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# Triangular cyclic rotaxanes: Size, fluctuations, and switching properties

Edited by J. Fraser Stoddart, Northwestern University, Evanston, IL, and approved December 22, 2017 (received for review September 7, 2017)

## Significance

Among the most important advances in chemistry in the last 60 y is the synthesis of molecules featuring a mechanical bond, i.e., the mechanical interlocking of components. As demonstrated in linear rotaxane switches, the relative positions and orientations among interlocked components can be altered to produce molecular-scale changes in size and shape. Here, we describe the conformations of a simple molecule made cyclic with mechanical bonding. Although the molecule, a cyclic [3]rotaxane, forms a cycle that is always a triangle and hence planar, we show that with only three mechanical degrees of freedom, characteristics such as interior cycle area, perimeter, and inscribed radius fluctuate significantly. Our model, based on statistical mechanics, suggests that the cyclic rotaxane can be used as the host in an inclusion compound, with the potential to switch sizes of the interior cycle or “pore.”

## Abstract

We examine one of the simplest cyclic rotaxanes—a molecule made from three rods with variable length between 0 and L. This [3]rotaxane, unlike a traditional molecule, shows significant size and shape fluctuations. We quantify these using a number of different measures. In particular, we show that the average angles are

Elucidation of molecular structure is a classic problem in molecular science. Many molecules have fixed shape. Some have partial freedom and convert between different structures through rotation about single bonds: The textbook example is the chair and boat conformers of the fully covalent cyclohexane. The conformers of larger molecules with many covalent bonds are central to biochemistry and materials science. These include biopolymers that form stable, but labile, folded conformers that mediate biological processes and the conformations of solvated synthetic polymers that follow the statistics of a random walker, where the fluctuations are large. The concerted actions of the many covalent degrees of freedom give rise to complex and fluctuating shapes and sizes of these fully covalent molecules.

However, a molecule featuring a mechanical bond (1) or the mechanical interlocking of components has additional degrees of freedom, namely the relative position and orientation of the interlocked components, that can have dramatic and controllable effects on the shape and size of the molecule. An example of such an interlocked molecule is a rotaxane (1), composed of a ring threaded with an axle that is stoppered at each end to prevent the ring from falling off. As the ring and axle are not covalently linked, the ring is free to translate along the length of the axle, providing an additional degree of freedom to the molecule. That is, the mechanical bond imparts an additional set of conformations or mechanical conformers (2).

Synthetic chemists have designed attractive “stations” on the axle, directing the ring to reside at one or another station. Switching between such mechanical conformers has allowed scientists to alter surface active properties of the molecule, to elongate/shorten the molecule for muscle actuation (3, 4), and to predict an isotropic–nematic phase transition of a lyotropic liquid crystal composed of a solution of these linear switches (5). Furthermore, scientists have hypothesized that many rings interlocked and freely translating on an axle provide a continuum of accessible mechanical conformers, providing a significant degree of conformational entropy, which can be unlocked to do work, as in a drug delivery system (6) or to dissipate energy in a molecular shock absorber (7).

Using mechanical bonds to design shape- and size-changing molecules has been limited mostly to the elongation of a single rotaxane molecule or a daisy-chain rotaxane (8, 9) of polymeric length scale. However, rotaxanes can be designed to reveal more complex shape and size changes. Here we describe an extension of a linearly extending daisy-chain rotaxane to a cyclic rotaxane constructed with the same covalent components and the shape and size fluctuations of this molecule. Our model focuses exclusively on mechanical bonding and does not consider the conformational degrees of freedom associated with covalent bonds: Consequently, atomistic representation is not required. We limit our cyclic rotaxane to just three components, i.e., a cyclic [3]rotaxane (10⇓⇓–13), whose mechanical conformers are all planar conformers. We show that with just three mechanically linked components, we can achieve significant fluctuations in the shape and size of the molecular cycle, for both mechanical bonds that are unbiased (freely mobile rings or an entropy-dominated molecule) and those biased with attractive stations, i.e., with added enthalpic control.

## The Model

We introduce a simple model to investigate the structure of a cyclic rotaxane. Each mechanical component of the rotaxane is an axle of unit length *,* there are many possible mechanical conformers. For simplicity, we assume that the rings are infinitely thin and wobble freely on the axle so that their positions are completely defined by a single value,

To develop the distributions of ring positions, and other distributions describing the shape and size of the cyclic rotaxane, it is instructive to first consider a single ring sliding along an isolated end-capped rod. The ring has position

To construct our distributions, it is useful to define a triangle function,

The probability density

## Geometric Measures of a Cyclic [3]Rotaxane

The most fundamental question for these cyclic [3]rotaxanes is “How long is a side of the triangle formed by the rings?” The probability density for the length of one side of the triangle is determined analytically from *SI Appendix: The Bond and Angle Distributions*), as shown in Fig. 5. The probability density of the longest side is simply

Other quantities of interest that we calculate exactly are the mean perimeter and mean of the interior area of the cycle. The probability density of the (interior) perimeter,

Of central interest to structural chemistry are the covalent bond lengths and bond angles of molecules. Here we look at the mechanical bond angle or angle between mechanically linked components. The angle distribution is determined using the law of cosines. We cannot obtain closed analytic expressions for the angle distributions, but they are readily calculated numerically and are plotted in Fig. 8. The average angles are

The cyclic [3]rotaxane molecule has one obvious feature—a fluctuating hole or cavity within which a “guest” molecule may be sequestered. Our model of a cyclic [3]rotaxane is purely geometric and does not include chemically specific interactions, but we can hypothesize that the size of a circle inscribed by the triangle is a simple measure of the potential of this cyclic rotaxane to act as a breathable inclusion complex. The radius of the inscribed circle,

## From Purely Geometric to Switchable Cycles

Thus far, we have studied a rotaxane system where there are no forces between the component parts, aside from those which keep the molecule from falling apart. As the rings are free to translate along its threaded axle, constrained only by the triangle inequalities, the cyclic [3]rotaxane exhibits large fluctuations. A major emphasis in the rotaxane literature is to incorporate attractive “stations” along the axle to which rings are attracted and which favor stable mechanical conformers with limited fluctuations. These conformers can be switched by turning the stations off or on using external controls such as light or solvent conditions. As a simple demonstration of switchable cyclic rotaxanes, we incorporate attractive stations on each axle, placing them far away from the stopper (Fig. 10). When these attractive stations are turned on, the rings preferentially reside at these stations, minimizing the length of a triangle side but still constrained by the triangle inequalities imposed by mechanical bonding.

When the attractive stations are switched on, the triangle becomes smaller, on average. The effect depends on the length of the stations,

The stations also affect the bond angles, as shown in Fig. 14, but rather weakly, as these are only slightly dependent on the bond lengths; i.e., if all of the bond lengths are scaled down by the same factor, then the angles are unchanged. The inclusion radius (Fig. 15) is, however, strongly affected, and switching the molecule would result in the expulsion of the guest from the inclusion complex.

## Conclusion

Here we have described the structure of a cyclic [3]rotaxane using methods of statistical mechanics and a geometric model which accounts for mechanical conformers. We explored the probability densities of side lengths, perimeter, area, and inscribed radii for a purely geometric (or entropic) mechanical conformer, as well as for mechanical conformers with a specifically configured set of attractive stations. Even with attractive stations, the mechanical bonds are responsible for significant fluctuations in the shape and size of the cyclic [3]rotaxane which are reminiscent of flexible polymers where mean and variance of molecular size are of the same order. However, unlike these synthetic polymers, the cyclic [3]rotaxane is always planar, with a clear triangular shape.

Finally, although cyclic rotaxanes have been synthesized, their properties have not yet been investigated thoroughly, and those that have been made have rather small side lengths. Furthermore, while synthesis of rotaxanes is focused upon the molecular scale, mesoscopic-scale rotaxanes, formed from colloidal constituents such as carbon nanotubes and colloidal scale rings, as well as self-assembled proteins, may hold promise.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: D.Williams{at}anu.edu.au.

Author contributions: P.R., E.M.S., and D.R.M.W. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1715790115/-/DCSupplemental.

Published under the PNAS license.

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