Flory radius

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... 

Rfo is the Flory radius describing the coil size at concentrations below c. ... 

Flory exponent for the polymer chain size Flory radius... 

This represents the length of the polymer if it is not squeezed, but constrained to a certain diameter d. The Flory radius for a polymer in a mushroom configuration is defined by the number of segments N, (in this case 45) and the length of the monomeric unit, a, (in this case, 3.5 A) and is written as... 

The Peclet number is seen to increase as the cube of the Flory radius showing the relative increase in importance of viscous forces with increasing polymer length. 

This is exactly the Peclet number defined by Eq. (5.3.25), which measures the characteristic rotational Brownian diffusion time to the time scale defined by the reciprocal of the shear rate. It is the same measure found for dilute polymer solutions with the particle radius here replacing the Flory radius for the polymer. 

In the dumbbell model, a polymer chain in a solvent is pictured as two massless spheres of equal size connected by a frictionless spring. The spheres experience a hydrodynamic drag proportional to their size, characterized by the Flory radius. Assume that the displacement of the spring generated by the thermal energy is also characterized by the Flory radius. Write the equation of motion for the dumbbell and show that the characteristic relaxation time for the chain deformation is that given by Eq. (9.2.1). 

In the lower limit of surface coverage, de Gennes adopted the oversimplified view that each chain, composed of N segments each of length /, can be represented by a hemisphere that is centred on the point of attachment of the terminus of the chain to the surface. Its radius is the Flory radius of a coil in a good solvent, i.e. For the chains to be well spaced and not... 

The important feature of this expression is the fact that the brush height depends on iV in a linear way, rather than as the square root, as for an ideal chain, or the 3/5 power, as it does for an isolated chain in a good solvent. This means that the chains are strongly stretched this is most easily seen by rewriting (6.1.5) in terms of the Flory radius of gyration of an isolated chain Rp and the average distance between grafting points D. This yields... 

A local correlation length f(x) was defined (the length over which excluded volume effects are screened), the polymer layer at the interface then resembling a network of variable mesh size (x). When the hydrodynamic equation is integrated up to a distance correspjonding to the longest mesh size Rp (the Flory radius of polymers in solution), it is found that the hydrodynamic thickness scales... 

Cylindrical Morphology. Fredrickson [32] studied surfactant-induced lyotropic behavior in polymer solutions. If the degree of polymerization of the backbone is N, the number of surfactant molecules is ctN, where a is a dimensionless parameter defined as the number of surfactant molecules per monomer unit. If the degree of polymerization of the surfactant is M, the Flory radius for surfactant molecules is... 

A similar law is expected to hold for swollen gels. Here the analog of g is the number of monomers per chain N in the network, and the analog of is the Flory radius Rp N ) given in terms of the excluded volume parameter V (measuring the quality of the solvent) by eq. (IV.49). We take N > N to ensure that the chain is indeed trapped into a tube. Then we expect a reptation time of the form ... 

The described distribution function refers to the asymptotic limit of large degrees of polymerization. It is important to note that, as for ideal chains, p R) includes one parameter only, now the quantity called in the literature Flory-radius . Rp is a measure for the diameter of the volume occupied by the expanded polymer chain, with the identical definition as for ideal chains... 

This is Flory s result. We identify R with the Flory radius Rp and write... 

The last equation relates the second virial coefficient to the Flory radius and the degree of polymerization of the chains. 

Fig. 3.8. Pair distribution function of an individual chain in a semi-dilute solution, exhibiting different regions with specific power laws 47rr g (r/Rp) Rf denotes the Flory-radius in the dilute state. The dotted line gives the function 4nr (cm). The dashed line indicates the pair distribution function for all monomers, 47rr g, which deviates from 47rr g for r > s-...

Considering these results, we can now see how the change from the expanded chains in dilute solutions to the ideal chains in a melt is accomplished With increasing concentration the screening length decreases continuously, and if it comes down to the thermic correlation length excluded volume effects disappear. Simultaneously the chain size shrinks, from the Flory radius Rp in the dilute solution down to the radius Rq of the Gaussian chains in the melt. 

Flory radius size of one coil in good solvent condition end-to-end distance of coil transmission factor absolute temperature... 

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