Shape control in nanocrystal growth is the control of the shape of nanocrystals (crystalline nanoparticles) formed in their synthesis by means of varying reaction conditions. This is a concept studied in nanosciences, which is a part of both chemistry and condensed matter physics. There are two processes involved in the growth of these nanocrystals. Firstly, volume Gibbs free energy of the system containing the nanocrystal in solution decreases as the nanocrystal size increases. Secondly, each crystal has a surface Gibbs free energy that can be minimized by adopting the shape that is energetically most favorable. Surface energies of crystal planes are related to their Miller indices, which is why these can help predict the equilibrium shape of a certain nanocrystal. Because of these two different processes, there are two competing regimes in which nanocrystal growth can take place: the kinetic regime, where the crystal growth is controlled by minimization of the volume free energy, and the thermodynamic regime, where growth is controlled by minimization of the surface free energy. High concentration, low temperatures and short aging times favor the kinetic regime, whereas low concentration, high temperatures and long aging times favor the thermodynamic regime. The different regimes lead to different shapes of the nanocrystals: the kinetic regime can give anisotropic shapes which are often connected to the kinetic Wulff construction, whereas the thermodynamic regime gives equilibrium, isotropic shapes, which can be determined using the Wulff construction. The shape of the nanocrystal determines many properties of the nanocrystal, such as the band gap and polarization of emitted light.
Miller indices and surface energy The surface energy of a solid is the free energy per unit area of its surface. It equals half the energy per unit area needed for cutting a larger piece of solid in two parts along the surface under examination. This costs energy because chemical bonds are broken. Typically, materials are considered to have one specific surface energy. However, in the case of crystals, the surface energy depends on the orientation of the surface with respect to the unit cell. Different facets of a crystal thus often have different surface energies. This can be understood from the fact that in non-crystalline materials, the building blocks that make up the material (e.g., atoms or molecules) are spread in a homogeneous manner. On average, the same number of bonds needs to be broken, so the same energy per unit area is needed, to create any surface. In crystals, surfaces exhibit a periodic arrangement of particles which is dependent on their orientation. Different numbers of bonds with different bond strengths are broken in the process of creating surfaces along different planes of the material, which causes the surface energies to be different. The type of plane is most easily described using the orientation of the surface with respect to a given unit cell that is characteristic of the material. The orientation of a plane with respect to the unit cell is most conveniently expressed in terms of Miller indices. For example, the set of Miller indices (110) describes the set of parallel planes (family of lattice planes) parallel to the z-axis and cutting the x- and the y-axis once, such that every unit cell is bisected by precisely one of those planes in the x- and y-direction.
Generally, a surface with high Miller indices has a high surface energy. Qualitatively, this follows from the fact that for higher Miller indices, on average more surface atoms are at positions at a corner instead of a terrace, as can be seen in the figure. After all, corner atoms have even fewer neighbours to interact with than terrace atoms. For example, in the case of a 2D square lattice, they have two instead of three neighbours. These additionally broken bonds all cost energy, which is why lower Miller indices planes generally have lower surface energies and are as a consequence more stable. However, the comparison is in fact somewhat more complex, as the surface energy as function of the Miller indices also depends on the structure of the crystal lattice (e.g., bcc or fcc) and bonds between non-next nearest neighbours play a role as well. Experimental research on noble metals (copper, gold and silver), shows that for these materials, the surface energy is well-approximated by taking only the nearest neighbours into account. The next-nearest neighbour interactions apparently do not play a major role in these metals. Also, breaking any of the nearest neighbour bonds turns out to cost the same amount of energy. Within this approximation, the surface energy of a certain Miller indices (hkl) surface is given by
γ h k l = N h k l 3 γ 111 {\displaystyle \gamma _{hkl}={\frac {N_{hkl}}{3}}\gamma _{111}}
with N h k l 3 {\displaystyle {\frac {N_{hkl}}{3}}} the ratio of the number of bonds broken when making this (hkl) plane with respect to making a (111) plane, and γ 111 {\displaystyle \gamma _{111}} the surface energy of the (111) plane. For any surface of an fcc crystal, N h k l {\displaystyle N_{hkl}} is given by
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