Supramolecular Polymers and Assemblies. Andreas Winter

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Figure 1.10 (a) Schematic representation of Kuhn's concept of effective concentration (c_eff) for a heteroditopic oligomer (i.e. having two different end groups, A and B) [74]. In solution, the end group A will experience an effective concentration of B, if the latter one cannot escape from the sphere of radius l, which is identical to the length of the stretched chain. Thus, the intramolecular association between the termini becomes favored for c_eff values higher than the actual concentration of B end groups. (b) Illustration of how the equilibrium concentration of chains and macrocycles can be correlated to the total concentration (c_t) of a ditopic monomer in dilute solution; such a ring‐chain supramolecular polymerization typically features a critical concentration.

      Source: de Greef et al. [26]. © 2009 American Chemical Society.

      The toolbox of polymer physics, in particular utilizing random‐flight statistics, enables one to calculate c_eff as a function of the length of the polymer chain [75]. In reasonably good approximation, the distribution function for random‐coil polymers is of Gaussian shape [62]; however, this model only holds true for long, flexible chains [76]. In the same context, a particle‐in‐a‐sphere model was utilized by Crothers and Metzger [77]. In a more realistic approach, Zhou employed a worm‐like chain model to determine c_eff for short and, thus, semi‐flexible polypeptides [78, 79].

For practical reasons, the rather theoretical concept of effective concentration, which basically relies on concentrations calculated from the physical properties of the terminal functionalities, is often replaced by a more empirical concept using effective molarities [80–85]. The effective molarity (EM) is defined as the ratio of intra‐ and intermolecular equilibrium constants (i.e. K_inter and K_intra, Eq. (1.4), see also Figure 1.9): cyclization is basically preferred for EM > 1, whereas linear chains are obtained for EM < 1. In addition, EM can be considered as a pure entropic correction, which becomes relevant when an intramolecular process replaces the analogous intermolecular one (however, this only applies to unstrained, flexible chains linking the end groups) [86].

(1.4)

      where EM: effective molarity, K_intra: dimensionless equilibrium constant for the intramolecular reaction, K_inter: association constant (M⁻¹) for an intermolecular reaction.

      In the case of a supramolecular polymerization in which a heteroditopic AB‐type monomer is used, EM defines the limit monomer concentration below which the (macro)cyclization pathway dominates the linear chain growth. This empirical approach allows one to predict the different cyclization reactions and, even more importantly, gives an absolute measure for a monomer's cyclization ability at the cost of its polymerization (valid only for reversible, non‐covalent interactions).

      For thermodynamically controlled step‐growth polymerizations, Jacobsen and Stockmayer predicted a critical concentration limit [87]: the system is exclusively composed of cyclic species below this value; above this value, an excess of monomer exclusively gives linear chains while the concentration of cyclic species stays constant (Figure 1.10b). These authors related the equilibrium constant for the cyclization to the probability for, thus directly connecting EM and c_eff. It was additionally shown that this constant would decrease with N^−5/2; in other words, a macrocycle composed of N subunits can reopen in N different ways. This study was extended by Ercolani et al., who also considered the size distribution of macrocycles under dilute conditions; thereby, a broad range of K_a values for the supramolecular macrocyclization were taken into account [83]. According to this, only for high K_a values (>10⁵ M⁻¹) can a critical concentration limit be observed.

The later model is particularly suited for describing the equilibrium, which is established between cyclic and linear species during a supramolecular polymerization (they are typically conducted in relatively dilute solutions). In contrast to an IDP (vide infra), which commonly features K as the only thermodynamic constant, Ercolani's ring‐chain model involves two such constants (Figure 1.9): K_inter and K_{intra(n‐mer)} (the latter represents the intramolecular binding constant for the n‐th ring closure). Considering all cycles as unstrained and obeying Gaussian statistics, the EM_n‐mer values can simply be expressed as a function of EM₁ (EM₁: effective molarity of the bifunctional monomer itself; Eq. (1.5)). An additional aspect that needs to be briefly mentioned is the role of the solvent: thus, volume effects cannot be neglected, and the exponent in Eq. (1.5) needs to be adjusted [62,88–90].

(1.5)

      where EM_n‐mer: effective molarity of the n‐mer, K_{intra(n‐mer)}: intermolecular binding constant for the n‐th ring closure, K_inter: association constant (M⁻¹) for an intermolecular reaction, EM₁: effective molarity of the bifunctional monomer.

Due to an additional parameter, which СКАЧАТЬ