Alexander-de Gennes scaling

The dependence of h on M and a is rather weaker than that predicted by Alexander-de Gennes scaling laws and by the self-consistent field theory of Milner, Witten and Cates (h A self-consistent field theory of... 

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... 

In the case of crew-cut micelles, //corona < Rcok and the logarithm in (22) can be expanded up to the term linear in //corona// core, to give / corona/ B — Z/corona/i / = //corona/. The thickneSS Of the corona, //corona, scales as //corona = In the framework of the Alexander-de Gennes blob model [51, 52], the micellar corona (the planar brush) can be envisioned as an array of closely packed blobs with size = 5 /, equal to the average distance between the coronal blocks. We note that a constant size of the blobs implies //corona Na. The number of coronal blobs per chain //corona/ is proportional to the free energy of the interchain repulsion that equals fcorona/kBT = ... 

The thickness of the corona Hcorona scales as Hcorona- Na (s/fl )" In the framework of the Alexander-de Gennes blob model, 2 (he micellar corona (the planar bmsh) can be envisioned as an auay of closely packed blobs of size equal to the average distance between... 

Numerous theories, models and mathematical approaches have been developed over the years in order to describe the micellisation process and the dependence of fundamental structural parameters of the micelles, like cmc, aggregation number (Nagg), overall size (Rm), core radius (Rc) and corona thickness (L), on the molecular characteristics of the block copolymer, with respect to the degrees of polymerisation of the constituent blocks (Na and N ), as well as the Flory-Huggins interaction parameters x between the blocks and between the blocks and the solvent. Some of these approaches use the minimisation of the total free energy of the micellar system so as to extract relations between the copolymer and micelle features, while others are based on the scaling concept of Alexander-de Gennes and... 

As long as D/Rg > 10 and a uniform brush gets formed all along the pore length, the measured adsorbed amounts are identical to those measured on flat substrates and also follow the same scaling law as a function of chain molecular weight (T This indicates that the Alexander-de Gennes... 

A successful theoretical description of polymer brushes has now been established, explaining the morphology and most of the brush behavior, based on scaling laws as developed by Alexander [180] and de Gennes [181]. More sophisticated theoretical models (self-consistent field methods [182], statistical mechanical models [183], numerical simulations [184] and recently developed approaches [185]) refined the view of brush-type systems and broadened the application of the theoretical models to more complex systems, although basically confirming the original predictions [186]. A comprehensive overview of theoretical models and experimental evidence of polymer bmshes was recently compiled by Zhao and Brittain [187] and a more detailed survey by Netz and Adehnann [188]. 

Rabin Y, Alexander S (1990) Europhys Lett 13 49 de Gennes PG (1979) Scaling Concepts in Polymer Physics, Cornell University Press Ithaca, NY JoannyJF (1992) Langmuir 8 989 Subramanian G,Williams DRM,Pincus P (1996) Macromolecules 29 4045 Milner ST (1991) Science 251 905 Semenov AN (1995) Langmuir 11 3560 Brochard-WyartF (1993) Europhys Lett 23 105 Brochard-Wyart F, Hervet H, Pincus P (1994) Europhys Lett 26 511... 

The first theories that implemented a proper balance of intramolecular interactions and conformational elasticity of the branches were developed by Daoud and Cotton [21] and by Zhulina and Birshtein [22-24]. These theories use scaling concepts (the blob model), originally developed by de Gennes and Alexander to describe the structure of semidilute polymer solutions [64] and planar polymer brushes [65, 66]. Here, the monomer-monomer interactions were incorporated on the level of binary or ternary contacts (corresponding to good and theta-solvent conditions, respectively), and both dilute and semidilute solutions of star polymers were considered. Depending on the solvent quality and the intrinsic stiffness of the arms, the branches of a star could be locally swollen, or exhibit Gaussian statistics [22-24]. 

For the corona contribution, some care has to be taken. In order to calculate the free energy of this part, the number of blobs has to be calculated for each morphology, taking into accotmt that the radial dependence of the density changes with the curvatore. For a completely planar surface, de Gennes and Alexander [36,37] showed that the blob size wDl be constant and scale by s, where s is the area available per chain. For a cuiwed micellar core, the surface per corona chain will naturally increase... 

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 ... 

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