Mechanical Conformers of Keyring Catenanes - The Journal of

Subscriber access provided by Kaohsiung Medical University

A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Mechanical Conformers of Keyring Catenanes Todd C. Harris, Edith M. Sevick, and David R.M. Williams J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b08646 • Publication Date (Web): 24 Oct 2018 Downloaded from http://pubs.acs.org on October 31, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Mechanical Conformers of Keyring Catenanes† T. C. Harris,‡ E.M. Sevick,∗,‡ and D.R.M. Williams∗,¶ Research School of Chemistry, The Australian National University, Canberra, and Research School of Physical Sciences & Engineering, The Australian National University, Canberra E-mail: [email protected]; [email protected]

†

Conformers of Keyring Catenanes To whom correspondence should be addressed ‡ The Australian National University ¶ The Australian National University ∗

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Interlocked molecules exhibit structural isomerisation which is different from that of molecules whose connectedness is solely through covalent bonds. A mechanical bond, or the interlocking of components, provides a rich conformational landscape. The ability of synthetic chemists to design directional motion between these mechanical conformers suggests mechanical bonds as building blocks in the design of synthetic molecular motors and machines. Here we examine the complexity of mechanical conformers of radial catenanes with n anisotropically repulsive rings (coined ‘keyrings’) threaded onto a single central ring for n ≤ 10. For a given number of rings, n, the ratio of the key length to the main ring radius, λ, determines the mechanical conformer. We show that this system displays symmetrical in-plane conformers for short keys and co-conformers of lower symmetry where the keys lie out of the plane for longer keys.

Keywords Catenanes, Rotaxanes, Conformers

Introduction Among the most important advances in chemistry in the last sixty years is the synthesis of molecules featuring a mechanical bond. 1,2 An example is an [n]catenane, a molecule comprised of n covalently bonded cycles, mechanically linked or topologically interlocked into a single molecular structure. The notion of mechanically linked cycles predates Staudinger’s description of polymers as covalently bonded chains; 3 however the concept of interlocked molecules was far more difficult to realise in synthesis for many decades. In 1960, Wasserman 4 demonstrated the first synthesis of a [2]catenane relying upon the statistical threading of a cyclic precursor through a macrocycle. Such statistical syntheses, and subsequent directed syntheses, produced [2]catenanes in yields too low to excite promises of practical

2


Page 2 of 25

Page 3 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


applications. However in the 1980s, new synthetic methods based upon metal-templating 5 and donor-acceptor arrangements 6 produced catenanes in high yield, thus beginning an era of mechanically interlocked molecules (or MIMs). Today, synthetic chemists focus on MIMs as switches and potential molecular machines, as the mechanical bond (as opposed to the covalent bond) facilitates directable internal motion, over a length comparable to that of the molecule. Each mechanical bond provides at least one additional conformational degree of freedom for the molecule. 7 These mechanical degrees of freedom can be far more versatile than the degrees of freedom that arise from rotations about covalent bonds. A single mechanical degree of freedom can significantly change the size, shape, or architecture of a molecule; 8 and the variation in the potential energy with this mechanical degree of freedom can give rise to directionality of internal motion. 9,10 Researchers in mechanostereochemistry refer to this type of isomerisation through mechanical bonds as “co-conformation" and the mechanical isomers as “co-conformers" 11,12 to distinguish these structural isomers from conformers that arise from rotation about single bonds. These researcher have also created synthetic strategies to manipulate these mechanical degrees of freedom. For example, by incorporating attractive “stations” into an axle or main ring, chemists can entice the interlocked cycles or rings to occupy station positions using external stimulus say pH, solvency, or radiation in this way the MIM is designed to “switch" states. If the energy barriers separating mechanical conformers are asymmetric, then such switching can induce uni-directional internal motion, such as rotary motion of rings, 9,13 or threading of rings onto an axle. 10,14 This notion of chemically altering the conformational energy landscape of a single molecule to achieve directional internal motion is not limited to MIMs, but is also demonstrated by Feringa’s conformational molecular rotors. 15 The synthetic chemistry of molecular motors and rotors garnered the 2016 Nobel prize in Chemistry for Sauvage, Stoddart and Feringa. Here we demonstrate the range and complexity of mechanical conformational states of a more complex MIM - a type of radial catenane. A radial [n + 1]catenane is a molecule of one

3



Figure 1: Illustration of a [4]keyring (or radial [5]catenane), comprised of a main ring onto which 4 “keys” (or asymmetric minor rings) are threaded, with point charges at their ends. The structure resembles a keyring, hence the name. The coordinate system that describes the keyrings is shown, where θ angles describe the location of the keys and φ coordinates describe their orientation. Any motion that cannot be described by these coordinates (such as “wobble”) is not permitted by our model. Conformational degrees of freedom associated with covalent structure are also ignored as we are exclusively interested in the mechanical motion of these keyrings. The first key defines the point θ = 0 on the main ring given its rotational symmetry. The length of the key, r, relative to the radius of the main ring, R, is the parameter λ = r/R. It would be determined at synthesis, or it could be switched if the keys were comprised of [2] rotaxane switches. main ring onto which n minor rings are interlocked. 1 There exist a handful of reported syntheses of radial [n + 1]catenanes, starting with Sauvage’s classic synthesis of multicatenanes of n ≤ 6, 16 followed later by many other syntheses, amongst them “molecular necklaces” 17–20 Our focus is on the mechanical conformers of radial [n + 1]catenanes where the minor rings have some asymmetry, as for example, a pendant or a charged group. A simple way to model this asymmetry in the minor ring is invoke a force centre located a fixed distance from the minor ring centre; we represent this as a force centre at the end of a rod, directed outward from the major ring as represented in Fig 1. These asymmetric rings resemble “keys,” hence we refer to the MIM as an [n]keyring, where n is the number of keys on the major ring or keyring; i.e., there are a total of n + 1 interlocking components. Each key adds two mechanical degrees of freedom: translational motion of the key around the main ring described by a θ coordinate, and rotational motion lifting the key out of the plane of the main ring described

4


Page 4 of 25

Page 5 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


by a φ coordinate. We describe the [n]keyring with mechanical degrees of freedom only: we do not consider the degrees of freedom that arise from rotation about single bonds within the interlocked components. Consequently, we do not distinguish isomerisation that due to rotation about single bonds (or a “conformation") from that due to rotation/translation of interlocked components mechanical bonds (or a“co-conformation"), and we refer to the structural isomers associated with mechanical bonds with the generic “conformers". Using only mechanical coordinates, we construct a simple energy model and minimise it to determine the conformational behaviour of the keyrings. There are no stations that bias the location of the keys on the keyring; instead it is the interactions of the asymmetric keys (or force centres) which give rise to mechanical conformer states. We associate a conformer state of an [n]keyring as the 2n − 1 coordinates that minimise the conformational energy. This 2n − 1 dimensional energy landscape grows rapidly with n so that finding the minimum becomes difficult as the number of keys increases. Our predictions are reliable for a small numbers of rings, up to n = 6, and we can identify conformational trends in [n]keyrings up to n = 10. However, this description of conformer states is related to a class of problems first stated by JJ Thomson, 21 involving a number of repulsive particles confined to the surface of a sphere. In the limit of very long keys, or strongly asymmetric minor rings, the conformational state of the [n]keyring recovers this classic JJ Thomson result.

Results and Discussion The Model Fig 1 illustrates the coordinate system used to describe keyring conformation. There are n − 1 degrees of freedom associated with the θi coordinates. These describe the position of keys on the main ring relative to the first key, which is defined to be θ1 = 0. There are an additional n degrees of freedom associated with φi coordinates, which describe the rotation of the keys out of the plane of the main ring. We assume that the keys are free to move 5



Page 6 of 25

within the bounds of this coordinate system as permitted by the mechanical bonds between the main ring and the keys. However we also assume that the minor rings are mechanically tight, as illustrated in Fig 1, thereby preventing “wobble” or “twist” motion of keys relative to the main ring, or any non-azimuthal translation. In this paper we assume all the keys for a particular keyring have the same length, r, and the main ring has a uniform radius R. The last parameter needed to completely describe a keyring in our model is λ = r/R, the relative size of the keys to the main ring. The full description of the keyring in Fig 1 is therefore: λ = 1, {θ1 , θ2 , θ3 , θ4 } = {0, π/2, π, 3π/2}, {φ1 , φ2 , φ3 , φ4 } = {0, π/6, 0, 0}. As the keys are free to move without penalty, the energy of keyring conformers is determined solely by the pairwise interactions of the repulsive force centres at the end of the keys. The interaction between the ith and j th keys has an energy of Uij = αd−m ij , where α and the energy parameter m are positive constants, and dij is the distance between force centres i and j. 7 For example, m = 1 corresponds to a Coulombic potential between two similar force centres; a larger value of m decreases the range of this interaction. The conformational energy is then the sum of pairwise interactions amongst keys:

Un =

n−1 X n X

Uij = α

i=1 j=i+1

n−1 X n X

d−m ij .

(1)

i=1 j=i+1

The distance between the ith and j th keys, dij , is determined by the angles {θi , φi } and {θj , φj }: d2ij = (R + r cos φi )2 + (R + r cos φj )2 − 2(R + r cos φi )(R + r cos φj ) cos(θj − θi ) + r2 (sin φj − sin φi )2 . (2) We can simplify equation 2 and substitute in λ = r/R to give the following expression for

6


Page 7 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the dimensionless conformational energy of a keyring: n−1 X n h X Un un = √ = 1 − cos(θj − θi ) + λ (cos φi + cos φj )(1 − cos(θj − θi )) α( 2R)m i=1 j=i+1 i−m/2 + λ2 1 − cos φi cos φj cos(θj − θi ) − sin φi sin φj . (3)

√ For given values of n, m, and λ, we can minimise the dimensionless energy un = Un /α( 2R)m over the 2n−1 dimensional potential energy surface to determine the energetically favourable conformers of a range of keyrings. More complex models are possible, however to model wobble and twisting motion of the keys would require many new parameters and degrees of freedom. The advantage of our model is that it is dimensionless, and so our results are valid for keyrings of any size. As we consider only repulsive interactions, a dominant conformer is intuitively obvious: a mechanical conformation where the keys are distributed uniformly around the keyring (θi = 2(i − 1)π/n for 1 ≤ i ≤ n) with keys oriented within the plane of the keyring (φi = 0 for 1 ≤ i ≤ n). The symmetry of this flat conformer is characterised by the point group Dnh where n is the number of keys. For [n]keyrings with n = 2 and n = 3 keys, this flat conformer is the only minimal energy structure for all values of λ. All keyrings display this flat conformation at sufficiently small values of λ, i.e., where the keys are short in comparison to the radius of the main ring. However when the number of keys is four or higher, n ≥ 4, the keyring conformation cannot be so easily inferred and require minimisation of Eqn 3. The global minimisation of un for discrete values of λ is determined numerically using a Python script. We generate random initial coordinates {θ, φ}, and perform a bounded minimisation using the L-BFGS-B method - a method for finding local minima. The resulting energy and coordinates of the local minimum is temporarily stored and the process is repeated with different initial coordinates, resulting in a list of 1000 local minima. From this list, we identify the coordinates with the lowest energy (the global minimum) as the conformational state. This process is repeated for a range of λ values. These calculations are carried out for 7



Page 8 of 25

[n]keyrings with a moderate number of keys, 4 ≤ n ≤ 10 and with key sizes that are a fraction to the keyring size, to many times the keyring size, or 0.1 ≤ λ ≤ 5. We use energy parameter m = 2 (unless otherwise specified) due to its increased computational efficiency. As we later show, the value of m does not affect the qualitative co-conformational behaviour. As the number of keys, n, is increased, the dimensionality of the conformational space increases, and the 1000 local minima represent an increasingly sparse or dilute sampling of the potential energy surface. Thus, as we increase n, we become less confident that we have sampled the energy surface sufficiently to catch all local minima, of which one is the global minima. However the consistency and clear trends across ranges of λ values for different n indicates that the global minima has been successfully found in the cases of high number of rings (n = 10) that we explore.

Model parameters that arise from synthesis of radial [n]catenanes The parameters n and λ of Eqn 3 are also variables that naturally occur in existing synthetic design strategies of radial [n]catenanes (in this context n is the total number of interlocking rings, and λ is the ratio of the minor to major ring size). For example, the first syntheses of radial [n]catenanes achieved by Sauvage and colleagues produced a homologous series of radial [n]catenanes where 1 ≤ n ≤ 7 5 . The strategy, illustrated in Fig 2, involved the threading of a dimerisable, linear axle of length ∆ through a ring of radius ρ, aided by metal templation. Dimerisation of the n − 1 axles creates a central ring with n − 1 template threaded minor rings, as in Fig 2(A). This creates a series of [n]catenanes where n is coupled to λ ≈

ρ . (n−1)∆

If templation is not quantitative (that is, some axles do not have threaded

rings), then the central ring radius, R, can vary even though the number of minor rings is fixed at n. Thus, [n]keyrings self-assembled with keys of length ρ would lead to a series of keyrings of different λ = ρ/R values for a given n:

λ={

ρ ρ ρ , , , · · · }. (n − 1)∆ n∆ (n + 1)∆ 8


(4)

Page 9 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2: Illustration of Sauvage’s radial [n]catenane synthesis technique. Rings of radius ρ are threaded onto dimerisable linear axles of length ∆, aided by metal templation. As illustrated by (A), bringing together n − 1 of these axles creates radial [n]catenanes or, if the rings are keys of length ρ, [n]keyrings with λ ≈ ρ/(n − 1)∆. However, radial catenanes can be assembled as illustrated in (B), where some of the axles used in the synthesis do not have rings threaded onto them. The corresponding [n]keyrings would have a larger diameter keyring for the same number of keys, or a smaller λ A more recent synthesis of radial [n]catenanes similarly dimerises the ends of pseudo [2]rotaxanes, i.e., a ring-threaded linear axle which is capped with bulky end groups to discourage dethreading. The dimerisation of such pseudo [2]rotaxanes into cycles creates “molecular necklaces” as the linkage between pseudo-rotaxanes serves as a bulky knot beyond which the rings cannot easily pass, similar to the knots between pearls or beads on such a necklace. 17–20 Thus the minor rings localised on a molecular necklace do not translate freely on the major ring. This is in contrast to Sauvages’s metal-templated catenanes, where removal of the templating metal allows the minor rings to translate completely around the major ring, which is an important assumption of our model. 9



Finally, chemists have also successfully synthesized a variety of 2-state linear rotaxanes where the molecule can extend and contract its length in response to an external stimuli. These molecules switch between an elongated state and a contracted state, changing the length of the molecule by up to a factor of 6. Such 2-state rotaxanes could act as the rigid rod part of the keys on a keyring, thereby allowing λ to be controlled post-synthesis with the use of external stimuli. This would allow for the creation of keyrings that could switch between conformers with drastically different symmetries and sizes.

Conformers of a [n = 4]keyring and higher order even [n]keyrings As expected, minimisation of Eqn 3 confirms that n = 2 and n = 3 [n]keyrings exhibit the expected flat conformation at all values of λ. That is, the keys are evenly spaced around the ring and all point outward. The same conformation is observed for all keyrings at sufficiently small values of λ. This conformation has the highest symmetry of all the conformers seen in this paper, belonging to the D[n]h point group (where n is the number of keys). [4]keyrings are the lowest order keyrings to exhibit a non-trivial conformation. When λ is sufficiently large, the keys rotate out of the plane of the main ring to minimise the total pairwise repulsive interaction between key ends. Two keys turn above the plane and two turn below, all with the same absolute angle |Φ|, while their positions on the main ring remain unchanged as shown in Fig 3. This “even-split” conformation belongs to the D2d symmetry point group.

Figure 3: Minimum energy mechanical conformers of a [4]keyring. On the left is the λ = 1 case which exhibits the flat conformation with D4h symmetry. The keys are distributed uniformly around the main ring and oriented within the plane of the ring. On the right is the λ = 2 case showing an “even-split” conformation which has D2d symmetry. The keys are still distributed uniformly around the main ring, but they now tilt symmetrically above and below the plane of the ring by an angle ±Φ. The transition between the two conformers occurs at a critical λ∗4 which is determined analytically. 10


Page 10 of 25

Page 11 of 25

Fig 4 shows the variation of the key rotation angle |Φ| with the size ratio λ, as determined by energy minimisation of Eqn 3. At low values of λ, |Φ| = 0, indicative of the flat conformation. The transition to the even-split conformation is sharp but continuous, occurring at a critical value, λ∗4 , which depends on the energy parameter m. In contrast, m does not affect the qualitative behaviour of the keyrings, as they all go to the same limit when λ → ∞. In this limit, the main ring becomes infinitely small, essentially reducing the problem to four repulsive points connected at a central point. It is therefore unsurprising that in this limit, |Φ| → 0.615 radians, which corresponds to the four keys forming the vertices of a tetrahedron.

0.6

| | (radians)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.4

m = 0.5 m=1 m=2 m=3

0.2

0

0

1

2

3

4

5

Figure 4: Key rotation out of plane of main ring, |Φ|, versus relative key length, λ, for a [4]keyring, for different values of m in the generalised pairwise repulsive interaction U ∼ d−m . Larger m indicates a shorter range of the repulsive potential with m = 1 corresponding to the Coulombic repulsion between point charges. Points are the result of energy minimisation. The qualitative variation of |Φ| with λ does not depend upon m. At small key lengths, |Φ| = 0 which is indicative of the flat conformation; in the opposite limit of vanishing main ring radius, |Φ| → 0.615 radians regardless of m, indicative of a tetrahedron. The critical value, λ∗4 is the value of λ at which the flat conformer changes to the even-split conformer; this dependence upon m is also predicted by Eqn 7. Given the simplicity of the even-split conformation, the critical value λ∗4 can be cal11



Page 12 of 25

culated analytically. The substitution {θ1 , θ2 , θ3 , θ4 } = {0, π/2, π, 3π/2}, {φ1 , φ2 , φ3 , φ4 } = {Φ, −Φ, Φ, −Φ} can be applied to Eqn 3, which can then be simplified to u4 = 21−m/2 (1 + λ cos Φ)2

−m/2

−m/2 + 4 1 + 2λ cos Φ + λ2 2 − cos2 Φ .

(5)

The relationship between Φ and λ in this conformation can be determined by solving the minimal energy condition; however, to find λ∗4 we must solve for the point that both satisfies this condition as well as cos Φ = 1. The expression is relatively simple, du4 2(λ − 1) − 2−m/2 (λ + 1) = 2mλ = 0, 1+ m2 d cos Φ cos Φ=1 (1 + λ)2

(6)

and so too is the solution, m

λ∗4

21+ 2 + 1 . = 1+ m 2 2 −1

(7)

Figure 5: Minimum energy mechanical conformers of (Left) a [6]keyring at λ = 1, and (Right) an [8]keyring at λ = 0.75. Both conformers exhibit even-split conformation, with uniform distribution of rings around the main ring and key rotation out of the plain, alternating above and below the plane, with angle |Φ|. Eachconformer is characterised by the D n2 d symmetry group. As with 4-keyrings, the transition from the flat conformatoin to the even-split coconformation occurs at a critical value of λ∗n , as described by Eqns 8 and 9. The n = 6 and 8 [n]keyrings behave in qualitatively the same way: a flat conformation at low λ, then a sharp but continuous transition to even-split co-conformation, illustrated in Fig 5. These conformations belong to the point groups D3d and D4d respectively, or more generally any even-split conformation will belong to the D n2 d point group. In the limit λ → ∞, these conformers circumscribe an octahedron and square antiprism, respectively, again in line with expectation. Analytic expressions can be determined for λ∗6 and λ∗8 using 12


Page 13 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the method above. These are

λ∗6

and λ∗8

=

√

3+2 2−

2 m

2 2 +1

1 + 3m/2 1 + 2−(m+1) = 31+m/2 − 1

!−1 √ m/2 √ m 2− 2 + 2 2 + 2 2 2 +1 + 1 . √ m/2 √ m/2 m m 2− 2 + 2 2 +1 2 + 2 + 2 2 +1 + 1

(8)

m+7 2

(9)

Despite the simplicity of the conformers involved, the analytic expression for the [8]keyring transition is quite complex. Finding expressions for critical λ values for other keyrings would be a difficult endeavour, as the conformations become more complex. As such, from this point on we rely entirely on the numerical minimisation data to determine critical points.

Conformers of [5]keyrings and higher order odd [n]keyrings Due to their odd number of keys, the mechanical conformations of [5]keyrings are more complex than even keyrings, though it remains true that the trivial flat conformation is observed when the keys are small (low λ). Again there is a critical key length, λ∗5 , at which the keys rotate out of the plane of the main ring to minimise their energy; however for n = 5, not all the keys come out of the plane by the same angle, and in fact one key remains in the plane. This “wedge” conformation is illustrated in Fig 6. In this conformation, the keys are no longer distributed evenly around the main ring, though it still belongs to the C2 symmetry point group. If we label the in-plane key as i = 1 such that Θ1 = Φ1 = 0, then the first pair of nearest neighbour keys (i = 2 and i = 5) are located at ±Θ2 and the next nearest pair (i = 3 and i = 4) are located at ±Θ3 . The first pair protrude above and below the plane by some angle ±Φ2 and the second pair by ∓Φ3 . For all values of λ where this conformation is observed, Φ3 > Φ2 > Φ1 = 0, hence the name “wedge”. The minimal energy coordinates for the [5]keyring are plotted as a function of λ in Fig 7. In the limit λ → ∞, the keys are arranged as the vertices of a trigonal bipyramid: {Θ1 , Θ2 , Θ3 } = {0, π/2, π}, {Φ1 , Φ2 , Φ3 } = {0, 0, π/3}. 13



Page 14 of 25

Figure 6: Minimum energy mechanical conformer of a [5]keyring at λ = 1.5. This is the lowest order odd keyring that displays a non-trivial conformation. The transition from the flat conformation to the illustrated “wedge” conformation occurs at a critical key length, λ∗5 . The keys are distributed non-uniformly around the main ring, with angles ±Θ2 and ±Θ3 from the in-plane key, which is described by Θ1 = 0 and Φ1 = 0. These keys are alternately rotated above/below the plain of the main ring: keys nearest to the in-plane key are rotated ±Φ2 out of the plane of main ring. The keys on the far side are rotated ∓Φ3 out of the plane of the main ring. At all λ > λ∗5 , Φ3 > Φ2 > Φ1 = 0, hence the name “wedge”. The conformation belongs to the C2 symmetry point group. Higher order odd-keyrings, specifically n = 7 and n = 9, exhibit a similar co-conformation with C2 symmetry at larger λ. We can generally describe this family of “uneven-split” coconformations like so:

{Θi } = {0, Θ2 , Θ3 , · · · Θ(n+1)/2 , −Θ(n+1)/2 , −Θ(n−1)/2 , · · · − Θ2 }, {Φi } = {0, Φ2 , −Φ3 , · · · (−1)

n+1 2

Φ n+1 , (−1) 2

n−1 2

Φ n+1 , (−1) 2

n−3 2

Φ n−1 , · · · − Φ2 } 2

(10) (11)

But unlike the [5]keyring, these higher order uneven-split conformations do not exhibit ordered Φ angles. For example, [7]keyrings have Φ3 > Φ2 > Φ4 > Φ1 = 0. In the large λ limit, the uneven-split co-conformation persists for [7]keyrings and is circumscribed a pentagonal bipyramid. However, as shown in Fig 8, there is a region in between the flat conformation and the uneven-split co-conformation, though it is quite noisy. At the upper end of this region, [7]keyrings appear to have Cs symmetry. It may be the case that this Cs conformation is the minimum energy structure across the entire region, and that numerical imprecision has produced the noisy minimisation results, however that was not confirmed during this

14


Page 15 of 25

1.2

3

⇥2 ⇥3

1

2.5

2

2

0.8

1.5

0.6

1

0.4

0.5

0.2

0

0

1

2

3

4

5

(radians)

3

⇥ (radians)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

Figure 7: Key location (left, blue) and key rotation out of plane of main ring (right, red) versus relative key length, λ, for a [5]keyring. The conformer is fully described by {Θi } = {0, Θ2 , Θ3 , −Θ3 , −Θ2 }, and {Φi } = {0, Φ2 , −Φ3 , Φ3 , −Φ2 }. Smooth lines were added to connect data points and are not the result of actual minimisation. At small key lengths the [5]keyring is in the flat conformation. There is again a sharp but continuous transition to the wedge conformation, most easily measured by noting the rapid increase in the Φ angles, where the Θ angles change very slowly. The large λ limit is Θ2 → π/2, Θ3 → π, Φ2 → 0, and Φ3 → π/3. study. This is caused by the large number of variables that need to be minimised over, and hence the complexity of the energy landscape mentioned in the introduction. Fig 9 shows a similar Φ-λ plot for [9]keyrings. Again, for the smallest key lengths, λ < 0.2, the flat conformation is observed. As with [7]keyrings, there is a noisy region from 0.2 < λ < 0.55 where the conformation is not quite symmetrical, though some data points are very close to Cs symmetry. With larger key length, we observe the uneven-split coconformation with one in-plane key and the remaining 8 keys sharing 4 different rotational angles. However, unlike [7]keyrings, this conformation does not persist into the large λ limit.

15



1

1

0.5

0.5

0

0

0.5

0.5

1 (radians)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

0

1

1 0

1

2

1 0.4

3

0.6

0.8

1

0.905

(b) Noisy region.

(a) Full plot.

0.91

0.915

0.92

(c) Transition region.

Figure 8: Key tilt angles as a function of λ for a [7]keyring. Subfigure (a) shows the full plot, with the flat conformation at low λ and the uneven-split co-conformation at higher λ. There is a region in between these two limits where there is significant noise in the data, shown in subfigure (b). Toward the end of this region, the conformers exhibit Cs symmetry. There is then a sharp but continuous transition to the uneven-split co-conformation, shown in subfigure (c). There is a transition to a highly symmetric structure, illustrated in Fig 10a, and described by: {Θi } =

4π 5π 5π π π 2π 0, , , , π, π, , 3 3 3 3 3 3

,

{Φi } = {0, Φ2 , −Φ2 , 0, Φ2 , −Φ2 , 0, Φ2 , −Φ2 } .

(12) (13)

This conformation belongs to the D3h point group, and it persists in the limit of λ → ∞ where it is circumscribed by a tricapped trigonal prism. However, this conformation requires pairs of keys to overlap, which would be prevented by steric effects not accounted for in our model. Exactly how [9]keyrings would behave in this large λ is therefore subject to further study.

16


Page 17 of 25

1

0.5 (radians)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

0.5

1 0

0.5

1

1.5

2

Figure 9: Key rotation angles, Φ, versus relative ring length, λ, for a [9]keyring as determined by energy minimisation. The flat conformation is observed for λ < 0.2. As with [7]keyrings, there is a noisy region (0.2 < λ < 0.55) where the conformation lacks precise symmetry, though at times it is close to Cs symmetry. For 0.55 < λ < 1.2, the uneven-split conformation is observed with one in-plane key and the remaining 8 keys sharing 4 different rotational angles. For λ > 1.2, the minimal energy conformation has D3h symmetry, with 3 in-plane keys arranged triangularly, and the remaining 6 keys staggered with a single rotational angle (see Eqn 13).

Conformers of a [10]keyring So far we’ve described the low order even keyrings (n = 4, 6, 8) as having considerably simpler mechanical conformational states than the odd keyrings. This is because they exhibit only two simple conformations: flat and even-split. One might expect to see the same behaviour for [10]keyrings and above. However, our preliminary findings such that is not the case. Our minimisation found that there was a small noisy region around λ = 0.25 just as the keys came out of the plane, similar to what we saw with [7] and [9] keyrings. The [10]keyring settled into the even-split (D5h point group) conformation around λ = 0.3 where it persists 17



Page 18 of 25

over larger key asymmetry, λ. There was then another continuous transition in the region 2.8 < λ < 3.2 where conformations were asymmetric. At λ > 3.2, [10]keyrings were in a complex C2 conformation that could be circumscribed by a bicapped square antiprism in the large λ limit. This conformation is illustrated in Fig 10b. We should emphasise that this finding is tentative, for the reasons given earlier, for large n the number of variables is large, and the minimisation may not be as accurate as for smaller n. It is however included here for completeness.

(a) 9-keyring

(b) 10-keyring

Figure 10: Large λ limit conformations for [9] and [10]keyrings. Subfigure (a) shows the [9] keyring conformer at λ = 1.25 belonging to the highly symmetrical D3h point group. However, this conformer is physically prohibited as it requires keys to overlap. A small amount of key wobble would allow to a more physical keyring to approximate thisconformer. This conformer can be circumscribed by a tricapped trigonal prism. Subfigure (b) shows the [10]keyring conformer at λ = 4, which belongs to the C2 point group. It is clear that eight of the keys are pairing up and beginning to overlap, which again would not be possible were steric effects accounted for. In the limit λ → ∞, this conformer can be circumscribed by a bicapped square antiprism.

Conclusion In traditional chemistry, a molecule can convert into different structural isomers exclusively by rotations about single bonds and these conformers are distinct when there is an appre18


Page 19 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ciable, but surmountable energy barriers between minimal energy conformers. However, molecules with mechanical bonds have a rich, and more complex conformational landscape. Here we characterise the possible mechanical conformers of radial catenanes, or keyring molecules, using a simple physical model in the limit of negligible thermal fluctuations. This model is a topological one in that it includes only mechanical degrees of freedom and does not include the internal degrees of freedom associated with rotations about covalent bonds. A summary of the equilibrium conformations of [n]keyrings, 4 ≤ n ≤ 10 observed in this study is characterised against λ, the size ratio of key asymmetry to keyring, in Fig 11. There are some clear trends. First, as the number of keys is increased, the flat conformation becomes limited to small values of λ. That is, small keyrings , [2] and [3]keyrings, persist in flat conformations irrespective of the strength of the asymmetry of the minor rings, or length of the keys; however when the number of keys increase, this flat conformation becomes limited to keys of smaller length. Second, for even number of keys, namely [4], [6], and [8]keyrings, the flat conformation transitions to one where the keys alternate above and below the plane of the main ring, giving the keyring structural thickness. In contrast, for an odd number of keys, namely [5] and [7] keyrings, the alternating in-out of plane is frustrated, but still leads to a chiral “wedge-like" structural thickness with C2 point group symmetry. These equilibrium, minimal energy conformations are predicted from a model where asymmetry in the minor rings, modelled as repulsive interactions between force centres off the minor rings, dominates the conformational energy. In keyrings where the number of keys is limited, n

Mechanical Conformers of Keyring Catenanes - The Journal of

Recommend Documents