vs. T/Tm. In both plots, the curves obtained for different enthalpies are shown."/>
Figure 1.7 Illustration of the characteristic properties of a temperature‐dependent IDP according to van der Schoot's model: (a) fraction of polymerized material (ϕ) vs. the dimensionless temperature T/Tm; (b) <DP>N vs. T/Tm. In both plots, the curves obtained for different enthalpies are shown (ΔHp = −30, −40, and −50 kJ mol−1, respectively).
Source: van der Schoot et al. [57]. © 2005 Taylor & Francis.
Dudovich et al. introduced an alternative approach, commonly referred to as the “free association model,” which is based on a mean‐field incompressible lattice model derived from the Flory–Huggins theory (for the Flory–Huggins model, see [58, 59]). In this approach, the flexibility of the polymer chains and the van der Waals interactions between the monomer and solvent molecules (quantified by the parameter χ) are taken into account [50, 51]. A variety of temperature‐dependent properties can be calculated from the lattice model (e.g. <DP>N and the specific heat at constant volume [CV]). It has been shown that neither of these (as well as the Đ value) is sensitive to χ when the temperature is changed; however, the situation is different if the χ value for the polymer–solvent interaction is different from the one for the monomer–solvent interaction [50]. On the other hand, a variety of thermodynamic properties do show a strong temperature dependency of χ; these include the osmotic pressure and the critical temperature at which phase separation between monomer and solvent occurs. Two free energy parameters describe the reversibility of the supramolecular polymerization: the polymerization enthalpy (ΔHp) and entropy (ΔSp), which are both temperature independent. Representatively, the fraction of polymerized monomers (ϕ), as a function of the dimensionless temperature T/Tm, for a system that reversibly polymerizes upon cooling, is shown in Figure 1.8a [51]. In accordance with van der Schoot's model (vide supra), the curve is of sigmoidal shape and, with the values of ΔHp and ΔSp becoming more negative, the steepness of the curve becomes more pronounced. For fixed monomer concentrations, the CV vs. T/Tm plots show broad and highly symmetric transition (Figure 1.8b). This feature is indicative of an IDP in which the equilibrium constant K for the addition of each monomer to the growing polymer chain has always the same value. On the other hand, the temperature dependency of CV in ring‐chain or cooperative supramolecular polymerizations shows a much sharper transition (see also Sections 1.3.2 and 1.3.3).
Figure 1.8 Illustration of the characteristic properties of a temperature‐dependent IDP according to the “free association” model: (a) fraction of polymerized monomers (ϕ) vs. T/Tm (assuming fully flexible polymer chains and a cubic lattice); (b) heat capacity at constant volume (CV) vs. T/Tm. In both plots, the curves obtained for various enthalpy (ΔHp = −30, −40, and −50 kJ mol−1, respectively) and entropy values (ΔSp = −100, −133, and −166 J mol−1 K−1, respectively) are shown; in all cases, the initial volume fraction of the monomers has been set to 0.1.
Source: Modified from Dudowicz et al. [50]; Douglas et al. [51].
In the field of supramolecular polymers, independent of the nature of the involved non‐covalent linkage, their formation via IDP is by far the most common mechanism. Many examples involving hydrogen‐bonding (Chapter 3) or host–guest interactions (e.g. by crown ether or calixarene recognition; Chapters 6–10) as well as metal‐to‐ligand coordination (Chapter 4) are discussed there. It has to be pointed out that the determination of the molar mass of all these supramolecular polymers is generally nontrivial, since the established direct analytical methods commonly used for traditional, i.e. covalent, macromolecules (e.g. size‐exclusion chromatography [SEC] or mass spectrometry) can often not be applied due to the weak nature of the supramolecular bonds: already small changes in temperature, solvent composition, and concentration might lead to significant changes of the DP [60, 61]. However, several spectroscopic techniques (e.g. nuclear magnetic resonance (NMR) or UV/vis absorption), calorimetry, and analytical ultracentrifugation (AUC) can be applied in many cases to determine the molar masses [33, 36, 37]. A summary of the scope and limitations in characterizing supramolecular polymers is given separately in Chapter 12.
1.3.2 Ring‐Chain‐Mediated Supramolecular Polymerization
The so‐called ring‐chain‐mediated supramolecular polymerization represents the second main mechanism to describe the growth of supramolecular polymer chains (Figure 1.4b). In general, a heteroditopic monomer is polymerized reversibly; this monomer as well as its oligomers and, eventually, polymer chains feature an equilibrium between a linear and a cyclic species (Figure 1.9). Ring formation occurs via the intramolecular reaction of the end groups, whereas intermolecular reactions will accordingly give longer chains. Flexibility of the monomer represents, thus, a prerequisite for this type of mechanism; for instance, flexible alkyl or even polymer chains can be used to link the terminal supramolecular binding sites of such a monomer [62]. It is generally accepted that the covalent step‐growth polymerization of such monomers typically gives some wt% of macrocyclic oligomers (thereby, the polymerization can be performed under kinetic or thermodynamic control) [44, 63, 64]. Representatively, two classic cases in which also macrocyclic species are formed shall be named briefly: the bulk polymerization of triethylene glycol with hexamethylene–diisocyanate (polycondensation under kinetic control) [65] and, as an example for a thermodynamically controlled process, the catalyzed equilibrium polymerization of α,ω‐disubstituted siloxanes (in particular, the later system was widely investigated by Scott [66], Brown and Slusarczuk [67], Carmichael and Winger [68], as well as Flory and Semlyen [69]). The entropically driven ring‐opening metathesis polymerization (ROMP) of cyclic olefins [70] and the ring‐chain polymerization of liquid sulfur [71–73] are further representatives for covalent ring‐chain polymerizations under thermodynamic control. As a general characteristic for a step‐growth polymer, the reversibility of bond formation establishes an equilibrium between macrocyclic and linear species. With respect to supramolecular polymers, where the formation/cleavage of non‐covalent bonds is typically fast, the macrocyclization pathway occurs always under thermodynamic control.
