations. Pressures are given in GPa.

size increases? The mechanism underlying this transition can be understood by a simple bond-counting argument that follows from the analysis at the geometric structures of achiral and chiral shells. The transformation of an achiral anti-Mackay shell into a chiral shell takes place via the concerted rotation of the triangular islands that cover the facets of the core (see figure 13) by an angle close to 19.5◦ [95]). This rotation (which breaks all mirror symmetries) slightly displaces shell atoms from their anti-Mackay equilibrium sites on the facet, but at the same time it creates new Ag–Ag nearest-neighbour bonds along the facets’ edges, as shown in figure 14. For example, in Ag132 Cu147 , two new nearest-neighbour bonds are formed at each edge, but they are quite stretched, since the distance between the new neighbours is 3.17 Å (to be compared to the equilibrium distance in bulk Ag crystals, which is of 2.89 Å). However, this situation improves at larger sizes. For Ag212 Cu309 , the new bonds along facet edges are stretched by a smaller amount, being of 3.01 Å and 3.13 Å [32]. For Ag312 Cu561 the new bonds becomes closer to the optimal length, being of 2.91 Å and 3.03 Å. In summary, for Ag132 Cu147 , the energy gain due to the new stretched bonds is overcompensated by the loss due to the displacement of Ag atoms from their ideal adsorption anti-Mackay sites. For larger sizes, the energy gain of the new bonds prevails, causing the transition to the chiral structure. The same kind of behaviour is found also in Ni@Ag, Co@Ag, Co@Au and Ni@Au, that share common features

Figure 12. Lowest-energy structure of Ag162 Cu147 according to the SMTB model. It is an anti-Mackay icosahedron covered by a further Ag island of 30 atoms.

quite small amount of energy, less than 0.5 eV. Incomplete Mackay icosahedra are much higher in energy (more than 7 eV) [32]. In the chiral structure, the external Ag shell is transformed with respect to the anti-Mackay structure, assuming the shape of a snub icosahedron [93, 94]. All mirror symmetries are broken, but rotational symmetries of the icosahedral group are preserved. Increasing size to Ag212 Cu309 , the chiral structure becomes the lowest in energy, a result which is confirmed for the next magic number, Ag312 Cu561 . The energy difference in favour of the chiral icosahedron increases with size [32]. Why is symmetry breaking from the anti-Mackay to the chiral icosahedron energetically favourable as the nanoparticle 15

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Figure 13. Top row: (a) Mackay icosahedron of 309 atoms, for composition Ag162 Cu147 . An external single Ag shell covers a Cu icosahedral core. (b) Anti-Mackay icosahedron of 279 atoms, for composition Ag132 Cu147 . Its Cu core is the same as in the Mackay icosahedron. (c) Chiral icosahedron of 279 atoms, for composition Ag132 Cu147 . Its outer shell is obtained by rotating all triangular Ag islands of the anti-Mackay shell by the same angle so that all mirror symmetries are broken. Neglecting local relaxations, the Cu core preserves the achiral icosahedral symmetry of the previous nanoparticles, while the Ag shell assumes a clearly different structure. Bottom row: the stacking of the three outer atomic shells is shown for the nanoparticles of the top row. Ag atoms are represented by larger spheres (in grey). Two layers of Cu atoms are shown. The lowest layer is in yellow (lighter grey). The outer Ag shell is in fcc-like and hcp-like stacking for the Mackay and anti-Mackay nanoparticles, respectively. In the anti-Mackay nanoparticles, the Cu atoms of the third layer are covered by the Ag atoms of the external shell. In the chiral nanoparticle, the Ag triangles are rotated so that atoms are displaced from their anti-Mackay sites. Reprinted with permission from [32]. Copyright 2010 American Chemical Society.

with Cu@Ag. In these systems, the chiral icosahedron is already favourable for smaller sizes than in Cu@Ag, as confirmed by DFT calculations [32]. This derives from the fact that the transformation to the chiral structure implies a slight contraction of the shell, which is favoured by a smaller size of the core, i.e. by a more significant lattice mismatch. Among these systems, Cu@Ag has the smallest mismatch. The chiral icosahedron is not the only high-symmetry chiral structure with perfect core-shell chemical ordering and shell of monoatomic thickness. Global optimization searches in the Ag–Cu system have found chiral structures belonging to the C5 symmetry group, which has a single fivefold rotation axis. Magic numbers for there structures are Ag67 Cu39 and Ag107 Cu85 . The latter is shown in the top row of figure 15. Ag107 Cu85 is of size 192, which is the geometric magic size of a Marks decahedron [96]. However, the Marks decahedron is higher in energy by several eV than the C5 chiral structure [32]. Achiral symmetric structures in the same size range belong to the family of polyicosahedra. At variance of the polyicosahedra treated in section 4.1, these polyicosahedra are made of interpenetrating icosahedra of 55 atoms. Magic numbers for these structures are Ag90 Cu56 and Ag102 Cu75 , shown in the middle and bottom rows of figure 15, respectively.

6. Morphological instabilities in nanoparticles of B-rich compositions

In this section, we start considering a small core of A atoms included in the shell of B atoms, and analyze the lowest-energy structures as composition changes in such a way that the size of the core increases, but the total number of atoms is fixed. Since B atoms are in the majority and we often consider the case of a few A atoms inside the nanoparticle, the shell of B atoms is also called the matrix, and A atoms are mentioned sometimes as impurities. The preferential shape and placement of the core are discussed, with reference to their interplay with the overall geometric structure of the nanoparticle. We show that, in icosahedral structures, morphological instabilities of the core shape occur as the core size increases [28, 34]. This instability causes symmetry breaking in the structure. Even though this morphological instability is not occurring during the nanoparticle growth, it has a physical origin which bears some analogy with the onset of the Stranski–Krastanov (SK) growth mode in thin films [41, 97]. 16

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Figure 14. Atomic configurations along facet edges in Ag132 Cu147 for the anti-Mackay achiral structure (left) and for the chiral icosahedron (right). The transformation to the chiral icosahedron creates two new nearest-neighbour A–C and B–D bonds. These bonds are however quite stretched in this case. Stretching of additional bonds along edges is smaller in larger chiral icosahedra. Reprinted with permission from [32]. Copyright 2010 American Chemical Society.

By increasing the number of A atoms at fixed total nanoparticle size, the shape and placement of the core change significantly. Small cores are placed at the centre of the icosahedron and have a quasi-spherical shape. At magic compositions, these small cores are perfect Mackay icosahedra. However, when the core exceeds a critical size, its shape changes. It becomes much more irregular and off-centred, extending asymmetrically towards the nanoparticle surface and driving the icosahedron towards a quasi-Janus chemical ordering pattern [28, 34, 99]. Icosahedra with off-centre cores were found in Monte Carlo simulations of Ag–Ni [23]. These simulations considered size 309 and composition 75%at in Ag. In order to show how this phenomenon takes place in detail, we consider specific examples [28]. As a first case we consider a Mackay icosahedron with fixed n = 8, containing 1415 atoms in total. In this icosahedron there are a number of A atoms that correspond to the completion of k inner concentric layers, with k from 1 to 5. The results of the search for the optimal chemical ordering are shown in figure 16. For k = 1 (not in the figure), the single A atom is preferentially placed at the central site [98]. For k = 2, the 13 A atoms form a small icosahedron at the centre of the nanoparticle. But when k > 2, a different core shape starts to appear in Ni@Ag. The 55 Ni atoms are not all contained within the third concentric layer, but some of them are now placed in the 4th layer, being correspondingly substituted in the 3rd layer by Ag atoms. When k increases to 4 and 5, the irregular core shape appears also in Cu@Ag, Co@Ag and Co@Au, with the development of irregular shapes that extend asymmetrically from the centre towards the surface of the nanoparticle, filling some parts of the 6-th and 7-th concentric layers. This is another occurrence of a symmetry breaking phenomenon which is caused by the accumulation of strain. The asymmetric

6.1. Morphological instability of the cores in Mackay icosahedra

In single-component nanoparticles, the atom at the central site of an icosahedron is highly compressed [98] with local pressure P positive and quite large (several ten GPa). In this case, eliminating the central atom and leaving a vacancy at its place can be energetically favourable if the icosahedron is sufficiently large, since the pressure on the central atom increases with size. This has been shown by SMTB calculations [98] for noble metals, and then confirmed, in the case of Ag icosahedra, also at the DFT level [34]. By following the same line of reasoning that leads to the stabilization of core-shell polyicosahedra (see section 4.1) we conclude that introducing a smaller atom at the centre of the icosahedron can allow a notable strain release [64], as it happens when a vacancy is created, but with the advantage of not eliminating first-neighbour bonds. This notable strain release is obtained for Cu, Ni, Co in Ag or Au [28, 64]. The local pressure still remains positive (compressive strain), but strongly decreases compared to the case in which the centre is occupied by an atom of the same species as the matrix. For the small impurity, the central site is by far the most favourable place [21] in the nanoparticle. This specific feature of the icosahedron has important consequences on the structures of the cores. Searches for the optimal chemical ordering in B-rich Mackay icosahedra have been performed by the basinhopping algorithm with exchange moves only [28] within the SMTB model, for the systems Cu@Ag, Ni@Ag, Co@Ag and Co@Au. DFT calculations for Cu@Ag, Ni@Ag and Co@Ag [34] have subsequently confirmed these results, showing a quite good agreement with the atomistic results. The main result of these calculations is the prediction of a strain-driven morphological instability of icosahedral cores. 17

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Figure 15. Top row: chiral structure of Ag107 Cu85 . Middle row: achiral pentaicosahedral structure of Ag90 Cu56 . Bottom row: achiral eptaicosahedral structure of Ag102 Cu75 . From left to right, for all structures the first and second snapshots are side views (the second showing the structure of the inner Cu core), and the third is a top view. Reprinted with permission from [32]. Copyright 2010 American Chemical Society.

is associated to a significant distance between the geometric centre of the whole nanoparticle and the geometric centre of the core [28]. The term morphological instability does not mean that centred cores are unstable configurations. Indeed, these are locally stable minima, whose energy is however much higher than the energy of the optimal configuration. The term morphological instability means that the centred morphology does not persist as the lowest-energy morphology when the proportion of A atoms increases in the nanoparticle, i.e. the term instability is referred to the overall core morphology and not to the local minimum corresponding to the centred core. This resembles the use of the term morphological instability in the case of some crystal growth phenomena. In crystal growth, compact shapes are unstable during the growth

core extends very close to the nanoparticle surface because A atoms gain in occupying subsurface positions with respect to internal off-centre positions [21, 27]. On the other hand, none or very few A atoms appear at the nanoparticle surface. The morphology of the overall nanoparticle is therefore quasiJanus, with a core that is placed asymmetrically within the nanoparticle and is covered by a very thin shell of B atoms on one of its sides. On the other hand, if the number of atoms in the shell is increased while keeping the size of the core fixed, the morphological instability is gradually suppressed, recovering the centred and more compact core shape for sufficiently thick shells (see figure 17). We note that one can distinguish a weak form of morphological instability, when only sites of the (k +1)th layer are occupied by A atoms, from a fully developed form, which 18

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Figure 16. Lowest-energy configurations of icosahedral A@B nanoparticles of fixed size (number of layers n = 8, total number of atoms

N = 1415) for increasing size of the core. From top to bottom: core of 13 atoms (number of layers k = 2), of 55 atoms (k = 3), of 147 atoms (k = 4), of 309 atoms (k = 5). B atoms are shown as small spheres so that A atoms (bigger spheres) are visible. From [28]. Copyright 2013 by The American Physical Society.

process, developing mounds and dendrites [41, 100]. Further analogies between the core morphological instability and phenomena occurring in crystal growth are discussed below. As anticipated, the morphological instability has a physical origin which is related to strain release. This is evident in the pressure maps of figure 18, in which atomic pressures in icosahedra with centred symmetric and with offcentre asymmetric cores are reported, for a core size at which the off-centre core is energetically favourable. The pressure map of the nanoparticles with centred core shows that the atoms of the outer concentric layer of the core have a strongly negative pressure, corresponding to a strong tensile strain. For the nanoparticle of size 1415, the negative pressure extends also to inner parts of the core. Moreover,

the atoms of the inmost concentric layer of the shell present a significantly positive pressure. This shows that the atoms on the two sides of the core-shell interface present a large pressure gradient. The strain on these interface atoms is indeed anisotropic. By diagonalizing the strain tensor on each atom, it is possible to separate with good approximation a radial and a tangential contribution to the strain and therefore to the pressure. The results concerning the anisotropy of strain are reported in figure 19. Form the calculation of τi (equation (13)) it follows that there is a strong anisotropy for the Ag atoms at the interface with the core. This reveals that these Ag atoms feel a notable compressive tangential strain, whereas the radial component of the strain is small. This behaviour is due to the radial 19

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Figure 17. Lowest-energy configurations of icosahedral A@B nanoparticles of fixed core size (number of layers k = 4, corresponding to 147 core atoms) for increasing number of shell atoms (and size of the nanoparticle). From top to bottom: N = 561 (number of layers n = 6), N = 923 (n = 7), N = 1415 (n = 8). Shell atoms are shown as small spheres so that core atoms (bigger spheres) are visible. From [28]. Copyright 2013 by The American Physical Society.

contraction of the Ag atoms to adapt to the small Ni core, which causes a tangential compression of the Ag atoms in the concentric layer at the interface. In the structures with asymmetric off-centre cores strain is almost completely released with most Pi close to zero (see the pressure maps in figure 18). No significant strain anisotropy is present any more. The stability of centred and off-centre cores is therefore determined by two competing factors. Off-centre cores much better release the strain if they are sufficiently large. On the other hand, in off-centre cores there are fewer A–A bonds and more A–B bonds than in centred cores. This has an energetic cost, because A–A bonds are stronger in these weakly miscible systems. Besides the gain due to strain release, this energetic cost is also compensated by the advantage of putting

a considerable number of A atoms in subsurface positions, which are quite favourable in these systems [21]. The morphological instability of the cores bears some analogy with the SK instability occurring in heteroepitaxial thin film growth [41, 97]. In heteroepitaxial growth, atoms of species B are deposited on a substrate of species A. As the number of deposited layers increases, strain (due to lattice mismatch) accumulates so that the film breaks up in its top part. The growth mode changes from layer-by-layer to threedimensional, with the formation of mounds. The SK instability occurs when there is a (free) energy gain in starting the (k + 1)th layer before completing the kth layer [97]. This explanation of the SK instability therefore relies on purely energetic (equilibrium) arguments and does not involve the growth kinetics. 20

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is energetically preferable to place further A atoms in the (k + 1)th concentric layer rather than completing the kth layer. Therefore both SK instability and core instability in icosahedra share a common origin. There are however some differences. In the SK instability, the strain corresponds to either compression or stretching, depending on the sign of the lattice mismatch. In the core instability, the strain changes sign, from compression, which is specifically inherent to the sites in the central part of the icosahedron [101], to the stretching which originates with the increase of the number of atomic layers in the core. Therefore the core instability requires a specific sign of the lattice mismatch, with A atoms being smaller than B atoms. Moreover, in the SK instability, the strain is caused by the lattice mismatch at the bottom layer of the thin film, and it is released by breaking the thin film in its outer layers. In the core instability, strain is originated by lattice mismatch at the interface between core and shell. This strain is released by breaking the symmetry of this interface. 6.2. Off-centre cores in fcc nanocrystals and in decahedra

At variance with icosahedra, fcc nanocrystals and decahedra do not present heavily compressed central sites. The preferential placement of small impurities is in subsurface sites, especially in subvertex sites, as demonstrated by both atomistic and DFT calculations [21, 27, 28]. Metal surfaces have a tendency to contract. Substituting small impurities in subsurface sites allows surface atoms to better contract. Optimization of chemical ordering thus leads naturally to quasi-Janus patterns, as shown in figure 20. This has been verified by searches within atomistic models for a series of systems with large lattice mismatch, such as Cu@Ag, Ni@Ag, Co@Ag, and Co@Au. The inner interface of these cores has been found to present preferentially close-packed facets, an effect attributed to the smaller size of core atoms [102]. However, there are indications that the same effect is not limited to systems with large size mismatch, but it appears also in systems with quite small lattice mismatch, when there are other physical effects that favour the subsurface placement of impurities. In recent calculations [103], Ir@Pt fcc truncated octahedra have been considered. Pt–Ir is a weakly miscible system presenting a lattice mismatch of about 2%, so that strain relaxation energy is estimated to be small [103]. Pt–Ir was modelled using a rigidlattice model. The free-energy concentration expansion method was used to model the system thermodynamics, with DFT-based coordination-dependent bond-energy variations as the input. Site-specific concentrations in the Pt–Ir truncatedoctahedron nanoparticles showed the stabilization of off-centre asymmetric configurations at low temperatures (see figure 21). These configurations were 2D subsurface clusters and 3D quasi-Janus patterns at low and higher Ir content, respectively. They were attributed to the preferential strengthening of surface-subsurface Pt–Ir bonds, due to specific electronic effects occurring at the DFT level. The transition from 2D subsurface clusters to 3D quasiJanus patterns with increasing A atoms content in fcc A@B

Figure 18. Cross sections of Ag–Ni icosahedra with centred symmetric and off-centre low-symmetry cores. In the left column we show the arrangement of the atomic species, drawing them with different colours (grey for Ag and red for Ni). In the right column we show the corresponding local pressure maps, according to the SMTB calculations. Negative, zero and positive pressures correspond to blue, green and red atoms, respectively. (a1) and (a2): Ag414 Ni147 with centred symmetric core. (b1) and (b2) Ag414 Ni147 with off-centre core. (c1) and (c2) Ag1106 Ni309 with centred symmetric core. (d1) and (d2) Ag1106 Ni309 with off-centre core. In both cases, the strain release obtained with off-centre low-symmetry cores is notable. Reprinted with permission from [34]. Copyright 2013 American Chemical Society.

In the morphological instability of icosahedral cores there is no growth, but a change in composition. When increasing composition, strain in the core is accumulated so that it 21

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Figure 19. (a) Map of the anisotropy parameter of pressure τi (equation (13)) for Ag414 Ni147 with centred symmetric core (a cross section of the nanoparticle is shown in figure 18(a1)). The atoms with strongly anisotropic pressure are marked in red, those with isotropic pressure in green. Intermediate anisotropies are marked in yellow. (b) Map of radial pressure. (c) Map of tangential pressure. In (b) and (c), negative, zero and positive pressures are in blue, green and red, respectively. The colour scale in this figure is scaled by a factor 1/3 with respect to figure 18. Reprinted with permission from [34]. Copyright 2013 American Chemical Society.

Figure 21. Low-temperature chemical ordering in a truncated octahedron of composition Pt159 Ir42 . Ir atoms are in blue and Pt atoms in dark red. From [103]. Reproduced by permission of the PCCP Owner Societies.

Figure 20. (a) Optimized chemical ordering in a truncated

octahedron of composition Ag2206 Cu200 . (b) Optimized chemical ordering in a decahedron of composition Ag750 Cu250 . In both cases, quasi-Janus structures are formed. The nanoparticle is heavily distorted on the side of the off-centre core to accommodate the lattice mismatch between the elements.

The physical effects causing this different behaviour of Au–Pt with respect to Ag–Cu have been analyzed in [28]. It has been verified whether it is possible to obtain stable onion-like structures in Ag–Cu by artificially changing the interaction parameters between the atoms. The result is that this behaviour is ruled by a rather subtle balance of the atomic interactions. In fact, increasing by 5% the parameter that rules the strength of the attractive part of the mixed interactions, it turns out that the lowest-energy chemical ordering presents non-compact aggregates of Cu atoms filling the subsurface layer, especially below vertices, edges and (1 0 0) facets. These aggregates are shown in figure 23.

nanocrystals was previously studied in [104], where Pt@Au fcc truncated octahedra of 586 atoms were simulated by using an embedded-atom potential. A quite interesting transition was found by changing composition (see figure 22). At low Pt content, Au forms disconnected 2D subsurface clusters. When the Au content increases, there is a transition to an onion-like pattern Au@Pt@Au, i.e. with an intermediate Pt shell. When the Pt content increases further to about 20%, quasi-Janus patterns with Pt inside are formed. These quasi-Janus patterns are of the same kind of those in Ag–Cu etc, but the nanoparticle distortion is smaller because of the lower lattice mismatch (about 4%). Onionlike structure in Ag–Cu and Ag–Ni were observed during growth simulations, but they were shown to be metastable [21]. Moreover, at variance with Au–Pt, subsurface clusters are already 3D when they are very small [28].

7. Temperature effects

The results shown in the previous sections were related either to the search of the lowest-energy configurations of nanoalloys, 22

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Figure 22. Low-temperature chemical ordering patterns in an Au@Pt truncated octahedron of size 586 and varying composition. From left to right: 1%at, 7.5%at 20%at Pt content. Pt atoms are represented by big circles. The snapshot were taken at T = 300 K, but they are representative of the lowest-energy configurations in this system. Reprinted with permission from [104]. Copyright 2012 American Chemical Society.

separation between the global minimum and the first excited isomer. With increasing temperature, p0 decreases. When p0 becomes smaller, a phase change may take place. In the case of nanoalloys, changes in the chemical ordering pattern can take place with increasing temperature while the nanoparticle is still at the solid state. A change in the chemical ordering pattern can be better singled out by defining an appropriate order parameter, which takes different values for different types of patterns. Further phase changes include the melting transition, which may be complicated in finite systems by surface melting or other premelting phenomena. The computational study of phase changes is mainly made by two different methods, molecular dynamics and Monte Carlo simulations, although other lattice-based methods are sometimes used. Global optimization searches give the correct structure to start with at low temperatures, even though some methods like multi-canonical Monte Carlo start from hightemperature liquidlike configurations [83]. Both canonical and (semi)grand-canonical schemes have been used in the literature for the study of phase transitions in nanoalloys [23, 52, 99, 104, 109, 112, 113]. In the following we restrict our overview to results concerning weakly miscible systems.

Figure 23. Lowest energy structure of Ag1189 Cu100 for the case in which the mixed Ag–Cu interactions are artificially reinforced. Cu atoms fill subsurface positions below (1 0 0) facets, forming 2D instead of compact aggregates. Compact aggregates are found when mixed interactions are not artificially reinforced. From [28]. Copyright 2013 by The American Physical Society.

7.1. Transformations at the solid state

which are representative of their equilibrium structures in the limit of low temperatures, or to low-temperature results obtained within Monte Carlo simulations and lattice model calculations. It is therefore interesting to study the influence of raising the temperature on the structures analyzed so far, in order to determine what their thermal stability is. The methods for studying this problem in nanoalloys range from molecular dynamics [64, 83, 105–107] to Monte Carlo simulations [104, 108, 109], to lattice models [42, 43, 103] and to the use of an appropriately modified Wulff construction [110, 111]. The latter method however, has not yet been applied to weakly miscible systems. At any finite temperature, a nanoalloy has a non-zero probability of sampling other basins than that of the basin of the global minimum. This probability (p ∗ = 1 − p0 , with p0 being the occupation probability of the global minimum) is very small for T  E1 /(kB ), where E1 is the energy

The effect of temperature on chemical ordering has been computationally studied by different groups for some weakly miscible systems such as Au–Pt, Pt–Ir Ag–Cu and Ag–Co. In [104] the phase diagram of an Au–Pt truncated octahedron of 586 atoms (see figure 24) was calculated by Monte Carlo simulations. The low-temperature configurations at different Pt content are shown in figure 22. Transitions between different chemical ordering patterns with increasing temperature are possible in this system while it is still in the solid state, for example from quasi-Janus to onion-like threeshell at composition 20%at in Pt. This transition is due to the higher entropy of onion-like patterns. A transition of the same kind was found also in simulations of Ir@Pt [103] in the Pt-rich regime. The low-temperature 2D subsurface clusters and 3D quasi-Janus patterns transformed upon heating into a centred symmetric three-shell structure with Ir mostly occupying subsurface sites. 23

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Figure 24. Phase diagram for an Au–Pt truncated octahedron of 586

atoms according to the model of [104]. At composition 20%at in Pt a transition from quasi-Janus to onion-like three-shell takes place with increasing temperature. Reprinted with permission from [104]. Copyright 2012 American Chemical Society.

Figure 25. Cu/Ag site occupancies in an icosahedral nanoparticle of composition Ag469 Cu150 at T = 300 K. The sites with Cu occupancies higher than 20% are shown as spheres, while the rest of the sites are only marked as dots. The orientation of the nanoparticle is the same as in the inset showing the lowest-energy configuration of the nanoparticle. The main effect of temperature is to increase Cu occupation of those sites along the fivefold axes that are not occupied in the lowest-energy configuration. From [99].

In a series of papers [42, 43, 112, 114, 115], the effects of the finite size of nanoparticles on the segregation in systems with large lattice mismatch such as Ag–Cu were computationally studied. Both rigid-lattice models (including energetic terms that take into account lattice mismatch) and SMTB calculations (with local relaxation) were used to model segregation in icosahedral and cuboctahedral nanoparticles. The latter are fcc structures having large (1 0 0) open facets and triangular (1 1 1) close-packed surfaces. The segregation of Ag was found to be stronger in the icosahedron than in the cuboctahedron, due to the expansion of the surface bonds in the icosahedron [114]. It was found that Ag superficial segregation in cuboctahedra involves the vertices first, then the edges, then (1 1 1) and finally (1 0 0) surfaces [115]. This is not what would follow from bond-breaking arguments (that would lead Ag to populate (1 0 0) before (1 1 1) facets) and it is a consequence of lattice mismatch in favour of Ag. Segregation isotherms at different chemical potentials (in the semi-grand canonical ensemble) were calculated. The nanoparticle phase diagram of Ag–Cu was studied by Monte Carlo simulations within the SMTB model for sizes between 1000 and 2000 atoms [99]. That study was focused on determining the mutual solubility of the two elements in nanoparticles of crystalline and icosahedral structures in the temperature range below their melting point. The main result is that nanoscale and bulk phase diagrams differ not only quantitatively but also qualitatively. In fact, in a bulk binary system with two coexisting phases, the dissolution concentrations of the two phases do not depend on the overall composition of the bulk system. In nanoscale systems there can be deviations from this behaviour, due to a combination of size-dependent factors such as surface segregation, and other effects due to the non-homogeneous nature of the nanoparticles (such as strain in non-crystalline structures). Because of the small system sizes, the number of atoms affected by such conditions become statistically significant. Deviations from this behaviour are more evident in icosahedral nanoparticles because of the less uniform distribution of the internal stress,

which favours placement of Cu atoms along fivefold axes (see figure 25). In fact, the simulation results indicate some enhanced solubility on the Ag-rich side, an effect which is significant but not big, and more evident in icosahedra (see the phase diagrams in figure 26). Enhanced solubility was previously predicted by the calculations in [22], up to the conclusion that below size of 270 atoms, phase separation in Ag–Cu should not be possible at all. This was obtained by Monte-Carlo simulations within an atomistic effective-medium potential and also by the solution of a semi-analytical model taking into account bulk mixing energies and the finite size of the nanoparticle. This prediction has not been confirmed by more recent simulations. The main point is that the calculations in [22] were not considering the fact that at small sizes the most favourable structures are polyicosahedral or icosahedral, in which strain effects cannot be neglected and complete intermixing is strongly hindered. Moreover, both SMTB and DFT calculations do not confirm the impossibility of phase separation in small clusters [32, 46, 112]. Even in small (distorted) fcc nanoparticles phase separation is indeed obtained [112]. A further confirmation of the dominant character of non-mixing patterns in Ag–Cu in the whole size range follows from the results of [39]. There, free-energy differences between Janus, intermixed, Cu@Ag and Ag@Cu configurations were computationally evaluated by means of the Bennett’s method [116] for a broad size range at 50–50% composition. Nanoparticles were modelled by the SMTB potential. The results of [39] show that non-mixing Janus and core-shell Cu@Ag patterns dominate over mixing patterns down to sizes of the order of 2 nm at least. Moreover, it was found that perfect Janus structures should prevail over Cu@Ag structures in the whole diameter range from 2 to 24

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1400 (a) fcc clusters

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Figure 26. Phase diagrams for Ag–Cu fcc (a) and icosahedral nanoparticles (b). The dashed lines correspond to the bulk phase diagram.

From [99].

20 nm. This last result can indeed be attributed to a series of approximations in [39]. The geometry of the nanoparticles was far from the optimal one, which is of the icosahedral family when size is close to 2 nm. In icosahedra, core-shell structures can be found as lowest-energy structures in the small-size range [31, 32]. Another approximation was neglecting quasiJanus morphologies, which can prevail over the pure Janus arrangement at least up to sizes of 5 nm. The structures of Ag–Co nanoparticles at different temperatures have been studied by Monte Carlo simulations within an atomistic embedded-atom model [108, 109] for several compositions and sizes in the 2–3 nm range. In the Co-rich part, Ag atoms were found to decorate lowcoordination sites. In the very Ag-rich case, Co atoms were found to form very small separated subsurface aggregates. With increasing temperature, these aggregates condense into a single Co off-centre cluster, which dissolves upon further temperature increase, but with Co still occupying subsurface sites [108]. However it is unclear whether the low-temperature

configurations were actually reflecting the lowest-energy types of chemical ordering. 7.2. Nanoparticle melting

The melting temperature in nanoalloys can be very sensitive to changes in composition. A striking example is given by the melting of icosahedral clusters with a single impurity at the central site [64]. Substituting the central atom in a pure Ag icosahedron by a single Ni or Cu impurity causes an upward shift of the melting temperature which is still notable up to unexpectedly large nanoparticle sizes. A single Ni impurity causes an increase of more than 50 K of the melting temperature of Ag icosahedra of 55 and 147 atoms (see figure 27). Even at size 561 a single Ni impurity causes an increase of 20 K. This stabilization of the icosahedron by small impurities at the central site is another effect that can be explained by strain release. Atomic pressure on the central site notably 25

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Figure 29. The black triangles represent the probability that Ag32 Ni13 is in the basin of its global minimum. The red dots represent the probability that the Ni core is in the basin of the 13-atom icosahedral structure. The former probability drops at significantly lower temperature (by almost 200 K) than the latter. From [105]. Copyright 2008 by The American Physical Society.

In general, high-symmetry core-shell polyicosahedra present a quite high melting temperature. Cu@Ag and Ni@Ag high-symmetry polyicosahedra melt higher than both Ag and Cu pure clusters in the same size range (see for example Ag27 Cu7 and Ag27 Ni7 [46] compared to pure Ag, Cu, Ni clusters of sizes 34 and 38). This behaviour is a consequence of the fact that high-symmetry core-shell structures are well separated in energy from their higher isomers. This can be true also for core-shell Mackay icosahedra. In fact, a Mackay Co@Au icosahedron of composition Au42 Co13 (global minimum structure of the same kind as that shown in figure 6(b)) melts higher than pure icosahedra of the same size [83]. Core-shell nanoalloys are well suited for studying surface melting phenomena. Surface melting is expected to occur in core-shell structures [109, 117, 118] because its main driving force, which is the lower coordination of surface atoms, is enhanced by the fact that the shell metal usually melts lower than the core metal. This issue has been investigated by molecular-dynamics simulations for Cu@Ag, Ni@Ag and Co@Ag anti-Mackay icosahedra [105, 107]. A clear example of surface melting is found in the anti-Mackay Ag32 Ni13 (see the global minimum structure in figure 3(c)). Figure 29 reports the temperature dependence of the occupation probability of the global minimum p0 , together with the probability of finding the Ni core in its unaltered icosahedral configuration. There is a rather wide temperature interval in which p0 is significantly smaller than 1 (indicating that the shell has changed its configuration) but the Ni core preserves the original lowtemperature structure. The same kind of behaviour is also found for Ag32 Co13 , Ag72 Ni55 and Ag72 Co55 [105, 107]. However, the occurrence of shell melting is ruled by a rather delicate balance. In fact, in Ag32 Cu13 , premelting phenomena involve both core and shell at the same time. This difference between the systems is due to specific features of

Figure 27. Caloric curves of the melting of Ag icosahedra

containing a single impurity obtained by molecular dynamics simulations. The quantity plotted is E = E(T ) − E(0) − 3(N − 1)kB T , where E(T ) is the average cluster internal energy, E(0) is the minimum energy at 0 K, and 3(N − 1)kB T is the harmonic part of the energy. In the upper panel, size 55 is considered and all systems (pure Ag, Ag–Cu, Ag–Ni, Ag–Pd and Ag–Au) are shown. In the lower panel, size 147 is considered and only pure Ag, Ag–Cu, and Ag–Ni are shown. Crosses refer to pure Ag, solid circles to Ag–Ni, squares to Ag–Cu, diamonds to Ag–Pd, and asterisks to Ag–Au. From [64]. Copyright 2005 by The American Physical Society.

Figure 28. Atomic stress in Ag icosahedra as a function of size N for different impurities. Crosses refer to pure Ag, solid circles to Ag–Ni, squares to Ag–Cu, diamonds to Ag–Pd, and asterisks to Ag–Au. From [64]. Copyright 2005 by The American Physical Society.

decreases after substituting a small impurity (figure 28). Ni and Cu impurities are by far more effective in decreasing the pressure than Au and Pd impurities. The strain release effect of small impurities inside icosahedra can cause transition from fcc structures to icosahedra upon the substitution of a small percentage of impurities [63]. 26

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Figure 30. Growth sequences from simulations of the deposition of Ag atoms on Cu seeds. The initial Cu seeds are fcc truncated octahedra of 201 atoms, and 200 Ag atoms are deposited in total at a rate of one atom each 2.1 ns. Top row: growth at 300 K. Bottom row: growth at 500 K. In both cases, Ag shells of monoatomic thickness are formed. After deposition, the core retains a fcc structure, but it is heavily distorted from the initial perfect truncated octahedral shape. From [72]. Copyright 2002 by The American Physical Society.

the energy landscapes of these nanoalloys [107]. In fact, in the first excited isomers of Ag32 Cu13 both core and shell change shape with respect to the global minimum structure, while in the first excited isomers Ag32 Co13 and Ag32 Ni13 only the shell changes its shape. To recover a surface melting behaviour in Cu@Ag, one has to consider anti-Mackay icosahedra with incomplete shell, for example for composition Ag27 Cu13 [106]. In this case, the weakening of the cohesion of the shell allows a surface melting phenomenon that is not observed when the shell is complete. It is worth noting that all these clusters undergo important changes well below the temperature of the main peak in the caloric curve. These changes simply cause shallow bumps in the caloric curve and cannot be ascribed to solid–solid morphology transitions because they are associated with steep increases of the atomic mobility. Another interesting observation is that the surface segregation of Ag in Ag–Ni also persists in the liquid clusters [107]. Also the chiral icosahedra reveal a remarkable thermal stability, as verified by molecular-dynamics simulations [32]. For composition Ag212 Cu309 the simulations show that the nanoparticle keeps its low-temperature shape up to about 640 K, with a perfect chiral shell. Between 640 and 700 K, the external Ag shell is still perfectly chiral for most of the time, but sometimes a single vertex atom escapes from its position and diffuses shortly above the cluster surface, before being retrapped again at the vertex. More evident surface melting phenomena appear above 700 K. A few diffusing Ag atoms are always present, but the external shell still has an overall chiral shape with some vacancies that are most likely (but not always) placed at vertices. The structure is identifiable as a defective chiral icosahedron. Exchanges between Ag and Cu atoms begin to take place above 800 K. From this temperature on, the structure is no longer identifiable as a defective chiral icosahedron, even though its Cu core is still icosahedral with defects and the shell partially keeps the arrangement of the

chiral structure. The overall melting of the cluster takes place in between 920–960 K, with a clear jump in the caloric curve. The melting of an icosahedron of Ag–Ni with off-centre core was studied in [23], finding that there is a surface melting transition associated with the transformation of the nanoparticle to a more prolate shape. These phenomena were associated to the most prominent peak in the heat capacity. Increasing temperature further caused the melting of the core and some intermixing between Ag and Ni. 8. Non-equilibrium effects: growth and coalescence

The gas-phase growth has been simulated for several kinds of nanoalloys. Here we focus on the growth of the weakly miscible systems Ag–Cu, Ag–Ni and Ag–Co [85]. The computational technique used in these simulations is molecular dynamics [119]. Two types of growth simulations have been made in the literature. A first type of growth simulations is denoted as direct deposition [70, 72]. The initial core is made of element A (either Cu, Ni or Co) that we have seen to have a natural tendency to occupy the nanoparticle core. Then, atoms of element B (i.e. Ag) are deposited on the core one by one. In the inverse deposition [21], the initial cluster is made of B atoms, and A atoms are deposited on top of it. Deposition intervals τ are in the range from one atom each 2.1 ns to one atom each 14 ns. In each simulation, temperature is kept constant. Different growth temperatures are considered, in the range 300–600 K. As initial cores, both icosahedral and fcc truncated-octahedra were considered. In the direct deposition of Ag on Cu cores, due to the rather high barriers at facet edges, the initial stage of the growth at 300 K shows the nucleation of separated Ag atomic islands on different facets (see figure 30, upper row). At increasing Ag coverage, these islands merge into a single outer Ag shell of monoatomic thickness, with few defects. The second Ag layer 27

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Figure 31. Growth sequence from a simulation of the deposition of Cu atoms on an Ag icosahedral seed of 147 atoms. In total, 162 Cu atoms are deposited at a rate of one atom each 7 ns. The temperature is 500 K. In the bottom panel, the radial distribution function of the two species is shown for the final snapshot of the simulation. The full and dotted lines refer to Ag and Cu atoms, respectively. Both the snapshots and the radial distribution function indicate the formation of a well-defined Cu@Ag arrangement, which is analogous to the arrangement produced in the direct deposition simulations. From [21]. Copyright 2003 by The American Physical Society.

should in principle be quite fast in rearranging their shapes to equilibrium. In summary, the direct deposition simulations always produce core-shell nanoparticles, often retrieving global minimum structures or isomers that are closely related to them. However, in some cases, there are evident kinetic trapping effects, leading to growth structures that are not related to the global minima. In the inverse deposition, results are more surprising and the final result crucially depends on the structure of the initial seed. Cu, Ni or Co atoms deposited on an icosahedral Ag seed incorporate and diffuse fast towards the cluster centre, so that Cu or Ni cores are nucleated in the central part of the icosahedron. This means that on the icosahedron, core-shell structure are formed, in both direct and inverse deposition. Growth snapshots are shown in figure 31. Continuing the deposition, the Cu or Ni cores grow asymmetrically to form the quasi-Janus structures that are expected from the global optimization. Structures of this kind, obtained for Ag–Co [26], are shown in figure 32. Cu or Ni atoms deposited on a truncated octahedral Ag seed incorporate fast, but instead of diffusion towards the nanoparticle centre, they stop in the subsurface layer and

does not nucleate before the first shell is completed, even at this temperature. For higher growth temperatures the Ag shell has even fewer defects, due to faster kinetics that allow a better rearrangement. Since Ag and Cu do not show any tendency to intermixing in the temperature range of the simulations, higher temperatures simply allow a better rearrangement of the external pure Ag shell. Growth sequences as those of figure 30 rather closely reproduce the sequence of equilibrium structures [31]. For sufficiently high temperatures, the growth of the monoatomic Ag shell occurs through the nucleation of a single island which gradually enlarges to cover the Cu core, showing thus ball-and-cup configurations that evolve to core-shell configurations. In the growth of Ag on Co cores of [70], Ag atoms are deposited with a rate of one atom each 14 ns on Mackay icosahedral Co seeds of 13 and 55 atoms. On the icosahedron of 13 atoms, the anti-Mackay icosahedron is formed only if the growth temperature is sufficiently high. At lower temperatures, the deposition of 32 atoms produces core-shell structures of the polyicosahedral family, which are however higher in energy than the antiMackay icosahedron. These results confirm the importance of kinetic effects even in the case of rather small clusters, which 28

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Figure 32. Excess energy  for the growth after deposition of Co atoms (yellow) on an Ag (red) truncated octahedron of 201 atoms (top row) and on an Ag icosahedron of 309 atoms (bottom).  is given for T = 450, 550, and 650 K). Growth snapshots are reported after deposition of 34 and 64 atoms of Co on the Ag truncated octahedron and for 55 and 139 atoms of Co on the Ag icosahedron. Here, the formation of a quasi-Janus motif is very clear up to a 45 Co atoms.  is defined as  = (E − Ebulk )/N 2/3 , where E is the cluster energy, Ebulk is the energy of NAg Ag bulk atoms and NCo cobalt bulk atoms and N = NAg + NCo is the total number of atoms. Reprinted with permission from [26]. Copyright 2010 American Chemical Society.

trigger the nucleation of an intermediate shell, thus producing a three-shell onion-like B@A@B structure (see figure 33) [21]. This structure is metastable, because upon annealing it transforms into a core-shell A@B arrangement, or into a quasi-Janus arrangement. The formation of three-shell onion-like arrangements can be rationalized in terms of the energetics of single Cu or Ni atomic impurities in a host matrix of Ag, which shows that the most favourable site in the icosahedron is the subsurface site, whereas in the fcc nanocrystals, subsurface sites are favoured [21, 27]. Coalescence is another process that may occur in nanoparticle growth. This happens when already formed clusters collide and rearrange into a single nanoparticle. Molecular-dynamics simulations are well-suited to study this process [120–124]. The works in [120, 121] treated the coalescence of weakly-miscible systems (pure Au colliding with pure Pt, or pure Cu colliding with pure Ag), considering head-to-head collisions first, and then varying the impact parameter. An embedded-atom potential was used to model the interaction between atoms. Depending on composition, different final structures were obtained in Au–Pt. The collision of Au95 with Pt95 clusters produced Pt@Au nanoparticles with the Au shell of monoatomic thickness. This happened rather quickly, since the resulting clusters were analyzed just 0.2 ns after the collision. The collision of Au537 with Pt537 resulted in Pt@Au structure with thicker shell core-shell configurations. On the other

hand, after the collision of Au95 with Pt537 the Au shell was incomplete (ball-and-cup configuration). Finally, the colliding Au537 with Pt95 produced core-shell structures with off-centre cores. The results of the collision simulations were in qualitative agreement with the experimental results, in which Pt@Au were produced by a variety of techniques [125–127]. The tendency to produce off-centre cores is in agreement with the results shown in section 6.2. In Ag–Cu, coalescence simulations either produced Cu@Ag core-shell arrangements or three-shell Ag@Cu@Ag onion-like patterns in agreement with the results of growth simulations [21, 72]. 9. Discussion and comparison with the experiments

In this section the computational results shown in the previous sections are compared to the available experimental results and to the predictions of the simple analytical model of section 2.1. Ag–Cu is one of the systems that have been more widely studied in experiments focused on determining chemical ordering patterns [24, 25, 27, 81, 128–132]. The comparison between experiments and simulations for small clusters (below sizes of 100 atoms) has been already discussed in section 4.1, so that here we focus on larger sizes. A first point that emerges from the experiments is the segregation of Ag at the nanoparticle surface, which is quite significant even in the Cu-rich regime. In [129], Ag–Cu free nanoparticles in the size range 103 –104 atoms and in 29

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Figure 33. Growth sequence from a simulation of the deposition of Cu atoms on an Ag truncated octahedral seed of 201 atoms. In total, 201 Cu atoms are deposited at a rate of one atom each 7 ns. The temperature is 500 K. In the bottom panel, the radial distribution function of the two species is shown for the final snapshot of the simulation. The full and dotted lines refer to Ag and Cu atoms, respectively. Both the snapshots and the radial distribution function indicate the formation of a well-defined Ag@Cu@Ag arrangement. The surface Ag shell is of monoatomic thickness. From [21]. Copyright 2003 by The American Physical Society.

shell nanoparticles was also confirmed in the coalescence of 12 nm Cu nanoparticles with 6 nm Ag nanoparticles [131]. Going towards the Ag-rich regime, the experiments in [27] show a transition from core-shell towards quasi-Janus structures, in which the Cu part is covered by a very thin Ag shell. Transmission-electron microscopy (TEM) images of these structures are shown in figure 34, in which Cu@Ag nanoparticles are imaged (with average composition 1@2). This result is in very good agreement with the computational findings [27, 28, 34]. We note however that there is a transition that has been predicted by the calculations but is still to be experimentally detected. This transition is back to centred core-shell nanoparticles, and should occur only in icosahedra when the composition becomes so rich in Ag that the morphological instability of the core is suppressed (see [28] and figure 17). The nanoparticles in [27] were obtained by pulsed laser deposition on an amorphous carbon substrate under ultra-high vacuum (UHV) conditions. Very recently, quasiJanus configurations have been observed also for Cu@Ag nanoparticles synthesized in solution [132]. Finally, a study of the ageing of core-shell Cu@Ag nanoparticles in colloidal suspensions has revealed in some

a wide range of compositions were produced in a beam by a gas-aggregation source. Magnetron-based sputtering was used for vaporizing the primary solid materials. The free nanoparticles were analyzed by photoelectron spectroscopy, finding that Cu appears on the surface of nanoparticles only at low Ag concentrations. The analysis of the photoelectron spectra showed also that in the Cu-rich case, the coordination of Ag with Cu is high. This excludes truly Janus configurations in favour of core-shell ones, with the evidence that surface Ag atoms are well coordinated with Cu neighbours. This points in favour of one-layer thick Ag shells, in which Ag atoms are coordinated with subsurface Cu atoms, and even indicates intermixing within the surface layer in the most Cu-rich cases. These experimental results are in very good agreement with the calculations reported in sections 5 and 7, which were showing that, for Cu-rich compositions, core-shell structures with onelayer thick shells prevail as global minima and persist to high temperature [31, 32]. Some intermixing of Ag with Cu at the very surface layer is also predicted (see [28, 43, 112] for fcc nanoparticles and [114] and section 5.1 for icosahedra). Note also that monolayer-thick Ag shells are predicted to be ubiquitous, being found not only in core-shell nanoparticles, but also in quasi-Janus nanoparticles. The formation of core30

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Figure 34. Left: TEM observations of an Cu@Ag sample with Ag-rich composition (1@2 on average). The Ag elemental map obtained by energy-filtered TEM using the three-window technique is shown. Right: high-resolution TEM image of a quasi-Janus Cu@Ag nanoparticle. From [27]. Reproduced by permission of The Royal Society of Chemistry.

of cluster formation on TiO2 surfaces [134] were showing a behaviour close to that of Au–Ni. The direct deposition (Au on Pt) produced nanoparticles with Au surface, while in the inverse deposition, the surface was still containing 50% Pt, indicating an even stronger kinetic barrier for the incorporation of Pt. These results are again in reasonable agreement with the computational results that indicate a preference for Au surface segregation. However calculations of kinetic effects and simulations of nanoparticle growth are still lacking. In summary, the experimental results substantially confirm the picture arising from the computational results, with the only possible exception of Au–Co. Transitions from coreshell to quasi-Janus patterns at increasing B content, and the formation of core-shell structures in the A-rich cases (with very thin B shells) have been confirmed. A–B intermixing in the outermost surface layer has been observed. The preference for low-symmetry structures with off-centre cores has been confirmed. The last part of the discussion concerns the comparison of the computational results with the predictions of the analytical model of section 2.1. The model has the merit of showing very easily that composition-dependent changes in the most stable chemical ordering pattern can be expected in nanoalloys. When γAB > γA − γB the sequence A@B → Janus → B@A may be observed with increasing A content. On the other hand, if γAB < γA − γB , the A@B configuration is always the most favourable, but a higher probability of off-centre cores is expected. This preference is due to a purely statistical effect, not to the energetics. Since this preference is sizeable when the core is small, and essentially irrelevant for large cores, the behaviour in the model is a smooth transition towards more and more centred cores as the A content increases. Even though this transition seems to be in qualitative agreement with the quasi-Janus → A@B transition experimentally observed in Cu@Ag [27], the model completely misses its driving force, which is the minimization of strain. This gives a strong energetic preference for offcentre cores [34] that are covered on one side by a very thin Ag

cases an Ag dewetting phenomenon, whose occurrence may be influenced by the presence of ligands on the nanoparticle surface [130]. In what concerns Ag–Ni, Au–Ni and Ag–Co (nanoparticles of diameters between 2 and 5 nm), an indirect indication in favour of core-shell structures with Ag or Au shells was obtained from the analysis of their optical properties [5, 133], and for Ag–Ni was also confirmed directly by lowenergy ion scattering data [5]. Au–Ni nanoparticles grown in UHV conditions on TiO2 surfaces showed that depositing Au on Ni preformed clusters leads to nanoparticles whose surface contains only Au [134]. On the other hand, depositing Ni on preformed Au under the same conditions leads to nanoparticles whose surface is mostly Au, but with about 20% of Ni. This can be interpreted by assuming that the Ni@Au configuration is the most favourable, but there are some kinetic barriers that are not easy to overcome when performing the inverse deposition. These results are thus in qualitative agreement with the calculations. More complete experimental investigations about core shapes and placement, and of the transition from core-shell to Janus or quasi-Janus patterns are not yet available for these systems. The situation for Au–Co is somewhat more complex. In fact, even though all calculations on free clusters would favour Co@Au structures [47, 83], the inverted Au@Co nanoparticles have been stabilized by depositing Co on Au [135] and even intermixed nanoparticles have been obtained by synthesis in solution [136], for diameters of 2–3 nm. There are however examples of the synthesis of Co@Au nanoparticles [137]. Calculations taking into account environment and kinetic effects are likely to be necessary to bring the computational results to some sort of agreement with the experiments producing Au@Co or intermixed nanoparticles. In Au–Pt, both Pt@Au, Au@Pt and intermixed nanoparticles were produced by reduction in liquid solution [125, 138]. The Pt shells in Au@Pt were ragged [125]. UHV experiments 31

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investigation, both by experiment and computation/modelling. Moreover, the study of the interplay of the nanoparticles with different environments is still at the embryonic stage, and waits for very interesting developments.

layer. Also the latter feature is completely missed by the simple model. We note that when there is considerable size mismatch, even attributing a definite value to γAB is problematic, due to volume contributions to the excess energy, and therefore the model loses predictivity. Moreover, the simple model does not reproduce at all the behaviour of chemical ordering in icosahedra, where strain is inherent to the structure itself. The same kind of comments apply to fcc Ir@Pt nanoalloys, in which strain is absent but the energetic preference for off-centre core positioning is caused by specific electronic effects [103]. Finally, the transition to B@A configurations, predicted for γAB > γA − γB , has not been observed yet, neither in experiments nor in simulations, to our knowledge. Also the occurrence of pure Janus nanoalloys (not quasi-Janus) needs some further investigation. The model suggests that these are in principle possible, but we believe that they might more likely be observed in systems with weak size mismatch.

Acknowledgments

Support from the COST Action MP0903 NANOALLOY is acknowledged. The author thanks G Rossi for a critical reading of the manuscript. References [1] Ferrando R, Jellinek J and Johnston R L 2008 Nanoalloys: from theory to applications of alloy clusters and nanoparticles Chem. Rev. 108 845–910 [2] Banadaki A D and Kajbafvala A 2014 Recent advances in facile synthesis of bimetallic nanostructures: an overview J. Nanomater. 2014 985948 [3] Jung N, Chung D Y, Ryu J, Yoo S J and Sung Y-E 2014 Pt-based nanoarchitecture and catalyst design for fuel cell applications Nano Today 9 433–56 [4] Bao Y, Calderon H and Krishnan K M 2007 Synthesis and characterization of magnetic-optical Co–Au core-shell nanoparticles J. Phys. Chem. C 111 1941–4 [5] Gaudry M et al 2003 Size and composition dependence in the optical properties of mixed (transition metal-noble metal) embedded clusters Phys. Rev. B 67 155409 [6] Lima F H B, de Castro J F R and Ticianelli E A 2006 Silver-cobalt bimetallic particles for oxygen reduction in alkaline media J. Power Sources 161 806–12 [7] Shin K, Kim D H, Yeo S C and Lee H M 2012 Structural stability of Ag–Cu bimetallic nanoparticles and Their application as a catalyst: a dft study Catal. Today 185 94–8 [8] Hernandez-Fernandez P, Rojas S, Ocon P, Gomez de la Fuente J L, San Fabian J, Sanza J, Pena M A, Garcia-Garcia F J, Terreros P and Fierro J L G 2007 Influence of the preparation route of bimetallic Pt–Au nanoparticle electrocatalysts for the oxygen reduction reaction J. Phys. Chem. C 111 2913 [9] Zhang J, Sasaki K, Sutter E and Adzic R R 2007 Stabilization of platinum oxygen-reduction electrocatalysts using gold clusters Science 315 220 [10] Selvarani G, Selvaganesh S V, Krishnamurthy S, Kiruthika G V M, Sridhar P, Pitchumani S and Shukla A K 2009 A methanol-tolerant carbon-supported Pt–Au alloy cathode catalyst for direct methanol fuel cells and its evaluation by DFT J. Phys. Chem. C 113 7461 [11] Abrams B L, Vesborg P C K, Bonde J L, Jaramillo T F and Chorkendorff I 2009 Dynamics of surface exchange reactions between Au and Pt for her and hor J. Electrochem. Soc. 156 B273 [12] Prieto G, Beijer S, Smith M L, He M, Au Y, Wang Z, Bruce D A, Spivey J J and de Jong P E 2014 Design and synthesis of coppercobalt catalysts for the selective conversion of synthesis gas to ethanol and higher alcohols Angew. Chem. Int. Edn 53 6397 [13] Toledo-Antonio J A, Ch´avez A, Cort´es-J´acome M A, Cuauht´emoc-L´opez I, L´opez-Salinas E, P´erez-Luna M and Ferrat-Torres G 2012 Highly dispersed Pt–Ir nanoparticles on titania nanotubes Appl. Catal. A 437 155 [14] Hwang S J, Yoo S J, Jeon T-Y, Lee K-S, Lim T-H, Sung Y-E and Kim S-K 2010 Facile synthesis of highly active and stable Pt–Ir/C electrocatalysts for oxygen reduction and liquid fuel oxidation reaction Chem. Commun. 64 8401 [15] Mahara Y, Ishikawa H, Ohyama J, Sawabe K, Yamamoto Y, Arai S and Satsuma A 2014 Enhanced Co oxidation

10. Conclusions

The concept of symmetry breaking naturally emerges when analyzing the computational results on weakly miscible systems with size mismatch. Several transitions analyzed in the previous sections are examples of symmetry breaking: • The transition involving A@B icosahedra in the A-rich regime, in which the symmetry of the shell is broken with increasing nanoparticle size, either by A atoms beginning to appear at the surface or by restructuring of the shell itself towards an anti-Mackay arrangement (section 5.1) • The transition from anti-Mackay to chiral shells in the Arich regime, in which all mirror symmetries are broken but rotational symmetries are kept (section 5.2) • The morphological instability of icosahedral cores in the B-rich regime, in which centred symmetric cores change shape to off-centre low-symmetry shapes (section 6.1) • The stabilization of off-centre low-symmetry cores in decahedral and fcc nanoparticles (section 6.2) • The stabilization of chiral clusters in Co@Au and of asymmetric capped decahedra in Pt@Au clusters (sections 4.2 and 4.3). In most cases, the occurrence of symmetry breaking can be rationalized by strain accumulation and its subsequent release in the lower-symmetry structure. Several computational results agree very well with the experimental findings (for example, the transition from coreshell to quasi-Janus nanoparticles with increasing B content), while other computational findings are still at the stage of prediction (the stabilization of small centred cores in icosahedra, followed by the morphological instability after a certain critical core size and the transition from anti-Mackay to chiral icosahedra), and some systems experimentally show behaviours that are not yet explained by the simulations (for example, the formation of intermixed Au–Co nanoparticles). This shows that there are still several phenomena in this quite specific subfield of nanoscience in which interesting new physics can emerge. These phenomena are worthy of further 32

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Symmetry breaking and morphological instabilities in core-shell metallic nanoparticles.

Nanoalloys are bi- or multi-component metallic particles in the size range between 1 and 100 nm. Nanoalloys present a wide variety of structures and p...
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