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Statistical Ring Catenation under Thermodynamic Control: Should the Jacobson−Stockmayer Cyclization Theory Take into Account Catenane Formation? Stefano Di Stefano*,† and Gianfranco Ercolani*,‡ †

Dipartimento di Chimica, Sapienza Università di Roma and Istituto CNR di Metodologie Chimiche (IMC-CNR), Sezione Meccanismi di Reazione, c/o Dipartimento di Chimica, Sapienza Università di Roma, P.le A. Moro 5, 00185 Roma, Italy ‡ Dipartimento di Scienze e Tecnologie Chimiche, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy S Supporting Information *

ABSTRACT: An extension of the Jacobson−Stockmayer theory is presented to include the reversible formation of [2]catenanes in a ring−chain system under thermodynamic control. The extended theory is based on the molar catenation constant, measuring the ease of catenation of two ring oligomers, whose expression was obtained in a previous work. Two scenarios have been considered: that of “thick” (hydrocarbon-like) chains and that of “thin” (DNA-like) chains. In the case of “thick” chains, the formation of catenanes can be neglected, unless in the unlikely case of a very large value of the equilibrium constant for linear propagation (K ≈ 108 mol−1 L, or larger). For K tending to infinity, the system becomes a chain-free system where only ring−catenane equilibria occur. Under this condition, there is a critical concentration below which only rings are present at equilibrium and above which the ring fraction remains constant, and the excess monomer is converted only into catenanes. In the case of “thin” chains, the formation of catenanes cannot be neglected even for values of K as low as 102 mol−1 L, thus justifying the use of the extended theory.

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INTRODUCTION Catenanes are fascinating mechanically interlocked molecules whose study has led to significant advances in supramolecular chemistry, with applications ranging from biology to nanoscience.1−3 They are among the favorite building blocks for the construction of artificial molecular machines,3,4 whose design and synthesis was recognized by the Nobel Prize in Chemistry 2016 to Sauvage, Stoddart, and Feringa. The first catenane synthesis reported in 1960 by Wasserman resulted from statistical threading of a C34 ring by a C34 chain with the subsequent macrocyclization of the latter.5,6 However, the catenane was isolated in poor yield of about 1%, leading to the rapid dismissal of statistical methods in favor, initially, of covalent-bond-directed syntheses by Schill and Lüttringhaus,7,8 and, successively, of template-directed syntheses pioneered by Sauvage.9 Template-directed synthesis has been the basis for the preparation of a plethora of catenanes ever since.1−3 More recently, dynamic combinatorial chemistry10−14 has demonstrated to be a particularly powerful tool for the discovery and synthesis of new catenanes.15−17 Accordingly, catenane synthesis is increasingly performed using reversible chemistry, which ensures that catenane production occurs under thermodynamic control. Although the statistical method proved to be unsatisfactory for the synthesis of catenanes under kinetic control, it could be promising when very large polymeric rings are formed under © 2017 American Chemical Society

thermodynamic control. The reversible formation of large polymeric rings was studied theoretically by Jacobson and Stockmayer (JS) as early as 1950,18 but for many years their theory was largely confined among the specialists of polymer chemistry.19−21 In the 1990s, owing to the increasing interest in macrocyclic chemistry, their theory was re-evaluated and popularized to make it more understandable to chemists of other disciplines.22,23 However, the JS theory considers only ring−chain equilibria and neglects the formation of catenanes, knots, and other topological complex molecules that, in the experimental practice, could remain hidden within the mass of the supposed linear polymer. Jacobson tried to correct the JS theory for the presence of catenanes,24,25 but his model largely overestimates catenane formation.26 Recently, we have critically reviewed models and simulations of the catenation process and obtained a reliable expression for the molar catenation constant.26 With this expression in our hands, we are now in a position to carry out a meaningful correction of the JS cyclization theory. The results of such a theoretical study are reported here with the aim of giving an answer to the title question. Received: December 7, 2016 Revised: December 29, 2016 Published: January 3, 2017 649

DOI: 10.1021/acs.jpcb.6b12323 J. Phys. Chem. B 2017, 121, 649−656

Article

The Journal of Physical Chemistry B

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RESULTS AND DISCUSSION Let us consider a long Gaussian chain M1 endowed with two end functional groups capable of reacting with each other in a reversible addition reaction. The monomer, present in solution at the initial concentration [M1]0, is converted at equilibrium into a mixture of a virtually infinite number of oligomeric chains Mi, rings Ci, and [2]catenanes CCij with i ≥ j, so that the mass balance equation can be written as eq 1. ∞

∞

∞

the ring opening of a cyclic i-mer. According to the theory, elaborated by Kuhn,33 and further refined by JS,18 and Flory and co-workers,34−36 the factor B, equal to EM1, can be calculated by eq 9, where NA is the Avogadro constant, σ is a symmetry factor equal to either 1 for chains of the type A−B or 2 for chains of the type A−A, υ is the number of skeletal bonds of length l, measured in cm, per repeating unit, and C∞ is the characteristic ratio in the limit of high chain length, which measures the stiffness of a chain.36

i

∑ i[Mi] + ∑ i[Ci] + ∑ ∑ (i + j)[CCij]

[M1]0 =

i=1

i=1

B=

i=1 j=1

(1)

K

[Mi] =

(2)

xi K

(3)

K ij = α

22,27−32

The equilibrium effective molarity (EMi), also known as the molar cyclization constant, is a measure of the propensity of a given chain to undergo cyclization. It is defined as the equilibrium constant of the back-biting process in which a linear oligomer, say Mi+j, splits into a cyclic oligomer Ci and a linear oligomer Mj (eq 4).18 (4)

Taking into account eqs 3 and 4, the equilibrium concentration of any cyclic i-mer is easily obtained as eq 5.22,27 [Ci] = EMix i

(5)

The molar catenation constant, Kij, is defined as the equilibrium constant for the reversible interlocking of the pair of ring oligomers, Ci and Cj, to yield the corresponding [2]catenane, CCij (eq 6). Of course, such an equilibrium is only possible if rings can undergo reversible ring opening. K ij

Ci + Cj ⇌ CCij

(6)

Taking into account eqs 5 and 6, the equilibrium concentrations of [2]catenanes are given by eq 7. [CCij] = K ij EMiEMjx i + j

(7)

Linear polymers adopt random-coil conformations in solution and, provided that they are of sufficient length and flexibility, they obey Gaussian statistics. As a result, the JS cyclization theory gives the following expression for the EM of an i-meric ring18 EMi = Bi−5/2

2[1 − INT(j / i)] 1/2 (i + j1/2 )3 σB

(10)

In the case of rigid chains for which the Kuhn segment thickness is negligible with respect to its length (“thin” chains), that is, in the absence of excluded-volume effects, the parameter α has been evaluated as (i) 0.0305,26 on the basis of the theory of Frisch and Wasserman;6 (ii) 0.0332,26 on the basis of the theory of Wang and Schwartz;37 and (iii) 9.6 × 10−3,26 on the basis of the results of Monte Carlo simulations by Degushi et al.38 The three figures are similar, although the latter evaluation appears to be the most reliable26 when compared with the available experimental catenation constant for the intertwining of 186 DNA and λcI857 DNA rings.37 The DNA chain is indeed a good model for the absence of excluded-volume effects, as its aspect ratio, that is, the ratio of the segment length to the double helix geometrical diameter, is as large as 66.39 However, if the Kuhn segment thickness is not negligible with respect to its length (“thick” chains), as in the case of a flexible polymethylene chain, its co-volume causing a steric hindrance to catenation cannot be neglected and the evaluation of the parameter α is more problematic. The effect of excluded volume on the catenation process has been simulated by Degushi et al. by using a rod−bead model.38 The simulations support the functional form of eq 10, but the estimate of parameter α for the hydrocarbon chain obtained on the basis of such simulations is about 1 order of magnitude too large.26 At present, the best estimate of the parameter α is obtained empirically as 6.6 × 10−5 by fitting the yield of the Wasserman catenane synthesis.5,26 If all chains follow Gaussian statistics, taking into account eqs 3, 5, 7, 8, and 10, the mass balance equation, eq 1, takes the form of eq 11, where S1 and S2 are functions of x given by the series in eqs 12 and 13, respectively. It appears immediately that the ratio of catenanes to rings is independent of B, and thus, is independent of the size and stiffness of the monomer ring, but only depends on α/σ and x.

EM i

Mi + j HoooI Ci + Mj

(9)

On the basis of the results of theories and simulations, we have shown previously that the molar catenation constant for rings following Gaussian statistics can be conveniently expressed by eq 10, where INT is a function that rounds a number down to the nearest integer, so that 2[1−INT(j/i)] is equal to either 1 when i = j or 2 when i > j, and α is a parameter that takes into account the steric hindrance to catenation.26 Equation 10 was obtained by assuming that the ring catenation process is dictated by entropy only, and thus, it holds for interlocked rings large enough to be noninteracting.

If it is assumed, as usual, that the equilibrium constant for linear propagation, K, is independent of the length of the chains to which the end groups are attached (eq 2), then it can be easily demonstrated that the equilibrium concentration of any linear imer is given by eq 3, where x is the extent of reaction in the chain fraction.22,27 Mi + M j ⇌ Mi + j

⎞3/2 1000 ⎛ 3 ⎟ ⎜ NAσ ⎝ 2πC∞υl 2 ⎠

(8)

Equation 8 points out that for a series of i-meric rings, the equilibrium EM varies inversely with the 5/2 power of the oligomerization degree. Factor i−5/2 may be regarded as the product of i−3/2 and i−1. The former relates to the probability that a Gaussian chain of i repeating units has its ends coincident and the latter to the number of equivalent bonds available for

[M1]0 = 650

1 x Bα + BS1 + S2 2 K (1 − x) σ

(11)


Article

The Journal of Physical Chemistry B ∞

S1 =

(12)

i=1 ∞

S2 =

are present at equilibrium and above which the ring fraction remains constant, and the excess monomer is converted only into chains.18−23,27 It is interesting to investigate how the presence of catenanes can modify this picture. To this end, we have built distribution curves of the fractions of chains, rings, and catenanes as a function of the initial monomer concentration for the two scenarios of “thick” and “thin” chains. The easiest way to build such curves is to (i) fix the values of the parameters σ, B, α, and K, (ii) make a list of x values in the range 0 < x < 1 such as that shown in Table 1, (iii) for each x, evaluate the three terms on the right-hand side of eq 11, corresponding to the weighted concentrations of chains, rings, and catenanes, respectively, and the sum of the three terms corresponding to the initial monomer concentration, (iv) plot the weighted concentrations of chains, rings, and catenanes against the initial monomer concentration. Let us consider first the case of a “thick” (hydrocarbon-like) unsymmetrical chain A−B (σ = 1, α = 6.6 × 10−5) for which, for the sake of simplicity, B = 1 mol L−1 and investigate the effect of increasing values of K, that is, K = 102, 103, 105, and 108 mol−1 L (Figure 1). By inspecting Figure 1A−C, it can be seen that the catenane fraction is negligible up to K values as large as 105 mol−1 L, implying that up to such values, the classical JS theory is perfectly suited to treat such a system accurately. In particular, from Figure 1C, it can be seen that the phenomenon of the critical concentration (equal to 2.612 mol L−1, as indicated by the arrow on the abscissa axis) is clear-cut. Indeed, as soon as the ring fraction is approaching its plateau value (equal to 2.612 mol L−1, as indicated by the arrow on the right ordinate axis), the chain fraction begins to grow steeply. It is interesting to note, however, that for K values larger than 105 mol−1 L, as exemplified by Figure 1D, the catenane fraction begins to significantly increase at the expense of the chain fraction. Such an increase occurs only in correspondence to the critical concentration; below the critical concentration, neither the chains nor the catenanes compete effectively with the ring fraction. It is evident that in contrast to the ring−chain system, the chains can completely disappear at any initial monomer concentration, provided that K is large enough. Thus, we can extrapolate the picture by considering the case for K tending to infinity. This case, illustrated in Figure 2, actually corresponds to a system of ring−catenane equilibria without the presence of chains. In other words, the plots in Figure 2 were calculated by dropping the first term on the right-hand side of eq 11. In this case, x has no longer the meaning of the extent of reaction in the chain fraction because under this condition the chain fraction is absent. It can be considered as a variable measuring the progress of the reaction and defined, according to eq 5, as x = [C1]/B. From Figure 2, it can be seen that in the case of thick chains, the ring−catenane system behaves similarly to the ring−chain system in the limit of very large K values. In other words, there is a well-defined critical concentration, approaching 2.612B (in this case B = 1 mol L−1), below which only rings are present at equilibrium and above which the ring fraction remains constant, and the excess monomer is converted only into catenanes. This behavior seems counterintuitive because the formation of a catenane involves a negative entropy change and cannot be more favored than the formation of the two component rings.26 However, one should not forget that in a system under thermodynamic control, the number of different catenanes is much larger than the number of different rings, for example,

∑ i−3/2x i i

∑ ∑ 2[1 − INT(j /i)](i + j)(i1/2 + j1/2 )3 (ij)−5/2 x i+ j i=1 j=1

(13) 40

The series S1 has been studied by Truesdell. It is rapidly convergent for x values near 0 but slowly convergent for x values approaching 1. When x = 1, S1 converges to 2.612.40 The series S2 is also slowly convergent for x values near 1 but, in contrast to the series S1, is divergent for x = 1. The evaluation of S1 and S2 is discussed in the Supporting Information (SI). Values of S1 and S2 for selected values of x are reported in Table 1. A more extended table is reported in the SI. Table 1. Numerical Values of S1−S3 Series x

S1

S2

S3

0.0500000 0.1000000 0.2000000 0.3000000 0.4000000 0.5000000 0.6000000 0.7000000 0.8000000 0.9000000 0.9500000 0.9700000 0.9900000 0.9950000 0.9990000 0.9995000 0.9999000 0.9999500 0.9999900 0.9999950 0.9999990 0.9999995 0.9999999 1.0000000

0.0509 0.1037 0.2159 0.3383 0.4734 0.6248 0.7982 1.0031 1.2586 1.6144 1.8842 2.0381 2.2717 2.3687 2.5017 2.5338 2.5771 2.5874 2.6012 2.6045 2.6088 2.6099 2.6113 2.6124

4.1953 × 10−2 1.7639 × 10−1 7.8536 × 10−1 1.9908 4.0495 7.3910 1.2802 × 101 2.1928 × 101 3.9053 × 101 8.1605 × 101 1.4713 × 102 2.1827 × 102 4.9919 × 102 8.5765 × 102 3.3495 × 103 6.2780 × 103 2.8746 × 104 5.6315 × 104 2.740 × 105 5.447 × 105 2.701 × 106 5.392 × 106 2.797 × 107 ∞

2.0643 × 10−2 8.5333 × 10−2 3.6602 × 10−1 8.8881 × 10−1 1.7190 2.9522 4.7378 7.3318 1.1244 × 101 1.7835 × 101 2.3674 × 101 2.7373 × 101 3.3788 × 101 3.7015 × 101 4.3070 × 101 4.5316 × 101 5.0125 × 101 5.2096 × 101 5.6559 × 101 5.845 × 101 6.282 × 101 6.469 × 101 6.905 × 101 ∞

In the classical JS theory only ring−chain equilibria are considered. As a result, the third term on the right-hand side of eq 11 is dropped, and such a reduced mass balance equation together with eqs 3, 5, 8, and 9 represents the core of the JS cyclization theory. The behavior of ring−chain equilibria has been discussed at length in the literature.18−23,27 It is well known that on increasing the initial monomer concentration, x tends to 1, and the concentration of each ring tends to a plateau value corresponding to its EM (eq 5). The overall cyclic fraction, represented by the second term in eq 11, converges to 2.612B.18,40 This result translates into the physical fact that only a finite number of monomer units can be hosted in the cyclic fraction. In contrast, under the same conditions, the chain fraction, represented by the first term in eq 11, is divergent, meaning that it can host, in principle, an infinite number of monomer units. If K is large enough, say larger than 105 mol−1 L,22 then there will be a well-defined critical concentration of the initial monomer equal to 2.612B, below which only rings 651


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Figure 1. Weighted equilibrium concentrations of chains (dotted line), rings (dashed line), and catenanes (solid line) for a “thick” (hydrocarbonlike) chain. Distributions are reported as a function of the initial monomer concentration for B = 1 mol L−1, σ = 1, α = 6.6 × 10−5, and K = 102 mol−1 L (A), 103 mol−1 L (B), 105 mol−1 L (C), and 108 mol−1 L (D). Arrows on the abscissa and ordinate point to the value of 2.612 mol L−1 (see text).

from just three ring oligomers, C1, C2, and C3, six catenanes can be obtained, namely, CC11, CC21, CC22, CC31, CC32, and CC33. Even if the concentration of a catenane is lower than that of the constituting rings, the sum of the weighted concentrations of all catenanes can become much larger than the weighted concentrations of rings. The reason is the fact that only a finite number of monomer units can be hosted in the cyclic fraction, and thus, in a ring−catenane system, when the ring

fraction has reached its limit value (2.612B), the excess monomer is forced to increase the catenane fraction that can host, in principle, an infinite number of monomer units. From the above considerations, it appears that the best conditions for the preparation of catenanes correspond to the chain-free case illustrated in Figure 2. To be more specific, let us consider an unsymmetrical polymethylene chain with 34 carbons per monomer ring. For such a ring, a value of B = 4.4 × 10−2 652


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The preparation of catenanes under thermodynamic control appears more promising in the case of “thin” chains for which excluded-volume effects can be neglected. The case of a “thin” (DNA-like) unsymmetrical chain A−B (σ = 1, α = 9.6 × 10−3) for which B = 1 mol L−1, as a function of increasing values of K, that is, K = 102, 103, 105, and 108 mol L−1 has been investigated and the results are shown in Figure 3. In this case, significant amounts of catenanes are formed even at lower values of K. Catenanes compete effectively with chains as well as rings, and the most striking result is that the phenomenon of critical concentration is not observed neither for chains nor for catenanes. By inspecting Figure 3D, it can be seen that for K = 108 mol−1 L, the chain fraction is negligibly small, thus, the curves of rings and catenanes are the same as that would be obtained for K tending to infinity. It is noteworthy that under this condition, the ring fraction has not reached its limit value even at the higher initial monomer concentrations. As to the distribution of catenane concentrations, Figure 4A reports the ratios [CCij]/α for B = 1 mol L−1 and σ = 1 versus the extent of reaction in the chain fraction, for i = 1−5 and j = 1. Catenane concentrations increase on increasing x, but for any value of x, catenane concentrations decrease on increasing the overall oligomerization degree, i + j. In particular, it can be easily demonstrated from eqs 7, 8, and 10 that for i = j and x ≈ 1, the concentrations of the catenanes CCii decrease as i−7/2, at variance with the ring concentrations Ci that decrease as i−5/2. The factor i−7/2 may be regarded as the product of i3/2 and the square of i−5/2. The former relates to the probability of overlap of the spherical domains of the two parent rings,26 as measured by the molar catenation constant, and the latter to the product of their probability of formation, as measured by the EM. Figure 4B reports the ratios [CCij]/α versus x for catenanes with the same overall oligomerization degree (i + j = 6), illustrating the effect of the relative size of the two constituting rings. It can be seen that lower catenane concentrations are obtained as the size of the two constituting rings becomes more even. The number-average polymerization degree of the catenane fraction is given by eq 14, where S3 is a function of x given by the series in eq 15

Figure 2. Weighted equilibrium concentrations of rings (dashed line) and catenanes (solid line) for a “thick” (hydrocarbon-like) chain as a function of the initial monomer concentration, calculated for B = 1 mol L−1, σ = 1, α = 6.6 × 10−5, and K tending to infinity. Arrows on the abscissa and ordinate point to the value of 2.612 mol L−1 (see text).

mol L−1 can be calculated by eq 9, using a value of l = 1.54 × 10−8 cm for the Csp3−Csp3 bond length and a value of C∞ = 6.7.41 By assuming a density of 0.8 g mL−1 for the neat hydrocarbon, its molar concentration can be estimated as [M1]0 ≈ 1.7 mol L−1. It is useful to divide both sides of eq 11 by B so that we can still use the curves reported in Figures 1 and 2, with the caveat that the curves now refer to the product KB instead of K, and the axes refer to the ratio concentration/B instead of concentration. Using the normalized concentration of [M1]0/B ≈ 38.6, and interpolating from the data related to Figures 1 and 2 (not shown), we can obtain at equilibrium the following yields: (i) for K = 2.3 × 103 mol−1 L, x = 0.9835, %chains (% M) = 94.30, %rings (%C) = 5.65, %catenanes (%CC) = 0.06; (ii) for K = 2.3 × 104 mol−1 L, x = 0.9947, %M = 93.74, %C = 6.12, %CC = 0.14; (iii) for K = 2.3 × 106 mol−1 L, x = 0.99946, %M = 92.43, %C = 6.56, %CC = 1.02; (iv) for K = 2.3 × 109 mol−1 L, x = 0.99996, %M = 74.12, %C = 6.71, %CC = 19.17; (v) for K → ∞ mol−1 L, x = 0.999995, %M = 0.00, %C = 6.75, %CC = 93.25. From these data, it appears that to obtain significant yields of catenanes from “thick” chains under equilibrium conditions, huge values for the constant of linear propagation are required. Such values are hardly accessible experimentally. Alternatively, some special catalysts should be envisaged capable of catalyzing the process of catenation intramolecularly without passing through the formation of the free linear polymer. A catalyst capable of performing reversible ring-expanding polymerization without the intermediacy of linear chains has been devised by Grubbs and co-workers,42−44 but there is no evidence that the mechanism of action of such a catalyst is capable of producing catenanes as well.

∞

∞

S3 =

i

∑i = 1 ∑ j = 1 (i + j)[CCij]

(DPn)CC =

∞ i ∑i = 1 ∑ j = 1 [CCij]

=

S2 S3

(14)

i

∑ ∑ 2[1 − INT(j /i)](i1/2 + j1/2 )3 (ij)−5/2 x i+ j i=1 j=1

(15)

The evaluation of S3 series is discussed in the SI. Values of S3 for selected values of x are reported in Table 1. A more extended table is presented in the SI. Figure 5 reports the number-average polymerization degree of the catenane fraction, as well as that of the ring fraction,18 as a function of x. On increasing x, the average polymerization degree of both fractions increases, but that of the catenane fraction increases more rapidly due to the fact that larger rings are more likely to undergo catenation. In particular, in correspondence to x tending to 1, the average polymerization degree of the ring fraction converges to about 1.95, whereas that of the catenane fraction becomes infinitely large. 653


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Figure 3. Weighted equilibrium concentrations of chains (dotted line), rings (dashed line), and catenanes (solid line) for a “thin” (DNA-like) chain. Distributions are reported as a function of the initial monomer concentration for B = 1 mol L−1, σ = 1, α = 9.6 × 10−3, and K = 102 mol−1 L (A), 103 mol−1 L (B), 105 mol−1 L (C), and 108 mol−1 L (D). Arrows on the abscissa and ordinate point to the value of 2.612 mol L−1 (see text).

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constant for linear propagation (of the order of 108 mol−1 L or larger). For a chain-free system, a significant amount of catenanes would form above the critical concentration. However, such a system, in which only ring−catenane equilibria occur, would require the action of a special catalyst, not yet devised, capable of promoting the process of catenation intramolecularly without passing through the formation of the

CONCLUSIONS

The answer to the title question is “It depends”. Synthetic organic polymers are typically constituted by flexible chains having aspect ratios (Kuhn step size/chain diameter) of about 2−3.39 If such “thick” polymers are engaged in ring−chain equilibria, the formation of catenanes can be neglected, unless in the unlikely case of a very large value of the equilibrium 654


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Figure 4. Distribution of catenane concentrations for B = 1 mol L−1 and σ = 1, as a function of the extent of reaction in the chain fraction. All curves are calculated by eqs 7, 8, and 10. (A) Effect of increasing the overall oligomerization degree; (B) effect of the relative size of the two constituting rings at constant overall oligomerization degree.

[2]-catenanes. Of course, the formation of higher order catenanes is less probable than that of [2]catenanes, and thus, it can be safely neglected in most of the cases. However, under conditions where catenanes constitute the main fraction, the formation of higher order catenanes, possibly leading to the formation of a gel, cannot be excluded.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b12323. Details regarding the evaluation of series S1, S2, and S3 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (S.D.S.). *E-mail: [email protected] (G.E.). ORCID

Stefano Di Stefano: 0000-0002-6742-0988 Gianfranco Ercolani: 0000-0003-2437-3429 Notes

The authors declare no competing financial interest.

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Figure 5. Number-average polymerization degree of catenane fraction and ring fraction as a function of the extent of reaction in the chain fraction.

REFERENCES

(1) Molecular Catenanes, Rotaxanes and Knots; Sauvage, J.-P., Dietrich-Buchecker, C. O., Eds.; Wiley-VCH: Weinheim, 1999. (2) Gil-Ramírez, G.; Leigh, D. A.; Stephens, A. J. Catenanes: Fifty Years of Molecular Links. Angew. Chem., Int. Ed. 2015, 54, 6110−6150. (3) Bruns, C. J.; J. Stoddart, J. F. The Nature of the Mechanical Bond: From Molecules to Machines; John Wiley and Sons: Hoboken, New Jersey, 2017. (4) Erbas-Cakmak, S.; Leigh, D. A.; McTernan, C. T.; Nussbaumer, A. L. Artificial Molecular Machines. Chem. Rev. 2015, 115, 10081− 10206.

free linear polymer. In contrast, the formation of catenanes cannot be neglected in the case of “thin” polymers having large aspect ratios. For such chains, the extended JS theory illustrated above should be used independently of the magnitude of the linear propagation constant. Finally, it is worth pointing out that the above theory takes into account only the formation of 655


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