Chains confined

A more accurate analysis of this problem incorporating renormalization results, is possible [86], but the essential result is the same, namely that stretched, tethered chains interact less strongly with one another than the same chains in bulk. The appropriate comparison is with a bulk-like system of chains in a brush confined by an impenetrable wall a distance RF (the Flory radius of gyration) from the tethering surface. These confined chains, which are incapable of stretching, assume configurations similar to those of free chains. However, the volume fraction here is q> = N(a/d)2 RF N2/5(a/d)5/3, as opposed to cp = N(a/d)2 L (a/d)4/3 in the unconfined, tethered layer. Consequently, the chain-chain interaction parameter becomes x ab N3/2(a/d)5/2 %ab- Thus, tethered chains tend to mix, or at least resist phase separation, more readily than their bulk counterparts because chain stretching lowers the effective concentration within the layer. The effective interaction parameters can be used in further analysis of phase separation processes... 

Hashimoto T., Shibayama M., Kawai H., and Meier D. J. Confined chain statistics of block polymers and estimation of optical anisotropy and domain size. Macromolecules, 18, 1855, 1985. 

Constraint release (CR). This takes place if a confining chain moves out of the way of a given chain and thus opens some freedom for lateral motion. This phenomenon is an intrinsic many-body effect and for monodisperse polymer melts becomes significant mainly in the creep regime. 

R. A. Vaia, B. B. Sauer, O. K. Tse, and E. P. Giannelis, Relaxations of confined chains in polymer nanocomposites Glass transition properties of poly(ethylene oxide) intercalated in montmorillonite, J. Polym. Sci. B Polym. Phys. 35(1), 59-67 (1997). 

Sikorski and Romiszowski455 study confined branched star polymers by on-lattice MC simulation. Attractive forces are excluded and only excluded volume accounted for, thus making the simulations relevant for chains in a good solvent. Contrary to expectation, they find that the diffusion constant is very similar for either moderate or highly confined chains and scales approximately as A 1, though a more accurate representation is suggested by... 

Constraint release mechanism when chain B reptates away, it releases the constraint on chain A. Later, this constraint is replaced by chain C. which confines chain A in a displaced tube. 

More recently, a theory based on confined chain statistics (CCS) predicted that Al should be a decreasing function of Z. Thus, modification of Eq 4.4 resulted in an expression valid for X Z > 20 [Spontak and Zielinski, 1993] ... 

For the computation of segment density profiles in polymer solutions near interfaces one can use the fact that there is a close analogy between the diffusion of a Brownian particle and the flight of a random walk [20, 21]. A dififusion-like equation can be derived to evaluate the partition function of polymer chains. Given the boundary condition this diffusion equation can be solved. The partition function z h) of one confined chain is given by [1, 22, 23]... 

Since the ideal chains do not interact the partition function for N confined chains can be written as... 

Torres and co-workers (304) studied the dependence of Tg on film thickness, and found both increases and decreases in the apparent Tg of thin films, depending on whether the film is free or deposited on an attractive substrate. Rranbuehl and co-workers (305) found that Tg decreases with confinement by noninteracting surfaces consisting of planes (one dimensional, 1-D), tubes (2-D), and spheres (3-D). Perhaps the surface-to volume ratio, and the influence of free volume at the constraint surface, creates additional freedom of motion for the chains in these types of confinement. Consistent with this view is the observation that when chains are attracted to a surface, and thereby immobilized, the Tg increases (304). Similar conclusions were reached by Aoyagi and co-workers (306) in a spring-bead model simulation of confined chains. 

The presence of a confined interfacial layer, with specific rheological behavior, is proposed to explain this complex behavior. The low stiffness of PDMS allows a competition between the (low) cohesion and the confined chain layer at the PDMS surface and the adhesion level (interfacial interactions between PDMS and substrates). At low speeds, interfacial interactions have a significant effect and partly govern the friction, and at high speeds the influence of the substrate surface becomes negligible and friction is then governed by the polymer s intrinsic viscoelastic behavior. Experimental results underline the subtle competition between interfacial interactions and polymer rheological properties, especially for PDMS samples. Comparison... 

Figure 2.61. Density profile of the end of the Gaussian chain when the other end is at z = dj 2. The density is compared for the confined chain (solid line) in a slit of walls at z = 0 and z = d and the unconfined chain (dashed line). R, = d/A was assumed.

We consider a real chain consisting of N monomers of size b and confined to a cylindrical pore of diameter d. When the chain dimension R in the free solution is smaller than the pore size, the chain does not feel much of the effect of the pore wall. As exceeds d, the chain must adopt a conformation extending along the pore because of the excluded volume effect. As R increases further, the confined chain will look like a train of spheres of diameter d (see Fig. 2.64). The excluded volume effect prohibits the spheres from overlapping with each other. Therefore, the spheres can be arranged only like a shish kebab. The partial chain within each sphere follows a conformation of a real chain in the absence of confinement. The number of monomers in the sphere is then given by... 

The linear dimension of the chain in the slit is different from the counterpart in the cylindrical pore. Because the confined chain follows the conformation of two-dimensional excluded-volume chain,... 

Note that Equation 6.4 predicts that for a given interanchor distance s, brush thickness L is an increasing function of cylinder radius R, which is a counterintuitive result if we recall the simple argument that smaller curvatures lead to less confined chains that should also be less extended. 

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