Ring polymer simulations with global radius of curvature

We simulate three-dimensional flexible off-lattice ring polymers of length $L$ up to $L=4000$ for various values of the global radius of curvature $R_{grc}=0.25$, 0.48, and 1.0 and $R_{grc}=2.0$. We utilize two different ensembles: one with a $\delta$-function constraint on the radius, and the other with a $\theta$-function. For both cases the global radius of curvature provides a valid regularization of polymers with thickness $D=2R_{grc}$. The Flory-type critical exponent ${\nu }_{\mathit{SAW}}$ of self-avoiding rings at $D=2$ is found to be ${\nu }_{\mathit{SAW}}=0.5869\left(5\right)$ from the radii of gyration chain length scaling, while other $D$ values produce consistent results. For our current implementation, the numerical effort of chain thickness calculations is bounded by a number $\mathcal{O}\left(L\ln L\right)$ per single update. We also study low-temperature configurations of spatially dense Lennard-Jones homopolymers on a ring and identify some conformational building blocks.

Figure 1

The solution set $\mathbf{z}$ of the inequality Eq. (2.11) for $R_0=1.4$. The positions of $\mathbf{x}$ and $\mathbf{y}$ are marked by solid circles and are associated with two nodes on a chain. Any third point $\mathbf{z}$, which neither falls into ${\Omega }_1$ nor ${\Omega }_2$ solves Eq. (2.11). A possible solution is indicated by the single cross in the figure. The areas ${\Omega }_1$ and ${\Omega }_2$ are bounded through solid and dashed circle sections, respectively, as drawn in the figure. Consequently there is a small lens-shaped area allowed along the chains backbone, provided $R_0$ is not too small. In 3D the excluded volume is generated upon rotating the figure around the 1-axis. The surface of the 3D excluded volume has two parts, an outer surface generated by the longer circle segments and an inner surface generated by the shorter ones. For discretized tubes at $a=1$ and $R_0<0.5$ the excluded volume vanishes.

Figure 2

Excluded volume for a node at $\mathbf{z}$ in front of an $L=9$ open-ended chain in a plane. The symbols ${\Omega }_1$ and ${\Omega }_2$ have the same meaning as in Fig. 1. The tube thickness is $D=2.8$ at $R_0=1.4$. There are now 9 $R_0$ circles whose rotations generate the excluded volume in ${\mathcal{R}}^3$, again similar to the construction in Fig. 1. The cross denotes the position of a possible solution $\mathbf{z}$.

Figure 3

Computer times ${\tau }_{1\mathit{MEGA}}$ in seconds for the calculation of ${10}^6$ global radii of curvature as a function of $L$ at $D=1$ in (a) and $D=2.8$ in (b). CELL list algorithm data are marked by triangles, while PROJ algorithm data are given by circles. The curves through triangles and circles describe a $L\ln L$ singularity. Panel (b) also contains computer times for the evaluation of ${10}^6$ Lennard-Jones energies (crosses), which as indicated by the dashed curve increase $\propto L^2$.

Figure 4

(Color online) Two ring configurations of tubes for $L=1000$ at $R_{grc}=1.0$ (on the left) and $R_{grc}=2.0$ (on the right). The authors have inspected these configurations by eye. The finding is that these rings are in the trivial knot class, even though the nonlocal updates are capable of achieving nontrivial knots and actually sum over all knot classes.

Figure 5

All $R_g≔\sqrt{R_g^2}$ data of Eq. (3.1) in units of $R_{grc}$ at $R_{grc}=0.25$, 0.48, and 1.0 and $R_{grc}=2.0$ with $D=2R_{grc}$ as a function of $L$. The scaling curves use ${\nu }_{\mathit{SAW}}=0.5874$ and are adjusted to match the data points at largest $L$. Around $D\approx 1\text{to}2$ the polymer swelling is $\mathrm{const}\times D$.

Figure 6

Effective exponent data of Eq. (3.4) at $R_{grc}=0.25$ (crosses), $R_{grc}=0.48$ (triangles), and $R_{grc}=2.0$ (circles) with $D=2R_{grc}$ as a function of $1∕\sqrt{L}$. The dashed horizontal lines are at ${\nu }_{\mathit{SAW}}=0.5874$. These data points either show large scaling deviations or even indicate a nonmonotonic trend in the final approach of ${\nu }_{\mathit{eff}}\left(L\right)$ towards $L\to \infty$. The inlay magnifies the large $L$ behavior of $R_{grc}=0.48$ and $R_{grc}=2.0$ data.

Figure 7

Effective exponent data of Eq. (3.4) at $R_{grc}=1.0$ or $D=2$ from $Z_{\theta }$ simulations as a function of $1∕\sqrt{L}$. The horizontal line is at ${\nu }_{\mathit{SAW}}=0.5874$. A three-parameter fit of corresponding ${\nu }_{\mathit{eff}}$ data to the form Eq. (3.5) yields the parameters in Eq. (3.7). The curve corresponds to the fit and the data point at zero corresponds to the extrapolated ${\nu }_{\mathit{SAW}}$ value. The plotted rectangle marks the outer limits of a higher resolution figure: Fig. 8.

Figure 8

Finer-scale representation of effective exponent data at $R_{grc}=1.0$ or $D=2$ from $Z_{\theta }$ simulations as a function of $1∕\sqrt{L}$.

Figure 9

Effective exponent data of Eq. (3.4) at $R_{grc}=1.0$ or $D=2$ from $Z_{\delta }$ simulations as a function of $1∕\sqrt{L}$. The horizontal line is at ${\nu }_{\mathit{SAW}}=0.5874$. A three-parameter fit of corresponding ${\nu }_{\mathit{eff}}$ data to the form Eq. (3.5) yields the parameters in Eq. (3.6). The curve corresponds to the fit and the data point at zero corresponds to the extrapolated ${\nu }_{\mathit{SAW}}$ value.

Figure 10

Low-temperature energy densities from $T=0.1$ simulations (circles) and near-ground-state energy densities at $T\approx 0$ (triangles) for the $R_{grc}=1.4$ Lennard-Jones ring homopolymer as a function of $L$. The data points at $T=0.1$ stem from Monte Carlo simulations of the partition function Eq. (2.21), where in long runs the minimal potential energy value was recorded. Near-ground-state energy densities at $T\approx 0$ were obtained from an extensive Monte Carlo search in phase space as explained in Sec. 2C. We cannot easily quote statistical error bars, as in both cases the observable arrives as the minimum value of a Monte Carlo time series. The scattering of $T\approx 0$ energy density values at large $L$ reflects the inability of the simulation to obtain the exact ground state.

Figure 11

Low-temperature conformations of a single Lennard-Jones ring homopolymer from $T=0.1$ simulations at $R_{grc}=1.4$ for $L=95$. The top left corner (at position 1) displays the complete ring, while as counted clockwise from the top left corner, positions 2, 3, 4, and 5 display substructures of the same ring.

Figure 12

Low-temperature conformations of a single Lennard-Jones ring homopolymer from $T=0.1$ simulations at $R_{grc}=1.4$ for $L=105$. The top left corner (at position 1) displays the complete ring, while as counted clockwise from the top left corner, positions 2–5 display substructures of the same ring.