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We have investigated a polymer growth process on the triangular lattice wherethe configurations produced are self-avoiding trails. We show that the scalingbehaviour of this process is similar to the analogous process on the squarelattice. However, while the square lattice process maps to the collapsetransition of the canonical interacting self-avoiding trail model (ISAT) onthat lattice, the process on the triangular lattice model does not map to thecanonical equilibrium model. On the other hand, we show that the collapsetransition of the canonical ISAT model on the triangular lattice behaves in away reminiscent of the $\theta$-point of the interacting self-avoiding walkmodel (ISAW), which is the standard model of polymer collapse. This implies anunusual lattice dependency of the ISAT collapse transition in two dimensions.  By studying an extended ISAT model, we demonstrate that the growth processmaps to a multi-critical point in a larger parameter space. In this extendedparameter space the collapse phase transition may be either $\theta$-point-like(second-order) or first-order, and these two are separated by a multi-criticalpoint. It is this multi-critical point to which the growth process maps.Furthermore, we provide evidence that in addition to the high-temperaturegas-like swollen polymer phase (coil) and the low-temperature liquid drop-likecollapse phase (globule) there is also a maximally dense crystal-like phase(crystal) at low temperatures dependent on the parameter values. Themulti-critical point is the meeting point of these three phases. Ourhypothesised phase diagram resolves the mystery of the seemingly differingbehaviours of the ISAW and ISAT models in two dimensions as well as thebehaviour of the trail growth process.

physical review. e, statistical, nonlinear and soft matter physics

Identification of a polymer growth process with an equilibrium  multi-critical collapse phase transition: the meeting point of swollen,  collapsed and crystalline polymers