Supramolecular Polymers and Assemblies. Andreas Winter

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СКАЧАТЬ href="#ulink_ae83c613-9d4c-576e-8426-e8179bd94c2f">Figure 1.5 Schematic representation of the IDP in which the intermolecular equilibrium constant (K) is independent of the length of the assembly (the mechanism is shown for a bifunctional monomer of the Ia‐type, see also Figure 1.2).

      Source: Winter et al. [39]. © 2012 Elsevier B.V.

According to the rules of thermodynamics, the free energy of the system constantly decreases when the monomeric units are successively added to the growing polymer chain; this, in turn, further supports the assumption that binding of a monomer to the terminus of a polymer chain is independent of its length (an idealized energy diagram, in which kinetic barriers within the self‐assembly process are neglected, is depicted in Figure 1.6a) [26].

Figure 1.6 (a) Schematic drawing of an energy diagram for an IDP (i: size of the oligomer, ΔG⁰: free energy in arbitrary units). (b) Evolution of the number‐ and weight‐averaged DP (<DP>_N and <DP>_W) and the dispersity (Đ) as a function of equilibrium constant and total concentration of monomer (K·c_t).

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

The number‐ and weight‐averaged DPs (i.e. <DP>_N and <DP>_W, respectively) can be derived from the monomer concentration and equilibrium constant K according to Eq. (1.2) (though only valid for K·[monomer] < 1) [43]. In the ideal case, Đ converges to the limiting value of 2.0, and thus, the monomer concentration approaches 1/K (Eq. (1.2)); this scenario is comparable to a standard step‐growth polymerization as known from traditional polymer chemistry [27, 33]. The correlation of these parameters with the dimensionless concentration K·c_t, where K represents the equilibrium constant and c_t the total monomer concentration, is shown in Figure 1.6b. Apparently, high DPs can only be reached for high K·c_t values; thus, high monomer concentrations and high K values are both required. Disadvantageously, the intrinsically poor solubility of monomers often excludes high concentrations, and thus, the equilibrium constants must be very high (K > 10⁶) to compensate for this when aiming for supramolecular polymers with high molar masses. As a typical feature of IDP‐type processes, increasing c_t automatically leads to a gradual and simultaneous increase of the concentration of monomers and polymer chains; thus, the monomer and polymer chains of various length coexist in solution. Finally, the equilibrium concentration of monomers converges to its maximum value, corresponding to K⁻¹, when increasing the concentration further. Thereby, the monomer remains the most abundant species in solution, independent of the values of K and c_t. As for their covalent counterparts, the precise stoichiometry of the functional groups in an IDP represents a prerequisite to obtain polymers with high molar masses: self‐complementary AB‐type monomers inherently bear the ideal stoichiometry, whereas complementary monomers (i.e. using a combination of AA and BB) require an exact 1 : 1 ratio. Moreover, the molar masses of the resultant polymers can be adjusted by the addition of appropriate chain‐stopping agents [47–49].

(1.2)

      where <DP>_N: number‐averaged DP, <DP>_W: weight‐averaged DP, Đ: dispersity, K: equilibrium constant, [monomer]: monomer concentration.

Besides the concentration dependency, the influence of the temperature on the IDP also needs to be addressed. Basically, any type of supramolecular polymerization using a bifunctional monomer represents the polymerization 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).

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