Polymer brush assumption

Another approach to study the penetration depth is to use computer simulations of simple shear flow. Computer simulations can check the validity of certain assumptions implicit in the theories, such as the assumption that the shear flow does not distort the density profile. However to do this solvent molecules must be included explicitly in the simulation. Almost all previous simulations of polymer brushes have modeled the solvent as a continuum to save CPU time. 

Another technique that was used to estimate the solvent content and the number of solvent molecules per EG monomers of PLL- -PEG coatings was recently developed by Pasche et al. and involves coUoidal-probe APM surface force measurements. The main assumption made in this technique is that the unperturbed PEG layer is compressed by the colloidal probe from a fully solvated state to a solvent-free, dry state. Thus, the decrease in the layer thickness upon compression is likely to reflect the amount of solvent absorbed within the polymer brush. The results of that study are in reasonable agreement with the findings of the present work. 

If the length of the most probable chain configuration z is assumed to be equal to the thickness of the brush L, eq 14 provides the same dependence between the graft density and thickness as the Alexander theory. Whereas eq 14 accounts explicitly for the persistence length of a polymer (its degree of flexibility), with the simplified assumptions that l a, Ni N, v a3 the Alexander scaling law is completely recovered ... 

Here we outline a mean field Flory-type model introduced by de Gennes [230] and developed by Leibler [231] and Aubouy and Raphael [232]. This approach is less detailed than SCMF models but it captures the main features of the physics of segregated copolymers. Even though it makes a number of assumptions, which are a simplification in comparison with the SCMF models, its predictions of the main features (such as, e.g., variation of mean brush height L vs size and surface density o of the diblocks) agree [226] well with those of more detailed SCMF calculations [236-238]. Because of clearness and simplicity it has been used as a basic framework for many experimental papers on brush conformation [240-245] and segregation properties of end-adsorbing polymers [246-255]. 

A fully close-packed brush, with the whole area of the interface taken up by the grafting sites, would have a value of a of unity and thus a volume fraction inside the brush of unity. However, this invalidates our assumption that the concentration inside the brush is in the semi-dilute regime recall that, for volume fractions of polymer greater than a crossover value 0 v/a, where V is the excluded volume parameter, we have a concentrated solution, for which... 

To obtain useful theoretical results for the concentration profile, we need to go beyond these simple scaling arguments. Luckily, at least for the situation of relatively dense, strongly stretched, brushes, we can expect self-consistent field theories to work rather well in such a dense brush the basic mean-field assumption that any polymer chain will interact with its neighbours more than it will with itself should be well obeyed. Niunerical mean-field theories of the kind described in chapter 5 are very well suited to this kind of calculation the earliest results, due to Hirz (these results are still unpublished but some were reproduced by Milner et al. (1988)) showed profiles very different in character from those found for adsorbed chains. Rather than a concave concentration profile, the curves were notably convex, with the concentration dropping rather abruptly to zero on the outside of the brush. In fact it turns out that the profiles are rather well described by a parabolic form (see figure 6.7). It soon turned out that there was a remarkably good analytical solution to the self-consistent mean-field equations which provided an explanation for these parabolic profiles. 

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