Swollen coils

For a semi-flexible tube in a dilute environment, local repulsive potentials among parts of the fiber induce a self-avoiding random walk configuration (swollen coil [151]). In a crowded environment, the depletive action may dominate and the fiber will tend to collapse on itself, forming a globular phase. We know from standard statistical physics of polymers that this latter phase... 

Since this chapter is concerned with block copolymers in dilute solution, it is useful to include a definition of the dilute regime for polymer solutions in the Introduction. This regime extends up to a volume fraction above which swollen coils overlap (de Gennes 1979) ... 

The degree of the orientational ordering in the swollen coil can be estimated using the Edwards equation (see Ref.24))... 

In the following two sections we first consider colls In solutions which are so dilute that coll-coll Interaction does not play a role. Section 5.2a deals with random-walk chains where the Interaction between the units within one chain may be neglected. We denote these as ideal chains. Section 5.2b treats swollen coils In which the monomeric units repel each other due to so-called excluded volume effects. 

The results obtained are the following In the absence of salt, the macromolecule has the swollen coil conformation due to the repulsive electrostatic interactions independently of the quality of the solvent. 

Studies on the effect of the conformational transition of PMA on RGG-RB energy transfer provide further information on the intermediate states that lie between compact coil and extended polymer chain. These studies confirm that the conformational transition of PMA induced by pH is a progressive process over several pH units, and that the polymer is partly swollen coil at pH 4-5. 

At the same time the data for PPQX macromolecular coil in chloroform, estimated according to the Eq. (4), are disposed lower than the dependence D d) for swollen coils, although the Eq. (12) of Chapter 1 supposes minimum value for them (6 =0). The dependence D (d) for swollen coils (the curve 2 in Fig. 72) does not correspond to the experimental data. Let us consider the Eq. (18) of Chapter 1 derivation, adduced in Ref [25], for this discrepancy explanation. It is supposed, that the fiactal solvent molecules screen the excluded volume interactions, that is, reduce the level... 

You can see that Yp 1 if p > 3. This indicates that many-body collisions are really rare. Even the number of three-body collisions in a swollen coil is of order 1. So they cannot seriously affect the conformation of the coil. In contrast, the number of simultaneous pair collisions is about This... 

Now let s concentrate on the entropy contribution Heff (a) to the free energy. In the case of a swollen coil, this contribution was described by Equation (8.10) resulting from (7.5). Would it be valid for a < 1 as well Let s think. Equation (7.5) gives the free energy of an ideal coil whose end-to-end distance is of order R. This is the only condition on the coil s... 

We were talking in Chapter 8 about the swelling of a real (not ideal) pol Tner coil — due to the fact that every monomer is not an infinitesimal point, but a body of maybe small, yet still finite, size. We have seen that the size of a swelling coil is R N /. A swollen coil is therefore also a fractal, with a fractional dimensionality df 5/3. 

S. Why, they ll learn that such things as a Gaussian coil, a swollen coil, and a randomly branched polymer are all fractals. This is interesting in its own right. Mind you, there are more things in life than polymers. It might be interesting to hear about other fractals, the scale invariance of different objects, and the mathematical idea of fractional dimensionality. 

We presume you might be interested in going through the same kind of proof for a swollen coil (which we discussed in Chapter 8) we proved then that R = hN /, where b is proper N-independent, that is, associated with the monomer, length scale — see formula (8.14)), as well as for a random tree R = feiV / ). You would then be able to see for yourself that the power laws do indeed correspond to self-similar objects, that is, to those which have, say, a g-unit organized in the same way as the whole thing (obeying the same power law as the whole chain). 

As the first approximation the authors [132] supposed, that the smallest stirring intensity (500 rpm) did not deform macromolecular coil to some extent significantly and therefore the value for polyarylate D could be determined in that case according to the Eq. (4). In its turn, the dimension ds for swollen coil with excluded volume interactions appreciation can be determined according to the Eq. (39). 

As before, we have for a > 1 and < 0.5 the swollen coil with R oc AP y when the particle concentration nJNv sufficiently high, we get the swollen coil even for X > 0-5 because the second term prevents the collapse. We may say the great excluded volume of the hard spheres hampers collapsing of the chain even tmder poor solvent conditions. 

In a dilute solution, when the polymer is in a coiled state (Figure 5(a)), the diffusion of hydrophobic particles into the coil is normally faster than the chemical reaction." In this case, the local concentration of partides H inside the swollen coil is practically the same as in the bulk. Therefore, we expect that at... 

Fig. 2 Schematic phase diagram of a single flexible polymer chain in the thermodynamic limit (Af —> Qo) as a function of temperature T and range of attractive monomer-monomer interaction X. For 2 > At, there occurs a transition at T = 6 X) from the swollen coil to the collapsed fluid globule. At TcystCiV = < ) the globule crystalhzes. Due to slow crystallization kinetics, this transition may be undercooled and at FcystW the collapsed globule freezes into a glassy slate. Since it was assumed that the transition lines vary linearly with the interaction volume A, A rather than A has been chosen as an abscissa variable. Adapted from Binder et al. [4]...

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