Wave transport in stealth hyperuniform materials: the diffusive regime and beyond AIP/123-QED Wave transport in stealth hyperuniform materials: the diffusive regime and beyond Élie Chéron, [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF] Simon Félix, [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF] Jean-Philippe Groby, [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF] Vincent Pagneux, [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF] and Vicente Romero-García 1, a) Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique -Graduate School (IA-GS), CNRS, Le Mans Université, Le Mans, France (*Electronic mail: Jean-Philippe.Groby@univ-lemans.fr) (Dated: 11 July 2022) By varying the degree of correlation in stealthy hyperuniform (SHU) materials, the continuous evolution from uncorrelated disorder to periodic media is possible and allows, as such, to study the fate of the bimodal distribution, characteristic of a diffusive transport.

Considering the wave transport through a SHU distribution of a given number of scatterers and at a given frequency, the transition from a diffusive to a transparent medium is clearly observed only below the Bragg frequency. This transition is characterized by a threshold value of the stealthiness, at the vicinity of which the material abruptly changes from diffusive to transparent. Contrastingly, no such clear transition is observed at or above the Bragg frequency and, surprisingly, a seemingly-bimodal distribution of the transmission eigenvalues still characterizes the SHU materials, even when strongly correlated.

a) Also at Instituto Universitario de Matemática Pura y Aplicada, Departamento de Matemática Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022, València, Spain 1 Wave transport in stealth hyperuniform materials: the diffusive regime and beyond Interferences play an essential role in wave transport through complex heterogeneous media.

They give rise to phenomena that prevail in the scattering properties of such media and open up numerous applications for wave control. Enhanced backscattering, conductance fluctuations, or the Anderson localization in disordered media [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF][START_REF] García-Marín | Statistical properties of wave transport through surfacedisordered waveguides[END_REF][START_REF] Rotter | Light fields in complex media: Mesoscopic scattering meets wave control[END_REF] , as well as the band structure of the transmission spectrum in periodic media [START_REF] Joannnopoulos | Photonic crystals. Molding the flow of light[END_REF][START_REF] Deymier | Acoustic metamaterials and phononic crystals[END_REF] , are well-known examples of interference induced phenomena.

Another striking example is the bimodal distribution of the transmission eigenvalues (TEV) in the transport through a diffusive disordered medium [START_REF] Dorokhov | On the coexistence of localized and extended electronic states in the metallic phase[END_REF][START_REF] Imry | Active transmission channels and universal conductance fluctuations[END_REF][START_REF] Mello | Macroscopic approach to multichannel disordered conductors[END_REF][START_REF] Nazarov | Limits of universality in disordered conductors[END_REF][START_REF] Beenakker | Random-matrix theory of quantum transport[END_REF][START_REF] Choi | Transmission eigenchannels in a disordered medium[END_REF][START_REF] Goetschy | Filtering random matrices: The effect of incomplete channel control in multiple scattering[END_REF][START_REF] Gérardin | Full transmission and reflection of waves propagating through a maze of disorder[END_REF][14] . The TEV distribution P(τ) indeed exhibits two peaks corresponding to closed, almost fully reflected, eigenchannels (τ → 0) and open, almost fully transmitted, eigenchannels (τ → 1). Of particular interest are the latter, which existence implies, that, given a sufficiently controlled pattern of the incident wave, it can be transmitted with almost no energy loss through an otherwise opaque medium. This counter-intuitive effect has given rise to the wavefront shaping technique, following a first experimental evidence by Vellekoop and Mosk [START_REF] Vellekoop | Focusing coherent light through opaque strongly scattering media[END_REF] .

Between the limit cases of fully disordered or perfectly crystalline media, correlated materials, that is, scattering systems which disorder displays spatial correlations, have emerged as new possibilities to control waves [START_REF] Vynck | Light in correlated disordered media[END_REF][START_REF] Davy | Experimental evidence of enhanced broadband transmission in disordered systems with mirror symmetry[END_REF][START_REF] Zhang | Experimental demonstration of luneburg lens based on hyperuniform disordered media[END_REF][START_REF] Zhang | Hyperuniform disordered distribution metasurface for scattering reduction[END_REF] and design functional materials [START_REF] Weijs | Emergent hyperuniformity in periodically driven emulsions[END_REF][START_REF] Chremos | Hidden hyperuniformity in soft polymeric materials[END_REF] . Also, one may wonder about the fate of the bimodal property when introducing correlations, especially since this property will no longer be observed in strongly correlated, crystalline, structures, which will rather be fully opaque or transparent due to Bragg scattering.

A good candidate to investigate the continuous transition from diffusive to transparent materials is the class of hyperuniform materials [START_REF] Torquato | Local density fluctuations, hyperuniformity, and order metrics[END_REF] . Hyperuniform materials are made of a discrete distribution of scatterers on a correlated point pattern, the long-range density fluctuations of which vanish. In Fourier space, this translates in a vanishing structure factor S(q) when |q| → 0. A particular class of hyperuniform materials, which we will consider in this paper, is that of stealthy hyperuniform (SHU) materials, for which the structure factor vanishes on a finite domain |q| < q c , the bound of which depends on a stealthiness parameter, χ (see below), that allows us to continuously tune the material from fully disordered to perfectly ordered [START_REF] Batten | Classical disordered ground states: Super-ideal gases and stealth and equi-luminous materials[END_REF][START_REF] Florescu | Designer disordered materials with large, complete photonic band gaps[END_REF][START_REF] Zhang | Ground states of stealthy hyperuniform potentials: I. entropically favored configurations[END_REF][26][27][28][START_REF] Froufe-Pérez | Band gap formation and Anderson localization in disordered photonic materials with structural correlations[END_REF][START_REF] Torquato | Hyperuniform states of matter[END_REF][START_REF] Bigourdan | Enhanced absorption of waves in stealth hyperuniform disordered media[END_REF][START_REF] Aubry | Experimental tuning of transport regimes in hyperuniform disordered photonic materials[END_REF][START_REF] Rohfritsch | Impact of particle size and multiple scattering on the propagation of waves in stealthy-hyperuniform media[END_REF][START_REF] Sheremet | Absorption of scalar waves in correlated disordered media and its maximization using stealth hyperuniformity[END_REF][START_REF] Romero-García | Wave transport in 1d stealthy hyperuniform phononic materials made of nonresonant and resonant scatterers[END_REF] . As a consequence, the material is transparent to long-wavelength incident waves under the assumption of single scattering. Experimental evidences of this kind of structures has been recently shown for airborne acoustic [START_REF] Romero-García | Stealth acoustic materials[END_REF][START_REF] Chéron | Experimental characterization of rigid scatterer hyperuniform distributions for audible acoustics[END_REF] as well as for electromagnetic waves [START_REF] Davy | Experimental evidence of enhanced broadband transmission in disordered systems with mirror symmetry[END_REF][START_REF] Aubry | Experimental tuning of transport regimes in hyperuniform disordered photonic materials[END_REF][START_REF] Man | Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids[END_REF] .

In this work, we investigate the transition from uncorrelated disorder to periodic media by terers in a quasi-one-dimensional waveguide. For the sake of simplicity and computational efficiency, the scatterers (red squares) are located on a regular grid, and a full wave numerical solution [START_REF] Chéron | Experimental characterization of rigid scatterer hyperuniform distributions for audible acoustics[END_REF] gives the scattering matrix of the L-length disordered slab.

considering the transmission of waves in a quasi-one-dimensional disordered waveguide. On a L-length segment of an otherwise homogeneous waveguide with unit width (Fig. 1), local heterogeneities are created by changing the material parameters on a set of randomly chosen sites of a regular grid (red squares).

Namely, the wave equation reads

div(a(r)∇ψ) + k 2 b(r)ψ = 0, (1)
with a = b = 1 in the background medium of wavenumber k and a 1, b 1 in the scatterers.

Small values of a and b are chosen so that the scatterers behave as acoustically rigid obstacles (or as perfectly conducting obstacles if ψ is a TM-polarized magnetic field travelling in the waveguide), hence without resonant behaviour. With the waveguide supporting N propagating modes, the N × N transmission matrix T of the disordered slab is computed and used to characterize the wave transport (see [START_REF] Chéron | Experimental characterization of rigid scatterer hyperuniform distributions for audible acoustics[END_REF] for details on the numerical computation).

The scatterers are located on the grid as follows: (i) a 2D SHU distribution of N s points on a L s ×L s square area is first generated following the procedure proposed by Froufe-Pérez et al. 28 (see below), (ii) this distribution is then scaled to a L × L area, so as to keep constant the characteristic length d = L/ √ N s , that is, the typical distance between the points, (iii) a subset of the point distribution that belongs to a L × 1 rectangular area is extracted, (iv) the distribution of the nearest square sites on the grid is associated to the point distribution. Note that the mesh size is taken small enough (typically, 10 -2 ) to ensure that this shifting of the points on the regular grid has no significant effect on the SHU pattern properties.

The algorithm proposed in 28 starts from a random distribution of points {r j }, j ∈ [1, N s ], in a square box and uses a simulated annealing relaxation scheme to find a pattern with a minimized structure factor in the reciprocal domain |q| < q c :

χ = 0.60 (c) (d) -2 -2 -1 -1 0 0 1 1 2 2 q x /q B q y /q B (e) -2 -2 -1 -1 0 0 1 1 2 2 q x /q B q y /q B (f) -2 -2 -1 -1 0 0 1 1 2 2 q x /q B q y /q B FIG. 2.
S(q) = 1 N s N s ∑ j=1 e iq•r j 2 < ε, (2)
with, typically, ε = 10 -6 . The degree of positional correlation of the generated pattern can be encoded by the stealthiness χ:

χ = M(q c ) 2(N s -1) , M(q c ) = 1 2 πq 2 c (2π/L s ) 2 , (3)
which is the ratio of the number of constrained degrees of freedom, M(q c ), over the total number of degrees of freedom, 2(N s -1) (upon removing the translational degrees of freedom). The lower bound χ = 0 corresponds to an uncorrelated disordered distribution. Figures 2(a-c into a square lattice when q c = q B = 2π/d, corresponding to a maximum value of the stealthiness χ max π/4 for large N s (sets of approximately 6000 points were generated for the following numerical results).

Figures 2(d-f) show the corresponding structure factor S(q) as estimated by the spatial Fourier transform of the L × 1 distribution of scatterers aligned to the grid, with L = 3 (it is thus made of approximately 2000 scatterers). For a low degree of correlation (χ = 0.20), the constrained region |q| < q c clearly appears and the surrounding region displays a global isotropy. SHU structures remain isotropic up to χ 0.5, while S(q) locally increases around |q| = q B , as a precursor signature of the Bragg scattering characteristic of periodic media, see Fig. Let us place the generated disordered distributions of scatterers in a waveguide, as shown in Fig. 1, and analyze how the disorder correlation affects the transmission, depending on the frequency of the incident wave. To do this, three frequencies are chosen as depicted on Figs. 2(d-f) by the pink, red, and dark red circumferences. Note that these circumferences depict the frequencies in a reduced form k/k B , with k B = q B /2 = π/d, as a consequence of the von Laue condition for scattering [START_REF] Ashcroft | Solid State Physics[END_REF] (see, e.g., [START_REF] Kuznetsova | Stealth and equiluminous materials for scattering cancellation and wave diffusion[END_REF] for details). For χ small enough, the three chosen frequencies "lie" in the unconstrained region, see Fig. 2(d) and thus a classical diffusive transport is expected. For larger values of χ, the transport in a strongly correlated medium, and consequently the transition from disorder to order, is more sensitive to the frequency, and in particular to its relative value to the Bragg frequency k B .

The first frequency, k = 0.6k B (pink circumfrence), is chosen below the Bragg frequency, such that, for χ large enough (namely, above χ = 0.6 2 χ max 0.3), this frequency lies in the constrained region and the medium is then expected to be transparent. This is indeed what is observed when plotting the Landauer conductance g = Tr(TT † ) from Ref.

[41] as a function of the stealthiness, see Fig. 3(a), where T represents the transmission matrix of the system.

The conductance is first relatively small ( g /N 0.3, here averaged over 100 realizations of the scatterer distribution), as typically observed in the diffusive transport, and reaches a high plateau for larger values of χ. Note that the conductance does not reach its maximum value, g /N = 1, although transparency is expected. This is due to the alignment of the point pattern on the regular grid (see above), the consequence of which is a structure factor that is not perfectly zero in the constrained region, hence a non perfect transmission. Between these two limits (diffusive and transparent), the transition is abrupt. Note that a scale of this transition width with χ can be deduced from the initial point pattern. Indeed, consider an integrated structure factor

F(q, χ) = ∑ |q|=q S(q, χ), (4)
the inverse of which is expected to be low in regions of strong scattering and large in the regions of weak scattering, as is the transmission. Figure 4(a) shows that F -1 (q, χ) displays the same abrupt transition for k < k B (analogously, q < q B ) and compares well with the conductance.

For a weakly correlated disorder, a characteristic property of the diffusive transport is that the TEV follow the bimodal distribution

P(τ) = N L + 1 τ √ 1 -τ , (5)
with the transport mean free path, as shown in Fig. 3(b), or, equivalently, in Fig. 3(c), with the TEV following

τ n = 1 cosh 2 (n/ n) , (6)
with n adjusted to meet ∑ N n=1 cosh -2 (n/ n) = g [START_REF] Dorokhov | On the coexistence of localized and extended electronic states in the metallic phase[END_REF][START_REF] Choi | Transmission eigenchannels in a disordered medium[END_REF]42 . Above the threshold value χ 0.3 -an example is given in Figs. 3(b-c) for χ = 0.4 -the TEV distribution no longer follows the bimodal distribution and no closed eigenchannel is observed, resulting in the strongly increased transparency of the medium. Figure 3(d) shows the distributions of TEV, τ n , as functions of the index n and stealthiness χ: the abrupt transition from the diffusive to the transparent regime clearly appears.

Close to the Bragg frequency k B (red circumference in Fig. 2), the effect of increasing the disorder correlation on the transmission significantly differs from that observed at lower frequencies. Figure 3(e) shows the evolution of the averaged conductance with the stealthiness. While the medium still behaves as a diffusive medium for low values of χ, with the conductance following the Ohm's law g /N = /(L + ) 43 , the Bragg scattering, consequence of the progressive crystallization of the medium, makes the averaged transmission decrease near χ 0.5. Above χ 0.6, the anisotropy of the hyperuniform medium makes the transmission strongly dependent on the incident field direction, or, in our case, on the realization of the randomly generated slab in the waveguide: the medium can be either strongly reflecting or almost transparent. This relates to the structure factor shown in Fig. 2(f). Near |q| = q B (red circumference) the structure factor may be close to zero or have larger values (Bragg peaks), depending on the orientation of the wavevector. When averaging over the angles, the integrated structure factor F(q q B , χ) decreases with increasing χ in the crystalline regime (χ > 0.6, see Fig. 4(b)). This decrease can be associated to the increase of the average conductance in Fig. 3(e). Remarkably, a bimodal distribution of the TEV is still observed when varying the stealthiness, see Figs. 3(f-h). Regardless of the disorder correlations, the eigenchannels are predominantly closed or open.

In the last case considered (dark red circumferences in Fig. 2), when the frequency is higher than the Bragg frequency, the wavevector always lies in the unconstrained region. A consequence is that the value of the conductance remains roughly that in a fully diffusive medium, until the transition from disordered to crystalline, that is, around χ 0.5, see Fig. 3(i). Then, the same averaging induced effect as discussed above leads to an increase of the transmission with the stealthiness in crystallized SHU media. Note that, above q = q B and for large χ, the evolution of the structure factor no longer follows that of the conductance above k = k B , see Figs. possibly overestimates this scattering. Note, finally, that a bimodal distribution of the TEV is still observed when k > k B , regardless of the stealthiness. Thus, although the bimodal distribution is usually known as a characteristic of the diffusive transport in a fully disordered medium, it seems that, considering SHU media, fulfilling a bimodal law is the rule, and breaking it, the exception.

Figures 5(a) and (b) show two snapshots for k < k B of the acoustic field through two SHU distributions with χ = 0.10 and χ = 0.48 respectively. As discussed before in Figs. 3 and4, the random scattering for the SHU distribution with χ = 0.10 at the k < k B makes the material almost opaque, as the conductance and, as a consequence, the transmission is very low. However, for the SHU distribution with χ = 0.48 at the k < k B , the material is transparent before the transition from diffusive to transparent media.

We have analyzed the continuous transition from the diffusive transport through an uncorrelated disorder to the transparency or Bragg scattering in an ordered, periodic, medium. This transition is achieved by using stealthy hyperuniform distributions of rigid scatterers in a waveguide, with a controlled and adjustable stealthiness. A first remarkable observation is, at sufficiently low frequency -namely, below the Bragg frequency -an abrupt transition from diffusive to transparent is observed. A threshold value of the stealthiness, frequency dependent, separates media that are mostly opaque for the incident wave, as illustrated in Fig. 5(a), from media that are almost transparent, see Fig. 5(b). The mechanism and typical scale of this sharp transition, although not fully explained with the present work, can be related to the structure factor of the spatial distribution of the scatterer locations. The results shown here can be used for the material design as elements to control both the diffusivity and the transparency of the material. It is also noticeable that the bimodal distribution of the transmission eigenvalues appears as a general property of the propagation through the SHU medium and not solely as a characteristic of the diffusive transport.

1 FIG. 1 .

 11 FIG. 1. Schematic representation of the scattering of a wave impinging on a random distribution of scat-

  FIG. 2. (a,b,c) typical configurations of two-dimensional SHU point patterns for increasing values of the stealthiness (χ = 0.2, 0.48, and 0.6), obtained as detailed in Ref. 28 . (d,e,f) corresponding structure factor S(q), as estimated by the spatial Fourier transform of a pattern having the dimensions (L, 1) of the slab in Fig. 1, with L = 3 (it is thus made of approximately 2000 scatterers). On each plot, the pink, red and dark red circumferences represent the chosen observation frequencies, k/k B = 0.6, 1.08, and 1.28, see Fig. 3.

FIG. 3 .

 3 FIG. 3. Transmission through a slab of scatterers with correlated disorder, at a frequency below the Bragg frequency k B (first row, a-d), at k B (second row, e-h), and above k B (third row, i-l). First (left) column: conductance, averaged over 100 realizations, as a function of the stealthiness χ. Second column: TEV distribution P(τ) (normalized) for selected values of χ, shown as dashed blue lines in the first column plots. The black solid line shows the theoretical bimodal distribution 5. Third column: TEV τ 2 n for the same values of χ, ordered by decreasing values, with comparison to the bimodal relation 6. Fourth column: TEV τ 2 n , as a function of the stealthiness χ and index n.

FIG. 4 .

 4 FIG.4. Averaged conductance, as shown in Fig.3(a,e,i), compared with the inverse of the integrated structure factor F(q = αq B , χ), with (a) α = 0.6, (b) α = 1.08, (c) α = 1.27, 1.28, 1.29.

  2(e). A second local increasenear |q| = 2q B is observed. Above χ = 0.5, the structures are no longer isotropic, see Fig.2(f), and a discrete pattern, characteristic of a crystalline structure, gradually appears.

  FIG. 5. (a) Amplitude of a typical wavefield in a SHU medium with low stealthiness and a a frequency below the Bragg frequency, illustrating a characteristic diffusive transport. (b) At the same frequency, amplitude of a typical wavefield in a SHU medium with a larger stealthiness such that the medium is almost transparent.The incident wave, ψ in , is a plane wave.

G. Zhang, F. H. Stillinger, and S. Torquato, "Ground states of stealthy hyperuniform potentials.ii. stacked-slider phases," Phys. Rev. E 92, 022120 (2015).

O. Leseur, R. Pierrat, and R. Carminati, "High-density hyperuniform materials can be transparent," Optica 3, 763 (2016).

ACKNOWLEDGMENTS

This work has been funded by the project HYPERMETA funded under the program Étoiles Montantes of the Région Pays de la Loire, by the ANR-RGC METARoom (ANR-18-CE08-0021) project. This work is also part of the R&D&I project/grant PID2020-112759GB-I00 funded by MCIN/AEI/10.13039/501100011033/.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

(1999). 43 Note that the "Ohm" conductance varies with the frequency (see, in Figs. 3(a,e,i), the values of the conductance for χ 1), since the transport mean free path itself depends on the frequency.


Christine Goyhenex

Rationalization of the sub-surface segregation in nanoalloys of weakly miscible metals †

The origin of the stability of sub-surface precipitates in core-shell bimetallic nanoparticles is investigated from the perspective of atomic size effects for systems where the core atoms have a size equal to or lower than the shell atoms. In the aim of providing more general assessments a systematic study is proposed by considering three model systems combining weakly miscible metals: IrPd (negligible lattice mismatch, ∆r/r Pd = -1%), AuRh (moderate lattice mismatch, ∆r/r Au = -7%) and AuCo (large lattice mismatch, ∆r/r Au = -13%). The main driving forces for sub-surface segregation and the characteristic core morphologies are quantified from the combination of Monte Carlo and Quenched Molecular Dynamics Simulations. The preferential occupation of the sub-surface shell by an impurity of Co or Rh (respectively Ir) in an Au nanoparticle (respectively Pd), in particular at the sub-vertex sites, is found to be a common feature in these dilute nanosystems. With the help of a model of the decomposition of the segregation enthalpies, it is shown that the dominant driving forces leading to the preferential subsurface segregation at vertex sites can be very different from one system to another: atomic size (AuCo, large lattice mismatch), coupled alloy-size-cohesion (AuRh, moderate lattice mismatch) or coupled alloy-cohesion (IrPd, negligible lattice mismatch) effect. As a consequence in core-shell nanoalloys, in the first stage of enrichment of an Au nanoparticle (respectively Pd) with Co or Rh (respectively Ir) core atoms, all the equilibrium structures consist of similar off-center solute clusters anchored at sub-vertex sites, and this regardless of the lattice mismatch.

Introduction

Since about two decades nanoalloys have been widely investigated, both experimentally and theoretically, in the perspective of many technological applications [START_REF]Nanoalloys: From Theory to Applications[END_REF][START_REF] Entel | [END_REF][3][4] among which nanomedecine and catalysis have given rise to a very abundant literature because of nowadays essential societal reasons. [START_REF] Mcnamara | Nanoalloys[END_REF][START_REF] Kamp | [END_REF][7][8][9][10] The mixing of at least two metallic species in one nanoparticle offers the possibility of tuning its properties and possibly to improve the known properties of the pure constituents. A major concern with nanoalloys is their stability after synthesis related to their thermodynamically stable structure towards which the system could evolve over time while possibly losing the targeted properties. It is therefore essential to be able to rationalize this structure in terms of the driving forces towards thermodynamical equilibrium: atomic size or lattice mismatch between components, chemical ordering/demixing tendency, difference in surface or cohesion energies, particle size and shape... The possible relative importance between these characteristics and their intricate correlation makes it difficult to predict atomic arrangements in a nanoalloy through a simple model and explains the long standing efforts of research in this area. This complexity is finally illustrated by the occurence of a great variety of nanoalloys structures in terms of their chemical arrangement, going from random or fully ordered to phase separated (Janus nanoparticles). 11 In the latter class, one finds the so-called core-shell nanoparticles which are obtained by associating weakly miscible elements in the bulk phase. In core-shell nanoalloys, a shell of the surface-segregating element covers a core of the other element. The core can take various morphologies: centered, off-centered, multi-shell. 11 Although a considerable number of studies have been carried out on core-shell nanoalloys, a rationalization of the link between segregation and precipitate morphology in a nanoparticle has been only partially achieved. France; E-mail: christine.goyhenex@ipcms.unistra.fr 1

The greatest deal of theoretical effort in this sense has been made in the case of systems where the atomic size mismatch between the two metallic species is very large, like in AgCu, AgCo, AuCo and AgNi. [12][13][14][15][16][17][18][19] In this case, the largest element is also the one corresponding to the greatest surface energy and more attention has been paid to its surface segregation, particularly in dilute systems where it is the impurity. On the other side sub-surface segregation was much less studied in these systems. Even scarcer studies aiming at building nanoalloys phase diagrams can be found for systems with much lower size mismatch like AuPt, IrPt or PdRh. 16,[20][21][22] However the knowledge and understanding of the segregation behavior in a nanoalloy is of considerable importance for predicting the evolution of the core morphology and its stability, which could have a strong implication on properties such as gas sorption and catalysis which are very sensitive to composition changes. 23,24 One missing aspect, finally, is the rationalization in terms of atomic size or lattice mismatch of the segregation behaviour in the core-shell systems which should involve the comparison, in a single systematic study, between systems having similar surface segregation behavior and mixing tendency but differing in the atomic size mismatch between the constituting elements. In this context three model systems have been investigated for the present theoretical work, IrPd (negligible lattice mismatch), AuRh (moderate lattice mismatch) and AuCo (large lattice mismatch). Atomistic simulations were used to first explore the segregation behavior in the most dilute configuration where one atom of a monometallic nanoparticle is substituted by one foreign metallic impurity atom. In particular, the segregation enthalpy is calculated using Quenched Molecular dynamics simulations by considering the variation of energy for an impurity located in the center of the nanoparticle and at all the other different possible sites under the surface shell. The results have been interpreted thanks to a model of segregation driving forces involving three distinct terms of atomic size, alloying and cohesion effects (and possibly their coupling) and which was until now mostly applied to AgCu nanoalloys. 15,25,26 In the following, it will be shown that this accurate study enables a better description of the first stages of enrichment of a nanoparticle in another metal, in the case where phase separation occurs in the bulk corresponding alloy and where the core atoms have a size equal to or lower than the shell atoms. The characteristic configurations of small precipitates were derived from off-site equilibrium Monte Carlo simulations at low temperature (200 K) while the associated local atomic stress was obtained from the Quenched Molecular Dynamics algorithm in the same way as for the impurity.

Materials and Methods

IrPd, AuRh and AuCo nanoalloys

The following notations will be used depending on the addressed configuration of the studied AB alloy system constituted by A and B metallic species:

• AB : general formula for an AB (nano)alloy system,

• A(B) : dilute (nano)alloy constituted by an A matrix containing one impurity of B,

• B@A : core-shell nanoalloy made of an A shell coating a B core.

The initial pure Pd and Au nanoparticles and the further modelled nanoalloys have an ideal Wulff shape corresponding to a face centered cubic (fcc) truncated octahedron (TOh in the following). The fcc TOh is an archetypal shape for metallic nanoparticles and nanoalloys. 3,27 It relates on the Wulff equilibrium shape of transition fcc metal crystals driven by the relative surface energy associated to the exposed facets, mainly (100) and (111). It is worth noticing that many other geometrical shapes can be developed, especially at very small size (≈ 20-200 atom or diameter < 1 nm), and lead to many homotops for nanoalloys. [START_REF] Johnston | Metal Nanoparticles and Nanoalloys[END_REF][START_REF] Montejano-Carrizales | [END_REF] The fcc truncated octahedral shape has been observed experimentally in the size (or diameter) range 1.5-4 nm in IrPd 30 and AuRh 31 samples of nanoparticle catalysts. For AuCo, experimental works report more rounded shapes for nanoalloys (but still with an fcc structure) than geometrically well defined ones. [32][33][34][35] In this case, the TOh shape can be considered as the geometrical one closest to a sphere. On the point of view of the theoretical approach, the most important is to have in the model different sites with characteristic coordination numbers in order to rationalize the phenomena of segregation at the atomic scale. The used particle contains 405 atoms distributed on the fcc lattice. The corresponding diameter is ≈ 2 nm, an usual size in experiments. 7,30,36 Fig. 1 shows this typical TOh nanoparticle. It can be viewed as a structure of concentric shells each having the same geometry as shown in the cross sectional view of Fig. 1 (right hand side). In the following, the concentric shells will be denoted by S i where i = 0, 1, 2 indicates the level of the concerned shell starting from the surface. The index i = 3 relates to the remaining central atoms. Starting from pure Au and Pd nanoparticles, Au or Pd atoms are then replaced by Ir, Rh and Co atoms to form respectively IrPd, AuRh and AuCo nanoalloys with different concentrations. The phase diagrams of the corresponding bulk alloys present a large miscibility gap and pure Pd and Au materials have both a much lower surface energy than Ir, Rh and Co leading to a strong surface segregation of Au and Pd in the considered nanoalloys. These systems differ mostly by their atomic size or lattice mismatch. This mismatch is usually quantified by relating the lattice parameter mismatch between the fcc structures of the pure materials. The bulk lattice mismatch relatively to the matrix element (here Au or Pd), ∆r/r Au,Pd , is the highest for AuCo (-13%), the lowest for IrPd (-1%) and intermediate for AuRh (-7%). The minus sign relates to the smaller atomic size of Co, Ir and Rh relative to that of the matrix element.

Atomistic simulations

The A(B) dilute systems, implying the introduction by substitution of one impurity B in a nanoparticle of species A, were studied with a classical molecular dynamics algorithm. More specifically, in the present work, a quenched molecular dynamics (QMD) method has been used as a relaxation procedure to optimize the crystal structure at T = 0 K. It means that the equations of motion are solved for atoms i while cancelling their velocity v i , when the product -→ F i . -→ v i becomes negative. Such atomistic simulations enable to obtain the total energy of a configuration while including realistic atomic relaxations. They also enable some site specific analysis, in particular through local pressures which should be useful to explore local stress effects due to the atomic size difference between the impurity and its matrix. The local pressure is given by the variation of the interaction energy E i j as a function of the local deformations of the interatomic distances r i j :

P i = - 1 3 ∑ j dE i j dr i j r i j . (1)
According to this definition, a positive value of the local pressure corresponds to an atomic compressive stress while a negative one corresponds to an atomic tensile stress. The local pressure is calculated on each site of the system after the relaxation within the QMD algorithm. Following the goal of this work to rationalize the sub-surface impurity segregation behaviour and its link with the subsequent core morphology evolution when increasing the concentration of the species B in a nanoparticle of A, it was required to have systems with an optimal atomic/chemical arrangement as a function of the concentration. For this, Monte Carlo (MC) simulations were implemented in the canonical ensemble. With this method one starts from a binary nanoalloy with an arbitrary distribution (random for instance) of the two species on the fcc lattice of the initial TOh structure, random atomic exchanges are then successively proposed as well as small random atomic displacements (≈ 0.1 Å). In a canonical simulation the concentration remains constant so that several configurations with different concentrations are considered in order to have a representative set of systems. Using the Metropolis sampling one reaches a Boltzmann distribution of the atomic/chemical configuration of a system at equilibrium owing that a sufficient large number of Monte Carlo iterations are performed. Around N=300 000 macrosteps were therefore performed where each macrostep includes an exchange proposition and 1 to 3 atomic displacements repeated N times, N being the number of atoms in the particle. At each microstep, a trial (exchange or displacement) is accepted if it leads to a lowering of the total energy of the system. Otherwise, the new tested configuration can still be accepted according an acceptation probability P = exp-( ∆E kT ). ∆E is the energy difference between the new proposed configuration and the previous one, k is the Boltzmann constant and T the temperature.

In this work the study of equilibrium structures was performed at a low temperature, T = 200 K, for specific concentrations. This simulation temperature has been chosen in order to characterize the low temperature part of the phase diagram which should correspond to the ground state. This choice could be questionable for nanoalloys where chemical or structural transitions can occur at much lower temperatures than in the bulk because of size effects. 37 For nanoalloys containing between 400 and 1000 atoms, it has been however shown that the ordering or the distribution of the different chemical species in the particles is only affected at temperature higher than 400 K. [37][38][39] In this work, some benchmark simulations have been performed between 100 K and 300 K in order to verify that, apart from small local fluctuations of atomic positions, there are no significant changes in the obtained equilibrium configurations. The structures obtained in MC simulations can then be used further in the MD procedure to again evaluate the local pressures for instance or other energy quantities. One serious advantage of combining both methods, MC and MD, is that they can be implemented using the same energy model insuring a good consistency between the results. This energy model is based on a semi-empirical many-body interatomic potential derived from the electronic structure in the tight-binding second moment approximation. It was initially proposed by Gupta 40 and by Rosato et al. 41 and was widely and successfully used for studying the structure of alloys and alloy surfaces. As a matter of fact it remains nowadays extensively used for nanoalloys (see for instance, in a non exhaustive list of examples, refs. 16-18,42,43). Within the tight-binding second moment approximation (TB-SMA), the total energy at a site i having j neighbors of a system of atoms is written as the sum of an attractive band term E b i , derived from the electronic structure, and a repulsive Born-Mayer one (E r i ):

E i = E b i + E r i , (2)
E b i = -∑ j ξ 2 IJ exp[-2q IJ ( r i j r IJ 0 -1)] 1/2 , (3)
E r i = ∑ j A IJ exp[-p IJ ( r i j r IJ 0 -1)]. (4)
where I and J indicate the chemical species and j the neighboring atoms over which are made the sums of the interactions, typically until the second next nearest neighbors. For homoatomic interactions (I = J), r IJ 0 is the next nearest neighbor distance at equilibrium in the considered pure material. For mixed interactions (I = J) it is taken as the arithmetic average between the values of the two pure corresponding materials. The parameters A, ξ , p, q are usually fitted in order to reproduce some bulk structural properties of reference metallic materials. The parameters for IrPd and AuCo are taken respectively from the works of Andrihamiharintsoa et al. 44 and Chado et al. 45 , whereas those for AuRh have been fitted especially for this study following the fitting procedure described in the work of Chado et al. 45 In brief, the main requirement for pure materials is to reproduce at best the cohesion energy, the lattice parameter, the bulk modulus and some elastic constants. In the presence of surfaces or facets in nanoparticles, the relative surface energies for each considered pair of materials should be also well reproduced in order to obtain afterwards the correct segregation effects. 46 This specific constraint may lead to slightly different parameters for a same pure material associated in different couples of metals. For alloys, the main requirement is that the miscibility tendency should be well reproduced. The set of parameters obtained for the bimetallic systems used in this work are gathered in Table 1. Then the main characteristic quantities of the bimetallic (α,β ) system are gathered in Table 2: size factor or lattice mismatch ∆r/r, relative cohesion energy or cohesion factor ∆E α-β coh /E β coh , relative surface energy ∆γ α-β /γ β and solution energy for an impurity in an fcc matrix of atoms. The latter drives the (de)mixing tendency in the alloy system and is also used to fit the mixed interactions. In the present case the solution energies are all positive (demixing tendency) and very close together so that all the systems have a very similar mixing tendency in the bulk. The relative surface energies for each considered pair of materials largely favor the surface segregation of the matrix element, here either Pd or Au (this surface effect is also related to the relative values of the cohesion energies). We will see in the following how alloying, cohesion, surface and lattice mismatch effects are at play in nanoparticles of weakly miscible metals, from dilute systems to systems with small precipitates.

α β ∆r α-β /r β ∆E α-β coh /E β coh ∆γ α-β (100) /γ β ∆γ α-β (111) /γ β E sol (α in β ) E sol (β in α) Ir Pd -1% 0

Results and discussion

Sub-surface impurity segregation: energetics and local pressures

In the dilute systems both total energies and local pressures can easily be mapped by substituting one atom of the pure Pd or Au TOh 405 by one impurity atom of respectively Ir and (Rh,Co), and repeating the procedure for all the inequivalent sites of the nanoparticle. This was achieved by using the QMD procedure. MC simulations have been only used in a preliminary stage to verify that the occupation rate of the impurity at the surface (shell S 0 ) of the nanoparticle is always 0 for any system at T = 100-300 K. In addition these simulations show that the occupation of the least coordinated sub-vertex sites prevails for all systems at 100 ≤ T ≤ 300K. The analyses can therefore be related only to the inner sites, from sub-surface to the center. These sites can themselves be classified in five main categories: sub-vertex, sub-edge1, sub-edge2, sub-(100) facet and sub-(111) facet. The names sub-edge1 and sub-edge2 correspond respectively to the position below the edge joining a (100) and a (111) facets and the edge joining two (111) facets. The characteristic sites are marked with numbers (Z coordination with upper shell atoms) in the graphical representation of Fig. 2a for S 1 (the surface shell S 0 is also represented in this figure). Taking as reference state the substitution by one impurity of the central atom of the nanoparticle, the variation of total energies after substitution and relaxation at the characteristic sites of sub-shells S 1 and S 2 have been calculated and reported in the graph of Fig. 2b (energy variations in the S 3 region are negligible). These variations can be directly assimilated to segregation energies. The sub-surface S 1 sites are mostly the most favorable one whatever the considered materials, particularly under edges and vertices. An interesting fact is that going from the center of the nanoparticle to sub-vertex sites (at S 1 ) the variations are much more important for the Pd(Ir) and Au(Rh) systems which have a moderate or low lattice mismatch, the strongest variation being obtained for the least mismatched Pd(Ir) one. So, whatever the size mismatch, the sub-surface positions are the most favorable, with increasing energy variations going from the central position to a sub-vertex one, the latter site being definitely the most favorable for the substitution. Let us however note that segregation to S 2 shell has significant energy values for some sites of the highly size-mismatched Au(Co) dilute system although they are close to zero for Pd(Ir) and Au(Rh). This results will be discussed in more details in section 3.2. At this stage, it is hardly possible to establish a clear and general rule about the driving forces for sub-surface segregation towards the single preferential S 1 sub-vertex site, in particular regarding the lattice mismatch effect. For other systems like AgCu, AgNi, AgCo or AuCo nanoalloys, which are all weakly miscible and exhibit very large absolute values of lattice mismatch (> 10%), studies have focused on the atomic size effect for explaining sub-surface segregation and/or off-center core formation. 14,16,18,39 For these systems a good correlation is found between the energetics and the local pressure acting on a substituted impurity. 14 When the size of the atom impurity is much lower than the atomic size of the matrix element, the local pressure is always negative and reaches its lower absolute values in the initially most compressed sites of the matrix nanoparticle, located under the surface due to the inward relaxation of surface atoms. In this description the site under vertex is obviously the most favorable for the substitution of an impurity of lower atomic radius than the matrix atoms. Fewer systems with smaller lattice mismatch were investigated so far. One first example is the weakly miscible AuPt (lattice mismatch ∆r/r Au = -4%). In the form of a nanoparticle, at low Pt concentration (<1%) Monte Carlo simulations have clearly shown the favored sub-surface segregation which was also interpreted as a local minimization of the stress. 20 Going to systems with no or almost no lattice mismatch, off-centered Ir core configurations in IrPt nanoalloys (lattice mismatch ∆r/r Pt = -2%) has been studied at the DFT (Density Functional Theory) level and it was concluded in this case that the preferential strengthening of surface-sub-surface Ir-Pt bonds was due to specific electronic features. 21 In the present work, it is shown that the energy balances are very similar whatever the lattice mismatch, in particular for sub-surface segregation. The sub-surface segregation can be even stronger for systems with negligible lattice mismatch like IrPd. 1). The variation is relative to the energy of one impurity placed in the center of the nanoparticle. The x label corresponds to the number of bonds Z with the next upper shell atoms above the site where is located the impurity.

Impurity at sub-surface: quantification of the driving forces for segregation

From the previous observations, it is fundamental to question the relative importance of the atomic size effect for the sub-surface segregation, in order to estimate its relevance and to possibly evaluate the actual driving forces leading to the same favorable sites whatever the lattice mismatch in core-shell metallic nanoparticles. As mentioned before, the local pressure should be a relevant quantity for exploring the atomic size effect. Local pressure at one impurity atom of Ir (respectively Rh and Co) in a Pd (respectively Au) nanoparticle (TOh 405 ) has been calculated after relaxation within QMD using eqn. 1. The results are presented in the bottom graph of Fig. 3, the top one representing the reference values in pure Au and Pd nanoparticles.

Clearly the energetics and the local pressure are very well correlated only for the Au(Co) system. The numerical values for this system are also in very good agreement with the ones obtained in the work of Bochicchio et al. 14 In this case the local pressure is always negative because of the tensile strain related to the hosting of an impurity atom of much smaller size than the matrix atoms so that the strain lowering seems to be the main driving force for favoring the most compressed site under surface vertex (see top graph of Fig. 3) in good agreement with the previous works on systems with large lattice mismatch. 14 When the absolute value of the lattice mismatch decreases from AuCo (∆r/r=-13%) to AuRh (∆r/r=-7%) and IrPd (∆r/r=-1%), the correlation with the local pressure is completely lost although the energetics always favors the site under surface vertices. For AuRh the lowest absolute values of local pressure correspond to the substitution sites under (100) and (111) facets and under the surface edge joining two (111) facets. Then the largest local pressure (compressive) is obtained at the site under the surface vertex. For Ir@Pd the local pressure remains positive whatever the substitution site and reaches its larger value under the surface vertex site. These observations led to a more detailed consideration of the driving forces for sub-surface segregation and their relative contributions as a function of the lattice mismatch which is important to predict the further core morphology when a metal nanoparticle is progressively enriched with another metal, in the case of weakly miscible metals. Segregation in bimetallic systems, as well at bulk defects as at surfaces, is of course not a new subject but remains less rationalized in nanoalloys. One efficient approach initially proposed for interpreting the segregation enthalpy in dilute systems (bulk and surfaces) is based on the tight-binding formalism. 25 In this approach it is proposed to reconstruct the segregation enthalpy within three independent (or quasi independent) contributions: the cohesion, the alloy and the size effects. The main quantity involved in this reconstruction is the permutation enthalpy which writes as the sum of three terms: 25 ∆H p perm = ∆H p perm,coh + ∆H p perm,alloy + ∆H p perm,size .

where ∆H p perm is the permutation enthalpy of an atom B of the matrix into a solute atom A for a site p. The segregation enthalpy to a site p is then the difference between the permutation enthalpy at site p and the one at a reference bulk site (it will be the most central site for a nanoparticle). The determination of each permutation enthalpy is based on the use of the SMA-QMD method in three special situations depending on the treated effect and well detailed in the work of Berthier et al. 25 Briefly, the three terms are:

∆H p perm,coh = H p A -H p B , (6)
∆H p perm,alloy = -∑ k Z p k V p k , (7)
∆H p perm,size = H p tot [B(B * )] -H tot (B). (8)
In eqn. (6) H p (A,B) is the energy of the site p in the pure metal A or B. In eqn. (7), Z p k is the coordination number of the k th neighbors. V p k corresponds to the effective pair interactions calculated up to the k th neighbors using the SMA-QMD procedure 25,47 (typically interactions are taken up to the 2 nd next nearest neighbors). In brief, this is done by determining the energy difference between a system containing two isolated impurities in the initial state and two impurities in the 1 rst and the 2 nd next nearest neighboring position in the final state. Finally in eqn. ( 8), H p tot [B(B * )] is the total energy of the dilute B(B * ) system where B * designates an impurity differing from the B atoms matrix only by its atomic size, the latter being taken as the atomic size of the A element of the considered A(B) system for which H p perm,size is evaluated. 47 Using this so-called three-effects rule, it has been in particular possible to successfully describe segregation in various AgCu based systems including nanoalloys. 12,15 The possibility of decomposing the sub-surface segregation energy into these three components should help to better interpret the results on the sub-surface segregation in the Pd(Ir), Au(Rh) and Au(Co) diluted nanoalloys. It has been however shown that one limitation of this model decomposition of the permutation energy is that the alloy contribution is more or less strongly coupled to the cohesion and/or size contribution which comes from the N-body character of the SMA interatomic potential and the atomic relaxations. 26 It was then proposed to explicitly reintroduce this coupling in order to reconstruct more accurately the segregation energy. 26 In a first step of the present work, the separate three contributions to the segregation energy and their sum have been calculated at the three inner shells S 1 , S 2 and S 3 using eqn. 6, 7 and 8 and taking as reference the permutation energies at the central core site.

The corresponding results for the segregation at sub-shells 1 and 2 (energy variations in the S 3 region are still negligible) are reported in Fig. 4 for the A(B) dilute systems where B is the impurity. Let us note that a negative (respectively positive) sign of the segregation energy indicates a preferential segregation for B over A (respectively A over B) at p site. Looking first at the curves for S 1 in the upper graphs of Fig. 4, one clearly sees that the reconstruction of the sub-surface segregation energy with the initial version of the three-effects rule is really efficient for the system Au(Co) which has a large lattice mismatch, a result which was already obtained in earlier studies of Ag(Cu) another system with a similar large lattice mismatch. For Au(Rh) the reconstruction still predicts the right segregation tendency even though there is a significant difference between the curves of ∆H sum seg and ∆H QMD seg . Finally, the different effects are weak for Pd(Ir) (negligible lattice mismatch) and the simple reconstruction fails for this system. As can be also seen in the graphs the two red plots representing ∆H sum seg and ∆H QMD seg are splitting up in an increasing way going from the center (coordination, Z=12) of the nanoparticle to the sub-vertex site (coordination with the surface shell atoms, Z=6). Therefore the coupling of the alloy term with cohesion and/or size effect cannot be neglected and has to be by some means reintroduced. In the present work, instead of an explicit calculation like the one performed in the work of Creuze et. al, 26 a pragmatic approach has been applied, based on the assessment that ∆H p perm,coh and ∆H p perm,size are unambiguously determined since they do not involve mixed A-B interactions in their formulation. The total segregation energies at sub-surface sites, presented in Fig. 2b, being given by the total energies calculations within the SMA-QMD procedure, the effective alloy term ∆H p * seg,alloy , including the coupling with size and cohesion, can be estimated by making the simple difference:

∆H p * seg,alloy = ∆H p seg,SMA-QMD -(∆H p seg,coh + ∆H p seg,size ). (9)
The results for ∆H * seg,alloy are plotted in Fig. 4 with the other results of segregation contributions. While the coupling is finally weak for Au(Co) with values of ∆H * seg,alloy of the same order of magnitude as the values determined from eqn. (7), it becomes stronger for the other two systems for which the values of ∆H * seg,alloy are even the largest ones of all effects contributions. Looking second at the curves for S 2 sub-shell in lower graphs of Fig. 4, the reconstruction of the segregation energy with the initial version of the three effects rule is still efficient for the Au(Co). Then for Au(Rh), the coupling is no more negligible and a positive value of ∆H * seg,alloy is obtained to recover the correct segregation energy on this shell. Then segregation effects at S 2 in Ir(Pd) are very small and the reconstruction using the corrected alloy term is mainly useful to recover the correct sign (positive) of the segregation energy at the sites of this shell. It is worth noticing that the used definition of the effective alloy enthalpy (eqn. 9) may seem somewhat artificial but it has the merit here of revealing the complexity of the alloy contribution to the segregation energies. The further analysis of the results in terms of the corrected three effects model 26 is indeed very informative and enables to elucidate the relative contributions of the possible driving forces for sub-surface segregation. One common feature for all systems is the positive sign for the cohesion segregation term at any site of the matrix nanoparticle implying that the cohesion effect favors always the segregation of the element matrix relatively to the center of the nanoparticle (i.e. ≈ bulk). It is easily understandable since the matrix elements Au and Pd have much lower cohesion and surface energies than the element taken as impurity, Co, Rh and Ir or in other words the considered bimetallic systems have a rather large positive cohesion factor (see table 2). Then, for a given nanoparticle, the contribution ∆H seg,size to the segregation energy increases when going from the center to the sub-surface shell, the maximal contribution being always at the site under the surface vertex which is the most compressed in the nanoparticle as already discussed in the previous section. However, as already suggested by the local pressures calculations, the size effect alone cannot explain the preferential segregation at the sub-surface vertex apart from the systems with a very large lattice mismatch like Au(Co). Then in Ir(Pd) the size effect term is very small and the major contribution to the preferential segregation at S 1 , in particular at the sub-vertex site, is clearly coming from the effective alloy term ∆H * seg,alloy . The results, including this effective alloy effect (looking only at ∆H * seg,alloy ), can be examined closer following an increasing order of lattice mismatch starting therefore with Au(Co). In this case, it is clear that the size effect characterized by a large negative value is the dominant one. It largely outweighs the alloy and cohesion effects, the latter giving rise to positive segregation energies favoring the segregation of A on all sites of sub-shells 1 and 2 when taking as reference the center of the nanoparticle. The alloy effect has the weakest importance in the case of this system even taking into accounts its coupling with cohesion and size effects. It is worth noticing that there is a great similarity between these results and the one obtained for Ag(Cu) dilute systems (including nanoparticles). 12,15,25,26 More generally, other known nanoalloys of weakly miscible metals and with a large lattice mismatch (|∆r/r| > 10%), like AgNi and AgCo (where Ag is the matrix element), have shown a similar sub-surface segregation behavior which could be unambiguously attributed to a dominant size (strain) effect. 14,16 Au(Rh), the second system to examine, corresponds to a so-called moderate lattice mismatch (∆r/r = -7%). The size effect is still important but less than the alloying one. Both effects contribute to strongly favor subsurface segregation at S 1 and the preferential segregation at the sub-vertex site. It can then be concluded that the driving force for segregation is a coupled alloy-size-cohesion effect. Finally Pd(Ir) can be considered as an archetype for a system with no atomic size effect or a negligible lattice mismatch. The atomic size contribution to the total sub-surface segregation energy is very small. However, the main contribution to the segregation comes from the effective alloying effect characterized by a very large negative value of ∆H * seg,alloy . Owing to the fact that the size effect is negligible, the segregation is clearly driven by the alloying effect and its coupling with the cohesion effect. The last interesting feature that should have an impact on the morphology of the core, when increasing the concentration beyond impurity, is that values of ∆H QMD seg at S 2 sites can be significant and of the same order of magnitude as in S 1 , which occurs only for Au(Co) as previously mentioned (section 3.1). For Au(Rh) some values are still negative on S 2 but close to 0 eV despite an important atomic size effect. For Pd(Ir) values on S 2 are positive. This suggests that when growing a precipitate of Ir in the Pd nanoparticle it could tend to be more confined in the first sub-surface shell than the two other systems AuRh and AuCo. Further simulations with MC of such precipitates are presented in the next section.

Evolution of the core solute morphology in equilibrium precipitates

Different stages of the core enrichment in the TOh 405 are addressed within equilibrium MC simulations in the canonical ensemble at fixed temperature (T=200K) for four different compositions: A N B 405-N where N = 25, 50, 80 or 100 which corresponds to concentrations c A = N × 100/405 ranging between 6% and 25% (A = Co, Rh or Ir; B = Au or Pd). Some final snapshots of the obtained structures are represented in Fig. 5. Let us recall here that the considered core-shell nanoalloys are referred as Co@Au, Rh@Au and Ir@Pd where the surface shell S 0 is always occupied by Au or Pd segregated species as confirmed by MC simulations. In the chosen range of composition, a biphase structure is always obtained where the precipitate of the B element is anchored at the sub-surface leading to an off-center solute core inside the matrix nanoparticle of element A. Some observations can be made for the Au or Pd matrix phase. In the case of Ir@Pd and Rh@Au where the lattice mismatch is negligible or moderate the nanoparticle keeps a regular TOh shape. In the case of Co@Au, the Au matrix nanoparticle undergoes distortions around the Co core which can be attributed to Fig. 5 MC simulations at 200 K: 3D snapshots of the Ir@Pd, Rh@Au and Co@Au nanoalloys. Four compositions are considered. N c is the number of solute core atoms (Ir, Rh or Co). The color code is: red for Ir, grey for Pd, brown for Rh, golden for gold and pink for Co. The orientation of the nanoparticles has been selected in order to visualize (if possible) the underneath core solute atoms and the possible modifications in the geometry of the facets.

the large mismatch between the Co lattice and the Au one leading to incoherent interfaces between the two materials. This can be easily seen in the snapshots of Fig. 5 where the Co@Au nanoparticles are represented with the actual atomic size ratio between Co and Au. In addition, also visible in Fig. 5, a structural transition is observed at the Au surface where square facets (100) reconstruct into diamond ones as they become more and more in contact with Co atoms. This phenomenon was already observed and investigated into detail for Cu@Ag nanoparticles 13,48 and various bimetallic systems with large lattice mismatch. 49 In these studies it was shown that an initial A@B TOh structure with a monolayer of B at the surface often transforms, through relaxations in atomistic simulations, from the TOh into a structure where the square (100) facets distort into a rhombus or diamond shape, with close-packed arrangement of the surface atoms. This transformation has been shown to reduce the compression on the B atoms of the surface in contact with the A core atoms. 49 It can be seen in the images of Fig. 5 that this transformation also occurs in Co@Au nanoalloys already at lower Co concentration (biphase regime here) as soon as some Co is in contact with a square (100) facet. This phenomenon is also very similar to the (111) reconstruction experimentally observed with scanning tunneling microscopy on some (100) metallic surfaces like Au(100) 50 or highly compressed Cu(100). 51 Coming to the core description, it was observed that the Rh and Co precipitates extend themselves from the S 1 to S 3 shells on the entire investigated concentration range. In comparison Ir core extends preferentially over the internal Pd surface shell in a bilayer morphology and the S 3 shells starts to be occupied only from N c = 100. The shape of the Ir precipitate remains close to a hollow shell contrary to Co and Rh ones which are more compact. This description has been correlated to a more quantified approach performed by calculating in the MC simulations the average site occupation rate by Ir, Rh and Co in their matrix nanoparticle. The Ir, Rh and Co occupation probabilities per site are represented in the cross sectional views of Fig. 6 using a color code going from 0 (blue) to 1 (red). It can be seen that the first description is indeed well corroborated by the site occupation rate representation. It can be seen also that there is a different filling profile of the S 1 to S 3 shells depending on the systems showing a competition between the preferential sub-surface (S 1 ) occupation and the demixing tendency of the investigated bimetallic systems.

Therefore, the results have been also gathered in the curves of Fig. 7, this time by reporting the rate of shell occupation by Ir, Rh and Co atoms from S 0 (surface) to S 3 (central part of the nanoparticle). The obtained evolution of the shell occupation rate as a function of the number of solute atoms is rather similar for Rh@Au and Co@Au with an equal probability of filling for S 1 , S 2 and S 3 at low concentration (at N c = 25 for Co@Au and until N c = 50 for Rh@Au) followed by a gradual splitting of the curves with a greater increase of the occupation probabilities going from S 1 to S 3 . The latter effect is more pronounced in the case of Co@Au and this translates into the structures by a more important extension of the core solute towards the inner part of the nanoparticle (see Fig. 5). The strengthening of the mixed bonds under the surface is clearly the most important in the Ir@Pd system for which the curve of the S 1 shell occupation rate (red curve in Fig. 7) is always above the others, in agreement with the found strong alloy-cohesion coupling for the impurity located under the surface shell (section 3.2). A clear difference between Ir@Pd and the two other systems is visible for the filling of the S 3 shell which occurs only from a certain concentration of Ir (here for N c = 80) again showing the strong preference for Ir to decorate the sub-surface shell, leading to a bi-layer hollow shell shape in the biphase systems where small off-center precipitates are formed. Otherwise Co and Rh solute core have a three-layer shape from the beginning and more compact shapes. One can try to return to what could be inferred from the study of diluted systems, although it is not trivial to extrapolate results from a dilute system to a concentrated one. Thereby the increasing extension of the core solute towards the inner shells going from Ir@Pd to Co@Au can be qualitatively related to the fact that the segregation energies of the impurity have still rather large negative values on shells below S 1 for Co@Au, still some small negative values for Rh@Au while all values are positive below S 1 for Ir@Pd. Then, the occupation of S 2 in the bilayer Ir core and of S 3 in the trilayer Rh and Co cores can then be related to the demixing effect favoring homoatomic bonds.

Finally, following the approach adopted for dilute systems in section 3.2, a general view of the local strain in the nanoalloys is given in Fig. 8 where cross sectional views of the final MC snapshots are represented. For each composition, the right hand side image shows the mapping of the local pressure at each site of the left hand side bimetallic nanoparticle. For rather similar core morphologies the strain landscapes are different depending on the lattice mismatch. Comparing Ir@Pd and Rh@Au, the highest absolute value of pressure remains under vertices with a major difference since it is in the Ir solute for the first and in the Au matrix nanoparticle for the second. In the Co@Au system the strain landscape in the core is rather homogeneous with the highest local pressure at site under the vertices of the Au matrix nanoparticle not occupied by Co atoms similarly to AuRh.

Conclusion

The preferential occupation of the sub-surface shell by an impurity in a nanoparticle, in particular at the sub-vertex sites, was found to be a common feature for weakly miscible bimetallic nanosystems in which the impurity has an atomic size equal to or smaller than the matrix atoms. Remarkably, the origin of this unique phenomenon at the sub-vertex site can be very different from one system to another. Three main effects have been highlighted in the present work from the study of three systems with very different lattice mismatch: strong alloy-cohesion coupling (Ir@Pd, no lattice mismatch), mixed size-alloy-cohesion coupling (Rh@Au, moderate lattice mismatch) and strong atomic size effect (Co@Au, large lattice mismatch). If in this latter case local pressures maps are well correlated with the segregation energies, it is no more the case for smaller lattice mismatches. Already for ∆r/r = -7% (case of Rh@Au in this work), the correlation is lost which is visible in local pressure curves where the lower absolute value of the local pressure (≈ 0) is obtained under the facets and the edges joining two (111) facets while the largest local pressure (compressive) is obtained at the site under the surface vertex although this latter site corresponds to the most favorable for the substitution by a Rh impurity. For a lattice mismatch ∆r/r ≈ 0 (case of Ir@Pd), the same segregation energy hierarchy is obtained as for nanoalloys with large lattice mismatch while this time the variation of the local pressures show an increasing positive absolute value from the center to the sub-surface of the Pd matrix nanoparticle.

Equilibrium Monte Carlo simulations at different compositions, in the low concentration range in the solute element, revealed rather similar precipitates morphology for the three studied model systems which can be described by a biphase structure showing an extension of an off-center core solute anchored under the surface at sub-vertex sites. This structure originates again from the different driving forces which could be roughly guessed from the study of the impurity solute. Some differences between the three studied systems were nevertheless noted in the core morphology evolution in this low concentration range. Indeed the strengthening of the mixed bonds under the surface is clearly the most important in the Ir@Pd system for which the S 1 shell occupation rate is dominant due to a strong alloy-cohesion coupling without any size effect at play. The main difference between Ir@Pd and the two other systems can be summarized as follows.

In Ir@Pd one obtains an extended bilayered structure with the filling of the next inner shell (fourth shell from the surface, S 3 , in the TOh shape) occuring only from a certain concentration threshold of Ir (here for N c > 80 in a TOh containing 405 atoms). On the other side the systems with non negligible lattice mismatch develop three-layered core solute from the first stages of enrichment and keep more compact shapes in their nanoparticle matrix. To conclude more generally, through atomistic simulations, it was possible to show that all equilibrium structures consist of similar off-center solute clusters anchored at sub-vertex sites, regardless of the lattice mismatch. Assessing the dominant driving forces leading to these similar morphologies appeared to be non trivial. They can indeed be very different from one system to another: atomic size (case of large lattice mismatch), alloy-size-cohesion coupling (case of moderate lattice mismatch) or alloy-cohesion coupling (case of negligible lattice mismatch).

a

  Institut de Physique et Chimie des Matériaux de Strasbourg, Université de Strasbourg, CNRS UMR 7504, 23 rue du Loess, BP 43, F-67034 Strasbourg, Cedex 2,

Fig. 1

 1 Fig. 1 Structure of a nanoparticle (TOh 405 ) containing 405 atoms. Left hand side: 3D perspective view. Right hand side: cross sectional view showing the concentric shells S i of the TOh. The indices 0 to 2 indicate the level of the concerned shell starting from the surface. The index 3 relates to the remaining central atoms.

Fig. 2 a

 2 Fig.2 a) Atomic representation of the surface shell (S 0 ) and the sub-surface one (S 1 ) of a TOh 405 nanoparticle. Numbers on the S 1 structure relate to the Z coordination (first neighbors) with atoms of the upper surface shell S 0 . b) Variation of the total energy as a function of the location of one impurity of Ir, Rh or Co in the Pd or Au nanoparticle, at first (S 1 ) and second (S 2 ) shells under the surface (see Fig.1). The variation is relative to the energy of one impurity placed in the center of the nanoparticle. The x label corresponds to the number of bonds Z with the next upper shell atoms above the site where is located the impurity.

Fig. 3

 3 Fig. 3 Top graph: local pressure at the characteristic sites of S 1 for pure Au and Pd nanoparticles, before substitution by one impurity. The points at Z = 0 correspond to the inner central site. Bottom graph: local pressure at one impurity atom respectively of Ir, Rh and Co in substitution in a Pd or Au nanoparticle (TOh 405 ). The points at Z = 0 correspond to the substitution at the inner central site. The dashed lines are just guidelines for the eye. As for Fig. 2 the x label corresponds to the number of bonds, Z, with the next upper shell atoms above the site where is located the impurity.

2 Fig. 4

 24 Fig. 4 Top graphs: S 1 , first sub-surface shell: decomposition of site segregation energies, seg on sub-surface sites for one impurity of Ir in Pd, Rh in Au and Co in Au. Values are given in meV. Bottom graphs : same as upper graphs for S 2 . As for Figs. 2 and 3 the x label corresponds to the number of bonds Z with the next upper shell atoms above the site where is located the impurity.

Fig. 6

 6 Fig.6 Cross sectional views of the nanoparticles of Fig.5. The color code is related to the Co, Rh or Ir occupation rate or probability on each site of the Au or Pd nanoparticle ranging from 0 (blue) to 1 (red). The orientation of the shown nanoparticles has been selected in order to visualize at best the core solute structure.

13

 13

Fig. 7 (

 7 Fig. 7 (Ir, Rh, Co) occupation rate in each shell of a Au or Pd TOh 405 nanoparticle from S 0 (surface) to S 3 (central part). The values were obtained from averages performed in MC simulations. The x axis indicates the number of Ir, Rh or Co core atoms.

Fig. 8

 8 Fig.8MC simulations at 200 K: cross sectional snapshots of the Ir@Pd, Rh@Au and Co@Au nanoalloys. Four compositions are considered. N c is the number of solute core atoms (Ir, Rh or Co). For each composition, the color code in the left hand side images is: red for Ir, grey for Pd, brown for Rh, golden for gold and pink for Co. The color code in the right hand side images relates to the local pressure (Mbar) going from most in tension (blue) to most in compression (red) sites. The orientation of the shown nanoparticles has been selected in order to visualize at best the core solute structure.

Table 1

 1 TB-SMA potential parameters for all the systems under study. A and ξ are given in eV

	α	β	A αβ	ξ αβ	p αβ	q αβ
	Au Au 0.189 1.743 10.400 3.867
	Co Co 0.106 1.597 10.867 2.36
	Au Co 0.141 1.614 10.634 3.11
	Au Au 0.208 1.812 10.140 4.008
	Rh Rh 0.241 2.465 10.346 3.444
	Au Rh 0.257 2.207 10.243 3.726
	Pd Pd 0.114 1.494 12.070 3.120
	Ir	Ir 0.139 2.431 15.000 2.864
	Pd Ir 0.127 1.904 13.535 2.992

Acknowledgments

Dr Guy Tréglia (CINaM, Marseille, France) is gratefully thanked for very fruitful discussions and his thorough reading of the manuscript.