Alexander-de Gennes model

In order to investigate the internal organization of the surface-anchored chains, we now consider the concentration profile of the layers when swollen by a good solvent. Such concentration profiles have been investigated in details theoretically, since the Alexander-de Gennes model, and we can hope to be able to dis-... 

Fig. 3.a Schematic representation of a polymer brush. L is the layer thickness and D the average spacing between two grafting points, b The monomer density profile vs distance from the grafting plane z according to the Alexander-de Gennes model. Figure adapted from [66]... 

The interaction between two surfaces each carrying a grafted polymer layer of thickness H is repulsive [15]. If the distance h between the surfaces is larger than 2H, the polymer layers do not overlap and the force between the surfaces owing to the polymer vanishes. If the distance h is smaller than 2H, the two brushes interact. It can be shown that they do not interpenetrate but that they compress. The repulsive force is then due to the osmotic pressure opposing the compression. In the Alexander-de Gennes model, the concentration in the compressed layers is c = 2cr/A and the osmotic pressure, n, which is the force per unit area between the surfaces is [6]... 

Silicon modified with 2-bromo-2-methylpropionyl bromide and 1-bromocarbonyl-1-methyl acetate was used as basal substrate. Estimated fi om the Alexander de Gennes model. 

Continuous line Repulsive force vs. thickness, measured at T = 23°C, upon compression of two facing brushes of grafting density 0.0009 chain/A. The ratio F/R, with R the radius of curvature, is plotted for direct comparison with predictions from the Alexander-de Gennes model. Dashed line Theoretical curve computed using an unperturbed thickness of 2L = 820 nm and the above grafting density. 

The other theoretical approach by Sevick [6] is essentially a variation of the Alexander-de Gennes model that takes into account the cylindrical geometry. The result for the brush thickness is given by... 

In fact, based on the above molecular picture and applying the Alexander°-de Gennes model, we could introduce reduced force and distance variables, both of which are essentially the quantities divided by the molecular weight of PS-block, and obtained universal force-distance profiles. This result suggests that our hypothesis is valid, at least for PVP-PS diblock copolymers having sufficiently long PVP blocks. 

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... 

The Alexander de Gennes [52,53] brush model considers the entropic situation in a brush. It is derived from an analogy to semi-dilute polymer solutions. There the screening of the excluded volume interactions leads to a blob structure with swollen chain sections inside the blobs [87]. In this approach each hair may be considered as a string of blobs reaching out from the surface. The blob size describes the screening of the excluded volume interactions due to the neighboring chains. Its size relates to the surface density of the blobs where ... 

However, the inaease in primary particle size to nearly 20 nm with ricinoleic acid (RA) as the surfactant can be explained using the deagglomeration model based on the Alexander de Gennes theory (AdG), and is given by Eq. (5) [24]. 

The distinctive properties of densely tethered chains were first noted by Alexander [7] in 1977. His theoretical analysis concerned the end-adsorption of terminally functionalized polymers on a flat surface. Further elaboration by de Gennes [8] and by Cantor [9] stressed the utility of tethered chains to the description of self-assembled block copolymers. The next important step was taken by Daoud and Cotton [10] in 1982 in a model for star polymers. This model generalizes the... 

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]. 

The important thickness and high monomer and charge densities encountered in polyelectrolyte brushes are expected to provide emulsion droplets with large steric and electrostatic repulsions. As a consequence, diblock polyelectrolyte surfactants are suitably designed to be effective emulsion stabilizers. Pincus [212] has proposed a description of polyelectrolyte brushes using an approach similar to that of Alexander and de Gennes for neutral brushes [75,213,214], The latter model is thus first presented before developing that of polyelectrolyte brushes. 

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]. 

---