Micha
Polak
* and
Leonid
Rubinovich
Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. E-mail: mpolak@bgu.ac.il
First published on 27th March 2015
A new coarse-grained layer model (CGLM) for efficient computation of axially symmetric elemental equilibrium configurations in alloy nanoparticles (NPs) is introduced and applied to chemical-order transitions in Pt–Ir truncated octahedra (TOs) comprising up to tens of thousands of atoms. The model is based on adaptation of the free energy concentration expansion method (FCEM) using coordination-dependent bond-energy variations (CBEV) as input extracted from DFT-computed elemental bulk and surface energies. Thermally induced quite sharp transitions from low-T asymmetric quasi-Janus and quasi ball-and-cup configurations to symmetric multi-shells furnish unparalleled nanophase composite diagrams for 1289-, 2406- and 4033-atom NPs. At even higher temperatures entropic atomic mixing in the multi-shells gradually intensifies, as reflected in broad heat-capacity Schottky humps, which become sharper for much larger TOs (e.g., ∼10 nm, ∼30000 atoms), due to transformation to solid-solution-like cores.
Previous theoretical–computational studies of NPs with separation tendency dealt mainly with cases of considerable size mismatch (>10%) between atoms of the two elements. Thus, in Cu, Ni or Co alloyed with Ag, the formation of side-separated “Quasi-Janus” configurations was attributed to the near-surface elastic strain release using the Gupta potential and computational global optimization.4,5 According to this and other methodologies (e.g., thermodynamics of spherical NPs) the tendency to form low-T asymmetric vs. symmetric configurations can depend on the composition and the nanoparticle size, as predicted for a number of nanoalloy systems, such as Au–Pt,6 Cu–Ag,7–9 and Cu–Ni.10
Recently, atomic site-specific chemical order in fcc-based 201- and 586-atom Pt–Ir truncated-octahedron (TO) nanoparticles was studied11 by us using near-surface coordination-dependent bond-energy variations (CBEV)12 as input to the statistical–mechanical free-energy concentration expansion method (FCEM),13 which takes into account analytically short-range order. Thus, in this weakly miscible bi-metallic alloy having a relatively small atomic-size mismatch, axially-symmetric quasi-Janus configurations were found to be stabilized at low temperatures mainly due to CBEV-induced preferential strengthening of Pt-surface/Ir-subsurface bonds, and the greatly reduced number of energetically unfavorable hetero-atomic bonds. At higher temperatures, the Pt–Ir quasi-Janus configurations transform into partially mixed central-symmetric multi-shell structures.
While this previous atomistic version of the FCEM could cope with small-medium structures comprising up to ∼1000 atoms, the present study aims at elucidating phase separation phenomena in significantly larger, more practical NPs. For this goal, the FCEM has to become computationally more efficient by devising a model that facilitates a reduction in the number of compositional degrees of freedom via “mapping” atomic sites into groups. A similar principle was used, for example, in mesoscale molecular-dynamics simulations of polymers, proteins, carbon nanomaterials etc.14,15 Pt–Ir has been chosen as a nanosystem model for the new “coarse-grained” FCEM version described below, while employing refined CBEV energetics, in an attempt to predict chemical-order transitions in medium-large nanoparticles consisting of thousands and tens of thousands atoms, with the ultimate goal of constructing the corresponding nanophase diagrams. The latter turn out to be highly involved and substantially distinct from our above-mentioned previous study for small-medium NPs, as well as from the Pt–Ir bulk diagram.
When applied to NPs consisting of geometrically nonequivalent atomic groups, numerical minimization of F (e.g., by MATLAB) yields all I-constituent equilibrium concentrations for the group number p, cIp. Geometric input parameters include the number of atoms in each group, Np, and the number of NN pairs belonging to p- and q-groups, Npq. The first term stands for most of the configurational entropies, the second one involves the elemental (homoatomic) interactions, wIIpq, the third term includes the heteroatomic effective pair-interactions, VIJpq, between constituents I and J , and the last term in F involves short-range order contributions (b ≈ 0.6 is a constant obtained from a best fit of FCEM predictions to Monte Carlo simulations17). The FCEM is advantageous compared to the simplistic Bragg–Williams approach that neglects short-range order. In spite of its limitations, the FCEM has at least one major advantage over numeric computer simulations, namely, its much higher efficiency due to the use of the above explicit analytical expression, which can furnish the full temperature-dependence of chemical order and thermodynamic properties even for medium-large NPs. Furthermore, FCEM computations exhibited reasonable accuracy when combined with energetics input based on the CBEV method (described below), expected especially for alloys with small atomic-size mismatch (for Pt–Ir NPs the roles of local strain and vibrational entropy were estimated to be minor11). Thus, computational results were found to be in general agreement with direct DFT computations for small-size TO clusters of Pt–Pd18 and Pd–Ir,19 as well as with MEIS experimental data for Pt25Rh75(111).20
In the previous version, here referred to as “fully atomistic FCEM” (FA-FCEM), the free-energy expression was minimized with respect to the number of all concentration variables, cIp, which amounts to the number of geometrically nonequivalent sites in central-symmetric or axially-symmetric nanostructures. Because of the rapid increase of this number with the NP size, making the computations extremely tedious (and feasible up to ∼1000 atoms), we decided to devise a method to reduce the number of FCEM variables without losing too much accuracy. The attainable size is significantly extended here using a coarse-grained layer model (CGLM), in which similar adjacent atomic sites within a layer are grouped together with a single assigned concentration variable. In particular, as demonstrated by Fig. 1, the method evaluates layer-by-layer subsurface and inner core “average concentrations”. (It should be noted that this grouping together of similar adjacent atomic sites differs substantially from a recently reported approach that puts atoms in equivalence sets defined on the basis of point group symmetry.21) For the 1289-atom TO core, the number of concentration variables goes down to 28 from hundreds of variables that have to be employed in the corresponding FA-FCEM. Due to the strong Pt surface segregation in this alloy, only four surface concentration variables, corresponding to sites having different coordinations, are used (namely, vertices, edges, (100) and (111) faces). It can be noted that although a common variable is assigned to every layer-specific subsurface entity (Fig. 1), its value is affected by an “integral” CBEV effect depending on the coordination of all neighboring surface sites. As an illustration, strong CBEV effects are expected to be associated with the NP central layer (#0) subsurface adjacent to many low-coordinated edge and vertex surface sites, as compared to layers #5 or #6 which intersect the surface mainly via (111) faces (Fig. 1).
Although this computationally advantageous method provides concentration variations along a chosen axis only, this information can be valuable, for example, in finding nanophase transitions involving symmetry breaking. (In some sense, the CGLM resembles certain experimental methods providing insights into concentration gradients along a chosen direction.)
The CBEV approach, introduced by us several years ago,20 derives the near-surface energetics input compatible with the FCEM by searching for a functional dependence of elemental bond-energy variations (δwpq) on the corresponding pair coordinations. This is done for intra-surface and surface–subsurface bonds (two-layer model) by employing DFT computed surface energies in the pair-bond expression,
δwpq = a1,0xpq + a2,0xpq2 + a0,2ypq2 + a3,0xpq3 + a1,2xpqypq2 + a4,0xpq4 |
In this way, bond-energy variations are treated as coordination-dependent functions rather than numerical values. The implied basic assumption concerning the dominant effect of coordination, reflected in the use of common polynomials with element-specific coefficients, helps to circumvent the transferability problems, namely, the need to repeat fitting of interactions for every site in a given nanoparticle surface structure. This makes the FCEM/CBEV combination highly efficient. Furthermore, as demonstrated below the high transparency of the CBEV helps to elucidate the origin of distinct bond-energy variation effects on site-specific segregation in alloy nanoclusters.
Compared to our recent work,11 elemental CBEV data are obtained here from refined DFT-computed elemental bulk and surface energies, as outlined elsewhere.19 According to both sets of energetics, surface sites are unfavorable for Ir atoms, which tend to segregate at the subsurface atomic sites since the strengthening of surface–subsurface Ir–Ir bonds (and presumably also Pt–Ir) is consistently larger than Pt–Pt bonds (Fig. 2). This trend is stronger in the present energetics compared to our previous study of 923-atom Pt–Ir cubooctahedra,12 and is qualitatively consistent with DFT studies of segregated structures on the Pt3Ir(211) surface22 and with a (slight) subsurface preference for a single Ir dopant atom in a 201-atom Pt TO.23
Fig. 3 Verification of the coarse-grained layer model (CGLM): asymmetric–symmetric transitions along the [111] axis in Pt376Ir210 TOs computed with the CGLM-FCEM vs. the fully-atomistic FA-FCEM. (a) Three distinct subsurface site concentrations (dashed lines) vs. one subsurface layer concentration computed using the CGLM (solid lines). In the quasi-Janus nanophase below the transition at Tsym: Pt-side – red, Ir-side – blue. The site positions are marked in the subsurface schematics (inset); (b) nanophase diagram computed in the CGLM (solid line), compared to the diagram obtained by the FA computation11 (■ – magic-number compositions). |
Fig. 4 CGLM-FCEM results obtained for a 1289-atom TO with a 28 at% Pt core: (a) thermally-induced variations of the Pt-side (red) and Ir-side (blue) inner core layer concentrations, which coincide (black) at the transition temperature Tsym. Insets: schematics of asymmetric (quasi-Janus) and symmetric (multi-shell) configurations below and above Tsym. (b) The corresponding configurational heat-capacity λ-peak. (c) Layer concentration profiles for T < Tsym, T = Tsym and T > Tsym. Layer numbering corresponds to Fig. 1. |
At higher Pt content (38 at% Pt core) the transition from quasi-Janus to multi-shell configurations is quite minor and occurs at significantly lower Tsym followed by thermally induced gradual mixing of the multi-shell structure (Fig. 5a). The latter contributes to a more significant heat capacity hump (Fig. 5b) with a shape characteristic of the “Schottky anomaly”.25 The hump maximum at TSck indicates the most intensive Pt–Ir mixing process. The Tsym lowering stems from geometric constraints that hamper the formation of significantly asymmetric nanophase at this composition, namely, the corresponding broader Pt maximum can shift only slightly upon approaching the transition (Fig. 5c). (This prediction is consistent with statistical arguments concerning the tendency to form asymmetric configurations when the separated region is small, while for the large one off-center preference should disappear.5) The shift is not accompanied by significant mixing and energy change, and therefore the heat-capacity exhibits just a weak feature at Tsym (Fig. 5b). It can be noted that in the previous case of the 28 at% Pt core actually Tsym and TSck coincide (Fig. 4). Furthermore, for both NP compositions the thermally-induced changes are accompanied by an increase in the number of Pt–Ir bonds (Table 1). However, this increase amounts to 284 bonds in a pre-transition range of 140 K for the 28 at% Pt core due to intense mixing that appears to be associated mainly with interphase broadening (Fig. 4c), whereas in the case of the 38 at% Pt core the corresponding increase amounts to only 69 Pt–Ir bonds, since the latter transition is associated mainly with weak mixing of the slightly shifted nanophase boundaries (Fig. 5c). Above the transitions multi-shell mixing is manifested in decreased/increased amounts of extra Pt–Ir bonds (per degree) in the case of 28% vs. 38% Pt cores (Table 1).
Fig. 5 CGLM-FCEM results obtained for a 1289-atom TO with a 38 at% Pt core (along the [100] axis): (a) thermally-induced variations of the Pt-side (red) and Ir-side (blue) inner core layer concentrations, which coincide (black) at the transition temperature Tsym. (b) The corresponding configurational heat-capacity Schottky hump with the maximum at TSck ≫ Tsym and a weak sharp feature related to the symmetry-breaking minor transition. Inset: schematics of a symmetric (multi-shell) configuration above Tsym. (c) Layer concentration profiles for T < Tsym, T = Tsym and T > Tsym. Layer numbering corresponds to Fig. 1. |
% Pt core | Temperature (K) | Number of Pt–Ir bonds |
---|---|---|
a Transition temperature, Tsym. | ||
28 | 900 | 2442 |
1040 | 2726 | |
1200 | 2827 | |
38 | 450 | 2169 |
590 | 2238 | |
1040 | 2495 |
Fig. 6 Subsurface layer concentration profiles computed for a 1289-atom TO. (a) Magic composition (∼57 at% Pt core): multi-shell structure without a transition; (b) ∼75 at% Pt core: asymmetric quasi ball-and-(subsurface)-cup at 300 K and symmetric profile at 1000 K. Layer numbering corresponds to Fig. 1. |
Fig. 7 Nanophase diagram of symmetry-breaking transitions in Pt–Ir TO NPs (with marked sizes) computed by the CGLM-FCEM. The left (I) and right (III) domes correspond to the inner-core quasi-Janus and subsurface quasi ball-and-cup transitions to multi-shell configurations, respectively. The intermediate region (II) involves an ideal multi-shell (no transitions) at “magic compositions” indicated by arrows (Pt-surface@Ir-subsurface@Pt-inner-core). The transitions presented in Fig. 4 and 5 are marked by small circles on the separation-coexistence dome of the 1289-atom TO. FCEM-computed bulk diagram miscibility gap is shown for comparison. |
(I) The low-Pt-core separation-coexistence line (dome) marks inner-core transitions from quasi-Janus to partially-mixed multi-shell configurations. With the increase of the NP size the dome widens, shifts to higher temperatures and its apex starts to turn towards the center in the direction of the bulk diagram. These variations are associated with the increase in the fraction of inner-core atoms with NP size, namely, ∼57%, ∼65% and ∼70% of the core atoms in 1289-, 2406- and 4033-atom TOs, respectively. It can be noted that Tsym and TSck coincide for all compositions up to the apex (see Fig. 4), whereas above it Tsym < TSck, because, as noted above, geometric constraints significantly hamper the asymmetric shift of the broad Pt-rich region, and most of the mixing takes place gradually at higher temperatures (see Fig. 5).
(II) This intermediate Pt concentration region below the magic-composition (Pt-surface@Ir-subsurface@Pt-inner-core) is characterized by greatly size-dependent, somewhat irregular variations in the quite low transition temperatures. For the 1289-atom TO, in addition to the magic composition (57% Pt core) Tsym = 0 also for the 44% Pt core that is another fully symmetric configuration, which involves Ir in sub-subsurface sites. This reflects again the subtle interplay of the geometric constraints, composition and near-surface energetics.
(III) The high-Pt-core dome that corresponds to intra-subsurface separation is lower than the dome in region I since this 2D “cup” configuration is naturally less stable than the 3D quasi-Janus separated structure. In addition, it gradually becomes narrower and shifts to the right with the increase of the NP size due to geometrical–compositional factors. Ultimately, this portion of the nanophase diagram is expected to disappear for very large particles (in the bulk limit). It can be noted that the predicted intra-subsurface separation, reflected in this second dome, occurs due to a strong CBEV-induced tendency of Ir atoms to segregate at all subsurface sites. With the former energetics this tendency is weaker and a side-separated Ir cluster starts to grow preferentially from one of the (111) subsurfaces11 and hence the diagram has just a single dome (Fig. 3).
The vast amount of output data enables the construction of a unique, perhaps the first of its kind, nanophase composite diagram comprising distinct composition ranges where the inner-core (quasi-Janus) and intra-subsurface (quasi ball-and-cup) configurations are stabilized at low T and transform into partially mixed multi-shells, which gradually mix further to form ultimately solid-solution-like cores. The intermediate region includes low-T “magic composition” multi-shell structures that do not undergo any thermally-induced nanophase transition.
According to preliminary computations for complementary TOs comprising up to tens of thousands atoms, the critical temperature (at the diagram apex) vs. NP size appears to exhibit power-law dependence characteristic of finite size scaling behavior,26 to be reported elsewhere. In addition to Pt–Ir that was chosen as a model system, the CGLM should be appropriate for medium–large nanoparticles of other weakly-miscible alloys and is not limited to the use of the CBEV energetics.
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