Theories for dilute block copolymer solutions

A simple scaling model of block copolymer micelles was derived by de Gennes (1978). He obtained scaling relations assuming uniformly stretched chains for the core radius, RB, of micelles with association number p.This model can be viewed as a development of the Alexander de Gennes theory (Alexander 1977 de Gennes 1976,1980) for polymer brushes at a planar interface, where the density profile normal to the interface is a step function. In the limit of short coronal (A) chains (crew-cut micelles) de Gennes (1978) predicted 

Tabic 3.5 Predictions of Zhulina and Birshtein (1985) for micellar characteristics in an AB diblock in a solvent selective for the A block  

It is worth emphasizing that all scaling theories (due to de Gennes, Daoud and Cotton, Zhulina and Birshtein, and Halperin) for block copolymer micelles with a small core and large corona predict that the association number and core radius are independent of the coronal chain length. 

It may be noted that, denoting the fraction of A monomers in the corona belonging to the copolymer by t], and the micellar association number by p, incompressibility conditions give simple relations between R, RB, p and t) (de Gennes 1978 Leibler el al. 1983)  

2 Mean field theories for block copolymer micelles 

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This chapter is organized as follows. The thermodynamics of the critical micelle concentration are considered in Section 3.2. Section 3.3 is concerned with a summary of experiments characterizing micellization in block copolymers, and tables are used to provide a summary of some of the studies from the vast literature. Theories for dilute block copolymer solutions are described in Section 3.4, including both scaling models and mean field theories. Computer simulations of block copolymer micelles are discussed in Section 3.5. Micellization of ionic block copolymers is described in Section 3.6. Several methods for the study of dynamics in block copolymer solutions are sketched in Section 3.7. Finally, Section 3.8 is concerned with adsorption of block copolymers at the liquid interface. 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

To date, there has been a very limited effort devoted to developing theory for ionic block copolymers. Gonziilez-Mozuelos and Olvera da la Cruz (1994) studied diblock copolymers with oppositely charged chains in the melt state and in concentrated solutions using the random phase approximation (RPA) (de Gennes 1970). However, this work has not been extended to dilute solutions. 

In dilute solutions using a solvent system good for all blocks, there is little intermolecular interaction, and the behavior of block polymers approaches that of homopolymers or random copolymers. Conventional dilute polymer solution theory may be applied with moderate success, as in osmometry and viscoraetry, provided that the block composition is taken into account. Some complicating features may arise because of the possibility of intramolecular block separation, for example, prevention of a random coil configuration, which is usually assumed in some conventional treatments (87). 

In Table 1 we present the Zp values determined in THF and two different THE/DME mixtures. These values, on the order of 1 rm, are comparable to those reported by Discher and coworkers [38,70] and by Bates and coworkers [71,72] for PEO-PI cylindrical micelles with a core diameter of 20 nm in water. Here PEG denotes poly(ethylene oxide). Bates and coworkers deduced their values of Zp from small-angle neutron scattering experiments, whereas Discher and coworkers determined the Zp values using fluorescence microscopy. The fact that the Zp values that we determined from viscometry are comparable to those of the PEO-PI cyUndrical micelles with similar core diameters again suggests the validity of the YFY theory in treating the nanofiber viscosity data. This study demonstrates that block copolymer nanofibers have dilute solution properties similar to those of semi-flexible polymer chains. 

Theory for the phase behaviour of block copolymers in semidilute or concentrated solution is less advanced than that for melts or dilute solutions due to the complexity of interactions between polymer and solvent. The two main... 

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