Free-energy Concentration Expansion Method Research Articles

The Thermal Stability of Asymmetric Separated Configurations inside Alloy Nanoparticles: Atomic-Scale Modeling of Pd-Ir Nanophase Diagrams.

Compared to alloy bulk phase diagrams, the experimental determination of phase diagrams for alloy nanoparticles (NPs), which are useful in various nanotechnological applications, involves significant technical difficulties, making theoretical modeling a feasible alternative. Yet, being quite challenging, modeling of separation nanophase diagrams is scarce in the literature. The task of predicting comprehensive nanophase diagrams for Pd-Ir face-centered cubic-based three cuboctahedra is facilitated in this study by combining the computationally efficient statistical-mechanical Free-energy Concentration Expansion Method, which includes short-range order (SRO) with coordination-dependent bond-energy variations as part of the input and with rotationally symmetric site grouping for extra efficiency. This nanosystem has been chosen mainly because of the very small atomic mismatch that simplifies the modeling, e.g., in the assessment of vibrational entropy contributions based in this work on fitting to the Pd-Ir experimental bulk critical temperature. This entropic effect, together with SRO, leads to significant destabilization of low-T Quasi-Janus (QJ) asymmetric configurations of the NP core, which transform to symmetric partially mixed nanophases. First-order and second-order intracore transitions are predicted for dilute and intermediate-range compositions, respectively. Caloric curves computed for the former case yield the NP-size dependent transition latent heat, and in the latter case critical temperatures exhibit a specific scaling behavior. The computed separation diagrams and intracore solubility diagrams reflect enhanced elemental mixing in smaller QJ nanophases. In addition to these diagrams, the revealed near-surface compositional variations are likely to be pertinent to the utilization of Pd-Ir NPs, e.g., in catalysis.

Thermally-induced chemical-order transitions in medium–large alloy nanoparticles predicted using a coarse-grained layer model

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, ∼30,000 atoms), due to transformation to solid-solution-like cores.

Comparative modelling of chemical ordering in palladium-iridium nanoalloys.

Chemical ordering in "magic-number" palladium-iridium nanoalloys has been studied by means of density functional theory (DFT) computations, and compared to those obtained by the Free Energy Concentration Expansion Method (FCEM) using derived coordination dependent bond energy variations (CBEV), and by the Birmingham Cluster Genetic Algorithm using the Gupta potential. Several compositions have been studied for 38- and 79-atom particles as well as the site preference for a single Ir dopant atom in the 201-atom truncated octahedron (TO). The 79- and 38-atom nanoalloy homotops predicted for the TO by the FCEM/CBEV are shown to be, respectively, the global minima and competitive low energy minima. Significant reordering of minima predicted by the Gupta potential is seen after reoptimisation at the DFT level.

Effects of Surface–Subsurface Bond-Energy Variations on Equilibrium Compositional Structures Evaluated for Pt–Ir Nanoparticles

The compositional structures of equilibrated Pt–Ir 923-atom cuboctahedron nanoparticles (NPs) are predicted by employing the recently introduced approach for deriving DFT-based coordination dependence of bond-energy variations (CBEVs) combined with the highly efficient statistical-mechanical free-energy concentration expansion method (FCEM). The roles played by preferential strengthening of intrasurface and surface–subsurface bonds in site-specific segregation are elucidated. While CBEV effects enhance Pt surface segregation to all NP surface sites, the driving force obtained for certain (111) sites exceeds the (100) values, in contrast to the simple bond-breaking model. Moreover, the CBEV induces Ir segregation to subsurface sites except the NP subvertexes. Computation of the complete temperature dependence of the compositional structure for low Pt content NPs indicates the occurrence of several smoothly varying atomic exchange processes between surface sites, which are reflected as distinct Schottky-typ...

Prediction of distinct surface segregation effects due to coordination-dependent bond-energy variations in alloy nanoclusters

The first implementation of a recently introduced method based on the extraction of the coordination dependence of surface bond-energy variations (CBEV) from density-functional theory (DFT) computed pure-metal surface-energy anisotropy is reported. In particular, polynomial functions fitted to DFT data computed previously for Pt, Pd, and Rh are used as input energetics for statistical-mechanical computations of Pt-Pd 923-atom cuboctahedron-cluster compositional structures (and Pt-Rh(111) as a test case) using the free-energy concentration expansion method (FCEM). The major findings concern the roles of preferential strengthening of intrasurface and surface-subsurface interlayer bonds leading to quite unique segregation characteristics: (i) strong Pt segregation at certain (111) surface sites of the Pt-Pd clusters, accompanied, at relatively high overall Pt composition, by weaker Pt segregation forming Pt-Pd ordered (100) structure, whereas Pd segregates mainly at the edge and vertex sites; (ii) dominant Pd subsurface segregation. The high computation efficiency of the CBEV/FCEM approach allows the determination of the complete temperature dependence of atomic-exchange processes among surface sites, as well as between subsurface and deeper sites, reflected in the corresponding configurational heat-capacity curves. Compared to other approaches, the high transparency of this method helps to elucidate the origin of the distinct bond-energy-variation effects on site-specific segregation in alloy nanoclusters.

Modeling effects of subsurface tension on segregation:Pt25Rh75(111)oscillatory profile used as a test case

Surface induced bond energy variations are incorporated as homoatomic interaction model in the statistical-mechanical free energy concentration expansion method (FCEM) for the prediction of temperature dependent segregation in alloys. This article focuses on the role of subsurface (and outermost) layer tensions in the emergence of oscillatory segregation profiles even in alloys with weak mixing or demixing tendency. As a test case for the proposed approach, surface segregation in ${\mathrm{Pt}}_{25}{\mathrm{Rh}}_{75}(111)$ is computed by FCEM with Naval Research Laboratory (NRL)-tight-binding (TB) data of surface-induced bond energy variations. A distinct two-layer oscillatory profile is obtained in fair agreement with previously reported experimental data. According to TB data, such subsurface bonding effects are anticipated also for some other Pt-based alloys. The FCEM/TB approach can be used also for predicting surface segregation phenomena in alloy nanoclusters.

Compositional structures and thermodynamic properties of Pd-Cu, Rh-Pd, and Rh-Pd-Cu nanoclusters computed by a combined free-energy concentration expansion method and tight-binding approach

The statistical-mechanical free-energy concentration expansion method in conjunction with tight-binding elemental bonding energetics are used in atomistic modeling of alloy cluster compositional structures. The study focuses on three systems of 923-atom cuboctahedron clusters, Pd-Cu, Rh-Pd, and Rh-Pd-Cu, each capable of inter-cluster atomic exchange. At low temperatures, cluster core ordering and surface mixed-type ordering are predicted for Pd-Cu compositional ``magic-number'' clusters, whereas the system of Rh-Pd clusters tends to separate into clusters exhibiting demixed order. In the former case, slight deviations from magic-number compositions strongly affect segregation levels. At high temperatures, Pd-Cu clusters exhibit surface segregation that affects order-disorder transitions, and at even higher temperatures a surface desegregation process, all of which are reflected in characteristic Schottky-type configurational heat capacity versus $T$ curves. Comparing Pd-Cu with Rh-Pd-Cu reveals distinct ternary alloying effects. The role of surface induced bond energy variations in segregation related order-disorder transitions is demonstrated.

Prediction of compositional ordering and separation in alloy nanoclusters

The statistical–mechanical free-energy concentration expansion method (FCEM) in conjunction with semi-empirical coordination-dependent energetic parameters was used for atomistic modeling of Ni–Cu–Al, Ni–Cu–Rh and the corresponding binary Ni–Cu, Ni–Al and Rh–Cu nanocluster systems, containing from 13 to 923 atoms with icosahedral and cuboctahedral shapes. The high efficiency of FCEM enables computations of such relatively large binary or ternary alloy clusters. Remarkable differences, governed by the opposite Ni–Cu and Ni–Al heteroatomic interactions, were noted in the surface segregated “magic-number” ordered compositional patterns of the two ternary clusters. Due to the strong Ni–Al interactions, the compositional ordering extends into the cluster core, and at elevated temperatures a sharp order–disorder transition occurs, depending on the cluster size, shape and composition. The computed site-specific atomic concentrations form the basis for evaluating pertinent thermodynamic functions. For all the alloy clusters a Schottky-type heat capacity anomaly is predicted and attributed to gradual desegregation excitation processes. In addition, inter-cluster compositional separation is computed for Rh–Cu clusters, and transition temperatures estimated from the disappearance of convexity in the free-energy vs. composition curves.

Prediction of intercluster separation and Schottky-type heat-capacity contribution in surface-segregated binary and ternary alloy nanocluster systems

Site-specific average atomic concentrations computed for systems of Rh-Cu, Ni-Cu and Rh-Ni-Cu icosahedral and cuboctahedral free nanoclusters of different sizes by the ``free-energy concentration expansion method'' are used for evaluating pertinent cluster thermodynamic functions. Compared to stable surface segregated ``magic-number'' structures, at intermediate overall compositions the cluster system is expected to separate into these ``phases.'' A Schottky-type heat-capacity anomaly is predicted and attributed to atomic exchange (e.g., desegregation) excitation processes. Its measurement can elucidate the cluster segregation energetics.

COMPUTATIONAL STUDY OF TERNARY ALLOY NANOCLUSTER COMPOSITIONAL STRUCTURES: Ni–Cu–Rh VERSUS Ni–Cu–Pd

The Free-energy Concentration Expansion Method (FCEM) was utilized for the prediction of compositional structures in Ni – Cu – Rh cubo-octahedron nanoclusters in comparison to recently reported Ni – Cu – Pd data. While both systems exhibit site-specific, sequentially competitive surface segregation (and resultant core separations), remarkable differences governed by the opposite heteronuclear effective interactions, were noted in the surface compositional patterns. Thus, at relatively low temperatures "mixed" Cu/Pd ordering takes place at the Ni – Cu – Pd cluster surface, whereas in the Ni – Cu – Rh cluster Cu and Ni populate separate low and high-coordinated surface sites, thus forming a kind of "demixed surface order". Dissimilarities in the temperature dependence are discussed in terms of the interplay of segregation and compositional order. Such findings may have implications in heterogeneous catalysis and other technologies based on highly dispersed alloyed particles.

Site-specific segregation and compositional ordering in Ni-based ternary alloy nanoclusters computed by the free-energy concentration expansion method

The free-energy concentration expansion method (FCEM), originally derived for the theoretical prediction of surface segregation in the presence of interatomic correlations (short-range order) in binary alloys, is extended to multicomponent alloy nanoclusters, and is applied to unravel the compositional structure of unsupported 309 atom Ni-Cu-Pd cubo-octahedrons. It is a first attempt, to the best of our knowledge, to compute ternary alloy nanocluster site-specific compositions, which are typically inaccessible by current experimental techniques. The calculations predict surface segregation of both solutes Pd and Cu leading to separation of Ni-rich cluster core (the ``finite size/matter effect''). Due to the core separation, even relatively low Ni content clusters are expected to be magnetic. The Ni enrichment and associated magnetic order can prevail up to temperatures noticeably higher than in bulk alloys of the same overall composition. Depending on temperature and overall composition, Cu and Pd form orderedlike patterns (``compositional oscillations'') or mix at the wrapping surface. As the overall Cu content in the cluster increases (on the expense of Ni), the two solutes exhibit surface multisite competition, namely, Cu displaces Pd sequentially from vertexes to edges and facets, and finally Pd desegregates from the surface sites into the core. FCEM can be conveniently applied also to calculation of ``surfaces'' of thermodynamic functions for all ranges of composition above the Gibbs triangle, including the currently defined ``cluster mixing functions.'' It is shown that sharp compositional-structural changes in Ni-Cu-Pd are reflected distinctly in the cluster mixing entropy surface, and the general shape of calculated free-energy surfaces indicates mixing properties of ternary alloy clusters that differ considerably from the corresponding bulk alloys.

On interatomic mixing and demixing phenomena in Cr–Fe: statistical–mechanical calculations based on composition-dependent interaction energy model

Variations of Fe(1 0 0) surface composition with Cr coverage at 53 K were evaluated by means of the “free-energy concentration expansion method” (FCEM). The calculations, complementing our previous study [Phys. Rev. B 65 (2002) 205404], are based on four energetics models: composition-dependent (or independent) and surface-enhanced (or uniform) interactions. A linear dependence of the mixing enthalpy and of the corresponding effective interactions on temperature was taken into account. Numerical free-energy minimization showed that only the first of the four models, namely composition-dependent surface-enhanced interactions that change sign at ∼10% Cr, is consistent with scanning tunneling microscopy (STM) and Auger electron spectroscopy (AES) studies reported before. Effects of temperature on alloying at the Cr/Fe(1 0 0) surface are weak in the mixing region of low Cr concentrations, while for higher coverages the transition to Cr segregated/separated surface layer, induced by the mixing/demixing interaction reversal, is diminished considerably at elevated temperatures. Mixing/demixing effects related to the interaction change seem to be manifested also in model calculations of the bcc part of the Cr–Fe bulk phase diagram.

Effects of composition-dependent interatomic interactions on alloying at the Cr/Fe(100) interface

Energetic factors that govern intermixing and alloying at the Cr/Fe(100) interface are discussed and applied quantitatively in a model based on locally composition-dependent, surface enhanced Cr-Fe interactions. The calculations, employing the free-energy concentration expansion method without any adjustable energetic parameters, are in good agreement with scanning tunneling microscopy results for Cr growth on Fe(100), as reported previously by Davies et al. [Phys. Rev. Lett. 76, 4175 (1996)]. Following a moderate, mixing related increase in Cr surface fraction, at $\ensuremath{\sim}1 \mathrm{ML}$ deposited Cr coverage, a sharp transition from a low $\ensuremath{\sim}10%$ value to $\ensuremath{\sim}30%$ Cr is predicted. This transition is closely associated with a change of sign in the composition dependent effective pair interactions reported for this system. According to the calculations for deposited coverages $>~2 \mathrm{ML},$ a pure Cr layer is separated at the surface, with 1--2 layer sharp interface and slightly Cr-mixed iron layers underneath. At submonolayer Cr deposited coverages only the latter, diffuse part of the interface prevails. Calculated short-range order pair probabilities at the alloy surface and in the bulk agree with experimental values.

Extension of the free-energy concentration expansion method to surface segregation in multi-component alloys and its application to Ni–Al–Cu

Statistical–mechanical description of multi-layer surface segregation in multi-component alloys, based on the free-energy concentration expansion method (FCEM), is introduced. The generalized FCEM provides simple analytical expressions for numerical minimization of the alloy free energy in the presence of short-range order (SRO). As an example, this theoretical approach is applied to unravel segregation with site competition at the (1 1 1) surface of a ternary Ni–8 at.%Al–4 at.%Cu(1 1 1) solid solution. Compared to the corresponding Ni–Al and Ni–Cu binary alloys, the calculations predict significant enhancement of Cu surface segregation and Al desegregation due to the relatively strong Ni–Al attractive interactions and SRO effects. A compositional phase transition marks the onset of Al segregation at high temperatures.

On the estimation of SRO effects on surface segregation

The free-energy concentration expansion method (FCEM) is introduced and applied to the elucidation of short-range order (SRO) effects on surface segregation in alloys. This statistical-mechanical analytical approach, based on the Ising model Hamiltonian energetics, is verified by Monte Carlo (MC) simulations. In particular, calculations performed for the (100) surface of fcc solid solution show that the FCEM agrees with MC results much better than the Bragg-Williams (BW) theory, including the prediction of an increase in equilibrium segregation level with temperature, associated with SRO/segregation competition. FCEM is quite accurate and simple to apply, demanding much less computational effort compared to MC simulations or the cluster variation method. It can replace the corresponding BW formulae in theoretical evaluation of various surfaces as well as bulk phenomena, allowing systematic studies of SRO effects in alloys.