  Oriented self-avoiding walks (OSAWs) on a square lattice are studied, with
binding energies between steps that are oriented parallel across a face of the
lattice. By means of exact enumeration and Monte Carlo simulation, we
reconstruct the shape of the partition function and show that this system
features a first-order phase transition from a free phase to a tight-spiral
phase at $\beta_c=\log(\mu)$, where $\mu =2.638$ is the growth constant for
SAWs. With Monte Carlo simulations we show that parallel contacts happen
predominantly between a step close to the end of the OSAW and another step
nearby; this appears to cause the expected number of parallel contacts to
saturate at large lengths of the OSAW.
