In a large particle the energy is dominated by the bulk bonding. The energy of the external surface where the atoms have less bonding is less important. The overall shape is the one which minimizes the total
surface energy, the solution of which is the
Wulff construction. When the size is reduced a significant fraction of the atoms are at the surface, and hence the total surface energy starts to become comparable to the bulk bonding energy. Icosahedral arrangements, typically because of their smaller total surface energy, can be preferred for small nanoparticles. For
face centered cubic (fcc) materials such as
gold or
silver these structures can be considered as being built from twenty different single crystal units all with three twin facets arranged in
icosahedral symmetry, and mainly the low energy {111} external facets. An fcc single crystal has both {111} and {100} surface
facets, and perhaps {110} if the energy of the latter is low enough. In contrast icosahedral twins normally have {111} and perhaps {110}, none of the higher energy {100}. there can also be a reconstruction of some of the surface atoms to a hexagonal coordination, which is called an anti-MacKay icosahedron. These different shapes have been found in experiments where the relative surface energies are changed with surface
adsorbates. There are several software codes that can be used to calculate the shape as a function of the energy of different surface facets. Packing rules for various types of icosahedra with multiple components are also known. Made out of single crystal fcc units, these structure cannot fill space and there would be gaps as shown in the
figure, so there are some distortions of the atomic positions, equivalent to an elastic deformation to close these gaps. an approach later extended to three dimensions by Elisabeth Yoffe. This leads to a compression in the center of the particles, and an expansion at the surface. shape is lowest in energy. The size when the icosahedra become less energetically stable is typically 10-30
nanometers in diameter, but it does not always happen that the shape changes and the particles can grow to micron sizes. The most common approach to understand the formation of these particles, first used by Shozo Ino in 1969, which can be a significant contribution. The sum of these three terms is compared to the total surface energy of a single crystal (which has no strain), and to similar terms for a decahedral particle. Of the three the icosahedral particles have both the lowest total surface energy and the largest strain energy for a given volume. Hence the icosahedral particles are more stable at very small sizes, the decahedral at intermediate sizes then single crystals. At large sizes the strain energy can become very large, so it is energetically favorable to have
dislocations and/or a
grain boundary instead of a distributed strain. There is no general consensus on the exact sizes when there is a transition in which type of particle is lowest in energy, as these vary with material and also the environment such as the gas environment and temperature; the coupling surface stress term and also the surface energies of the facets can both change significantly. In addition, as first described by Michael Hoare and P Pal and
R. Stephen Berry and analyzed for these particles by
Pulickel Ajayan and
Laurence Marks as well as discussed by others such as
Amanda Barnard,
David J. Wales,
Kristen Fichthorn and Francesca Baletto and Riccardo Ferrando, at very small sizes there will be a statistical population of different structures so many different ones will exist at the same time. In many cases nanoparticles are believed to grow from a very small seed without changing shape, and hence what is found reflects the distribution of coexisting structures. where the two families of structures are separated by a relatively high energy barrier at the temperature where they are in
thermodynamic equilibrium. This arises for a cluster of 75 atoms with the
Lennard-Jones potential, where the global
potential energy minimum is decahedral, and structures based upon incomplete
Mackay icosahedra are also low in potential energy, but higher in entropy. The free energy barrier between these families is large compared to the available thermal energy at the temperature where they are in equilibrium. An example is shown in the
figure, with probability in the lower part and energy above with axes of an
order parameter Q_6 and temperature T. At low temperature the
75 atom decahedral cluster (Dh) is the global free energy minimum, but as the temperature increases the higher
entropy of the competing structures based on incomplete icosahedra (Ic) causes the finite system analogue of a
first-order phase transition; at even higher temperatures a liquid-like state is favored. == Ubiquity ==