METHOD FOR THE DETERMINATION OF THE REPRESENTATIVE HOMOTOP OF A BINARY METALLIC NANOPARTICLE (AxB1-x)N AND METHOD FOR MANUFACTURING THE CORRESPONDING NANOPARTICLE

Abstract

A method for the manufacturing a representative homotop of a binary metallic nanoparticle (A.sub.xB.sub.1-x).sub.N with a given composition A.sub.xB.sub.1-x, number of atoms N and shape, and at a given temperature, including generating a plurality of homotops, calculating an energy of the generate homotops using formula:

[00001]$\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A& \left(1\right)\end{array}$

where E.sub.0(x, N) is constant for a given particle, ε.sub.BOND.sup.A-B(x) is related to an energy gain caused by the mixing of both metals, N.sub.BOND.sup.A-B is a number of heteroatomic bonds, ε.sub.CORNER,i.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on a corner of type i with an atom of type B in the nanoparticle interior, ε.sub.EDGE,j.sup.A(x) is an energy required for or gained from an exchange of an atom A on the nanoparticle edge of type j with an atom B in the nanoparticle interior, ε.sub.{LMN}.sup.A(x) is an energy required for or gained from an exchange of an atom A on a terrace on a nanoparticle facet with Miller indices {LMN} with an atom B in the nanoparticle interior, N.sub.CORNER,i.sup.A, N.sub.EDGE,j.sup.A, and N.sub.{LMN}.sup.A are numbers of atoms of type A on the respective corners, edges and terraces, determining the representative homotop, and manufacturing the nanoparticle.

Claims

1- Method of manufacturing a binary metallic nanoparticle (A.sub.xB.sub.1-x).sub.N comprising: [a] determining the representative homotop for each of a plurality of binary metallic nanoparticles (A.sub.xB.sub.1-x).sub.N with different compositions A.sub.xB.sub.1-x, numbers of atoms N or shapes and at a given temperature, by: [a1] selecting one of said plurality of binary metallic nanoparticles; [a2] generating a plurality of homotops of the nanoparticle of step [a1]; [a3] calculating an energy of the generated homotops of step [a2] using formula (1): $\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A& \left(1\right)\end{array}$ wherein E.sub.0(x, N) is constant for a given particle (A.sub.xB.sub.1-x).sub.N, ε.sub.BOND.sup.A-B(x) is related to an energy gain caused by the mixing of both metals A, B, N.sub.BOND.sup.A-B is a number of heteroatomic bonds; ε.sub.CORNER,i.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on a corner of type i of the nanoparticle with an atom of type B in an interior of the nanoparticle, given that N.sub.BOND.sup.A-B remains constant, ε.sub.EDGE,j.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on an edge of type j of the nanoparticle with an atom of type B in the nanoparticle interior, given that N.sub.BOND.sup.A-B remains constant, ε.sub.{LMN}.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on a terrace on a nanoparticle facet with Miller indices {LMN} with an atom of type B in the nanoparticle interior, given that N.sub.BOND.sup.A-B remains constant, N.sub.CORNER,i.sup.A, N.sub.EDGE,j.sup.A, and N.sub.{KLM}.sup.A are numbers of atoms of type A on corners of type i, edges of type j and terraces on nanoparticle facets with Miller indices {LMN}, respectively; [a4] determining the representative homotop of the nanoparticle of step [a1], [a5] repeating steps [a1] to [a4] for each of the plurality of binary metallic nanoparticles, [b] determining the physical and/or chemical properties of the resulting representative homotops, [c] selecting one of said representative homotops having the desirable combination of composition, number of atoms and shape, and [d] manufacturing of the corresponding binary metallic nanoparticle, wherein the manufacturing includes forming the nanoparticle according to the selected combination of composition, number of atoms and shape.

2- Method according to claim 1, wherein the manufacturing includes forming the nanoparticle by using at least one of molecular beams, chemical reduction, thermal decomposition of transition-metal complexes, ion implantation, electrochemical synthesis, radiolysis, sonochemical synthesis, biosynthesis, co-deposition of two metals on a support, co-precipitation of two metals from a solution or annealing.

3- Method according to claim 2, wherein the annealing step is performed at a predetermined annealing temperature and on a support which is thermally stable at said annealing temperature.

4- Method according to claim 2, wherein the annealing step is performed at a predetermined annealing temperature and surrounded by ligands that are thermally stable at said annealing temperature.

5- Method according to claim 1, wherein a term ε.sub.INTERFACE.sup.A(x)N.sub.INTERFACE.sup.A is added to formula (1) to describe support-induced segregation on an interface, $\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A+{\mathrm{.Math.}}_{\mathrm{INTERFACE}}^A\left(x\right).Math.N_{\mathrm{INTERFACE}}^A& \left(2\right)\end{array}$ wherein ε.sub.INTERFACE.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on the nanoparticle-support interface with an atom of type B not in contact with the support and with the same coordination number as the atom A, given that N.sub.BOND.sup.A-B remains constant, and N.sub.INTERFACE.sup.A is a number of atoms of type A on the nanoparticle-support interface.

6- Method according to claim 1, wherein a term ε.sub.LAYER(x)N.sub.LAYER is added to formula (1) to describe alloys with a layered structure, $\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A+{\mathrm{.Math.}}_{\mathrm{LAYER}}\left(x\right).Math.N_{\mathrm{LAYER}}& \left(3\right)\end{array}$ wherein ε.sub.LAYER(x) is an energy associated with a formation of monometallic layers of atoms, and N.sub.LAYER=Σ.sub.LAYERS|n.sub.k.sup.A−n.sub.k.sup.B| defines an arrangement of atoms in monometallic layers, where n.sub.k.sup.A and n.sub.k.sup.B are numbers of atoms A and atoms B, respectively, in layer k of a nanoparticle and a sum is taken over all layers.

7- Method according to claim 1, wherein values of ε.sub.BOND.sup.A-B(x), ε.sub.CORNER,i.sup.A(x), ε.sub.EDGE,j.sup.A(x) and ε.sub.{LMN}.sup.A(x) are calculated by fitting them with total energy E.sub.ES values of various reference homotops of a reference nanoparticle.

8- Method according to claim 7, wherein said total energy E.sub.ES values are calculated by density functional theory method.

9- Method according to claim 7, wherein electronic structure calculations used for fitting of formula (1) include a presence of adsorbates, in order to account for a reaction atmosphere.

10- Method according to claim 1, wherein the generating of step [a2] and the determining of step [a4] are done with a random walk using the Metropolis Monte-Carlo algorithm.

11- Method according to claim 10, wherein the random walk includes a multiple exchange algorithm that allows an exchange of N different pairs of atoms between one generated homotop and the next generated homotop, where N follows the probability distribution p(N)˜N.sup.−x with 1≦x≦2.

12- Method for determining a representative homotop of a binary metallic nanoparticle (A.sub.xB.sub.1-x).sub.N with a given composition A.sub.xB.sub.1-x, number of atoms N and shape, and at a given temperature, comprising: generating a plurality of homotops; calculating an energy of the generated homotops using formula (1): $\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A& \left(1\right)\end{array}$ wherein E.sub.0(x, N) is constant for a given particle (A.sub.xB.sub.1-x).sub.N, ε.sub.BOND.sup.A-B(x) is related to an energy gain caused by mixing of both metals A, B, N.sub.BOND.sup.A-B is a number of heteroatomic bonds; ε.sub.CORNER,i.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on a corner of type i of the nanoparticle with an atom of type B in the interior of the nanoparticle, given that N.sub.BOND.sup.A-B remains constant, ε.sub.EDGE,j.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on an edge of type j of the nanoparticle with an atom of type B in the nanoparticle interior, given that N.sub.BOND.sup.A-B remains constant, ε.sub.{LMN}.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on the terrace on a nanoparticle facet with Miller indices {LMN} with an atom of type B in the nanoparticle interior, given that N.sub.BOND.sup.A-B remains constant, and wherein N.sub.CORNER,i.sup.A, N.sub.EDGE,j.sup.A, and N.sub.{KLM}.sup.A are numbers of atoms of type A on corners of type i, edges of type j and terraces on nanoparticle facets with Miller indices {LMN}, respectively, and determining the representative homotop.

13- Method according to claim 12, wherein a term ε.sub.INTERFACE.sup.A(x)N.sub.INTERFACE.sup.A is added to formula (1) to describe support-induced segregation on an interface, $\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A+{\mathrm{.Math.}}_{\mathrm{INTERFACE}}^A\left(x\right).Math.N_{\mathrm{INTERFACE}}^A& \left(2\right)\end{array}$ wherein ε.sub.INTERFACE.sup.A(x) is an energy required for or gained from an exchange of an atom of type A on the nanoparticle-support interface with an atom of type B not in contact with the support and with the same coordination number as the atom A, given that N.sub.BOND.sup.A-B remains constant, and N.sub.INTERFACE.sup.A is a number of atoms of type A on the nanoparticle-support interface.

14- Method according to claim 12, wherein a term ε.sub.LAYER(x)N.sub.LAYER is added to formula (1) to describe alloys with a layered structure, $\begin{array}{cc}{E}_{\mathrm{TOP}}=E_0\left(x,N\right)+{\mathrm{.Math.}}_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}\left(x\right).Math.N_{\mathrm{BOND}}^{A.Math.\text{-}.Math.B}+\underset{i}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{CORNER},i}^A\left(x\right).Math.N_{\mathrm{CORNER},i}^A+\underset{j}{.Math.}.Math..Math.{\mathrm{.Math.}}_{\mathrm{EDGE},j}^A\left(x\right).Math.N_{\mathrm{EDGE},j}^A+{.Math.}_{\left\{\mathrm{LMN}\right\}}.Math.{\mathrm{.Math.}}_{\left\{\mathrm{LMN}\right\}}^A\left(x\right).Math.N_{\left\{\mathrm{LMN}\right\}}^A+{\mathrm{.Math.}}_{\mathrm{LAYER}}\left(x\right).Math.N_{\mathrm{LAYER}}& \left(3\right)\end{array}$ wherein ε.sub.LAYER(x) is an energy associated with a formation of monometallic layers of atoms, and N.sub.LAYER=Σ.sub.LAYERS|n.sub.k.sup.A−n.sub.k.sup.B| defines an arrangement of atoms in monometallic layers, where n.sub.k.sup.A and n.sub.k.sup.B are numbers of atoms A and atoms B, respectively, in layer k of a nanoparticle and a sum is taken over all layers.

15- Method according to claim 12, wherein values of ε.sub.BOND.sup.A-B(x), ε.sub.CORNER,i.sup.A(x), ε.sub.EDGE,j.sup.A(x) and ε.sub.{LMN}.sup.A(x) are calculated by fitting them with the total energy E.sub.ES values of various reference homotops of a reference nanoparticle.

16- Method according to claim 15, wherein said total energy E.sub.ES values are calculated by density functional theory method.

17- Method according to claim 15, wherein electronic structure calculations used for fitting of formula (1) include a presence of adsorbates, in order to account for a reaction atmosphere.

18- Method according to claim 12, wherein the generating of a plurality of homotops and the determining of the representative homotop are done with a random walk using the Metropolis Monte-Carlo algorithm.

19- Method according to claim 18, wherein the random walk includes a multiple exchange algorithm that allows an exchange of N different pairs of atoms between one generated homotop and the next generated homotop, where N follows the probability distribution p(N)˜N.sup.−x with 1≦x≦2.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Other advantages and features of the invention can be seen from the following description in which preferred non-limiting embodiments of the invention are described in reference to the attached drawings, where:

[0041] FIG. 1 Relative energy contributions (%) to the lowest-energy homotops of Pd.sub.70X.sub.70 NPs according to the topological energy calculated as ε.sub.iN.sub.i/Σε.sub.iN.sub.i. Since in Pd—Cu the only negative term is ε.sub.BOND.sup.Pd—Cu, the value of ε.sub.BOND.sup.Pd—CuN.sub.BOND/Σε.sub.iN.sub.i exceeds 100%.

[0042] FIG. 2 Core, subsurface and surface shells of the lowest-energy Pd.sub.70X.sub.70 (X=Au, Ag, Cu, Zn) homotops according to density functional calculations. Spatial dimensions of the NPs are also indicated (for Pd.sub.70Zn.sub.70 the dimensions are given in two directions). Pd atoms are displayed as light grey spheres; atoms X—as dark grey spheres.

[0043] FIG. 3 Dependency on the NP composition of a) ES calculated mixing energy per atom, and of the descriptors b) ε.sub.BOND.sup.Au—Pd, c) ε.sub.CORNER.sup.Au, d) ε.sub.EDGE.sup.Au, e) ε.sub.TERRACE.sup.Au in E.sub.TOP for Pd.sub.140-YAu.sub.Y (solid line) and Pd.sub.79-YAu.sub.Y (dashed line) NPs. Error bars represent 60% confidence intervals.

[0044] FIG. 4 Structures of Pd.sub.732X.sub.731 (X=Au, Ag, Cu, and Zn) NPs with optimized chemical ordering. Pd atoms are displayed as light grey spheres; elements X—as dark grey spheres.

DETAILED DESCRIPTION

[0045] Chemical ordering in Pd.sub.70X.sub.70 NPs of fcc structure for X=Au, Ag, Cu and tetragonal L1.sub.0 structure for X=Zn, which are interesting for heterogeneous catalysis applications, has been optimized employing the proposed method. Unsupported transition metal NPs of this size were shown to be representative models for catalytic studies (see S. M. Kozlov and K. M. Neyman, Top. Catal., 2013, 56, 867-873, cited above). In order to ensure finding the lowest-energy homotops, a multiple exchange algorithm was used, which allowed the inventors to overcome very big energy barriers in the configurational space of certain NPs. In principle, one would expect bulk segregation of Pd in all these structures, since Pd has the highest surface energy among the considered metals. Nevertheless, only for Pd—Au and Pd—Ag the inventors found high stability of the segregated structures, while Pd—Cu and Pd—Zn exhibited more complex morphologies. The inventors also performed the fitting of topological energies for Pd.sub.79-YAu.sub.Y and Pd.sub.140-YAu.sub.Y and found that the E.sub.TOP expressions depend much less on the NP size than on the composition. This observation allowed the inventors to apply the topological expressions obtained for Pd.sub.70X.sub.70 NPs (X=Au, Ag, Cu, and Zn) to the optimization of the chemical ordering in the respective 4.4 nm big Pd.sub.732X.sub.731 NPs, illustrating the power of the proposed approach.

Methodology: Optimization of Chemical Ordering in Nanoalloys

[0046] The inventors have observed that in bimetallic NPs A.sub.YB.sub.N-Y atoms of one element often prefer to occupy interior sites in the most stable structures, while atoms of the other element tend to stay at low-coordinated surface sites. The formation of heteroatomic bonds and layered structures during the alloying process is also important. To quantify these trends in the present examples of the invention, the inventors considered the following form of topological energies, E.sub.TOP, that depend only on the mutual positions of atoms of types A and B within a predetermined lattice,

E.sub.TOP=E.sub.0+ε.sub.BOND.sup.A-BN.sub.BOND.sup.A-B+ε.sub.CORNER.sup.AN.sub.CORNER.sup.A+ε.sub.EDGE.sup.AN.sub.EDGE.sup.A+ε.sub.TERRACE.sup.AN.sub.TERRACE.sup.A+ε.sub.LAYERN.sub.LAYER  (4).

[0047] Formula (4) is a specific case of formula (3). In formula (4), parameter E.sub.0 is required for the fitting to energies E.sub.ES of the electronic structure calculations and it is constant for a given particle A.sub.YB.sub.N-Y; N.sub.BOND.sup.A-B—number of heteroatomic bonds (nearest-neighbor A-B pairs of atoms) in the considered structure; N.sub.CORNER.sup.A, N.sub.EDGE.sup.A, and N.sub.TERRACE.sup.A numbers of atoms of type A on corners, edges and terraces, respectively. Other parameters, such as N.sub.BOND.sup.A-A, N.sub.BOND.sup.B-B or N.sub.CORNER.sup.B, N.sub.EDGE.sup.B, and N.sub.TERRACE.sup.B as well as N.sub.INTERIOR.sup.A depend linearly on the employed parameters for a given NP size and composition. Thus, they were not included in the E.sub.TOP. The term N.sub.LAYER=Σ.sub.LAYERS|n.sub.j.sup.A−n.sub.j.sup.B| accounts for possible atomic arrangements in monometallic layers and tetragonal distortion. n.sub.j.sup.A and n.sub.j.sup.B are the numbers of atoms A and atoms B, respectively, in layer j of a NP and the sum is taken over all layers. |n.sub.j.sup.A−n.sub.j.sup.B| is maximal for layers composed entirely of atoms A or B and is close to zero for layers composed of both atoms in equal proportions.

[0048] In formula (4) ε.sub.i are energetic parameters associated with each degree of freedom N.sub.i considered in the topological energy. They are referred to as descriptors. In contrast to parameters in many empirical methods, each descriptor ε.sub.i has a clear physical meaning. For instance, ε.sub.BOND.sup.A-B is related to the energy gain caused by the mixing of two metals. For example, the formation (mixing) energy of ordered L1.sub.0 A.sub.0.5B.sub.0.5 bulk alloy from separated bulk A and bulk B is 4ε.sub.BOND.sup.A-B per atom, since in this alloy each atom forms 8 heteroatomic bonds and each bond connects two atoms. In turn, ε.sub.CORNER.sup.A is the energy required for or gained from the exchange of an atom of type A on the corner with an atom of type B in the NP interior (given that the number of heteroatomic bonds remains constant). ε.sub.LAYER is a descriptor associated with the formation of layers in (tetragonally distorted) L1.sub.0 structure; the latter is favored when ε.sub.LAYER<0. For Au—Pd, Ag—Pd and Cu—Pd alloys ε.sub.LAYER values were calculated to be zero within statistical accuracy. Hence, the term ε.sub.LAYERΣ.sub.LAYERS|n.sub.j.sup.A−n.sub.j.sup.B| was neglected for description of these materials, which did not affect the accuracy of the E.sub.TOP expression. The model Hamiltonian that leads to the topological energy expression (4) for bimetallic nanocrystals with fcc structure is presented in the Electronic Supplementary Information.

[0049] For each nanocrystal with a given shape, size and composition, an individual topological energy expression is tailored via fitting the descriptors ε.sub.i to the DFT total energy E.sub.ES values of various homotops of this particular NP (obtained via local geometry optimization at DFT level). This way of fitting leads to a rather high accuracy of this approach compared to e.g. interatomic potentials despite the more complex structure of the latter with many more fitting parameters. Naturally, this way of fitting leads to different topological expressions for nanoparticles of different shape, size and composition. However, since each of these descriptors ε.sub.i determines certain interactions, changes in their values from system to system directly reflect the underlying changes in material properties.

[0050] In the present invention, the inventors calculated DFT energies E.sub.ES of 22 to 127 homotops to fit E.sub.TOP for every considered NP shape and composition via multiple linear regression. When several structures with the same set of N.sub.i degrees of freedom were present in the fitting set, only the structure with the lowest E.sub.ES was taken for the fitting. The electronic structure calculations of E.sub.ES for N.sub.FIT NP structures required for the fitting is the computationally demanding part of the method. Therefore, one should aim to keep the number of DFT calculations at a minimum. Nevertheless, insufficient size of the fitting set would lead to overfitting and poor statistical accuracy of the obtained descriptors ε.sub.i. The accuracy can be estimated as 95% confidence intervals via the bootstrap method. This method was applied since it seamlessly takes into account that ε.sub.i are not independent statistical quantities and may strongly correlate with each other. In practice, descriptors that significantly contribute to E.sub.TOP have rather small inaccuracies, while those not crucial for the fitting are determined less accurately. Therefore, the inaccuracy of the latter descriptors does not reflect the inaccuracy of the energy expression as a whole.

[0051] The precision of the topological expressions themselves was estimated as twice the residual standard deviation (RSD) δ between E.sub.ES and E.sub.TOP energies for a set of N.sub.TEST≧10 structures not included in the fitting procedure

[00007]$\delta =2.Math.\sqrt{\frac{.Math.{\left(E_{\mathrm{ES}}-E_{\mathrm{TOP}}\right)}^2-{\left(.Math.\left(E_{\mathrm{ES}}-E_{\mathrm{TOP}}\right)\right)}^2/N_{\mathrm{TEST}}}{N_{\mathrm{TEST}}-1}}$

[0052] According to this definition, (relative) E.sub.TOP values are within δ from the respective (relative) E.sub.ES values with >95% probability. In turn, the accuracy (trueness) of the topological energies, ΔE, was estimated as the energy difference between the lowest-energy structure according to the ES calculations and the global minimum structure within the topological energy optimization. Since many homotops yield the same E.sub.TOP value but somewhat different E.sub.ES, the energy difference ΔE was calculated by the topological expression to avoid any arbitrariness.

[0053] Once descriptors in formula (4) for a given system are determined, one may use this formula to perform optimization of the chemical ordering within the predetermined lattice. In the examples of the present invention, the inventors carried out Monte-Carlo simulations with only one kind of moves—simultaneous exchange of n random atoms of element A with n random atoms of element B. The number of atoms to be exchanged was chosen randomly with the probability p(n)˜n.sup.−3/2, which yields the probability of single exchange moves for big NPs around 1/ζ(3/2)˜38%, where ζ is the Riemann zeta function. This method makes it possible to overcome very big energy barriers that exist, e.g. in the configurational space of Pd—Zn NPs (see the respective discussion in Pd—Zn section).

[0054] The temperature in a Monte-Carlo simulation was chosen in such a way that a system spends <50% of time in the lowest-energy configuration. A configuration of A.sub.YB.sub.N-Y NP was considered a global minimum, if a move from it to a lower energy structure failed after 10Y(N-Y) multiple exchange moves. This means that the inventors applied every possible one of Y(N-Y) single exchange moves for the global minimum search with probability of

[00008]$1-{\left(1-\frac{1}{Y\left(N-Y\right)}\right)}^{0.38\times 10.Math..Math.Y\left(N-Y\right)}\sim 1-e^{-3.8}=98.Math.\%$

and a structure of lower energy could not be found.

[0055] Whenever a structure with lower energy than the previously calculated ones was found in Monte-Carlo simulations, its geometry was recorded. Out of these structures N.sub.TEST structures with the lowest E.sub.TOP energies were calculated by the chosen ES technique, and their ES energies were used to estimate the precision of the topological energy approach employed in the Monte-Carlo simulation. Thus, the test sets included diverse low-energy structures ranging from the predicted global minimum to structures located ˜N.sub.TEST multiple exchange moves far from it in the configurational space. If it is wanted to improve the precision δ of the E.sub.TOP estimated using this test set, then the test set has to be added to the fitting set and new descriptors are obtained. As a result of the global optimization with the new energy expression, a new test set was generated in the fashion described above and the precision of the new E.sub.TOP expression was estimated on the new test set, which had not been included in the fitting.

[0056] Such way of fitting allows for a better description of low-energy structures (prevailing in the fitting set). It suits the purpose of optimization focusing on finding the structure with the lowest possible energy. In the cases where calculated high-energy structures are qualitatively different from low-energy homotops, one may consider deliberately removing high-energy structures from the fitting in order to further improve the description of low-energy structures. One of such cases could be alloys of metals with considerably different atomic sizes, where high-energy and low-energy homotops may have notably different geometric structures due to the mechanical stress and concomitant relaxation.

[0057] The Monte-Carlo scheme also allows one to estimate thermal energy associated with the Boltzmann population of different homotops for a given NP structure. Thermal energies calculated in such a way (with fixed atomic positions) are used only to inspect the magnitude of chemical disorder at finite temperature and to put precision δ of the proposed approach into perspective. Other contributions to the thermal energy may be much bigger and, thus, more important. Nevertheless, they are not relevant to the analysis performed herein.

[0058] In the present examples of the invention, the inventors applied the just outlined method to the optimization of chemical ordering in PdAu, PdAg and PdCu NPs with fcc lattices as well as in PdZn NPs with tetragonally distorted L1.sub.0 lattice. It is important to emphasize that having slightly modified topological energy expression and/or electronic structure calculations one may apply the proposed method to a variety of materials, crystalline structures and reaction conditions. For instance, one may substitute ε.sub.TERRACE.sup.AN.sub.TERRACE.sup.A in formula (4) by ε.sub.{111}.sup.AN.sub.{111}.sup.A and ε.sub.{100}.sup.AN.sub.{100}.sup.A to account separately for segregation on {111} and {100} facets in NPs of certain shapes. Similar modifications can be done to distinguish different kinds of edges and corners. To account for NP-support interactions one may include the term ε.sub.INTERFACE.sup.AN.sub.INTERFACE.sup.A in formula (4) to describe support-induced segregation on the interface. In order to account for the reaction atmosphere there is no need to change the E.sub.TOP expression at all: It is sufficient to consider the presence of adsorbates in the respective electronic structure calculations.

[0059] Pd—Au

[0060] Alloys of Au and Pd have been intensively studied due to their numerous actual and potential applications in heterogeneous catalysis. They include H.sub.2O.sub.2 synthesis, CH.sub.4 conversion to methanol, C—C coupling, oxygen reduction reaction, and various hydrogenation and oxidation reactions. According to theoretical predictions the Au-shell Pd-core structure is the most thermodynamically favorable for Pd—Au NPs, while a rich variety of Pd—Au NP structures has been detected in experiment. It is important to note that the surface composition of Pd—Au systems may be altered by adsorbates such as CO and that even single Au or Pd atoms or dimers on the surface may significantly affect the overall catalytic performance of the system.

[0061] According to the inventor's analysis, the most significant contributions to the E.sub.TOP (here and in the following discussion E.sub.0 is subtracted from E.sub.TOP) for Pd.sub.70Au.sub.70 NP come from stabilization of Au atoms on low-coordinated sites (Table 1). The lower coordination number of the site, the bigger is the energy gain: 200 meV for 9-coordinated terrace atoms of Au, 301 meV for 7-coordinated edge atoms, and 404 meV for 6-coordinated corner atoms. The respective contributions to the global minimum energy calculated with the E.sub.TOP are 18%, 29%, and 39% (FIG. 1). The energy of a Pd—Au bond is calculated by E.sub.TOP to be only ˜13 meV; however, due to the large number of the heteroatomic bonds their contribution to the energy is sizeable, 14%.

TABLE-US-00001 TABLE 1 Descriptors.sup.a ε.sub.i in the topological energy expressions E.sub.TOP formula (4) for the Pd.sub.70X.sub.70 NPs with their precision, δ, and accuracy, ΔE, values (in meV) and number of structures used for the fitting, N.sub.FIT. X Au Ag Cu Zn ε.sub.BOND.sup.Pd-X  −13.sub.−6.sup.+4  −1.sub.−2.sup.+2 −26.sub.−5.sup.+5 −160.sub.−40.sup.+52 ε.sub.CORNER.sup.X −404.sub.−72.sup.+76 −361.sub.−68.sup.+50  95.sub.−33.sup.+36 −251.sub.−342.sup.+316 ε.sub.EDGE.sup.X −301.sub.−77.sup.+52 −289.sub.−129.sup.+78 147.sub.−45.sup.+46 −205.sub.−243.sup.+280 ε.sub.TERRACE.sup.X −200.sub.−64.sup.+52 −163.sub.−64.sup.+43 183.sub.−40.sup.+42  −90.sub.−234.sup.+231 ε.sub.LAYER — — — −105.sub.−38.sup.+29 N.sub.FIT  32  53 127  28 δ  115  150 360  348 ΔE  26  29 171   0 .sup.a95% confidence intervals of ε.sub.i are also given, e.g. −13.sub.−6.sup.+4 means that the interval is −19 ÷ −9.

[0062] The chemical ordering of the lowest-energy homotop found for Pd.sub.70Au.sub.70 reflects the magnitude of different terms in the topological energy expression (see FIG. 2). There, Au atoms occupy all the most energetically stable corner and edge positions and the remaining Au atoms are in surface terrace positions. The configuration of Au atoms on terraces may vary from facet to facet tending to maximize the number of Pd—Au bonds. However, in the lowest-energy structure found by DFT there is 260 heteroatomic bonds, while in the global minimum according to E.sub.TOP it is 262 (Table 2). This finding reflects the expected presence of other minor contributions (of the order of δ=115 meV) to the E.sub.ES that are not accounted for by the E.sub.TOP, formula (4).

[0063] Pd—Ag

[0064] Pd—Ag alloys are studied mostly because of their applications as hydrogenation, fuel cell and other catalysts, sensors and biosensors. Similarly to Pd—Au NPs, theoretical studies' predict Pd-core/Ag-shell structure of Pd—Ag NPs and surface segregation of Ag was also observed in experiment. Interestingly, several experimental studies report homogeneous Pd—Ag alloys or even Pd-shell/Ag-core structures. It was also found that the surface segregation may be affected by the presence of adsorbates such as atomic H.

[0065] The inventors have found interactions in Pd.sub.70Ag.sub.70 to be quite similar to those in Pd.sub.70Au.sub.70: the lower the coordination number of a site, the more energy is gained when it is occupied by an Ag atom. The most prominent contributions to the topological energy E.sub.TOP come from Ag atoms on corners (45%), edges (36%) and terraces (18%) (FIG. 1). The energy of Pd—Ag bonds was calculated to be essentially zero (−1±2 meV); thus, their contribution to E.sub.TOP does not exceed 1%.

[0066] In line with the topological energy expression for PdAg, the structure of Pd.sub.70Ag.sub.70 with the lowest E.sub.ES has all corner and edge positions occupied by silver atoms. The remaining Ag atoms are located on terraces, whereas the NP interior is composed of solely Pd (FIG. 2). The number of heteroatomic bonds in this structure is only 234, i.e. significantly less than 262 in the global minimum for the respective E.sub.TOP. The reason for this difference is the negligible energy associated with N.sub.BOND.sup.Pd—Ag, which probably compares in magnitude with other contributions to the E.sub.ES, not accounted for by the E.sub.TOP.

[0067] Pd—Cu

[0068] A lot of scientific effort has been devoted to Pd—Cu alloys since they catalyze oxygen reduction reaction, O-enhanced water-gas shift reaction (when supported on ceria), formic acid oxidation, water denitrification and several hydrogenation reactions. Early interatomic potential studies revealed two competing effects governing the structure of Pd—Cu NPs: the tendency to maximize the number of heteroatomic bonds and the tendency of Pd or Cu to segregate on the surface. In some studies the enrichment of the surface by Cu or Pd was found to depend on their concentration. In those studies the most energetically stable NP structures also featured higher concentration of surface Cu atoms on corner and edge sites rather than on terrace sites. Experimentally Pd—Cu NPs with Cu-rich surfaces and well mixed ordered or disordered Pd—Cu alloys were characterized. Note that CO-induced surface segregation of Pd was documented for Pd—Cu.

[0069] The inventors considered chemical ordering in Pd—Cu with fcc structure since for NPs of few nm it is more stable than bcc structure observed in Pd—Cu bulk. The energetic stability of Pd—Cu NPs comes mainly from the energy of heteroatomic bonds, which is twice of that in Pd—Au NPs (Table 1). Unlike the cases of Pd—Au and Pd—Ag, Cu prefers to stay inside the NP, whereas the surface of Pd.sub.70Cu.sub.70 is enriched by Pd. The reason is that Pd atoms are bigger than Cu atoms and, therefore, tend to segregate on the surface, where a part of the elastic stress is relieved. However, the employed density functional (as well as other local and gradient-corrected functionals) also favors Pd segregation on the surface, since it predicts the surface energy of Pd (111) to be slightly smaller than the surface energy of Cu (111) in disagreement with experiments. Curiously, the order of stability of Cu in different positions, interior>corner>edge>terrace, does not correlate with the coordination number of Cu in these sites. Since descriptors corresponding to Cu atoms on corner, edge and terrace positions are positive (reflecting that these positions are unstable for Cu with respect to interior ones), their destabilizing contributions to the E.sub.TOP of the global minimum are −29, −47 and −3%, respectively. Hence, to counteract that the contribution of the heteroatomic bonds to E.sub.TOP formally exceeds 100%.

TABLE-US-00002 TABLE 2 Structural properties of the homotops Pd.sub.70X.sub.70 (X = Au, Ag, Cu, Zn) with the lowest energies E.sub.ES and E.sub.TOP. Average coordination numbers of X by X, N.sup.X—X, X by Pd,.sup.a N.sup.Pd—X, and Pd by Pd, N.sup.Pd—Pd are given to facilitate comparison with experimental (e.g. EXAFS) data. X N.sub.BOND.sup.Pd—X N.sub.CORNER.sup.X N.sub.EDGE.sup.X N.sub.TERRACE.sup.X N.sub.SUBSURFACE.sup.X N.sub.CORE.sup.X N.sup.X—X N.sup.Pd—X N.sup.Pd—Pd Au ES 260 24 24 22 0 0 3.57 3.71 7.17 TOP 262 24 24 22 0 0 Ag ES 234 24 24 22 0 0 3.94 3.34 7.54 TOP 262 24 24 22 0 0 Cu ES 358 16 17 1 35 1 4.20 5.11 3.69 TOP 382 12 14 8 34 2 Zn .sup.b ES = 422 16 14 16 20 4 2.57 6.03 3.54 TOP .sup.aFor NPs with 1:1 composition, the average coordination number of X by Pd equals the average coordination number of Pd by X; .sup.b The same structure yields both the lowest E.sub.ES and E.sub.TOP for Pd.sub.70Zn.sub.70; in this structure Σ.sub.LAYERS|n.sub.i.sup.Zn − n.sub.i.sup.Pd| equals to 136.

[0070] The descriptor value for Pd—Cu bonds, −26.sub.−5.sup.+5 meV/bond, corresponds to the binding energy of −104.sub.−20.sup.+20 meV per atom in (fcc or bcc) Pd—Cu bulk, which agrees with the experimental value of −114 meV per atom for the bcc alloy.

[0071] The homotop with the lowest E.sub.ES of the Pd.sub.70Cu.sub.70 NP exhibits matryoshka-like (also called onion- or multishell-like) arrangement with Pd-rich surface shell, Cu-rich subsurface shell and Pd-rich core. This chemical ordering allows the formation of 358 heteroatomic bonds, while the number of Cu atoms is kept low on the surface, especially on terraces. The structure of the global minimum according to the E.sub.TOP features even more heteroatomic bonds, 382, more Cu atoms on terraces and less Cu on edges and corners.

[0072] Pd—Zn

[0073] Bimetallic Pd—Zn is actively studied (often in the form of surface alloys) because of its catalytic activity in (reverse) water-gas shift and hydrogenation reactions as well as potential application as selective and highly stable catalysts for methanol steam reforming. However, the employment of Pd—Zn catalysts is complicated by the significant dependence of their properties on the Zn/Pd ratio, NP size and the composition of the subsurface region. Further complications come from strong dependence of the structure and composition of Pd—Zn systems on environmental conditions.

[0074] Both bulk and nanoparticulate Pd—Zn are known to have tetragonally distorted L1.sub.0 crystal structure without pronounced surface segregation of any component. Experimental interatomic distances in bulk of 1:1 PdZn are Zn—Zn=Pd—Pd=289 pm and Pd—Zn=222 pm. There, Pd and Zn atoms in a distorted fcc-like lattice form monometallic layers and the distances between adjacent atoms in different layers are shorter than those within the same layer. The clear propensity of Pd.sub.70Zn.sub.70 NPs to build alternating Pd and Zn layers normal to one of the [001], [010] or [001] directions (equivalent in the fcc lattice) accompanied by NP compression along this direction is revealed by DFT calculations and topological ordering optimizations (FIG. 2). Energy of such compression is properly taken into account by the term ε.sub.LAYERN.sub.LAYER in formula (4). It noticeably increases the E.sub.TOP precision for Pd.sub.70Zn.sub.70 from 1294 meV to 348 meV. (The accuracy ΔE is 0, even when this term is neglected in the E.sub.TOP expression.) For alloys that do not tend to form layered structures, the term ε.sub.LAYERN.sub.LAYER does not improve the E.sub.TOP precision or may be even somewhat detrimental due to overfitting. For instance, for Pd.sub.70Cu.sub.70 excluding that contribution from E.sub.TOP increases its precision of the latter by 16%.

[0075] Heteroatomic bonds in Pd—Zn NPs are found to be an order of magnitude stronger than those in the other investigated alloys. This difference reflects the fact that composites of Au, Ag, Cu with Pd are alloys of d-elements, while Zn is an sp-element. Strong heteroatomic Pd—Zn bonds of polar character (the charge separation is estimated to range from Pd.sup.−0.2Zn.sup.−0.2 to Pd.sup.−0.4Zn.sup.+0.4) result in the prevalence of ordered structures of Pd—Zn in the phase diagram. On the contrary, alloys of d-elements are more prone to exhibit more random crystal structures, where the number of heteroatomic bonds is not maximal. Hence, Pd—Zn is better classified as an intermetallic compound rather than a bimetallic alloy.

[0076] In line with these considerations, the dominant E.sub.TOP contribution for Pd.sub.70Zn.sub.70 is given by Pd—Zn bonds. The descriptor ε.sub.BOND.sup.Pd—Zn=−160.sub.−40.sup.+52 meV yields the alloy formation energy of −640.sub.−160.sup.+208 meV per atom, in agreement with the measured value of −520 meV per atom. Hence, heteroatomic bonds define 75% of the E.sub.TOP of the global minimum, while the rest comes mostly from the energy associated with the formation of the layered structure (16%). Despite that the energies of Zn atoms on low-coordinated sites are only slightly smaller than the respective energies of Ag atoms in Pd.sub.70Ag.sub.70, their overall contribution is rather small (9%) compared to that of Pd—Zn bonds. Since the relative energies of Pd.sub.70Zn.sub.70 NPs do not strongly depend on the number of low-coordinated Zn atoms (compared to other characteristics), it is hard to accurately fit the respective descriptors. Therefore, the formal statistical inaccuracy of ε.sub.CORNER.sup.Zn, ε.sub.EDGE.sup.Zn, and ε.sub.TERRACE.sup.Zn exceeds 100%. Yet, this does not seem to affect the overall accuracy of the proposed approach, because these descriptors appear to be less important for the description of Pd—Zn NPs. To examine how sensitive the chemical ordering in the obtained global minimum is to the statistical inaccuracy of the descriptors for Pd.sub.70Zn.sub.70 (Table 1), its chemical ordering was re-optimized with 10 other sets of descriptors. These sets were generated by fixing each one of the 5 descriptors at the lowest or the highest value of its 95% confidence interval and re-fitting all other descriptors using for that the same homotops. Despite substantial variations of descriptors produced in such a way, global optimizations with all these 10 sets yielded the homotop with the same N.sub.i characteristics as the one depicted in FIG. 2.

[0077] The structures with the lowest E.sub.ES and E.sub.TOP are the same for Pd.sub.70Zn.sub.70 NPs due to the high accuracy of the topological energy expression. They feature the maximum possible number of heteroatomic bonds, 422, for the A.sub.70B.sub.70 NP of the considered shape. The N.sub.LAYER value is also the maximum possible, 136, for this particular stoichiometry and shape. As already mentioned, the energies of Zn atoms on the low-coordinated sites are of minor importance for the determination of the most stable Pd.sub.70Zn.sub.70 homotops. Hence, the amounts of Zn atoms on various types of low-coordinated sites have intermediate values. All in all, the most energetically stable homotop exhibits the layered L1.sub.0 structure, similar to PdZn bulk. Nevertheless, this structure also exposes Pd atoms on Zn-composed edges, due to the slight excess of Pd for the formation of perfect layered structure. Unlike monolayer thick Pd—Zn films on Pd (111), no zigzag-like structures are seen on {111} facets of Zn.sub.70Pd.sub.70 particles.

[0078] For Pd.sub.70Zn.sub.70 NP one could construct a homotop apparently very similar to the obtained global minimum by exchanging all Zn atoms with all Pd atoms at once. This homotop has the same number of heteroatomic bonds and the same layered structure as the global minimum, but less Zn atoms on corners and edges and more Zn atoms on terraces. Therefore, its E.sub.ES (E.sub.TOP) is 1765 meV (1389 meV) higher than that of the global minimum displayed in FIG. 2. Despite the apparent similarity, for the simulation code this homotop looks absolutely different compared to the global minimum structure, since the position of every atom has changed. The transition from one homotop to another via exchange of one random Zn atom with a random Pd atom at a time would go through the configurations with rather small amount of Pd—Zn bonds and, therefore, very high relative energy. It was impossible to overcome the transition state between these two homotops via single exchange moves even at Monte-Carlo simulation temperatures as high as 10000 K. However, the transformation between the discussed homotop and the global minimum does not pose any difficulty when multiple exchange moves are applied, that is, n random atoms are exchanged at a time. This illustrates the efficiency of the employed computational scheme, which ensures that the lowest-energy structures from the Monte-Carlo simulations are indeed the global minima within the studied topological framework.

[0079] Thermal Energies

[0080] Naturally, a system in thermodynamic equilibrium adopts exclusively its global minimum configuration only at zero Kelvin, while at any finite temperature the presence of other homotops is possible with a probability determined by the Boltzmann factor. The configurational space of Pd.sub.70Au.sub.70, Pd.sub.70Ag.sub.70 and Pd.sub.70Cu.sub.70 NPs features many homotops different from the global minima only by the number of heteroatomic bonds. The energies of these homotops are just slightly higher than the energies of the global minima and hence the population of these homotop states is notable even at relatively low temperatures. Consequently, these homotops can contribute to the thermal energy.

[0081] It is very instructive to compare the thermal energy accumulated by chemical (dis-)ordering to the precision δ of the topological expressions (Table 1). For example, the Pd.sub.70Au.sub.70 NP obtains (homotopic) thermal energy of 115 meV already at ˜140 K. (That is, at this temperature the average energy of the system in the Monte-Carlo simulations is 115 meV above the energy of the global minimum.) Therefore, despite that the structure of Pd.sub.70Au.sub.70 with the lowest found E.sub.ES (FIG. 2) may not yet be the global minimum for the chosen ES computational scheme, it is certainly feasible at 140 K and may serve as a representative model for the NP at this and higher temperatures. In a similar way one gets that the considered lowest-energy structure of the Pd.sub.70Ag.sub.70 is a representative homotop at ˜360 K, while for Pd.sub.70Cu.sub.70 this temperature is ˜220 K. Unlike the aforementioned three nanoalloys, PdZn features strong heteroatomic bonds with ε.sub.BOND.sup.Pd—Zn of 160 meV. Thus, there are not many low-energy homotops around the global minimum. In fact, the second most stable structure has the energy E.sub.TOP ˜205 meV higher than the global minimum. Therefore, below 500 K essentially the global minimum structure alone has to be present in the thermodynamic ensemble. Much higher temperatures are required to populate less stable homotop states, so the (homotopic) thermal energy reaches the precision of the topological expression, 348 meV, only at ˜1300 K. Thus, this high temperature is related mostly to the propensity of Pd—Zn to form regular nanostructures and to avoid any disorder, rather than to the low precision of the E.sub.TOP for this system. Note that Zn evaporates from Pd—Zn surface alloys at temperatures above 800 K. Therefore, it is safe to assume a very small degree of disorder in experimental samples of Pd—Zn close to the thermodynamic equilibrium.

[0082] Mixing Energies

[0083] Another way to quantify the binding strength of the metals A and B in A.sub.YB.sub.N-Y NPs is by means of their mixing energy (per atom):

E.sup.MIX=[NE(A.sub.YB.sub.N-Y)−YE(A.sub.N)−(N−Y)E(B.sub.N)]/N.sup.2,

where E(A.sub.YB.sub.N-Y) is the total energy of the A.sub.YB.sub.N-Y NP and E(A.sub.N), E(B.sub.N) are the energies of respective monometallic NPs with the same structure optimized with the same plane-wave basis cut-off as the A.sub.YB.sub.N-Y NP. Thus, e.g. for Pd.sub.70Zn.sub.70 the NPs Zn.sub.140 and Pd.sub.140 with fcc structure were considered as energy references. According to this definition negative E.sub.MIX values mean exothermic mixing. Since the presented approach allows calculating total energies of alloy NPs with precision δ, the respective precision of the calculated mixing energy per atom is δ/N.

TABLE-US-00003 TABLE 3 Mixing energies.sup.a E.sup.MIX (per atom, in meV) of the Pd.sub.70X.sub.70 homotops with the lowest calculated energy E.sub.ES. NP Pd.sub.70Au.sub.70 Pd.sub.70Ag.sub.70 Pd.sub.70Cu.sub.70 Pd.sub.70Zn.sub.70 E.sub.ES.sup.MIX −109.sub.−1.sup.+1 −108.sub.−1.sup.+1 −119.sub.−3.sup.+3 −498.sub.−2.sup.+2 E.sub.TOP.sup.MIX  −82.sub.−15.sup.+15  −55.sub.−10.sup.+10  −89.sub.−13.sup.+14 −484.sub.−135.sup.+100 .sup.aThe 95% confidence intervals for E.sub.ES.sup.MIX were calculated as δ divided by the number of atoms in the NP; the 95% confidence intervals for E.sub.TOP.sup.MIX were calculated with the bootstrap analysis.

[0084] The mixing energies (per atom) of the homotops with the lowest energy E.sub.ES, calculated using the respective topological expression and DFT are presented in Table 3. The magnitudes of ES mixing energies are found to be ˜110 meV for Pd—Au, Pd—Ag and Pd—Cu NPs, while for Pd—Zn it is almost 500 meV. The magnitudes of E.sub.TOP.sup.MIX energies resemble the respective E.sub.ES.sup.MIX values, except the case of Pd—Ag, for which E.sub.TOP.sup.MIX is almost twice smaller than E.sub.ES.sup.MIX. The reason is that the topological energy expression for Pd—Ag assigns almost zero energy to the heteroatomic Pd—Ag bonds and consequently predicts their essentially vanishing contribution to the mixing energy. In the rather similar Pd.sub.70Au.sub.70 NP heteroatomic bonds are responsible for 30% of the mixing energy calculated using E.sub.TOP, which can explain the 33% difference between E.sub.TOP.sup.MIX for Pd—Au and Pd—Ag.

[0085] Dependency of Descriptors on the Composition and the Size of Nanoparticles

[0086] For practical purposes it is very important to know if descriptors obtained for one system can also be used to represent a slightly different system. For instance, one may wonder if descriptors calculated for smaller NPs yield reasonable results when applied to bigger species, for which ES calculations are unfeasible. To evaluate the dependency of descriptors on the size and the composition of NPs the inventors constructed E.sub.TOP expressions and performed optimization of chemical ordering in Pd.sub.YAu.sub.79-Y (Y=6, 28, 40, 53, 71) and Pd.sub.YAu.sub.140-Y (Y=11, 20, 30, 35, 40, 49, 70, 91, 126) NPs. The results are summarized in FIG. 3, where the error bars represent 60% confidence intervals of the ε.sub.i calculated via the bootstrap analysis. Note that if such confidence intervals for two ε.sub.i values do not overlap, the probability that these descriptors are not different is less than ((1−0.6)/2).sup.2=4%.

[0087] The first observation is that the descriptors (and the mixing energies) significantly depend on the composition of the NPs. That is, the binding in Pd-rich Pd—Au NPs is quite different from that in Au-rich NPs. The latter feature stronger heteroatomic bonds, but less stable gold atoms on surface sites compared to Pd-rich NPs. These differences are probably related to the gradual changes in the electronic structure and average interatomic distances in the NPs with growing Au content. In most cases quantitative changes of the descriptors do not cause qualitative changes in the NP ordering. The only exception is that at very low Au concentrations Au atoms seem to prefer to occupy edges rather than corners of the Pd—Au NPs. This effect is more pronounced for Pd.sub.YAu.sub.79-Y than for Pd.sub.YAu.sub.140-Y NPs. The change in the relative stability of corner and edge positions for Au is reflected in the structure of the respective global minima.

[0088] Nevertheless, Pd—Au NPs prepared by galvanic displacement expose Au atoms on corners rather than on edges. However, according to the inventor's calculations corners are the most stable positions for Au only at moderate and high Au concentrations. The inconsistency between the presented and experimental results may also be due to kinetic limitations in the experimental setup or deficiencies of the employed exchange-correlation density functional.

[0089] One notices a rather limited dependency of the descriptors ε.sub.i and the mixing energies per atom on the NP size. Especially at high Au concentrations, differences between ε.sub.i values for Pd.sub.YAu.sub.79-Y and Pd.sub.YAu.sub.140-Y are barely visible and they are often within the statistical accuracy of the calculations. However, at lower Au content the binding was found to be slightly stronger in the smaller NPs. In numerous cases it was shown that many (physical) properties of NPs bigger than 1.5 nm already depend rather smoothly on their size and start to converge to a certain value. Hence, it is probable that the descriptors calculated for Pd.sub.YAu.sub.140-Y as well as for other Pd.sub.70X.sub.70 NPs may serve as a reasonable approximation for descriptors of bigger NPs or, at least, they will lead to qualitatively correct chemical ordering, when applied to bigger NPs. At the same time, descriptors may not work satisfactorily for very small bimetallic clusters, where the quantum nature of interatomic interaction is expected to be notable. The inventor's findings suggest that it is more important to use descriptors tailored for a particular composition than for a particular size.

[0090] One may ask, to what extent applications of the present topological method can be limited to such high-symmetry “magic” shapes of bimetallic crystallites, as truncated octahedral ones discussed so far. To address this question the inventors optimized the chemical ordering in a fcc NP Pd.sub.61Au.sub.61 of just C.sub.3ν symmetry with a shape reminiscent of typical shapes of supported Pd NPs. The individual topological descriptors, the overall picture of interactions as well as the chemical ordering in Pd.sub.61Au.sub.61 are very similar to those of the highly symmetric truncated octahedral NP Pd.sub.70Au.sub.70. The accuracy ΔE and precision δ values of the E.sub.TOP expressions for Pd.sub.61Au.sub.61 and Pd.sub.70Au.sub.70 are also very close. These findings strongly suggest that the method is applicable to reliably describe chemical ordering also in nanocrystallites with rather unsymmetrical shapes.

[0091] Extrapolation to the 4.4 nm Large Nanoparticles

[0092] Benefiting from the rather moderate dependency of descriptors on NP size, it was possible to apply the descriptors calculated for Pd.sub.70X.sub.70 NPs to bigger ˜4.4 nm Pd.sub.732X.sub.731 NPs as an illustrative example (see FIG. 4). The shape of these NPs is chosen to mimic the shape of Pd.sub.70X.sub.70 NPs, i.e. featuring small {100} facets composed of only four atoms. To simulate NPs with bigger {100} facets one would need to calculate the descriptor for X atoms on {100} terraces, which are absent in the M.sub.140 models. Using the proposed E.sub.TOP expression the inventors were able to perform efficient simulations of such ˜4.4 nm NPs with the speed of >10.sup.7 Monte-Carlo steps per hour on one Intel 2.66 GHz processor.

[0093] Both Pd.sub.732Au.sub.731 and Pd.sub.732Ag.sub.731 NPs have surfaces covered by Au and Ag, respectively. In turn, their subsurface shells are composed mostly of Pd atoms and only two Au or three Ag atoms, which allows the maximization of the number of heteroatomic bonds. Consequently, the cores of the NPs have stoichiometries of Pd.sub.332Au.sub.157 and Pd.sub.333Ag.sub.156. In order to maximize the number of heteroatomic bonds these Pd.sub.0.68X.sub.0.32 cores also develop L1.sub.0-like structure with partially formed layers of Au or Ag in Pd. The structure of the Pd—Cu NPs is more complicated due to two competing tendencies: maximization of N.sub.BOND.sup.Pd—Cu and bulk segregation of Cu. As a result, the surface shell has a stoichiometry of Pd.sub.412Cu.sub.160 and exhibits abundant Cu monomers as well as occasionally present Cu dimers on terraces and edges. Each corner of the NP has two Cu atoms on the opposite vertices of the small {100} facet. The subsurface shell of the NP is enriched in Cu (stoichiometry Pd.sub.87Cu.sub.315). Finally, the NP core has almost 1:1 stoichiometry, Pd.sub.233Cu.sub.256, and again features layer-like structure. As for the global minimum of Pd.sub.732Zn.sub.731, quite expectedly, it has almost a bulk-cut structure similarly to the Pd.sub.70Zn.sub.70 case.

[0094] Summary

[0095] The inventors have developed a method to optimize chemical ordering in bimetallic NPs using an energy expression related to topological degrees of freedom, E.sub.TOP. This expression depends on the topology of bonds between the atoms composing the NP, but not on the explicit coordinates of these atoms. Using this approach the inventors optimized the chemical ordering in truncated octahedral Pd.sub.YAu.sub.79-Y and Pd.sub.YAu.sub.140-Y NPs as well as in Pd.sub.70Ag.sub.70 and Pd.sub.70Cu.sub.70 NPs with fcc lattices and Pd.sub.70Zn.sub.70 with L1.sub.0 lattice; the chemical ordering in the fcc nanocrystal Pd.sub.61Au.sub.61 with a less symmetric shape has been also determined. This approach can be applied to bimetallic NPs of any given lattice type, up to the point, when the structure becomes disordered, e.g. at higher temperatures.

[0096] For every NP size and composition the energetic parameters (descriptors) in the E.sub.TOP expression were fitted to the energies of more than 20 NP structures obtained via density functional calculations. The precision of the topologic description tailored in such a way (i.e. their ability to predict results of the electronic structure calculations) was 115-360 meV for Pd.sub.70X.sub.70 NPs (X=Au, Ag, Cu and Zn) and the accuracy of the E.sub.TOP was at least twice better. For the Pd—Au, Pd—Ag, and Pd—Cu NPs the precision of the topological approach is comparable to the thermal energy associated with the population of low-energy homotops at temperatures of 140-360 K. Therefore, even if some of the lowest-energy structures mentioned above are not exactly the lowest-energy homotops (according to electronic structure calculations), they are representative homotops at very moderate temperatures.

[0097] A very useful advantage of the proposed approach is that the descriptors ε.sub.i in the E.sub.TOP expression have a clear physical meaning, e.g. the energy of heteroatomic bonds or the relative energy of X atoms on terrace, edge or corner positions of the NP (interior positions being the reference). Thus, the overall binding energy is inherently a sum of contributions from particular structural features. In turn, changes of these contributions from system to system reflect changes in their properties. Analyzing the structure of the topological energy expression the inventors were able to get valuable insights into the binding in Pd—Au, Pd—Ag, Pd—Cu, and Pd—Zn nanoalloys. Available experimental formation energies of bulk 1:1 PdCu and PdZn agree well with the descriptor values ε.sub.BOND.sup.Pd—X of heteroatomic bonds for Pd.sub.70Cu.sub.70 and Pd.sub.70Zn.sub.70 NPs, respectively. The analysis of descriptors for Pd.sub.YAu.sub.79-Y and Pd.sub.YAu.sub.140-Y NPs showed a notable dependency on the composition of the NPs, but much smaller dependency on their size. This allows one to use descriptors based on electronic structure calculations of relatively small NPs of e.g. 140 atoms to optimize chemical ordering in bigger species formed of thousands of atoms. Hence, the inventors applied their method to describe the chemical ordering in large Pd.sub.732X.sub.731 (X=Au, Ag, Cu and Zn) NPs, which are beyond the scale of conventional density functional calculations.

[0098] The optimization of Pd—Au and Pd—Ag NPs with E.sub.TOP yields Au and Ag atoms preferentially occupying positions with lower coordination numbers. The energy gain due to the formation of heteroatomic bonds is rather small for these materials and plays a secondary role in the determination of the NP ordering. On the contrary, the energy of heteroatomic bonds is the driving force for the alloying of Cu and Pd. In this case, the stability of Cu atoms is the highest inside the NP and the lowest on NP terraces. These two effects lead to the matryoshka-like structure of the lowest-energy Pd.sub.70Cu.sub.70 homotop, which has the surface shell enriched with Pd, the subsurface region enriched with Cu and the core composed mostly of Pd.

[0099] Unlike bimetallic Pd—Au, Pd—Ag, and Pd—Cu alloys formed by d-elements, the binding in intermetallic Pd—Zn involves the interaction of a noble d-metal with an sp-element. The result is a much higher energy gain due to the formation of Pd—Zn bonds and a much higher (in magnitude) mixing energy of Pd—Zn NPs compared to other considered nanoalloys. The preferential occupation of any particular type of sites by Zn atoms is much less important for the NP structure and energy in this case. The structure of the most energetically stable homotop is very close to the cut from bulk Pd—Zn with L1.sub.0 crystal structure. Just like the bulk, it features alternating Pd and Zn layers and tetragonal distortion. The term in the E.sub.TOP expression related to the formation of such a layered structure turned out to be responsible for 16% of the binding in Pd—Zn NPs.

[0100] The proposed method for the optimization of the chemical ordering in bimetallic particles paves the way to atomistic studies of several nanometer big bimetallic crystallites with known lattice structure. Fortunately, the latter can be determined by contemporary experimental techniques. Notably, the present new approach may be straightforwardly augmented to be applicable to heterometallic nanocrystals on a support or in a reaction atmosphere.

[0101] Methods

[0102] Electronic structure calculations were performed with the periodic plane-wave code VASP. The inventors used the PBE exchange-correlation functional found to be one of the most appropriate common functionals for the description of transition metals. The interaction between valence and core electrons was treated within the projector augmented wave approach. To moderate computational expenditures the inventors carried out calculations with the 250-280 eV energy cut-off of plane-wave basis sets, which yielded results very close to those obtained with the cut-off 415 eV. The one-electron levels were smeared by 0.1 eV using the first-order method of Methfessel and Paxton; finally, converged energies were extrapolated to the zero smearing. Calculations were performed only at the Γ-point in the reciprocal space. All atoms were allowed to move (relax) during the geometry optimization until forces on them became less than 0.2 eV/nm. The minimal separation between NPs exceeded 0.7 nm, at which the interaction between adjacent NPs was found to be negligible.