Brownian motion of flexibly linked colloidal rings

Ring, or cyclic, polymers have unique properties compared to linear polymers, due to their topologically closed structure that has no beginning or end. Experimental measurements on the conformation and diffusion of molecular ring polymers simultaneously are challenging due to their inherently small size. Here, we study an experimental model system for cyclic polymers, that consists of rings of flexibly linked micron-sized colloids with $n=4–8$ segments. We characterize the conformations of these flexible colloidal rings and find that they are freely jointed up to steric restrictions. We measure their diffusive behavior and compare it to hydrodynamic simulations. Interestingly, flexible colloidal rings have a larger translational and rotational diffusion coefficient compared to colloidal chains. In contrast to chains, their internal deformation mode shows slower fluctuations for $n\lesssim 8$ and saturates for higher values of $n$. We show that constraints stemming from the ring structure cause this decrease in flexibility for small $n$ and infer the expected scaling of the flexibility as function of ring size. Our findings could have implications for the behavior of both synthetic and biological ring polymers, as well as for the dynamic modes of floppy colloidal materials.

Figure 1

Flexibly linked colloidal loops. (a) The flexible rings are built from colloid-supported lipid bilayers (CSLBs). CSLBs consist of spherical silica colloids coated with a fluid lipid bilayer. DNA linkers are inserted into the bilayer using a hydrophobic anchor. Because of the fluid lipid bilayer, the linkers can diffuse on the surface of the particles and therefore, the particles can move with respect to each other whilst staying bonded. (b) The DNA linkers are functionalized with sticky ends A that are complementary to sticky ends B, so that particles functionalized with A-type linkers can only form bonds with particles coated with B-type linkers. [(c)–(e)] Confocal images of a tetramer (c, $n=4$), hexamer (d, $n=6$), and octamer (e, $n=8$) loop. (f–h) Bright field snapshots of a flexible tetramer (f, $n=4$), hexamer (g, $n=6$), and octamer (h, $n=8$) ring, which show shape changes that are more pronounced for the larger rings. Scale bars in panels [(c)–(h)] are $2\mu \mathrm{m}$.

Figure 2

Conformations of flexibly linked colloidal loops. (a) The most compact configurations for all rings studied here, from $n=4–8$. The opening angles ${\theta }_i$ are indicated by the blue arcs. [(b)–(f)] The free energy of (b) tetramer ($n=4$), (c) pentamer ($n=5$), (d) hexamer ($n=6$), (e) heptamer ($n=7$), and (f) octamer ($n=8$) rings, for $\circ$ experimental, $\diamond$ simulated, and permutation data (black line). The shaded area indicates free energy differences lower than the thermal energy. Blue lines indicate internal opening angles taken into account. The free energy was determined using Boltzmann weighing of the joint probability density function of all opening angles ${\theta }_i$. (g) Mean total curvature of the loops as function of $n$ for the experimental and simulated data.

Figure 3

Radius of gyration of colloidal rings. (a) The radius of gyration in units of the bond length $b$ scales as function of $n$ as predicted by Eq. (9), which is also shown in the inset, along with the values of the fitted parameters. (b) The ratio between the radius of gyration of rings and chains is close to the value predicted by renormalization theory [4, 37], as shown for the experimental and simulated data that are available for both the rings and chains. Additionally, the solid line indicates the ratio of the scaling found in panel a for rings and the previously determined scaling of chains, which takes into account all available ring ($n=4$ to 8) and chain ($n=3$ to 6) lengths. (c) The free energy in terms of the radius of gyration divided by its average value for all ring sizes. The distribution is asymmetric and covers a wider range of possible values for the larger ring sizes. Note that the radius of gyration of tetramer rings is constant and therefore not included here. The schematics show a circle with its radius equal to the radius of gyration of the most compact (left) and most extended (right) hexamer ring.

Figure 4

Diffusion of flexible tetramer loops. (a) An illustration of the coordinate system used to analyze the diffusivity of the tetramer rings, as defined in Sec. IID. (b) The displacement of the center of mass (c.m.) of a 10 min experimental measurement of a tetramer loop. The color indicates the instantaneous value of one of the opening angles $\theta$. [(c)–(f)] We have compared the diffusion tensor elements calculated from $\diamond$ simulated and $\circ$ experimental data. (c) The translational diffusion along the $x$ and $y$ directions is comparable in magnitude and shows little shape dependence. (d) The in-plane translational diffusion coefficient $D_T$. We find that $\langle D_{\mathrm{expt}}/D_{\mathrm{sim}}\rangle =1.10\pm 0.02$. (e) The rotational diffusivity, for which $\langle D_{\mathrm{expt}}/D_{\mathrm{sim}}\rangle =0.92\pm 0.04$. (f) Compared to the simulated flexibility, the experimental flexibility is much lower, namely $\langle D_{\mathrm{expt}}/D_{\mathrm{sim}}\rangle =0.20\pm 0.04$. All off-diagonal diffusion tensor elements are close to zero.

Figure 5

Shape-averaged translational and rotational short-time diffusivity. (a) The position of the center of mass of three experimental 10 min measurements of a tetramer, hexamer and octamer ring. (b) Translational diffusivity of the rings as function of ring size $n$. The experimental diffusivity is greater than for the simulated data, both follow the same scaling described by Eq. (11). (c) The orientation $\alpha$ with respect to the initial orientation of three experimental rings as function of time. (d) The rotational diffusivity of rings as function of $n$ shows the same behavior for the experimental and simulated data, which can be described by Eq. (12).

Figure 6

Shape-averaged flexibility. (a) Probability density of the angular displacements $\mathrm{\Delta }\theta$ of all loops (simulated data) averaged over all opening angles $\theta$. Solid lines show a fit of a Gaussian. The inset shows the standard deviation ${\sigma }_n$ of $\mathrm{\Delta }\theta$ vs $n$. (b) Flexibility of the rings as function of $n$. Solid lines are fits of Eq. (15). Dotted lines show the expected flexibility of chains and loops for $n\to \infty$. (c) Probability density of the angular displacements $\mathrm{\Delta }\theta$ of the tetramer loops (simulated data), binned per opening angle $\theta$ (see legend). Dotted lines show the mean angular displacement, solid lines show a fit of a Gaussian.