Coil, dimensions excluded volume

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). 

We can conclude that the dimensions of a macromolecular coil exhibiting excluded-volume effect are larger than those of the ideal coil. 

The subscript 0 on 1 implies 0 conditions, a state of affairs characterized in Chap. 1 by the compensation of chain-excluded volume and solvent effects on coil dimensions. In the present context we are applying this result to bulk polymer with no solvent present. We shall see in Chap. 9, however, that coil dimensions in bulk polymers and in solutions under 0 conditions are the same. 

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

Here a0 is a constant called the effective bond length of the chain, and as(z) is a dimensionless quantity called the linear expansion factor of the chain. The latter depends on long-range interactions between pairs of monomer units and chain length through the so-called excluded-volume parameter z. For details of these quantities characterizing the dimensions of random-coil polymers, the reader is referred to a recently published book by Yamakawa (40). At this place we simply note that as tends to unity in the absence of excluded-volume effect. 

It can be shown that the < 2> of an interrupted helical polypeptide is expressed by Eq. (C-3) for mean-square dipole moment of a random-coil unit. Precisely, this replacement is permissible if we neglect excluded-volume effects. Nagai (107) has shown theoretically that these effects on < 2> are virtually absent in randomly coiled macromolecules, even when they are appreciable on the molecular dimensions. It is our belief that Nagai s conclusion may apply to interrupted helical polypeptides as well. 

For dilute solutions in good solvents the net excluded volume is positive, and coil dimensions are expanded beyond their unperturbed values. The expansion... 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

Experiments due to neutron scattering by the labelled macromolecules allow one to estimate the effective size of macromolecular coils in very concentrated solutions and melts of polymers (Graessley 1974 Maconachie and Richards 1978 Higgins and Benoit 1994) and confirm that the dimensions of macromolecular coils in the very concentrated system are the same as the dimensions of ideal coils. It means, indeed, that the effective interaction between particles of the chain in very concentrated solutions and melts of polymers appears changes due to the presence of other chains in correspondence with the excluded-volume-interaction screening effect. The recent discussion of the problem was given by Wittmer et al. (2007). 

It is pertinent to consider separately the enhancement effect of salt on two steps the initiation step (onset of the flow) and the structured flow. The transport rates are related to the properties of the final structured flow and are contributed from the effects on both steps. The effect on the initiation step is clearly noticed since the critical PVP concentrations for the occurrence of the structured flow depended on the kind of salt. Effects of a salt on the cross diffusion constants of the two polymer components will be examined on both excluded volume and frictional effect. The effect on the excluded volume interaction between the two polymer components is expected to be small. This expectation is partly supported by the result that coil dimension of PVP was not influenced by the addition of a salt at 2 M in the cases of three salts LiCl, NaCl and CsCI, while these salts showed quite diverse effects on the trrmsport rates of PVP. Since viscosities vary with the kind and the concentration of salt, frictional coefficients are influenced by the presence of a salt. In this respect cross diffusion constants may be affected by salt through a change in viscosity of the medium. 

The above calculations assume that the gross chain conformations are those of a random walk, which is the case in the melt. However, for an isolated polymer molecule in a dilute solution, the average conformation is affected by excluded-volume interactions between one part of the chain and another. Because the chain must avoid self-intersection, the conformation of the chain will be that of a self-avoiding walk, rather than a random walk, if the solution is athermal—that is, if all interactions are negligible except excluded volume. Self-avoiding walks lead, on average, to more expanded coil dimensions, since expanded configurations are less likely than contracted ones to lead to self-intersection of the chain. Thus, in an athermal solution, the mean-square end-to-end dimension of a polymer molecule scales as... 

Using this approach SANS has been used to measure the dimension of the Gaussian coil structure of a single chain in melts, solution and blends, provided an affirmation of the screened excluded volume model, and a verification of scaling laws in polymer solutions, determined the structure of diblock copolymer aggregates, and established the relationship between the micro and macroscopic deformation in rubber elasticity. 

It is weU known that the dimensions of a pol3mer coil in dilute solution are influenced by two factors the polymer solvent interaction and the intramolecular excluded volume effect. The energy of interaction between polymer segments and the molecules of a good solvent tends to increase the chain dimensions, because expansion creates more polymer-solvent contacts. The intramolecular excluded volume effect also increases the dimensions. 

It can be proved that these unperturbed dimensions approximate to the statistically determined dimensions of the isolated chain under 0 conditions the excluded volume effect is compensated by a positive, i. e. unfavorable, polymer-solvent interaction enogy. The rmperturbed dimensions of polymer chains are, therefore, subject in general to direct experimental determination in dilute solution in appropriate solvents in which the volume exclusion effect upon coil dimensions is nullified. 

A discussion of random coil conformations is difficult because it perforce includes many deductions which involve excluded volume theories. This is a very controversial subject and extremely difficult to discuss in an easily grasped way. It must be emphasized however that this secticHi is not concerned with excluded volume theory, but rather with the problem of how unperturbed dimensions may be related to polyelectrolyte structure. 

Ideal solvents are necessarily poor solvents for the pol57mer in question. In good solvents the linear dimensions of the potymer coil exceed the tmperturbed dimensions by a factor a, due to the excluded volume effect. 

Excluded volume effects (2) in polymers are defined as those effects which come about through the steric interaction of monomer units which are remotely positioned along the chain contour. Each Individual interaction has only a small effect, but, because there can be many such interactions in a long polymer, excluded volume effects become very large. One consequence of excluded volume is to expand the polymer coil dimensions over that predicted from simple random walk models. The "unperturbed values of the... 

We may now consider the effect of excluded volume on dimensions of polymer coils in solution. It may be recalled that the mathematical model for evaluating is that of a series of connected vectors (representing the... 

The so-called 0 (theta) conditions, in marginally weak solvents, are usually preferred in solution viscosity measurements. Polymer chains are believed to manifest their "unperturbed dimensions" under 0 conditions. This is a result of the nearly perfect balancing of the effects of "excluded volume" (a consequence of the self-avoidance of the random walk path of a polymer chain in a random coil configuration) by unfavorable interactions with the solvent molecules. 

Figure 1 Examples of fractal dimensions of 2-D coils. The fractal dimension can be calculated as the exponent relating the number of monomers to the radius of gyration of the polymer. In (a), the coil is quasi-linear, the fractal dimension is 1. In (c), the monomers occupy all the surface, the fractal dimension is 2 (the dimension of the space concerned by the coil). In(b), the fractal dimension is intermediate between 1 and 2. A typical coil with an excluded volume geometry has a fractal dimension of 4/3 (in a 2-D space)...

This equation applies for polymeric solutions under theta conditions. Theta conditions are those at which excluded volume effects (expansion of the dimensions of the ideal coil) are exactly compensated by polymer solvent interactions (Chapter 25). The dependence between intrinsic viscosity and MW is given by the Mark-Houwink-Sakurada equation (see also Chapter 1) ... 

Flory (1953) has presented a celebrated theory of the excluded volume effect that relates the expansion factor to the thermodynamic properties of the polymer-solvent system. Basically what Flory calculated was the free energy of mixing of polymer segments with solvent that is associated with the expansion of the coil dimensions in a good solvent. Such expansion is opposed, however, by the loss of configurational entropy of the chain. The latter corresponds, of course, to an elastic contractive force. Expansion proceeds until the two opposing effects are in equilibrium. Flory s result was... 

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