Elastic interactions and complex patterns in binary systems

One of the many beauties of condensed matter physics is that it can reveal and illuminate how two systems or phenomena that at first appear to be quite different actually involve similar physics. This is an example of universality: for emergent phenomena, many details don't really matter. One example is the similarities between superconductivity and superfluidity. A consequence of universality is that the same concepts, techniques, toy models, and effective theories can be used to describe a wide range of systems.

The complex organometallic molecules, known by the misnomer "spin crossover" compounds, exhibit a rich range of phase transitions and types of spatial order. Key aspects of the physics are the following.

• Each transition metal ion can be in one of two possible states: low-spin or high-spin. 
• The size of each molecular complex depends on the spin state.
• Consequently, the molecules interact with their neighbours via elastic interactions.

A toy model that can describe this is expanding balls connected by springs. Various versions of this type of model are reviewed here. The simplest version is the chain model below.

It turns out there are other classes of systems described by similar models. As far as I am aware, this was first pointed out in Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids Layne B. Frechette, Christoph Dellago, Phillip L. Geissler

That paper is motivated by experiments on the growth of semiconductor quantum dots, by ion exchange, such as when CdSe is bathed in an Ag-rich solution and Ag2Se is produced with heterostructures (i.e., patterns of Ag and Se ions) that are different from the bulk crystal.

They consider the balls and springs model above on a triangular lattice.

They also point out how similar physics is relevant to binary metal alloys, e.g, AgCu, citing 

Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory, P. Fratzl and O. Penrose

Those authors consider a square lattice with elastic interactions associated with bond stretching along the edges and diagonals of the squares and bending of the square angles.

Frechette et al. also mention experiments on thin films of  DNA modified metallic nanoparticles. Compared to atomic systems these can tolerate larger lattice-mismatch before the formation of defects due to lattice strain.

Other systems (not mentioned) described by similar Ising models are metal-hydrogen systems, where the Ising pseudospin signifies whether a hydrogen atom is present at a particular site in the metallic crystal.

Frechette et al. start with the ball and springs model and "integrate out" the springs to obtain an effective Hamiltonian, which is an Ising model.

The spatial range of the interaction between Ising spins is shown in the colour-shaded plot below.

The interaction has two components.

One is an infinite range "ferromagnetic" part, seen as the light blue below.

The second is a short-range interaction which is mostly "antiferromagnetic" (i.e., red), but extends over several lattice sites. (Note, this interaction will be frustrated on the triangular lattice).

Using this toy model, Frechette et al. can obtain complex patterns (heterostructures) similar to those seen in quantum dots grown by ion exchange.

There is some subtle (and confusing) physics associated with deriving the Ising model from the ball and springs model. 

Due to the long-range nature of elastic interactions, the boundary conditions matter. 

The infinite range part of the Ising interaction arises from dealing with the lattice constant for the crystal, depending on the net "magnetisation" of the "spins". But that is a story for another day.