Elastic lattice polymers

We study a model of “elastic” lattice polymer in which a fixed number of monomers $m$ is hosted by a self-avoiding walk with fluctuating length $l$. We show that the stored length density ${\rho }_m\equiv 1-⟨l⟩/m$ scales asymptotically for large $m$ as ${\rho }_m={\rho }_{\infty }\left(1-\theta /m+\dots \right)$, where $\theta$ is the polymer entropic exponent, so that $\theta$ can be determined from the analysis of ${\rho }_m$. We perform simulations for elastic lattice polymer loops with various sizes and knots, in which we measure ${\rho }_m$. The resulting estimates support the hypothesis that the exponent $\theta$ is determined only by the number of prime knots and not by their type. However, if knots are present, we observe strong corrections to scaling, which help to understand how an entropic competition between knots is affected by the finite length of the chain.

Figure 1

Example of an elastic lattice polymer on a square lattice. This polymer is composed by $m=11$ monomers describing a SAW backbone of length $l=8$ (dashed area).

Figure 2

Stored length calculated from exact enumeration data for SAWs (bullets) and SAPs (empty circles) on the cubic lattice $\left(d=3\right)$ and on the square lattice $\left(d=2\right)$. The solid lines are the leading terms in the ${\rho }_m$ vs $1/m$ expansion when using the expected value for the exponent $\theta$. The asymptotic values ${\rho }_{\infty }=1/\left(1+\mu \right)$ for cubic and square lattices are shown as a dashed line.

Figure 3

(Color online) Examples of knots on the fcc lattice: (a) a $3_1$ knot (trefoil) with $l=15$ steps and (b) a composite $3_1\#4_1$ knot with $l=31$ steps (the notation $k_1\#k_2$ indicates a closed polymer ring with two knots, one of type $k_1$ and one of type $k_2$). These configurations have been used as backbones with stored length $m-l$ for starting the simulations of ELP with $m\ge l$ monomers.

Figure 4

(Color online) Plot of the average equilibrium stored length density ${\rho }_m$ as a function of the inverse monomer number $1/m$ for closed polymers with some fixed topology. The simulations were extended to polymers of lengths up to $m=2000$. From top to bottom the data refers to: unknotted ring, $3_1$, $4_1$, $5_1$, and $5_2$ knots. The two bottom data set correspond to configuration with two knots: $3_1\#3_1$ and $3_1\#4_1$, respectively. The horizontal dashed line is ${\rho }_{\infty }=0.28498$, as expected from Eq. (20). Inset: zoom of the asymptotic region. Straight lines represent ${\rho }_{\infty }\left(1-\theta /m\right)$: the dotted line corresponds to the conjectured value $\theta =-3\nu$, the dashed line to $\theta =-3\nu +1$, and the dot-dashed line to $\theta =-3\nu +2$.

Figure 5

(Color online) Plot of $f_m{m}^{1+\Delta }$ vs ${\text{log}}_{10}m$, with $\Delta =1/2$. The data tend to a constant for large $m$, which, as discussed in the text, is consistent with a correction-to-scaling exponent of $\Delta \approx 0.5$.

Figure 6

(Color online) (a) Sketch of an entropic competition: a portion of the ring polymer, with two knots, is constrained to stay on the left half-space (holes are small enough to forbid more than one monomer at a time to pass), the remaining part has one knot and is on the other side of the wall. The total length $N$ of the chain is constant but the lengths $m$ and $N-m$ of the two subchains can fluctuate. (b) The virtual version (without the wall) of the same competition: the two polymers swim in separate dilute solutions and are coupled via a “wormhole” trough which they can exchange a monomer (hence not a knot) at a time.

Figure 7

(Color online) Examples of probability distributions of loop lengths for two polymer loops exchanging monomers as in Fig. 6b, in the case of negative (a) and positive (b) entropic exponent $\theta$ for both loops. In (a) the competition is between two $3_1$ knots, the thick line (boundary of the shaded area) is the distribution for the limit of long $L$ while the other ones are for two short $L$’s. In (b) the competition is between a $3_1\#3_1$ knot and a $3_1\#4_1$ knot, with the same notation.

Figure 8

(Color online) Plot of stored-length densities vs $m$ and $N-m$ in the entropic competition setup, for four different knotted chains: (a) $3_1\#3_1$ vs $3_1\#4_1$, (b) unknot vs unknot, (c) $3_1$ vs $3_1$, and (d) unknot vs $3_1\#3_1$. The intersection points of the densities are highlighted by arrows and correspond to local maxima (filled arrows) or minima (empty arrows) of the total entropy of the two competing loops. Horizontal dotted lines indicate ${\rho }_{\infty }$.