Micelles scaling theories

In scaling theories, Rc, Rm> and Z are directly correlated to NA and Mb for the investigated micelles. Two limiting cases have to be distinguished, the starlike or hairy micelles with N < Nb and the crew-cut micelles with Na > Nb (Sect. 2.3). 

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

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. 

The scaling theory for spherical polymer brushes due to Daoud and Cotton (1982) (Section 3.4.1) has been applied to analyse the coronal density profile of block copolymer micelles by Forster et al. (1996). If the density profile is of the hyperbolic form r as found by FOrster et al. (1996) for the coronal layer of block copolymer micelles, the brush height scales as... 

Fig. 3.23 Coronal thickness, A a. plotted as a function of NX fv f°r micelles formed by PS-poly(2-cinnamoylethyl metharylate) diblocks in cyclopentane (Tao et at. 1997). This yields a straight line in accord with the predictions of scaling theory for the micellar radius (the core radius for these micelles was small enough to be neglected, then RA - R).

Fig. 3.24 Micellar core radius, RB, as a function of Ng for PS-poly(caesium acrylate) (A) and PS-poly(caesium methacrylate) ( ) in toluene (Nguyen el al. 1994). A linear relationship is anticipated by the scaling theories of Zhulina and Birshtein (1985) and Halperin (1987,1990) for type IV micelles.

Scaling Theory of Non-ionic Block Copolymer Micelles. 69... 

Scaling Theory of Micelles with Polyelectrolyte Corona. 78... 

We first review the main results of the scaling theory for micelles formed by block copolymers with a neutral (non-ionic) soluble block [37,45-50],... 

In the framework of the scaling theory, the corona of a spherical micelle can be envisioned [53-56] as an array of concentric spherical shells of closely packed blobs. The blob size, (r) = r/grows as a function of the radial distance r from the center of the core. Each blob comprises a segment of the chain within the local correlation length of the monomer density fluctuations [57], and corresponds to a contribution to the free energy of steric repulsion between the coronal chains. After calculating the total number of blobs in the micellar corona, one finds fhe free energy (per coronal chain) as ... 

Copolymers with quenched PE blocks form micelles that strongly respond to variations in the ionic strength in solution. To illustrate this behavior, which is already anticipated above from the scaling theory and the boxlike model, we show the micelle characteristics in Fig. 12. In these graphs, we plot the aggregation number, the CMC and the sizes of the core and corona as a function of the fractional charge along the corona block for various values of the salt concentration... 

The theory proposed by Halperin and Alexander (H-A theory) [60] is based on the structural scaling description of polymeric micelles outlined in Sect. 2.1.2. Using a combination of scaling theory and Kramers rate theory for diffusirai in an external potential [61], the expulsion rate for both crew-cut and star-like spherical micelles was derived. Moreover, Halperin and Alexander discussed different scenarios of chain exchange between micelles. 

The last reported diblock copolymer family that formed tubular aggregates in block-selective solvents was poly(phenylquinoline)-fc/ock-polystyrene or PPQ-PS, where PPQ was a rigid-rod block [47]. Such tubes are not discussed further for the following reasons First, the tubes had diameters of several micrometers and were not nanotubes. Second, the formation mechanism and chain packing in such tubes were not well understood at all. While Halperin [48] has developed a scaling theory for micelle formation from rod-coil diblock copolymers with the rod block forming the core, the theory did not apply to the PPQ-PS system as the block-selective solvents used were good for the rod PPQ block rather than the coil PS blocL... 

Figure 9 Dimensionless apparent molecular weight Mapp times the concentration c of giant surfactant micelles of hexadecyl trimethyl ammonium bromide in brine, obtained by means of light scattering measurements at a temperature of 306 K. Circles experimental data of Buhler and coworkers [50]. Line renormalization group theory for self-assembled flexible chains fitted to the data [51,52], Also indicated by the dashed lines are the predictions of scaling theory in the dilute and semidilute regimes [1,53], which agree well with results of Monte Carlo simulation [52],...

In scaling theories, two limiting cases have to be considered for a monodisperse AB diblock copolymer in a selective solvent for the A block [82], These are the hairy or starlike micelles, with Nj N and accordingly L>Rc, and the crew-cut micelles for which Va and Rc>L (see Fig. 2). For the case of hairy micelles, based on the star polymer theory of Daoud and Cotton [84], the following scaling relations for N gg and L have been derived ... 

For crew-cut micelles, the theories of Alexander and de Gennes for polymer brushes have been used, assuming micelles with uniformly stretched chains in the core and the corresponding scaling relations are ... 

Key words Scaling theory - micelle formation - polyoyethylene dodecyl ethers - coil-globule transition -surface tension... 

In order to further discuss the temperature effect by a scaling theory in polymer science, the quantity of log CMC was converted to the standard Gibbs free-energy change of micellization AG by eq. (1). 

Figure 7.8 (a and b) Cartoon representing the oriented wedge theory, as presented by Harkins et and Langmuird The monolayers covering emulsion droplets have different frustration energies, which favor one emulsion type over another. Note that the picture shows the macroscopic emulsion droplets and not the surfactant micelles. The theory is wrong because the monolayers are essentially planar on the molecular scale (c and d)... 

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