Solution of a model of SAW's with multiple monomers per site on the Husimi lattice

We solve a model of self-avoiding walks which allows for a site to be visited up to two times by the walk on the Husimi lattice. This model is inspired in the Domb-Joyce model and was proposed to describe the collapse transition of polymers with one-site interactions only. We consider the version in which immediate self-reversals of the walk are forbidden (RF model). The phase diagram we obtain for the grand-canonical version of the model is similar to the one found in the solution of the Bethe lattice, with two distinct polymerized phases, a tricritical point and a critical endpoint.Comment: 16 pages, including 6 figure

Grand canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice

We solve a model of polymers represented by self-avoiding walks on a lattice which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical endpoint and tricritical lines, as well as a multicritical point. From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical endpoints and a line of tricritical points, separated by the multicritical point, is always continuous. This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.Comment: 34 pages, including 10 figure

Drops with non-circular footprints

In this paper we study the morphology of drops formed on partially wetting substrates, whose footprint is not circular. This type of drops is a consequence of the breakup processes occurring in thin films when anisotropic contact line motions take place. The anisotropy is basically due to hysteresis effects of the contact angle since some parts of the contact line are wetting, while others are dewetting. Here, we obtain a peculiar drop shape from the rupture of a long liquid filament sitting on a solid substrate, and analyze its shape and contact angles by means of goniometric and refractive techniques. We also find a non--trivial steady state solution for the drop shape within the long wave approximation (lubrication theory), and compare most of its features with experimental data. This solution is presented both in Cartesian and polar coordinates, whose constants must be determined by a certain group of measured parameters. Besides, we obtain the dynamics of the drop generation from numerical simulations of the full Navier--Stokes equation, where we emulate the hysteretic effects with an appropriate spatial distribution of the static contact angle over the substrate