complex when compared to the previously discussed IDP. To account for this, the monomer fraction in linear species as well as the <DP>N and <DP>W values were calculated for a general ring‐chain equilibrium, which only involved unstrained macrocycles (i.e. various Kintra(1) values were considered, Kinter = 106 M−1) [26, 83]. As shown in Figure 1.11a, the transition between cyclic and linear species at the critical concentration becomes much sharper when Kintra(1) is increased. In addition, both <DP>N and <DP>W exhibit a steep increase for ct > EM1 (the sharpness of the transition still depends on Kintra(1)). In contrast, in an IDP, the DP gradually rises with increasing concentration. However, at high total concentrations, it is no longer possible to distinguish between the different modes of polymerization (i.e. the IDP or the ring‐chain equilibrium polymerization), and the obtained DPs are almost identical at given concentrations that are much higher than the EM1 value (Figure 1.11b).
Figure 1.11 (a) Illustration of the fraction of polymerized monomer as a function of Kinter·ct for three different EM1 values and a fixed value of Kinter (106 M−1). (b) Illustration of the evolution of <DP>N as a function of Kinter·ct for various EM1 values.
Source: Flory and Suter [91].
Dormidontova and coworker addressed the issue of the spacer's rigidity with respect to the ring‐chain equilibrium of supramolecular polymers [92]. Applying Monte Carlo simulations on such supramolecular polymerizations, these authors showed that the critical concentration was strongly dependent on the rigidity of the spacer (in these modeling studies, H‐bonding interactions were representatively studied). Keeping all further parameters constant (e.g. the length of the spacer or the energy for the interaction of the end groups), the critical concentration decreased in the following order: rigid > semi‐flexible > flexible. Thus, for rigid and semi‐flexible systems, the probability of their end groups meeting within a bonding distance and, thus, the formation of rings, is much smaller as for flexible systems.
Various groups have reported on critical temperatures in ring‐chain equilibria (Tc). These values define the transition between macrocyclic and linear species of high molar mass [71, 72, 93]. Like the supramolecular IDP elaborated in Section 1.3.1, one has to also distinguish two limiting cases for the ring‐chain equilibrium polymerization [56]:
1 Above a certain ceiling temperature, polymers of high molar mass are thermodynamically less stable than cyclic monomers or oligomers.
2 Below a certain floor temperature, polymers of high molar mass are thermodynamically less stable than cyclic monomers/oligomers.
In other words, a ceiling temperature can be found in those (supramolecular) polymerizations where negative changes in the enthalpy and entropy of propagation occur; in the second case, the changes in these measures are positive and, consequently, the floor temperature defines the limit below which (supramolecular) polymerization cannot occur.
Covalent ROPs typically involve the opening of strained cycles (e.g. the cationic polymerization of tetrahydrofuran [THF] and dioxolane [42]). In general, such polymerizations represent enthalpy‐driven processes for which ceiling temperatures can be observed (basically, all species are of cyclic nature above this value). Very few examples are known for ROPs exhibiting a floor temperature [94]. Examples for such processes that are characterized by a gain in entropy are the ROP of cyclic S8 in liquid sulfur [93] and the ROMP of unstrained, macrocyclic olefins [70].
Also in the “supramolecular world,” the ring‐chain equilibrium polymerization is a common feature, independent of the type of employed non‐covalent interactions. Representative examples in this respect are the formation of pseudorotaxanes (i.e. the supramolecular polymerization of crown ether derivatives equipped with a pending positively charged amine; Figure 1.12, see Chapter 6) [95–97], the polymerization of poly(dimethylsiloxane)s functionalized in α,ω‐position with carboxylic acids (see Chapter 2) [98], and the equilibrium between linear, tape‐like, and cyclic structures that can be observed in stoichiometric mixtures of cyanuric acid and melamine derivatives (see Chapter 3) [99].
Figure 1.12 Schematic representation of the formation of a poly(pseudorotaxane) via a ring‐chain equilibrium.
Source: Cantrill et al. [95]. © 2001 American Chemical Society.
1.3.3 (Anti)‐cooperative Supramolecular Polymerization
The third and last mechanism for supramolecular polymerization to be discussed herein involves (at least) two distinct stages, resulting in a cooperative or an anti‐cooperative growth of the polymer chains. At first glance, the mechanism of the cooperative supramolecular polymerization is reminiscent to the one for the IDP; however, the polymerization initially occurs via the reversible binding of monomers to the growing chain (as for the IDP, all these steps basically possess the same equilibrium constant Kn). At a certain DP, a nucleus is formed and, from this point on, the binding of monomers to the polymer chain features an association constant Ke, which is higher than Kn (Figure 1.13). In such a nucleation‐elongation polymerization (NEP) model, the supramolecular polymerization proceeds via a linear IDP. In this elongation phase, the actual association constant is now Ke rather than Kn [26, 43, 100, 101].
Figure 1.13 Schematic representation of a typical cooperative supramolecular polymerization reaction (nucleation‐elongation mechanism). Kn and Ke represent the association constants for the nucleation and the elongation phase, respectively (Kn < Ke).
Source: Winter et al. [39]. © 2012 Elsevier B.V.
The complex thermodynamics of the (anti‐)cooperative supramolecular polymerization have already been summarized by de Greef et al.; the reader is referred to this review for a more in‐depth discussion
