Flexible polymer coils

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

As we have seen, the phase behaviour of block copolymers consisting of flexible polymer coils is remarkably rich. If one of the blocks is rigid, the copolymer would be expected to exhibit even more complex phase behaviour. For example, the rigid block could be mesogenic. This leads to the possibility of self-assembly of structures consisting of domains of liquid crystalline material within a microphase-separated block copolymer superstructure. Diblock copo-... 

FIGURE 14.4 (a) Schematic diagram of a flexible polymer coil with one end contained in a small volume element (dv, dy, dz) and the other located at the origin of the coordinates, (b) Hypothetical sphere containing the coil. 

Polymers that possess long rigid sections in the backbone chain usually cannot be treated as random coils and exhibit characteristic properties peculiar to non-flexible systems. Some display liquid crystalline behaviour, others can be spun into strong fibres and interest in such structures is growing. Their very nature often makes them insoluble in common solvents and this complicates characterization procedures. When suitable solvents are found and precautions taken to overcome complicating features such as fluorescence in solutions, characteristic dilute solution parameters can be measured. Typically one observes high chain extensions but low virial coefficients, and values of p > 0.8, which is the accepted upper limit for flexible polymer coils. A few of the more recent studies are summarized in Table 2 and occasionally it can be seen that some behave like random coils. This tends to be exceptional and, when a suitable analysis can be made, the worm-like chain model seems to be an accurate description, particularly for cis syndiotactic poly (phenyl... 

Fig. 1. Spatial segment density distribution of a flexible polymer coil in a good solvent from a dilute to a concentrated solution (schematic).

A theoretical model has been proposed to treat the concentration dependence of the flexible polymer coil dimension in semidilute solutions by a perfectly reflecting spherical wall around each segment of the coil, which influences the random walk conformation. In a 0-solution the intersegmental interaction is expressed by a potential... 

The radius of the spherical reflecting wall around each segment of the coil in solution should be connected to the concentration of the solution. At least it is reasonable to assume a relation c in the semidilute region. The function ft2e(c)/h as a function is shown in Fig. 7, which should simulate the concentration dependence of the flexible polymer coil dimension in a 0-solution. It is interesting to note that the normalized coil dimension, A20(c)/h o decreases with increasing concentration to a minimum value of 0.69 at = 1 and then it will increase with further increase in concentration in a concentrated solution. The function h Q c)/ o given by (Eq. 4) approaches unity when t 0, i.e., c - < , in accord with the experimental fact that in an amorphous solid film the coil dimension is the same as that in a dilute 0-solution. 

Flory-Krigbaum Theory Case of Flexible Polymer Coils... 

Most biological polymers, such as proteins and nucleic acids and some synthetic polymers, have relatively inflexible chains. For rigid particles, the size is no longer of predominant importance, because the polymer chain is no longer in the form of a flexible random coil instead, shape becomes an important parameter. Following are some theoretical proposals for the estimation of the shape factor p from the viscosity measurement (table 4). The term f/fo is sometimes denoted as p, Perrin constant. 

Free-draining models were among the first to be considered [14-18]. For flexible polymer chains of sufficient length, [77] behaves as if the polymer coil occupied a spherical volume through which the solvent cannot flow. Under these conditions,... 

FIG. 2.14 The effect of fixed bond angle in restricting the flexibility of a polymer coil. For tetrahedral bonds, 6 = 109°. 

Accordingly, given the necessity from equilibrium coil dimensions that bt> 1, the shear rate and frequency departures predicted by FENE dumbbells are displaced from each other. Moreover, the displacement increases with chain length. This is a clearly inconsistent with experimental behavior at all levels of concentration, including infinite dilution. Thus, finite extensibility must fail as a general model for the onset of nonlinear viscoelastic behavior in flexible polymer systems. It could, of course, become important in some situations, such as in elongational and shear flows at very high rates of deformation. 

Changes in the flexibility of polymer coils owing to concentration - variation may effect the entropy. Huggins [21] introduced two corrections for the athermal entropy of mixing, that take into account the influence a second polymer has on the stiffness of the other polymer... 

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