  Two closely related models of oriented self-avoiding walks (OSAWs) on a
square lattice are studied. We use the pruned-enriched Rosenbluth method to
determine numerically the phase diagram. Both models have three phases: a
tight-spiral phase in which the binding of parallel steps dominates, a
collapsed phase when the binding of anti-parallel steps dominates, and a free
(open coil) phase. We show that the system features a first-order phase
transition from the free phase to the tight-spiral phase, while both other
transitions are continuous. The location of the phases is determined
accurately. We also study turning numbers and gamma exponents in various
regions of the phase diagram.
