A method has been proposed to assess cooperativity in self-assembly processes. The method is based on a clear distinction between intermolecular and intramolecular processes which are compared with the corresponding reference reactions. It has been applied to two classical cases, namely the self-assembly of helicates and of porphyrin ladders, by using data previously published by the groups of Lehn and Anderson, respectively. Contrarily to the conclusions of the authors, pointing out self-assembly processes driven by positive cooperativity, the method here presented indicates in both cases the absence of cooperative effects. The methods previously used to assess cooperativity, in particular Scatchard plot and/or Hill plot, are criticized as being inappropriate for self-assembly, because they are pertinent to a specific case only, namely the intermolecular binding of a monovalent ligand L to a multivalent receptor M, a case very different from self-assembly which involves both inter- and intramolecular interactions. The present method underscores the fact that positive cooperativity in artificial self-assembling systems is probably much more rare than it was previously thought.

A general treatment of macrocyclization reactions occurring under thermodynamic control is presented. The fundamental quantities on which the treatment is based are the effective molarities of the cyclic oligomers and the equilibrium constant for the intermolecular model reaction between monofunctional reactants (Kinter). Four typical cases have been considered, namely, addition and condensation of a monomer of the type A-B, addition of A-A, and addition of A-A + B-B. A critical comparison with the classical theory of Jacobson and Stockmayer is presented.It is shown that the phenomenon characterized by the critical monomer concentration (cut-off point) is a limiting phenomenon which would occur only for infinitely large values of Kin,,,. The treatment has been successfully applied to the DOS/DTC-induced cyclooligomerization of /3-propiolactone in CDC13 solution that yields well-behaved ringchain equilibrates closely adhering to the theoretical model. Best fit of the experimental product distributions to the general equations gave the equilibrium constant (Kinter) of the intermolecular model reaction, as well as the effective molarities (EMi) for the cyclic oligomers from trimer to octamer. The EMi values decrease in proportion of the -2.5 power of the oligomerization degree, thus providing a strong indication that the oligomeric polylactones are essentially strainless. The extremely low value of Kin,,, (2.5) is responsible for the absence of a cut-off point, which is usually present in ring-chain polymeric equilibrates.

The distinction between different types of cooperativity is essential for understanding the fundamentals involved. The three title cooperative effects arise from the interplay of intermolecular binding interactions, the presence of one or more intramolecular binding interactions, and, in the latter case, their possible interplay. A master equation for the stability of an assembly is outlined that takes into account all of the three possible types of cooperativit

Evaluation of statistical factors in self-assembly processes is not a firmly settled question. As a contribution to solve this problem, a critical re-examination of the symmetry number method and generalization of the direct count method are presented. The two approaches, producing the same results, mutually reinforce their role with respect to other discordant methods whose results cannot be independently checked. The direct count method moreover serves as a rationale for the apparently odd results the symmetry number method sometimes provides. The two methods thus turn out to be complementary to each other. Discussion of some exemplary cases points to the importance and subtlety of the role played by the geometrical features of assemblies involving intramolecular bonds.