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# Equilibrium shapes of nanoparticles

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### Equilibrium shapes of nanoparticles

1. 1. Equilibrium Crystal Shapes: free and supported nanoparticlesIf the surface energy is isotropic (as for a liquid) the problem is simple to minimize the surface and the solution is a sphere. In crystalline solids the surface energy is anisotropic and the energy-minimizing shape is found using the limiting planes of the lowest possible surface energy. review : A8.Morphology of supported nanoparticles_Henry, D2
2. 2. For a crystal grown at equilibrium See D2
3. 3. In 1901 Wullf introduced, withour proving, a theorem where he said: for anequilibrium crystal there is a point in the interior such that its perpendiculardistance hi from the ith face is proportional to the surface energy γi Wulff Plot See D2
4. 4. Theoretical Wulff shapes of TiO2 i ∈ {polyhedron facets} 1Average surface energy: γ = ∑Aγ ,i i Atot i rutile brookite anatase<γ>= 1.1 J/m2 <γ> = 0.7 J/m2 <γ> = 0.5 J/m2Ramamoorthy, Vanderbilt, King-Smith, PRB 1994;Lazzeri, Vittadini, Selloni, PRB 2001; Gong & Selloni, PRB 2007
5. 5. Different shapes can originate from different extension of the equivalent faces
6. 6. Equilibrium shape at T= 0 K: the surface energy anisotropy is maximalFCC: truncated octahedron BCC: rhombic dodecahedron
7. 7. Equilibrium shape at T≠0 K: Around (but below) their melting temperatures, crystals tend to have shapes which are pretty round: not a complete sphere, but with no regions which are flat (faceted). This is because at high temperature the atoms on the surface jiggle and wiggle more: they dont care so much which places are easier to sit because they have so much energy to spare. The facets T appear at lower temperatures, as the crystal is cooled: the first temperature at which a facet occurs is called the roughening temperature. Crystals grown at T>Trough do not form facets (e.g. most nanoparticles grown by solution methods at high T are spherical)Roughening Transition
8. 8. Equilibrium shape in the nanoworldSeveral factors can change theequilibrium shape when going tonanometer size range:♦First, both the surface energy and thesurface stress increase.♦ Second, different structures (e.g.,icosahedral structure) can become morestable.♦ Finally, the proportion of edges atomsbecomes no longer negligible. Even if thecrystal structure remains bulk-like, theequilibrium shape can change.
9. 9. This is the situation for naked nanoparticles What does it happen when they are supported ?
10. 10. Solid-solid interfaces
12. 12. Supported particles: Wullf-Kaichew construction the thermodynamic approachThe space around the particles no more isotropic !!with the hypothesis that there is no strain between particle and substrate. i.e.: the more is theaspect ratio: Eadh, the more theheight/lateral size particle is truncated s
13. 13. Deviations from Wullf-Kaichew previsions However, even for macroscopic supported crystals, several factors can modify the equilibrium shape: -the adsorption on foreign atoms or molecules -the presence of strain at the interface due to a misfit between the lattices of the support and of the deposited crystal. For non-zero misfit, the height-to-width aspect ratio can change. As an example if there is a compressive strain, the particle grows faster in height than laterally. The equilibrium shape then deviates from the Wulff–Kaischew case, giving larger aspect ratios (i.e., taller crystal).Qualitatively, one can understand this evolution because the crystal is strained at the interface (it can relax more easily at the top), and therefore prefers to decrease the interface area.
14. 14. Kinetics effectsIn practice, when we grow a crystal we are not at the equilibrium because thesupersaturation is larger than one.The supersaturation S is equal to the ratio of the (actual) pressure around the growingcrystal and the equilibrium pressure at the same temperature.If S is larger than one the crystal grows, and it evaporates if S is smaller than one.In general (especially at large supersaturations) the shape of thecrystal depends on the growth rate of the different facets. See Struttura e dinamica delle Superfici
15. 15. Conclusions: the morphology of nanocrystals depends on both kinetic (i.e., growth) and thermodynamic parameters. If the growth takes place far from equilibrium conditions (i.e., large supersaturation) the growth shape is not unique and depends on many parameters, such as: flux of growing material, structure of the support (if it is present), presence of defects (dislocations, twins), presence of impurities, confinement (i.e., template effect). If we grow particles close to the thermodynamic equilibrium(i.e., low growth rate, high temperature,but not too high to avoid Ostwald ripening) we can approach the equilibrium shape of the crystalline particles, which is unique for defined thermodynamic conditions. In the case of supported crystals the equilibrium shape is truncated in proportion to the adhesion energy (i.e., deposit/substrate interaction). Thus, choosing substrates with stronger adhesion energy will result in particles with smaller aspect ratios (height/lateral size).