Alexander model

The Alexander model is based on two assumptions that enable simple expressions for these two terms (1) The concentration profile of the layer is step-like. That is, the monomer volume fraction within the layer, (p Na3/d2L, is constant, independent of position (2) The chains are uniformly stretched. That is, all chain ends are positioned on a single plane at a distance L from the surface. [In this paper, we use the symbol to mean approximately equal to or equal to within a numerical factor of order one we use to mean proportional to .] The first assumption simplifies the calculation of Fin, while the second yields a simple expression for Fel. 

The essential idea of the Alexander model, a global balance of interaction and stretching energies, can be applied to other situations involving tethered chains besides the good solvent case. In theta or poor solvents, the interaction term must be modified to account for poorer solvent quality. A simple limit is precisely at the theta point [29, 30] where binary interactions effectively vanish (% = 1/2 or v = 0). The leading term in Fim now accounts for three-body interactions ... 

Daoud and Cotton [10] pioneered this geometrical analysis of tethered layers with spherical symmetry, which was later extended by Zhulina et al. [36] and Wang et al. [37] to cylindrical layers. The subsequent analysis is purely geometrical and requires no free energy minimization. The tethered layer consists of a stratified array of blobs such that all blobs in a given sublayer are of equal size, E , but blobs in different layers differ in size. This corresponds to the uniform stretching assumption of the Alexander model. 

The Alexander model and its descendants impose strong restrictions on the allowed chain configurations within the tethered assembly. The equilibrium state thus found is subject to constraints and may not attain the true minimum free energy of the constraint-free system. In particular, the Alexander model constrains the segment density to be uniform and all the chain ends to be at the same distance from the grafting surface. Related treatments of curved systems retain only the second... 

The importance of polydispersity is an interesting clue that it may be possible to tailor the weak interactions between polymer brushes by controlled polydispersity, that is, designed mixtures of molecular weight. A mixture of two chain lengths in a flat tethered layer can be analyzed via the Alexander model since the extra chain length in the longer chains, like free chains, will not penetrate the denser, shorter brush. This is one aspect of the vertical segregation phenomenon discussed in the next section. 

As described in Sect. 2.4, SCF calculations are useful in determining local details of density profiles. A more local examination of profiles is indeed necessary to study the question of interpenetration in more detail. The analytical SCF theory [56, 57] shares with the adapted Alexander model embodied in Eq. 35 the characteristic of impenetrability. The full numerical SCF theory is necessary to... 

The Alexander model allows a simple approach to this problem. Within this model, each tethered chain is, in effect, confined within a cylindrical capillary of diameter d. Combining Eq. 5 and 7, we can express the stretching energy as ... 

In a recent work, the original Halperin and Alexander model was, in light of new experimental data, extended for the case of high cOTicentrations and particularly for the case of overlapping coronal A-chains [64]. As noted, Eq. 34 is only approximately correct and several corrections should be included. In particular, as is evident from Fig. 4, Eq. 34 does not give a complete description of the activation barrier. In addition to the surface free energy of the exposed insoluble B-block, the expulsion process involves interactions with the corona chains. The free energy of the activated state must therefore be calculated in more detail. 

Figure 16 A schematic cross-section of a planar brush within the Alexander model. All the blobs, represented by circles, are of equal size. (Adapted from Ref. 29.)...

Our discussion is founded on the Alexander model . This Flory type model is based on two assumptions (i) The concentration profile is step like with a constant monomer volume fraction (j) = Na /La where N is the polymerization degree, a is the monomer size and L is the thickness of the layer, (ii) the chains are uniformly... 

Within the Alexander model, a chain in a brush behaves as if confined to a cylindrical capillary of cross section cf and height L (Figure 1). Blobologically, this is an especially important point since the diameter of this virtual capillary, (7 /, sets the blob size, In terms of our preceding discussion this follows because at equilibrium F / F nf or In turn, this leads to F and... 

The internal modes of the brush may be described by an approach due to deGennes This method enables the analysis of collective, elastic deformations of brushes of various geometries. Our presentation focuses on the breathing modes of a flat brush as described by the Alexander model. In this case the layer is transitionally invariant... 

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