5052 J Phys Chem B 2003 107 5052 5057 A Model for Self Assembly in Solution1 Gianfranco Ercolani Dipartimento di Scienze e Tecnologie Chimiche UniVersita di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma Italy ReceiVed December 27 2002 In Final Form March 12 2003 A model treating the competition under thermodynamic control between self assembly and nonlinear random polymerization is presented The fundamental quantities on which the treatment is based are the effective molarity EM of the assembly and the equilibrium constant for the intermolecular model reaction between monofunctional reactants Kinter Knowledge of these quantities allows the evaluation of the distribution curve of the self assembling complex In order for effective self assembly to take place the product KinterEM must be no lower than a limit value easily computable on the basis of simple structural parameters such as the number of molecules in the assembly N the number of bonds joining them B and the number of interaction sites in the monomers This limit decreases on decreasing N and on increasing B and the most obvious way to realize this condition is by increasing the degree of cyclicity of the assembly B N 1 The yield of an assembly with a high degree of cyclicity is very sensitive also to modest changes of KinterEM about its limit value Depending on the value of the monomers concentration the assembly could undergo either sharp disassembly denaturation or conversion into gel Introduction Self assembly consists of the spontaneous generation of a well defined discrete supramolecular architecture from a given set of components under thermodynamic equilibration 2 A variety of molecular architectures have been obtained in this way spanning from exotic structures such as ladders helicates and grids to two dimensional or three dimensional structures that closely resemble well known geometric shapes including triangles squares hexagons cubes triangular prisms octahedra cuboctahedra etc Such materials are interesting as artificial molecular scale containers or receptors in which novel synthetic chemistry electrochemistry photoluminescent chemistry supramolecular chemistry or catalytic chemistry inter alia can be carried out Despite the plethora of self assembling systems that can be found in the literature the physicochemical basis of selfassembly is however largely unexplored As a contribution to fill this gap here is presented a model of general applicability that addresses a number of issues that are crucial to the understanding of self assembly inter alia What are the factors governing self assembly and how can they be put on a quantitative scale What is the range of reactant concentrations in which a self assembling structure is stable What are the criteria to judge if a given assembly is more or less favored than another It is apparent that the answers to these and other related questions are of fundamental importance not only to allow the deliberate formation of desired architectures but also to extract information from a system under study in a consistent and significant way Theoretical Basis Self assembly of a discrete supramolecular structure always occurs in competition with the process of polymerization The E mail ercolani uniroma2 it self assembly of a two dimensional structure requires bifunctional monomers that react to yield a specific cyclic oligomer in competition with linear polymerization treatment of this case has been previously reported and experimentally tested 3 The case regarding the self assembly of a three dimensional structure is more complex because at least one of the reactants carries more than two reactive groups and thus the self assembling architecture is formed in competition with nonlinear oligomers displaying an enormous number of different topologies Classification of the various species that could form in solution is a hopeless task however as will be illustrated in the following such a level of detail is not needed to extract the necessary information In the vast majority of the cases self assembly of a threedimensional architecture requires two monomeric building blocks Let us indicate the two building blocks as L1 having l binding sites A and M1 having m binding sites B The two functional groups A and B are each capable of reacting with the other only in a reversible addition reaction When L1 and M1 are mixed and the equilibrium is attained the monomeric units can be considered as partitioned in two fractions in equilibrium between them one constituted by an infinite number of oligomers having one or more loops in their structure and including the assembly S and the other constituted by an infinite number of more or less branched oligomers devoid of loops Self assembly takes place when the monomers have a rigid structure predisposed4 in such a way that formation of the selfassembling complex is strongly favored over other cyclic or polycyclic species The latter are disfavored by one or more of the following i the presence of unreacted end groups ii strained loops involving a high enthalpy content and iii large loops involving a high entropy loss It is assumed therefore that S is the only significant species of the first fraction This assumption substantiated later in this paper is further justified by the fact that the most important theory of nonlinear 10 1021 jp027833r CCC 25 00 2003 American Chemical Society Published on Web 04 24 2003 Model for Self Assembly in Solution J Phys Chem B Vol 107 No 21 2003 5053 SCHEME 1 Now consider the infinite McLaurin expansion shown in eq 5 Figure 1 Self assembly of a cube by two predisposed building blocks In the example shown p 4 m 2 l 3 polymerization due to the contributions of Flory and Stockmayer completely neglects the formation of looped structures 5 The acyclic oligomers of the second fraction have the general formula LrMs They can be subdivided in families of oligomers Ri where i the degree of polymerization is given by i r s so that R1 represents the monomers L1 and M1 R2 the dimers and so on Schematically the equilibra to consider for the process of self assembly are shown in Scheme 1 The molecular formula of the assembly S is LpmMpl where p is a coefficient needed to account for the stoichiometry of L and M within S A cartoon showing the self assembly of a cube is reported by way of illustration in Figure 1 The mass balance equation in terms of the total number of monomer units is then given by L1 0 M1 0 pl pm S i