Phase separation of multicomponent liquid mixtures plays an integral part in many processes ranging from industry to cellular biology. In many cases the morphology of coexisting phases is crucially linked to the function of the separated mixture, yet
it is unclear what determines morphology when multiple phases are present. We developed a graph theory approach to predict the topology of coexisting phases from a given set of surface energies (forward problem), enumerate all topologically distinct morphologies, and reverse engineer conditions for surface energies that produce the target morphology (inverse problem).
Thermally fluctuating sheets and ribbons provide an intriguing forum in which to investigate strong violations of Hookes Law: large distance elastic parameters are in fact not constant, but instead depend on the macroscopic dimensions. Inspired by re
cent experiments on free-standing graphene cantilevers, we combine the statistical mechanics of thin elastic plates and large-scale numerical simulations to investigate the thermal renormalization of the bending rigidity of graphene ribbons clamped at one end. For ribbons of dimensions $Wtimes L$ (with $Lgeq W$), the macroscopic bending rigidity $kappa_R$ determined from cantilever deformations is independent of the width when $W<ell_textrm{th}$, where $ell_textrm{th}$ is a thermal length scale, as expected. When $W>ell_textrm{th}$, however, this thermally renormalized bending rigidity begins to systematically increase, in agreement with the scaling theory, although in our simulations we were not quite able to reach the system sizes necessary to determine the fully developed power law dependence on $W$. When the ribbon length $L > ell_p$, where $ell_p$ is the $W$-dependent thermally renormalized ribbon persistence length, we observe a scaling collapse and the beginnings of large scale random walk behavior.
