A molecule landing on a growing crystal surface has access only to its immediate neighborhood. It does not see the overall shape of the crystal. It does not know whether it will become a cube, an octahedron, or a needle. It settles where the local energy is minimal, nothing more.
And yet the crystal produces a coherent global form. The angles between faces are fixed, reproducible, measurable by goniometer to a fraction of a degree. Haüy's law (1784) establishes that these angles are consequences of the periodic structure of the lattice. The macroscopic form is inscribed in the unit-cell parameters, in the symmetries of the space group, in the intermolecular interactions at the angstrom scale. Everything is local. The result is not.
The problem is that of boundary conditions. In equilibrium crystallography, the final shape of a crystal is given by the Wulff construction (1901): the shape that minimizes the total surface energy for a given volume. This shape depends on the surface energies of each face, which depend on the crystallographic orientation, which depends on the lattice, which is fixed from the very first unit cell. The final shape is contained in the initial conditions.
But the initial conditions are not sufficient. The growth medium, the supersaturation, the temperature, the impurities, the convection flows modify the relative growth rates of the faces. A crystal that should be a cube according to Wulff becomes a parallelepiped, a dendrite, a spherulitic aggregate. The realized form depends on the path, not only on the endpoint.
Sunagawa (2005) distinguishes the equilibrium form (the one the crystal would have if it had the time) from the growth form (the one it actually has). Between the two lies the process. The crystal almost never reaches its equilibrium form. It is always on its way.
Doctrine
Each molecule decides locally. The global form emerges. It was in the unit-cell parameters from the beginning, and it will never be fully realized.
Vecteur ouvert
A crystal is a system whose final form is determined and inaccessible. The specification exists, inscribed in the physics of the lattice. The process does not reach it. What actually exists is always between the two.
