Recent years has witnessed extensive interest in nanocrystals due to their unique place bridging single atoms and bulk solids. A special attention has been devoted to spherical metallic nanoparticles, especially those on a suitable support. To understand the mechanisms of catalytic activity of nanoparticles, which was shown to depend strongly on their size and shape [1], a practical and reliable method of nanoparticle characterization is necessary. Recent developments in instrumentation, data analysis and modelling methods led X-ray absorption spectroscopy (XAS) to quickly become the premier technique for catalytic studies. XAS can elucidate the distances of atoms, reveal the average size of particles smaller than ca. 2 nm, provide details on the shape of the nanoparticles, and can be used under in situ or in operando conditions. Its analytical power is demonstrated to match or surpass common laboratory techniques, e.g. chemisorption, X-ray diffraction or transmission electron microscopy, in measuring particle size, shape and inner structure in the nanometre range. Because it is element-specific, XAS can further reveal the mole fractions of atoms in the nanoparticles [2, 3].

While there is no direct relationship between the coordination numbers and interatomic distances to the diameter of spherical nanoparticles, it was shown that the latter can be aproximated to that of a cuboctahedron [4]. For cuboctahedron nanoparticles an exact mathematical equation that correlates the coordination number to the total number of atoms is derived [5]. We have recently shown that the diameter D of the cuboctahedron nanoparticle is related to that of imaginary sphere having the same volume, and can be expressed as a function of the diameter d of the single atom and the number of atoms in the edge of cuboctahedron m, as D ≈ 1.65 m d, and the results were verified by independent experimental techniques [6]. Here we extend this method to estimate the diameter of nanoparticles composed of atoms of different elements forming bimetallic solid solutions, core-shell or aggregate mixtures. Sets of equations for coordination numbers of one component to the other, as well as the interatomic distances between the two elements in these different inner-strucutured mixtures are derived for each system. By testing the experimental data against these predictions, we identify the inner structure of the nanoparticle and apply the appropriate particle size equation. A solid solution of two metals can be recognized as their coordination numbers are in direct relationship to the mole fraction of the components, as well as that the total coordination number of one consitutent equals to that of the other [3]. The diameter of the nanoparticle can be estimated by inserting in the above equation the average atomic diameter d_av evaluated by Vegrad’s Law as the weighted mean of the two constituents’ lattice parameters. The relationships for solid solution do not hold for a core/shell system, and the latter can be distinguished by the the similarity of the total coordnation number of the core/shell nanoparticle to that of the monometallic nanoparticle of the same size composed of the core atoms only, so that the diameter of the core/shell nanoparticle can be found by inserting the the atomic diameter of the core atom, d_core in the above equation for monometallic shell. Similar conclusion is derived for core/bilayered shell nanoparticles. As the aggregate mixtures are composed of two completely separated nanoparticles with very little interaction between them, the structure can be recognized as the coordination number of one kind of atom to the other is close to zero, and their sizes can be evaluated using the equation for monometallic nanoparticle and the appropriate atomic diameter.

It was further shown that the set of equations can be applied to three-component systems using reasonable approximations. Triatomic core/shell particles with a solid solution of two components in the core and a monoatomic shell of the third component, as well as core/inner shell/outer shell nanoparticles in which every layer is composed of different atoms were prepared, and the particle sizes of these systems estimated by our approach are shown to match closely the experimental data obtained by independent techniques.

References:

[1] E. Roduner, Nanoscopic Materials: Size Dependent Phenomena, RSC Publishing, Cambridge, UK (2006).

[2] S. Calvin, EXAFS for everyone, CRC Press-Taylor and Francis Group, Boca Raton, FL (2013).

[3] A. Frenkel, Z. Kristallogr., 2007, 222, 605.

[4] A. Jentys, Phys. Chem. Chem. Phys., 1999, 1, 4059.

[5] R.E. Benfield, J. Chem. Phys. Faraday Trans., 1992, 88, 1107.

[6] N.S. Marinkovic et al, Zast. Mater., 2016, 57, 101.