Fractal domain morphologies: signatures and implications

Abstract

Multi-scale and fractal morphologies are ubiquitous in experiments. Accurate statements about sizes of domains and interfaces, roughness exponents, fractal dimensions, etc. can be obtained from the correlation function $$C(r)$$ and the structure factor $$S(k)$$ . Smooth morphologies are characterized by the Porod law. The signature of fractal domains and interfaces is a power-law decay of $$C(r)$$ and $$S(k)$$ with non-integer exponents. As typical experimental morphologies are smooth on some length scales and fractal on others, the structure factor is characterized by cross-overs from one form to another. We illustrate this with two examples: (i) ground state morphologies in dilute anti-ferromagnets; and (ii) droplet-in-droplet morphologies of double-phase-separating mixtures. The identification of fractal morphologies in these systems yields novel insights on the underlying micro-scale phenomena.