Nanoscale Effects on Phase Separation - ACS Publications

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Nanoscale effects on phase separation Juan-Pedro Palomares-Baez, Emanuele Panizon, and Riccardo Ferrando Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b01994 • Publication Date (Web): 11 Aug 2017 Downloaded from http://pubs.acs.org on August 14, 2017

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Nanoscale effects on phase separation Juan-Pedro Palomares-Baez,†,§ Emanuele Panizon,†,k and Riccardo Ferrando∗,‡,¶ Dipartimento di Fisica, Via Dodecaneso 33, Genova, I16146, Italy, Dipartimento di Chimica e Chimica Industriale, Via Dodecaneso 31, Genova, I16146, Italy, and CNR/IMEM, Via Dodecaneso 33, Genova, I16146, Italy E-mail: [email protected]

Abstract Classical nucleation theory predicts that a binary system which is non-miscible in the bulk should become miscible at the nanoscale when lowering its size below a critical size. Here we tackle the problem of miscibility in nanoalloys with a combination of ab-initio and atomistic calculations, developing a statistical-mechanics approach for the free energy cost of forming phase-separated aggregates. We apply it to the controversial case of AuCo nanoalloys. AuCo is non-miscible in the bulk, but a rich variety of nanoparticle configurations, both phaseseparated and intermixed, has been obtained experimentally. Our calculations strongly point to the permanence of an equilibrium miscibility gap down to the nanoscale, and to the nonexistence of a critical size below which phase separation is impossible. We show that this is due to nanoscale effects of general character, caused by the existence of preferred nucleation sites in nanoparticles, which lower the free-energy cost for phase separation with respect to bulk systems. ∗ To

whom correspondence should be addressed di Fisica, Via Dodecaneso 33, Genova, I16146, Italy ‡ Dipartimento di Chimica e Chimica Industriale, Via Dodecaneso 31, Genova, I16146, Italy ¶ CNR/IMEM, Via Dodecaneso 33, Genova, I16146, Italy § Present address: Facultad de Ciencias Químicas, Universidad Autónoma de Chihuahua. Circuito No. 1, Nuevo Campus Universitario; Chihuahua, Chih. México k Present address: International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy † Dipartimento

1 ACS Paragon Plus Environment

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Keywords: Nanoalloys, nanothermodynamics, gold, cobalt, simulations There are several binary metallic systems whose equilibrium miscibility in bulk crystals is very limited. These systems comprise AuCo, AgCu, AgNi, AgCo, AuNi and several others. 1 However, there have been suggestions that miscibility may drastically increase by downsizing the systems to the nanoscale. Nanoscale effects on miscibility have been the subject of several research efforts, stemming from the seminal paper of Christensen et al., 2 in which a semi-macroscopic model was used to derive a lower limit for the size of phase-separated nanoclusters. Christensen et al. noted that, for a nanoparticle of radius R, the gain in free energy upon phase separation decreases approximately as R3 , whereas the cost of creating an interface between the two phases decreases as R2 . The scaling of these competing terms may be expected to give rise to a limiting size R0 beneath which phase separation is not possible anymore. On the other hand, Shirinyan and Wautelet 3 analyzed the effect of depletion in finite systems in the framework of classical nucleation theory. In its simplest form, their line of reasoning considered the nucleation of a B-rich new phase in a spherical nanoparticle of radius R, with initial mole fraction xB in the parent phase. If the B mole fraction in the nucleating new phase is xB0 > xB , and the radius of the critical nucleus is r∗ , nucleation requires that B atoms are subtracted from a sphere of radius R∗ = r∗ (xB0 /xB )1/3 . If R∗ > R nucleation becomes impossible simply because the number of B atoms in the whole nanoparticle is not large enough to sustain the formation of the critical nucleus. Therefore, nanoparticles with R < R∗ can sustain only the presence of a single phase. Both lines of reasoning thus concur in setting a lower size limit to the possibility of phase separation. For what concerns the occurrence of nanoscale increased miscibility and of limiting sizes to phase separation, AuCo is a very interesting system. AuCo presents very limited equilibrium miscibility in solid bulk phases, 4 since already at 695 K, miscibility is below 1% at of Co in Au and below 0.05% at of Au in Co, 4 and these values decrease with decreasing temperature. In spite of this clear tendency towards phase separation in bulk crystals, AuCo nanoparticles have been recently synthesized with a variety of intermixed and phase-separated chemical ordering. Bhattarai et al. 5 chemically synthesized AuCo nanoparticles of diameters between 2.5 and 5 nm by the phase-


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transfer method, and analysed their structure and chemical ordering by Z-contrast imaging and energy dispersive X-ray spectroscopy used jointly with STEM finding that the nanoparticles presented inhomogeneous intermixing with minor segregation. Marbella et al. 6 produced intermixed AuCo nanoalloys of diameters in the range 2-3 nm by co-reduction in aqueous medium. Even the chemical synthesis of Au3 Co intermetallics (L12 ordered phase) has been achieved. 7,8 These ordered phase were predicted to be metastable in bulk AuCo crystals. 9,10 On the other hand, several groups have been able to synthesize Co@Au phase-separated nanoparticles, both by wet chemistry methods 11,12 and by gas-phase growth. 13–15 Finally, also the inverted core-shell Au@Co structures have been obtained, 13,16,17 which were however shown to be metastable upon annealing. 16 At least in some cases, the nanoparticle structures were strongly influenced by Co oxidation. 18 This collection of experimental data does not allow to draw a firm conclusion about the problem of miscibility vs non-miscibility of AuCo at the nanoscale, mainly because it is very difficult to disentangle non-equilibrium and equilibrium effects in experiments. With this respect, computer simulations can be very useful, since they allow to analyze separately these effects. Here we investigate the problem by using several different computational techniques, from DFT calculations to atomistic molecular-dynamics (MD) and global optimization simulations, to develop a statistical mechanics approach for calculating the free-energy cost of aggregate formation. Our results show that – Equilibrium miscibility in AuCo is not significantly enhanced at the nanoscale, so that intermixed AuCo configurations are indeed metastable up to temperatures well above room temperature. There is no critical size below which phase separation is impossible. – The lack of enhanced nanoscale miscibility is due a general feature of the structure and energetics in nanoalloys, i.e to the existence of preferred nucleation sites, which cause a nanoscale decrease of the free-energy cost for aggregate nucleation with respect to the corresponding bulk crystal. This in turn implies the decrease of the critical nucleus size for nanoscale phase separation, so that the depletion effect becomes negligible. – The scaling of free energy with size shows that the cost decrease is still non-negligible for


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nanoparticle diameters up to 10 nm. This behaviour is not expected to be limited to AuCo, but it should manifest in several systems including, for example, AgCu, AgNi, AuCo, and AuNi. Our results show that the use of macroscopic interface free energies and critical nucleus sizes when dealing with nanoscale objects requires some care, since it may lead to incorrect predictions. The problem of phase separation in nanosystems has been treated in a series of works. 19–21 Here we show that a full atomistic approach, taking into account the existence of preferred nucleation sites, may be necessary to tackle this problem. We use several types of calculations and simulations. Global optimization searches are performed by the Basin Hopping method 22 using an atomistic potential based on the second-moment approximation to the tight-binding model (SMATB potential, also known as Gupta potential), 23,24 whose parameters are taken from Refs. 25–27 Optimization runs are made by combining shapechanging moves, such as the shake and Brownian moves 28 together with exchange moves, 29,30 both of the random and of the tailored types. Atomistic Molecular-Dynamics (MD) simulations are made by using our own code, 31 with a time step of 5 fs and keeping temperature constant by an Andersen thermostat. DFT calculations are made by the QUANTUM ESPRESSO code 32 employing the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 33 with scalar-relativistic ultrasoft pseudopotentials. 34 Cutoffs of 30 and 900 Ry have been imposed on the wave function and the density, respectively, and a smearing of 0.002 Ry has been used. As a first step in our study we look for the optimal chemical ordering of AuCo clusters of different sizes and compositions, in order to check whether intermixed configurations become energetically favourable when decreasing cluster size. We perform these global optimization searches by using the atomistic SMATB potential, and then we validate the atomistic results by comparing them to DFT calculations on selected isomers. The first size that we consider is 147, corresponding to a diameter D of about 1.7 nm. This is a magic size for the Mackay icosahedron (referred to as Ih). We consider Au-rich compositions, because in the bulk Co solubility in an Au matrix is higher than Au solubility in a Co matrix, 4 and, as we will see, chemical ordering in Co-rich nanoalloys is dominated by Au surface segregation.


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Eb (eV)

Dh r-Ih Ih

(a) x

Co atoms Ih – SMATB r-Ih – SMATB Ih – DFT r-Ih – DFT

ΔE (eV)

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(b)

Dh energy

Co atoms

x

Figure 1: (a) Binding energy Eb (in eV) of the lowest-energy Au147−x Cox clusters (x = 0, ..., 20) according to the atomistic SMATB potential. Different symbols correspond to different structures, which are shown in the insets: squares for Dh, diamonds for r-Ih and circles for Ih. (b) Comparison between DFT and atomistic results for x between 0 and 4. The energy difference ∆E (in eV) of r-Ih and Ih with respect to Dh structures is reported, showing a Dh → r-Ih transition for x = 3 according to DFT and for x = 4 according to SMATB.


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(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Cross-sections of AuCo nanoalloys of size 201. The insets show the nanoalloy surfaces. (a) Au188 Co13 , cuboctahedral centered core; (b) Au188 Co13 , cuboctahedral subsurface core; (c) Au188 Co13 , two-layer subsurface core; (d) Au188 Co13 , intermixed; (e) Au180 Co20 , ordered L12 phase in the inner part; (f) Au180 Co20 , subsurface core. The results for size 147 are shown in Figure 1. In pure Au147 , the lowest-energy structure is not an icosahedron, but a 146-atom (3,2,2) Marks decahedron (Dh) 35,36 with an additional atom adsorbed on one of the five rectangular (100)-like facets. This structure is the global minimum also for compositions with one, two or three Co atoms, which are placed inside the cluster. 29 However, for Au143 Co4 , there is a transition to a global minimum of different geometry, an icosahedral structure with rosette reconstruction at some vertices 37,38 (see Figure 1(a)), which will referred to as r-Ih structure in the following. Co atoms form a compact core at the center of the r-Ih, which is by far the most favorable position for small impurities such as Co atoms. 39 With increasing Co content, the central compact core increases in size, and from Au139 Co8 on the rosette reconstruction disappears so that the global minimum is the usual Mackay icosahedron, whose stability is enhanced 6 ACS Paragon Plus Environment

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for composition Au134 Co13 , at which a central high-symmetry core is formed. The DFT results are in very good agreement with the SMATB results, as shown in Figure 1(b). In fact, the DFT calculations confirm that the decahedron is lower in energy than the r-Ih and Ih structures for pure Au. When inserting a few Co impurities, a transition to r-Ih structures is also found at the DFT level. According to DFT, the transition is already accomplished for three Co atoms, instead of the four of the atomistic calculations. For size 201 (see Figure 2 and Table 1), we investigate the preferential chemical ordering in fcc truncated-octahedral (TO) structures. We consider Au188 Co13 and Au181 Co20 compositions. For Au188 Co13 , the results of the atomistic global optimization simulations show that phase-separated arrangements are much lower in energy than intermixed ones. Compact Co inner cores are formed, whose optimal placement is not in the center of the cluster (as in Figure 2(a)), but in subsurface positions (Figure 2(b)-(c)) which allow a better stress release. 29 As for the Co core shape, a twolayer geometry is slightly more favourable than the cuboctahedral geometry. These results are very nicely confirmed by our DFT calculations, which show an even more marked preference for subsurface core placements. For Au181 Co20 , we compare a L12 ordered phase arrangement in the inner part of the cluster (Figure 1(e)) with a phase-separated arrangement with subsurface compact core (Figure 1(f)), the latter being the optimal chemical ordering according to atomistic calculations. Both DFT and SMATB results agree in preferring the phase-separated arrangement by large energy differences. Table 1: Energy differences (in eV) between the isomers in Figure 2, according to atomistic SMATB calculations (∆ESMAT B ) and to DFT calculations (∆EDFT ). isomer Au188 Co13 (a) Au188 Co13 (b) Au188 Co13 (c) Au188 Co13 (d) Au181 Co20 (e) Au181 Co20 (f)

∆ESMAT B 0.00 -0.53 -0.62 4.33 0.00 -3.71

∆EDFT 0.00 -0.94 -1.02 2.80 0.00 -5.34

Finally, for size 561 (Figure 3), we report the global optimization results for a series of com7 ACS Paragon Plus Environment

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(a)

(b)

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(c)

(e)

(d)

(f)

Figure 3: Lowest-energy structures of AuCo nanoalloys of size 561 according to SMATB calculations. (a),(b) Dh structures for pure Au and very low Co content; (c) r-Ih structure; (d) Ih structure with centered symmetric core; (e) Ih structure with off-center low-symmetry core; (f) fcc structure for a perfect Co@Au nanoalloy, with all Au atoms on the surface and all Co atoms in inner sites. positions, from pure Au to the composition of a perfect Co@Au structure, with all Co atoms in inner sites and all Au atoms on the surface. We find that optimal structures always present phaseseparated arrangements (with a single exception, the Dh Au559 Co2 , in which the two Co atoms occupy the sites below opposite vertices), and morphology transitions at increasing Co content from Dh, to r-Ih, to Ih (with centered-symmetric and off-center cores 29 ). Finally we find also some fcc structures. In summary, our results show that lowest energy structures are always phase-separated, with no indication of enhanced miscibility due to nanoscale effects. However these results reflect the nanoalloy behavior in the limit T → 0. In order to take temperature effects into account, we


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develop a statistical mechanics approach to calculate the free energy-cost ∆G for the formation of Co aggregates. We apply this approach to Co aggregates of N = 13 in fcc and icosahedral Au host nanoparticles, showing that this cost is considerably lowered with respect to the bulk crystal case due to specific nanoscale effects. In our approach we calculate the free-energy difference between states in which a specific aggregate is formed and states in which the atoms of the aggregate are dispersed in the host cluster. In the latter case we assume a concentration of impurities low enough that each can be considered independent. This involves estimating the energies and the multiplicities of homotops 40 of different types. We assume that the variation of volume between different homotops is negligible therefore ∆G = ∆E − T ∆S + p∆V ' ∆E − T ∆S = ∆F

(1)

∆F = Faggr − Fdiss = −kB T (ln Zaggr − ln Zdiss ),

(2)

Now

where Zaggr and Zdiss are the partition function of aggregated and dissociated configurations, respectively. In the following we neglect vibrational free-energy differences between homotops (see the supporting information for an evaluation of vibrational contributions), so that free energy differences depend on energy differences between local minima and configurational entropy differences, since vibrational contributions cancel out when calculating ∆F. We consider first fcc TO clusters, and then icosahedra. We consider N impurities dissolved randomly in a matrix TO nanoparticle of M N sites. Since single-impurity energies are different between different sites in a TO, we group these sites in ns = 7 classes. Within each class, differences in impurity energies are very small (with the exception of class 7, see below), so that they are neglected. Class k contains Matom,k sites, with energy Eatom,k . Class 1 contains inner sites (i.e. those that are neither surface nor subsurface sites). Classes from 2 to 6 contain subsurface sites. Subsurface sites must be treated in more detail, since the most favorable placements of single impurities are in some of these sites. Specifically classes 2 and 3 contains sites below (100) and


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(111) terraces, classes 4 and 5 contain sites below (100)/(111) and (111)/(111) edges, and class 6 contains sites below vertices. Finally class 7 contains all surface sites, which are the least favorable. Here we set Eatom,7 equal to the energy of an impurity in a terrace site of a (111) facet, because the majority of surface sites is of this type, and they are also the surface sites with the lowest energy. By this approximation, we slightly overestimate Zdiss . Considering the subdivisions of the impurities obtained by placing Nk atoms in class k in all possible ways (with ∑k Nk = N), the expression for Zdiss is N

Zdiss =

N−N1 −...−Nns −2

N−N1

∑ ∑

...

∑

N1 =0 N2 =0

Nns −1 =0

# ns Matom,k ! ∏ Nk !(Matom,k − Nk )! e−β ∑k=1 Nk Eatom,k k=1

"

ns

(3)

where Nk is the number of Co atoms placed in sites of class k and β = 1/(kB T ). On the other hand, if all sites are equivalent to internal sites (which is what one obtains when extrapolating bulk quantities to nanoalloys) one has

Zdiss,bulk =

M! e−β NEatom,1 . N!(M − N)!

(4)

Finally, if we take into account the Au surface segregation effect, assuming that Co can occupy only internal sites with the same probability, we have

s Zdiss,bulk =

(M − Matom,ns )! e−β NEatom,1 . N!(M − Matom,ns − N)!

(5)

Zaggr depends on which aggregate is considered. Here we focus on size 13 and consider an aggregate of cuboctahedral shape. Analogous results can be obtained considering a flatter aggregate (for example as the aggregate in Figure 2(c)) instead of a cuboctahedron. The position of the cuboctahedral aggregate is determined by the position of its central atom, which can occupy all subsurface and inner sites, but not the surface sites. Placing the center of the Co cuboctahedron in a subsurface site is unfavorable (because this exposes some Co atoms on the surface), whereas placing the center in the second atomic layer below the surface is energetically favourable, especially


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below (100) facets. We group the sites in na = 8 classes. Class 1 contains inner sites, for which the center of the cuboctahedton is at least three atomic layers below the surface. Classes from 2 to 6 refer to cuboctahedra whose center is two layers below the surface. Classes 2 and 3 refer to positions below (100) and (111) terraces, classes 4 and 5 to subedge positions for (100)/(111) and (111)/(111) edges, and class 6 to subvertex positions. In classes 7 and 8, the center of the cuboctahedron is in the subsurface layer, below (100) and (111) terraces, respectively. In these cases four and three Co atoms appear on the surface. Placements in which more than 4 Co atoms appear on the surface are neglected because of their higher energy corresponding to very small statistical weight. Zaggr is then obtained as follows na

Zaggr =

∑ Maggr,k e−β Eaggr,k ,

(6)

k=1

where Maggr,k and Eaggr,k are the number of sites and the energy of class k. If we assume that all placements of the cuboctahedron are equivalent (as in the bulk crystal) we obtain Zaggr,bulk = Me−β Eaggr,bulk

(7)

where Eaggr,bulk is the energy gain in assembling the cuboctahedron from separated atoms in inner sites. If we take into account the surface segregation of Au, forbidding the occupation of surface sites, then the center of the 13-atom cuboctahedron cannot be in surface or subsurface sites, so that s Zaggr,bulk = Matom,1 e−β Eaggr,bulk .

(8)

The free energy differences are

∆G = −kB T (ln Zaggr − ln Zdiss )

(9)

∆Gbulk = −kB T (ln Zaggr,bulk − ln Zdiss,bulk )

(10)

s s ∆Gsbulk = −kB T (ln Zaggr,bulk − ln Zdiss,bulk ).

(11)


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The values of Eatom,k , Eaggr,k , Matom,k and Maggr,k are given in the supporting information. The most important point about these parameters is that placing the aggregate in subvertex positions gives an energy gain of more than 0.6 eV over inner positions, according to SMATB calculations made on clusters of diameters D up to several nanometers. From the results in Table 1 follows that the energy gain of subvertex placements is even larger according to DFT calculations. In icosahedra, configurations in which the central site is occupied by a Co atom have a much larger statistical weight than other configurations, because the energy gain in placing the Co atom at the center of the cluster is of several eV, as verified both by DFT and atomistic calculations. Therefore we consider only homotops in which the central site is occupied by Co. In the following we consider the free energy cost ∆G for forming an icosahedral aggregate of N = 13 atoms. We will make two approximations which will both overestimate ∆G. The most favorable placement of the aggregate is at the center of the icosahedron. For calculating Zaggr we consider only this placement, thus underestimating the statistical weight of aggregated configurations. Therefore Zaggr = e−β Eaggr .

(12)

For calculating Zdiss we look for the best configuration in which the central site is occupied by a Co atom and the remaining N − 1 = 12 Co atoms are dispersed in the nanoalloy. We associate to all dissociated configurations the statistical weight of the best configuration, thus overestimating their statistical weight. This gives

Zdiss =

(M − N)! e−β Ediss,best (M − 2N + 1)!(N − 1)!

(13)

which means that 12 Co atoms are randomly placed in M −13 sites of the icosahedron of M sites (to avoid the vicinity of the central site), each configuration being associated with the energy Ediss,best of the best configuration. Therefore

Zaggr ∆G = −kB T ln Zdiss

(M − N)! = −kB T ln + (Eaggr − Ediss,best ). (M − 2N + 1)!(N − 1)! 12 ACS Paragon Plus Environment

(14)

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The values of ∆Eih = Eaggr − Ediss,best for icosahedra of different diameters D up to several nm are calculated by the SMATB potential. All these values are well below -3 eV. They are given in the supporting information. ∆G results for the TO are shown in Figure 4(a-c). In (a) and (b) we consider two different sizes, of 1289 (diameter D=3.6 nm) and 23178 atoms (D = 10 nm) and plot ∆Gbulk , ∆Gsbulk and ∆G for a T range from 300 to 800 K. For size 1289, ∆G is lower than ∆Gbulk by 0.45-0.62 eV in the whole T range, thus showing a notable lowering of the free-energy cost for aggregate formation due to nanoscale effects. ∆Gsbulk is in between, being closer to ∆Gbulk at low T and closer to ∆G at high T . Even taking surface segregation into account, other nanoscale effect induce a cost lowering of between 0.46 eV at 300 K and 0.27 eV at 800 K. For size 23178, nanoscale effects are decreased. The lowering of the free-energy cost is smaller but still non negligible. In (c), we plot the quantities

f∆G =

∆Gbulk − ∆G ∆Gbulk

f∆Gs =

∆Gsbulk − ∆G ∆Gsbulk

(15)

which measure the relative lowering of the free-energy cost due to nanoscale effects, as functions of the nanoparticle diameter D and for T = 500 and 600 K. Strong decreases of the free-energy cost for aggregate formation are found for D up to 5 nm, while they tend to become fully negligible for D ' 20 nm. The free-energy cost decrease is mostly due to the energy gain in placing the aggregate in subvertex and other subsurface positions. As we noted before, this energy gain is likely to be even larger at the DFT level, as it happens for size 201. ∆G results for icosahedra are given in Figure 4(d), for three sizes in the range 2.7 ≤ D ≤ 8.7 nm. In spite of the fact that our calculations overestimate ∆G in the icosahedron, we find that ∆G is much lower than in the fcc case, thus reinforcing the nanoscale effect which facilitates phase separation. The energy gain in forming a 13-atom Co aggregate at the center of the nanoparticle is so large that its free-energy cost is still negative well above room temperature even for the largest nanoparticle size. As a final check on the prevalence of phase separation at equilibrium we have simulated


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(b) TO, M=23178

ΔG [eV]

ΔG [eV]

(a) TO, M=1289

T [K]

T [K]

(c)

M=12431

ΔG [eV]

(d) Ih

fΔG

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M=3871

M=561

T [K]

D [nm]

Figure 4: (a)-(b) Free-energy cost ∆G (in eV) for forming aggregated cuboctahedral configurations from separated configurations of 13 Co atoms in fcc TO host nanoparticles, as a function of T between 300 and 800 K. Two cluster sizes M are considered: 1289 and 23178 atoms, corresponding to diameters D of 3.6 and 10 nm, respectively. The red top lines correspond to ∆Gbulk , the middle green lines to ∆Gsbulk and the bottom blue lines to ∆G (see Eqs.(9-11)). (c) Relative lowering of the free-energy cost (see Eq.(15) for the definitions) f∆G (full symbols) and f∆Gs (open symbols) in fcc TO as a function of the nanoalloy diameter D for T = 500 K (squares) and T = 600 K (circles). (d) Free-energy cost for forming a 13-atom central aggregate of Co in Ih nanoalloys (see Eq. (14)) of different sizes M as a function of T .


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annealing-freezing cycles of a few nanoalloys by molecular dynamics. We have considered size 201 (D = 1.8 nm), for Au188 Co13 and for intermediate compositions, and size 4033 (D = 5.5 nm) for intermediate compositions. The clusters were heated from room temperature to a temperature well above melting, then they were cooled down back to room temperature. Heating/cooling rates were of 1 K/ns and 0.1/ns for size 201 and 1 K/ns for size 4033. Snapshots from the simulations are given in Figure 5. For Au188 Co13 we performed 10 independent simulations starting from a randomly intermixed TO structure. After heating and cooling, the final result was always an Ih structure with compact Co core at its center. For sizes 201 and 4033 with intermediate compositions, we considered different initial configurations (randomly intermixed, Janus, Co@Au and Au@Co) always obtaining Co@Au structures with Co compact cores and practically no intermixing as final results. Our results thus indicate that the intermixed configurations obtained in some experiments are metastable, whereas the experiments obtaining Co@Au structures are closer to equilibrium. However, metastable intermixed configurations can present very long lifetimes at room temperature. Our MD simulations were indeed able to produce the transition to phase-separated configurations only when annealed close to the melting range of the nanoparticles. This suggests that growth/synthesis methods aiming at obtaining equilibrium configurations may require heating above room temperature and/or quite long times. In conclusion, according to our calculations, the free-energy cost for nucleating the same phaseseparated aggregate strongly decreases at the nanoscale with respect to the corresponding bulk crystal, so that the simple scaling laws depending on aggregate volume (∝ R3 ) and interface area (∝ R2 ) for bulk and interface contributions do not hold anymore. The free-energy decrease is due to the energy gain in placing aggregates in specific positions, such as subsurface positions in fcc nanoparticles and central positions in icosahedra, which act as preferred nucleation sites. The statistical weight of these preferred sites tends to vanish as the nanoparticle size increases, but the associated free-energy decrease is still quite large for sizes of ' 10 nm in a wide temperature range. The lowering of the free-energy cost for aggregate formation at the nanoscale implies a


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(a)

(b)

(c)

(d)

(e)

(f)

e) (g)

(h)

(i)

(j)

Figure 5: Final structures of AuCo nanoalloys of sizes 201 and 4033 (D = 1.8 and 5.5 nm, respectively) after heating/cooling cycles by molecular dynamics. Gold atoms are shown as small spheres so that Co cores are visible. The insets show the initial configurations. (a)-(b) Au188 Co13 with heating/cooling rates of 1 K/ns and 0.1 K/ns. (c)-(f) Size 201 and different compositions in the intermediate range. These compositions correspond to different initial configurations: (c) Au@Co; (d) Co@Au, (e) Janus; (f) intermixed. The heating/cooling rate is 1/K ns. (g)-(j) The same as in (e)-(d) but for size 4033.


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decrease of the critical nucleus size with nanoparticle size, so that phase separation is still possible at equilibrium down to the smallest sizes. This shows that predicting nanoscale behavior by using bulk parameters for free energies is not always possible. Finally, we note that our statisticalmechanics approach to calculate free-energy differences is of general character, provided that the specificity of each system is taken into account when defining the classes of sites. The results shown here are obtained for AuCo, but we expect that the same behaviour is shared by a series of systems such as for example AgNi, AgCo, AuNi, CuNi, AuPt, and AgCo, which present some of the key features of AuCo about bulk phase separation, surface segregation and lattice mismatch. This is supported by our simulation results for AgCu, AgNi and CuNi, reported in the supporting information, which show no evidence of a critical size below which phase separation is impossible. Supporting information available: energetic and geometric parameters for the calculation of free-energy differences; evaluation of vibrational entropy contributions to ∆G; simulation results for AgCu, AgNi, and CuNi. The authors declare no competing financial interests. J.-P. P-.B. acknowledges postdoctoral grant fellowship from CONACYT-Mexico, with number 238903.

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