Fluctuation-induced forces between rings threaded around a polymer chain under tension

We characterize the fluctuation properties of a polymer chain under external tension and the fluctuation-induced forces between two ring molecules threaded around the chain. The problem is relevant in the context of fluctuation-induced forces in soft-matter systems, features of liquid interfaces, and to describe the properties of polyrotaxanes and slide-ring materials. We perform molecular-dynamics simulations of the Kremer-Grest bead-spring model for the polymer and a simple ring-molecule model in the canonical ensemble. We study transverse fluctuations of the stretched chain as a function of chain stretching and in the presence of ring-shaped threaded molecules. The fluctuation spectra of the chains are analyzed in equilibrium at constant temperature, and the differences in the presence of two-ring molecules are compared. For the rings located at fixed distances, we find an attractive fluctuation-induced force between the rings, proportional to the temperature and decaying with the ring distance. We characterize this force as a function of ring distance, chain stretching, and ring radius, and we measure the differences between the free chain spectrum and the fluctuations of the chain constrained by the rings. We also compare the dependence and range of the force found in the simulations with theoretical models coming from different fields.

Figure 1

Snapshot of the system close to the rings, as given by the simulations. The rings are fixed at a distance $d$. The polymer chain fluctuates at thermal equilibrium in the canonical ensemble. A typical chain has $N=1024$ beads, and the chain ends are connected through periodic boundary conditions. The links show bond connections for rings and chain given by the FENE model (see Sec. 2).

Figure 2

Snapshots of a fragment of the polymer chain at different extensions and temperature $T=33.6\varepsilon /k_B$. $L^{*}$ accounts for the length as a fraction of the maximum possible chain extension in the FENE model, $L_{\text{max}}=1.5\sigma N$.

Figure 3

Mean bond length as a function of chain stretching for different temperatures in units of maximum FENE length. The dashed line represents bead distance in the limit of $k_{B}T=0$ and no excitations in the chain (athermal limit). The vertical dotted line indicates the extension at which the bond length increases its rate of change with chain stretching.

Figure 4

Mean distance among beads as a function of temperature for different chain extensions. A saturation at higher temperatures is observed in accordance with the fast increase of FENE bond energy.

Figure 5

Fourier spectrum of the polymer chain at different lengths for $T=33.6\varepsilon /k_B$. The inset shows the same data in log-log plot.

Figure 6

Exponent of the fitted decay of the fluctuation spectrum as a function of chain length for different temperatures. From the capillary-wave Hamiltonian, a coefficient $\alpha =2$ is expected (indicated with a dashed line). The exponent gets closer to $\alpha =2$ from $L^{*}=0.5$ toward higher stretching values.

Figure 7

Fourier spectrum for the chain without rings (open circles) and with the rings located at different distances (filled symbols). All the cases correspond to $T=33.6\varepsilon /k_B$ and $L^{*}=0.70$. The effect of the rings is only noticeable at small $q$ (long wavelengths) and it becomes unnoticeable at high $q$ values.

Figure 8

Difference of Fourier coefficient normalized sum for the case of the chain with rings ($C_r^2$) and the chains without rings ($C_{fc}^2$). The legends show the distance between rings at which the spectra were calculated. The vertical line indicates the $q$ value ($\lambda \sim 67\sigma )$ below which the presence of rings begins to influence the amplitude of the chain modes. The cutoff of modes at $q\lesssim 2\pi /d$ produced by the presence of the rings is clearly appreciated.

Figure 9

Color plot of the number density for the central zone of the chain. Panel (a) presents the chain without rings. The density with rings at distance $d=30\sigma$ is presented in panel (b), and the density with rings at $d=6\sigma$ is presented in panel (c).

Figure 10

Mean bond lengths $\langle \mathrm{\Delta }x\rangle$ as a function of bond number for different polymer lengths. The rings are located at $d=15\sigma$ with a chain stretching of $L^{*}=0.8$ and temperature $T=15\varepsilon /k_B$. The bond number is labeled such that the bond number $b_i$ is the bond distance between beads $i+1$ and $i$. Vertical dot-dashed lines indicate the approximate position of the rings. The error bars are almost the same size as the symbols.

Figure 11

Total mean force on rings $\langle f_{\text{ring}}^{\left(x\right)}\rangle$ in the chain direction as a function of ring distance. Each annular molecule is composed of two rings formed by 11 LJ particles located on a circle of radius $1.5\sigma$. The sample was set at $T=33.6\varepsilon /k_B$ and $L^{*}=0.70$. These are the fluctuation-induced or Casimir-like forces from the chain on the annular molecules, due to the restrictions that they impose in the natural fluctuations of the chain. For a given distance, the rings hinder the fluctuations of the chain with modes of wavelength $\lambda >d$.

Figure 12

Mean force between rings vs distance for different ring radius $r$. The parameters were set as $L^{*}=0.70$ and $T=33.6\varepsilon /k_B$. The inset shows a magnification of the case $r=6\sigma$, for which the minimal restrictions imposed by this big ring radius reduce significantly the fluctuation-induced force.

Figure 13

Upper panel: mean force vs ring distance for different chain stretchings and temperatures. The dashed line represents the harmonic model for the force $f\sim 1/d$. It indicates only the power-law decay. Lower panel: log-log plot for the same cases. The dashed line shows an idealized theoretical model that is consistent for medium to high stretching (see the text).

Figure 14

Fit of the fluctuation-induced forces vs distance for low stretching. They were adjusted with the model of Eq. (5) by Lehle et al. [9]. Panel (a) corresponds to the case $L^{*}=0.4$ and $T=5.05\varepsilon /k_B$, while panel (b) corresponds to $L^{*}=0.5$ and $T=15.0\varepsilon /k_B$.

Figure 15

Force between rings $F_x$ as a function of temperature for different chain lengths $L^{*}$. In all the cases, the rings were fixed at a distance $d=6\sigma .$ The lines come from a linear fit of the data point.

Figure 16

$F_x$ vs temperature for different ring distances. All the cases correspond to $L^{*}=0.56$, and the number of beads in the chain is $N=1024$. The lines are linear fits of the data for each case.

Figure 17

(a) Force vs distance for different numbers of beads in the chain at the same chain stretching $L^{*}=0.80$. (b) $F_x$ as a function of distance for $L^{*}=0.8$ at different temperatures. The insets in both panels show a detailed view of $F_x$ for short distances.