1. 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

2. For a crystal grown at equilibrium
See D2

3. 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
Wulff Plot
See D2

4. Theoretical Wulff shapes of TiO2
i ∈ {polyhedron facets}
1
Average 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/m2
Ramamoorthy, Vanderbilt, King-Smith, PRB 1994;
Lazzeri, Vittadini, Selloni, PRB 2001; Gong & Selloni, PRB 2007

5. Different shapes can originate from different extension of the equivalent faces

6. Equilibrium shape at T= 0 K:
the surface energy anisotropy is maximal
FCC: truncated octahedron BCC: rhombic dodecahedron

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
don't 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. 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.

9. This is the situation for naked nanoparticles
What does it happen when they are
supported ?

10. Solid-solid interfaces

11. Energia di adesione

12. Supported particles: Wullf-Kaichew construction
the thermodynamic approach
The space around the particles no more isotropic !!
with the hypothesis that there is no strain between particle and substrate.
i.e.: the more is the
aspect ratio:
Eadh, the more the
height/lateral size
particle is truncated
s

14. 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.

15. Kinetics effects
In practice, when we grow a crystal we are not at the 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 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 the
crystal depends on the growth rate of the different facets.
See Struttura e dinamica delle Superfici

17. 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).