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of monomers by equilibrium bond formation and features an ideal polymerization temperature (T_p⁰) [50–54]. The Dainton–Ivin equation, initially introduced to describe the thermodynamics of ROP and polyaddition reactions, correlates the enthalpy and entropy of propagation (ΔH_pr and ΔS_pr) as well as the initial monomer mole fraction to T_p⁰ (Eq. (1.3)) [55, 56]. There are two fundamental cases that one must distinguish:

      1 The polymerization only occurs at a temperature so high that the entropy term exceeds the enthalpy term and the system exhibits a floor temperature (ΔHpr, ΔSpr > 0).

      2 The polymerization represents an enthalpically driven process, which is only allowed below a certain ceiling temperature (ΔHpr, ΔSpr < 0).

      The so‐called polymerization transition line, separating monomer‐rich phases from polymer‐rich ones, can be constructed by plotting [M_i] vs. the polymerization temperature, which can be determined experimentally. However, this model is only valid in those cases where a sharp monomer‐to‐polymer transition can be found (in general, applicable only for ring‐opening, living, or cooperative polymerizations) [50]. For most of the reported IDPs, this transition is, however, very broad and the two phases rather coexist. Thus, for such a supramolecular polymerization, the polymerization transition line as a boundary appears less appropriate.

(1.3)

      where T_p⁰: ideal polymerization temperature, ΔH_pr: enthalpy of propagation, ΔS_pr: entropy of propagation, R: gas constant, [M_i]: initial mole fraction of a monomer.

Historically, the temperature dependency in isodesmic self‐assembly processes has been explained by means of statistical mechanics [52]. More recently, mean‐field models that are free of restrictions concerning the actual mechanism of chain have been applied for the same purpose. In such models, the chain growth can occur by either the addition of a single monomer or the linkage of two existing chains. van der Schoot proposed a model, where the temperature‐dependent melting temperature (T_m; in essence, the temperature at which the monomer mole fraction in the supramolecular polymer is 0.5) and the temperature‐independent polymerization enthalpy (ΔH_p) were considered [57]. As one example, a system that polymerizes upon cooling is analysed by plotting the fraction of the already polymerized material (ϕ) against T/T_m for various ΔH_p values; (Figure 1.7a) in such an IDP, the steepness of the transitions of the curves only depends on ΔH_p and contributions arising from cooperativity effects can be excluded. Moreover, a gradual increase of <DP>_N with decreasing temperature can typically be observed (Figure 1.7b).

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/T_m; (b) <DP>_N vs. T/T_m. In both plots, the curves obtained for different enthalpies are shown (ΔH_p = −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 [C_V]). 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 (ΔH_p) and entropy (ΔS_p), which are both temperature independent. Representatively, the fraction of polymerized monomers (ϕ), as a function of the dimensionless temperature T/T_m, 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 ΔH_p and ΔS_p becoming more negative, the steepness of the curve becomes more pronounced. For fixed monomer concentrations, the C_V vs. T/T_m 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 C_V 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/T_m (assuming fully flexible polymer chains and a cubic lattice); (b) heat capacity at constant volume (C_V) vs. T/T_m. In both plots, the curves obtained for various enthalpy (ΔH_p = −30, −40, and −50 kJ mol⁻¹, respectively) and entropy values (ΔS_p = −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

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