Equilibrium shapes of nanoparticles

Equilibrium Crystal Shapes:free and supported

nanoparticles

If 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

For a crystal grown at equilibrium

See D2

Wulff Plot

See D2

In 1901 Wullf introduced, withour proving, a theorem where he said: for an equilibrium crystal there is a point in the interior such that its perpendicular distance hi from the ith face is proportional to the surface energy γi

Theoretical Wulff shapes of TiO 2

anatase

<γγγγ>= 1.1 J/m2 <γγγγ> = 0.7 J/m2 <γγγγ> = 0.5 J/m2

Ramamoorthy, Vanderbilt, King-Smith, PRB 1994;

Lazzeri, Vittadini, Selloni, PRB 2001; Gong & Sel loni, PRB 2007

rutile brookite

{ }facetspolyhedron,1 ∈= ∑ iA

A ii

itot

γγAverage surface energy:

Different shapes can originate from different exten sion of the equivalent faces

Equilibrium shape at T= 0 K:the surface energy anisotropy is maximal

FCC: truncated octahedron BCC: rhombic dodecahedron

Roughening Transition

T

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 don't care so much which places are easier to sit because they have so much energy to spare. The facets appear at lower temperatures, as the crystal is cooled: the first temperature at which a facet occurs is called the roughening temperature.

Equilibrium shape at T ≠0 K:

Crystals grown at T>T rough

do not form facets

(e.g. most nanoparticles grown by solution methods at high T are spherical)

Equilibrium shape in the nanoworld

Several factors can change the equilibrium shape when going to nanometer size range:

♦First, both the surface energy and the surface stress increase.

♦ Second, different structures (e.g., icosahedral structure) can become more stable.

♦ Finally, the proportion of edges atoms becomes no longer negligible. Even if the crystal structure remains bulk-like, the equilibrium shape can change.

This is the situation for naked nanoparticles

What does it happen when they aresupported ?

Solid-solid interfaces

Energia di adesione

Supported particles: Wullf-Kaichew construction

the thermodynamic approach

with the hypothesis that there is no strain between parti cle and substrate.

i.e.: the more is the Eadh, the more the particle is truncated

s

aspect ratio:height/lateral size

The space around the particles no more isotropic !!

Deviations from Wullf-Kaichew previsions

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–Kais chew case, giving larger aspect ratios (i.e., taller crystal).

Qualitatively, one can understand this evolution becaus e the crystal is strainedat the interface (it can relax more easily at the top), and therefore prefers to

decrease the interface area.

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 m isfit

between the lattices of the support and of the depo sited crystal.

In practice, when we grow a crystal we are not at t he equilibrium because the supersaturation is larger than one. The supersaturation S is equal to the ratio of the (actual) pressure around the growing crystal and the equilibrium pressure at the same te mperature.

If S is larger than one the crystal grows, and it e vaporates if S is smaller than one.

Kinetics effects

In general (especially at large supersaturations) t he shape of the crystal depends on the growth rate of the different facets.

See Struttura e dinamica delle Superfici

Conclusions:

the morphology of nanocrystals depends on both kine tic (i.e.,growth) and thermodynamic parameters.

If the growth takes place far from equilibrium cond itions (i.e., large supersaturation)

the growth shape is not unique and depends onmany 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 eq uilibrium (i.e., low growth rate, high temperature,but not to o high to avoid Ostwald ripening)

we can approach the equilibrium shape of the crysta lline particles, which is unique for defined thermodynamic conditions.

In the case of supported crystals the equilibrium s hape is truncated in proportion to the adhesion energy (i.e., deposit/substrate int eraction). Thus, choosing

substrates with stronger adhesion energy will resul t in particles with smaller aspect ratios (height/lateral size).