Brownian motion of a rod threading through a ring with fixed ring-center

TL;DR

This work analyzes the Brownian dynamics of a rigid rod threading a ring with a fixed center, revealing an entropically governed, $s$-dependent equilibrium distribution and a Smoluchowski equation that couples sliding and rotational diffusion. By applying the Onsager variational principle, the authors derive a dimensionless Smoluchowski equation that predicts a metastable sliding regime and a sliding-relaxation time that scales as $\tau_s \sim α^{-1/2}$ for small $α$, with rotational relaxation lying between the center-fixed and end-fixed limits. The findings show how mass distribution along the rod (encoded in $α$) creates an effective energy barrier and modulates both sliding and rotational dynamics, providing insight into rotaxane-like polymers and sliding ring materials. The framework connects microscopic topological constraints to macroscopic relaxation behavior, suggesting avenues to include frictional effects and multi-ring configurations in future work.

Abstract

We study the Brownian motion of a rigid rod threading through a small fixed ring while the ring can freely rotate. We derive the distribution function for the sliding displacement and the unit vector along the rod both at equilibrium and non-equilibrium. The equilibrium distribution is quadratic in the sliding displacement and is controlled by the moment of inertia (mass distribution). Applying the Onsager variational principle, we derive a Smoluchowski equation in which sliding and rotational diffusion are coupled. The mean square displacement (MSD) of sliding shows a metastable plateau in a certain time range before it approaches the final equilibrium value. The longest sliding relaxation time decreases as $α^{-1/2}$ as the moment of inertia increases. The rotational relaxation time obtained from the orientational correlation function is longer than that of a rod with its center fixed but faster than a rod with one end fixed. These results may be useful in understanding the dynamics of polymers connected by sliding rings.

Figures (5)

• Figure 1: Illustration of a sliding rod of length $L$, where $\bm{R}_\mathrm{c}$ denotes the rod-center (also the center-of-mass of the rod). The position $\bm{R}_\mathrm{c}$ is represented as $\bm{R}_\mathrm{c}=s \bm{n}$, where $s$ is the distance (which can be positive or negative) between the rod-center and the ring, and $\bm{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ is the unit vector along the axis.
• Figure 2: Time evolution of the probability distribution of sliding distance $\tilde{\psi} (\tilde{s},\tilde{t})$ under different mass distributions ($\beta_\perp = 12$ and $\beta_\parallel = 6$ have been used): (a) $\alpha=1/4$ (mass concentrated at both ends), (b) $\alpha=1/12$ (uniform distribution along the rod), and (c) $\alpha=1/60$ (mass concentrated at the rod-center). The initial condition $\tilde{\psi}_\mathrm{in}(\tilde{s})$ at $\tilde{t}=0$ is a Gaussian distribution with $\mu=0$ and $\sigma^2=0.0001$ (see inset in panel (a)). Red solid lines show the distributions at the longest relaxation time $\lambda_{01}^{-1}$, and black dashed lines represent the equilibrium distributions $\tilde{\psi}_{\mathrm{eq}}(\tilde{s})$.
• Figure 3: (a) Mean square displacement (MSD) of the sliding distance for different mass distribution parameters $\alpha$. For comparison, the MSD for a fixed rod orientation is also shown, denoted by "Fixed orientation". (b) Longest sliding relaxation time $\tau_\mathrm{s}$ (scaled by $D_0$) versus $\alpha$, exhibiting a $-1/2$ scaling at small $\alpha$. The relaxation time for the fixed-orientation case is shown for comparison and is independent of $\alpha$. $\beta_\perp = 12$ and $\beta_\parallel = 6$ have been used.
• Figure 4: (a) Orientational correlation function $\langle\bm{n}(\tilde{t})\cdot\bm{n}(0)\rangle$ for a sliding rod with different mass distribution parameters $\alpha$. For comparison, results for a non-sliding rod with its center fixed at the origin ("Fixed rod-center") and with one end fixed ("Fixed rod-end") are shown. (b) Longest rotational relaxation time $\tau_\mathrm{r}$ (scaled by $D_0$) of the sliding rod as a function of $\alpha$. The corresponding times for the fixed-center and fixed-end cases are independent of $\alpha$ and are included for comparison. $\beta_\perp = 12$ and $\beta_\parallel = 6$ have been used.
• Figure 5: Mean square displacement (MSD) of the rod end for different mass distributions $\alpha$, where the rod-end is defined as $\tilde{\bm{R}}_\mathrm{e}(\tilde{t}) = [\tilde{s}(\tilde{t}) + 1/2]\bm{n}(\tilde{t})$. $\beta_\perp = 12$ and $\beta_\parallel = 6$ have been used.