A Model for Self-Assembly in Solution1Gianfranco Ercolani†Dipartimento di Scienze e Tecnologie Chimiche, UniVersita` di Roma Tor Vergata,Via della Ricerca Scientifica, 00133 Roma, ItalyReceiVed: December 27, 2002; In Final Form: March 12, 2003A model treating the competition, under thermodynamic control, between self-assembly and nonlinear randompolymerization is presented. The fundamental quantities on which the treatment is based are the effectivemolarity (EM) of the assembly and the equilibrium constant for the intermolecular model reaction betweenmonofunctional reactants (Kinter). Knowledge of these quantities allows the evaluation of the distribution curveof the self-assembling complex. In order for effective self-assembly to take place, the product KinterEM mustbe no lower than a limit value easily computable on the basis of simple structural parameters such as thenumber of molecules in the assembly (N), the number of bonds joining them (B), and the number of interactionsites in the monomers. This limit decreases on decreasing N and on increasing B, and the most obvious wayto realize this condition is by increasing the degree of cyclicity of the assembly (B - N + 1). The yield ofan assembly with a high degree of cyclicity is very sensitive also to modest changes of KinterEM about itslimit value. Depending on the value of the monomers concentration, the assembly could undergo either sharpdisassembly (denaturation) or conversion into gel.IntroductionSelf-assembly consists of the spontaneous generation of awell-defined, discrete supramolecular architecture from a givenset of components under thermodynamic equilibration.2Avariety of molecular architectures have been obtained in thisway, spanning from exotic structures such as ladders, helicates,and grids to two-dimensional or three-dimensional structuresthat closely resemble well-known geometric shapes, includingtriangles, squares, hexagons, cubes, triangular prisms, octahedra,cuboctahedra, etc. Such materials are interesting as artificial,molecular-scale containers or receptors, in which novel syntheticchemistry, electrochemistry, photoluminescent chemistry,supramolecular chemistry, or catalytic chemistry, inter alia, canbe carried out.Despite the plethora of self-assembling systems that can befound in the literature, the physicochemical basis of self-assembly is, however, largely unexplored. As a contribution tofill this gap, here is presented a model of general applicabilitythat addresses a number of issues that are crucial to theunderstanding of self-assembly, inter alia: What are the factorsgoverning self-assembly and how can they be put on aquantitative scale? What is the range of reactant concentrationsin which a self-assembling structure is stable? What are thecriteria to judge if a given assembly is more or less favoredthan another? It is apparent that the answers to these and otherrelated questions are of fundamental importance not only toallow the deliberate formation of desired architectures but alsoto extract information from a system under study in a consistentand significant way.Theoretical BasisSelf-assembly of a discrete supramolecular structure alwaysoccurs in competition with the process of polymerization. Theself-assembly of a two-dimensional structure requires bifunc-tional monomers that react to yield a specific cyclic oligomerin competition with linear polymerization; treatment of this casehas been previously reported and experimentally tested.3Thecase regarding the self-assembly of a three-dimensional structureis more complex because at least one of the reactants carriesmore than two reactive groups and thus the self-assemblingarchitecture is formed in competition with nonlinear oligomersdisplaying an enormous number of different topologies. Clas-sification of the various species that could form in solution isa hopeless task; however, as will be illustrated in the following,such a level of detail is not needed to extract the necessaryinformation.In the vast majority of the cases self-assembly of a three-dimensional architecture requires two monomeric buildingblocks. Let us indicate the two building blocks as L1, having lbinding sites -A, and M1, having m binding sites -B. The twofunctional groups -A and -B are each capable of reacting withthe other only in a reversible addition reaction. When L1andM1are mixed and the equilibrium is attained, the monomericunits can be considered as partitioned in two fractions inequilibrium between them, one constituted by an infinite numberof oligomers having one or more loops in their structure andincluding the assembly S, and the other constituted by an infinitenumber of more or less branched oligomers devoid of loops.Self-assembly takes place when the monomers have a rigidstructure predisposed4in such a way that formation of the self-assembling complex is strongly favored over other cyclic orpolycyclic species. The latter are disfavored by one or more ofthe following: (i) the presence of unreacted end groups, (ii)strained loops involving a high enthalpy content, and (iii) largeloops involving a high entropy loss. It is assumed therefore thatS is the only significant species of the first fraction. Thisassumption, substantiated later in this paper, is further justifiedby the fact that the most important theory of nonlinear†E-mail: [email protected] J. Phys. Chem. B 2003, 107, 5052-505710.1021/jp027833r CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 04/24/2003polymerization, due to the contributions of Flory and Stock-mayer, completely neglects the formation of looped structures.5The acyclic oligomers of the second fraction have the generalformula LrMs. They can be subdivided in families of oligomers,Ri, where i, the degree of polymerization, is given by i ) r +s, so that R1represents the monomers L1and M1,R2the dimers,and so on. Schematically the equilibra to consider for the processof self-assembly are shown in Scheme 1.The molecular formula of the assembly S is LpmMpl, wherep is a coefficient needed to account for the stoichiometry of Land M within S. A cartoon showing the self-assembly of a cubeis reported, by way of illustration, in Figure 1.The mass balance equation in terms of the total number ofmonomer units is then given bywhere the subscript 0 has the meaning of initial concentration.It is obvious that the maximum concentration of S is obtainedwhen the initial concentration of functional groups -A and -Bis equal, i.e., when the