Mobile Doubly Grafted Polymers and their Interaction

Doubly grafted polymers can, for example, be found in viral membrane proteins.
The interaction between doubly grafted polymers, essentially ring polymers,
on a two-dimensional substrate differs from the singly grafted polymer due to the entropic repulsion.
What does this entail as the grafting density increases and what
are the consequences for the viral membrane proteins? We study various quantities of
interest such as the order parameter, density profiles etc. varying the maximal distance
between the anchor points in the range [d_{min},d_{max}] (otherwise the anchor points
are mobile), the chain length and the grafting density. We show that there is no phase
transition from the disordered (isotropic) phase to an ordered (nematic) phase. However,
due to the interaction of loops, i.e. the entropic repulsion,
an increasing tendency for locally orientational order without long range order is observed.
As the grafting density is increased,
the transition from the mushroom regime to an almost upright conformations is observed.