Incollection,
Solutions of models of self-avoiding walks with multiple monomers per site on Bethe and Husimi lattices
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
In poor solutions, at sufficiently low temperatures, polymers undergo
a collapse transition, changing their configurations to more compact
ones. The temperature at which this transition occurs is called the
$\Theta$ temperature. This situation may be described by models where
the polymers are represented by self-avoiding walks on a lattice with
attractive interactions between monomers which are not consecutive
along a polymeric chain (ISAW's), thus in these models monomer-solvent
contacts are disfavored. The introduction of these interactions is a
major complication in such models. For example, the dimension of a
transfer matrix becomes much larger in the interactions are
considered. Recently, a model with one-site interactions only, but
with multiple monomers per site, inspired in the Domb-Joyce model, was
proposed to describe the collapse transitions in interacting polymers
J. Krawczyk et al., Phys. Rev. Lett. 96, 240603 (2006), and
the thermodynamic properties of this model on square and cubic lattices
were studied using simulations. Here we report studies of this model
on Bethe and Husimi lattices. The models are exactly solved on these
lattices in the grand-canonical ensemble, where the number of monomers
fluctuates and a site with $i$ monomers is associated to a statistical
weight $ømega_i$. The maximum number of monomers per site is $K$. The
solution of the model
on the Bethe lattice with at most $K=2$ monomers per site where
immediate reversals of the walk are allowed (RA model) displays a
phase diagram with a non-polymerized and a polymerized phase. The
transition between the two phases is of first order for a non-vanishing
statistical weight of doubly occupied sites ($ømega_2$). In the case
of the model where immediate reversals are forbidden (RF model) a
phase diagram with two distinct polymerized phases is found, one of
them with double occupied sites only. The transition between the
non-polymerized and the regular polymerized phases is continuous for
low values of the statistical weight $ømega_2$, but becomes of first
order as this variable is increased, thus the collapse transition is a
tricritical point in the phase diagram. The second order transition
line between the non-polymerized and the double occupied polymerized
phase ends at a critical endpoint, at which these two phases coexist
with the regular polymerized phase P. Serra and J. F.Stilck,
Phys. Rev. E 75, 011130 (2007). The phase diagram of this model
may be seen in the figure. In the solution of the $K=2$
RF model on the Husimi tree the doubly occupied polymerized phase is
no longer present and the tricritical point moves to a higher value of
$ømega_2$. Finally, we find no evidence of transitions between two
regular polymerized phases in the solution of the $K=3$ RF model on
the Bethe lattice in the parameter space $(ømega_1,ømega_2,ømega_3)$.
