Stretching of a Polymer below the $\theta$ Point

The unfolding of a polymer below the $\theta$ point when pulled by an external force is studied both in $d=2$ on the lattice and in $d=3$ off the lattice. At $T=0$ and for finite length chains, it is found that the globule unfolds via multiple steps, corresponding to transitions between different minima, in both cases. In $d=3$ one of these intermediates is a regular helix. In the infinite length limit, these steps have a qualitative effect only in $d=2$. The phase diagram in $d=2$ is determined via the transfer matrix. To rationalize these results, energy-entropy and renormalization group arguments are given.

Figure 1

(a) $d=2$: Schematic diagram of minima for polymers of different sizes (top) and low $T$ $⟨x⟩$ vs $f$ curve for $N=25$ (bottom) found with exact enumerations in $d=2$. (b) $d=3$: Same as in (a), except that the $⟨x⟩$ vs $f$ curve (bottom) is for $N=30$ and is found by simulated annealing.

Figure 2

Phase diagram for the stretching of a SAW on a $2d$ lattice, obtained with the TM technique. Results and error bars for even and odd data are shown separately. The solid line corresponds to globule-coil coexistence. Inset: plot of $\log \lambda$ vs $\log z$ for $T=0.7$.

Figure 3

Plot of $⟨x⟩$ as a function of $f$ with the data in Ref. 5. We considered only the free energy minimum. Longer chains display one single dominant step.