Coils excluded volume

In a very good solvent, a chain s segments are repelled from each other, which is formalized in the existence of segment excluded volume at the intramolecular level and of polymer coil excluded volume at the intermolecular level. In this case, very long chauns obey the universal scaling laws characteristic of the proximity of the critical point in general-type systems. 

In polymer theories, the length of a segment (molecular unit) a is the natural unit of the length scale at the molecular level, but experimentally measured quantities Q do not feel this molecular discrepancy. Hence mathematical expressions of the macroscopic quantity Q have to be well-defined in the limit a —> 0. Regarding the problem of the molecular coil excluded volume, this limit should be treated as a rejection with the consideration of the interactions of a segment with itself (self-excluded volume). 

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. 

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

We begin our attempt to reconcile these two expressions for the excluded volume of the random coil by reviewing some ideas about random coils from Chap. 1 ... 

If we disregard the term in brackets in Eq. (8.112)-which we shall call f(y) and which is close to unity for values of x that are not too different from 1/2-this expression gives the same result for the excluded volume of the coil as given by Eq. (8.104) from a comparison of Eqs. (8.97) and (8.103). We may regard f(y) as a correction factor which is required for Eq. (8.104) to be valid as the difference 1/2 - x increases. In the next section we shall discuss the implications of Eq. (8.112) in greater detail. 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

The parameter a which we introduced in Sec. 1.11 to measure the expansion which arises from solvent being imbibed into the coil domain can also be used to describe the second virial coefficient and excluded volume. We shall see in Sec. 9.7 that the difference 1/2 - x is proportional to. When the fully... 

One thing that is apparent at the outset is that polymer molecules in solution are very different species from the rigid spheres upon which the Einstein theory is based. On the other hand, we saw in the last chapter that the random coil contributes an excluded volume to the second virial coefficient that is at least... 

Such a coil is said to be nondraining, since the interior of its domain is unaffected by the flow. We anticipate using Eq. (1.58) to describe the molecular weight dependence of In view of this, we replace rg by (rg ) and attach a subscript 0 to the latter as a reminder that, under 0 conditions, solvent and excluded-volume effects cancel to give a true value. With these ideas in mind, the volume fraction of the nondraining coil is written... 

Universal SEC calibration reflects differences in the excluded volume of polymer molecules with identical molecular weight caused by varying coil conformation, coil geometry, and interactive propenies. Intrinsic viscosity, in the notation of Staudinger/ Mark/Houwink power law ([77]=fC.M ), summarizes these phenom-... 

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. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The conformations adopted by polyelectrolytes under different conditions in aqueous solution have been the subject of much study. It is known, for example, that at low charge densities or at high ionic strengths polyelectrolytes have more or less randomly coiled conformations. As neutralization proceeds, with concomitant increase in charge density, so the polyelectrolyte chain uncoils due to electrostatic repulsion. Eventually at full neutralization such molecules have conformations that are essentially rod-like (Kitano et al., 1980). This rod-like conformation for poly(acrylic acid) neutralized with sodium hydroxide in aqueous solution is not due to an increase in stiffness of the polymer, but to an increase in the so-called excluded volume, i.e. that region around an individual polymer molecule that cannot be entered by another molecule. The excluded volume itself increases due to an increase in electrostatic charge density (Kitano et al., 1980). 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

On macroscopic length scales, as probed for example by dynamic mechanical relaxation experiments, the crossover from 0- to good solvent conditions in dilute solutions is accompanied by a gradual variation from Zimm to Rouse behavior [1,126]. As has been pointed out earlier, this effect is completely due to the coil expansion, resulting from the presence of excluded volume interactions. 

An excluded-volume random-coil conformation will be achieved when the solvent quality exceeds the theta point, the temperature or denatu-rant concentration at which the solvent-monomer interactions exactly balance the monomer—monomer interactions that cause the polymer to collapse into a globule under more benign solvent conditions. A number of lines of small-angle scattering—based evidence are consistent with the suggestion that typical chemical or thermal denaturation conditions are good solvents (i.e., are beyond the theta point) and thus that chemically or thermally unfolded proteins adopt a near random-coil conformation. 

Of the 20 reduced or disulfide bond-free proteins for which the Rg of the chemically or thermally unfolded state has ben reported, 17 fall on a single curve (Fig. 4). Fitting the Rg of these 17 proteins produces a strong, statistically significant correlation (r 2 = 0.96) and an exponent, v = 0.61 d= 0.03, startlingly close to the expected value for an excluded-volume random coil chain. 

Why is this reconciliation important After all, a 6 M GuHCl solution hardly recapitulates the physiologically milieu, and proteins unfolded under more physiologically relevant conditions are often (though not always see Hoshino et al., 1997) much more compact than expected for an excluded-volume random coil. Still, the chemically or thermally... 

It is now useful to consider how we may define the concentration of polymers in solution. Dilute solutions are ones in which the polymer molecules have space to move independently of each other, i.e. the volume available to a molecule is in excess of the excluded volume of that molecule. Once the concentration reaches a value where there are too many molecules in solution for this to be possible, the solution is considered to be semi-dilute.8 The concentration at which this occurs is denoted by c. For convenience c is defined as the concentration when the volumes of the coils just occupy the total volume of the system, i.e. 

Tables of this sort are valid for Gaussian coils only. In thermodynamically good solvents the Gaussian behaviour of chain molecules is perturbed by what is called the excluded volume effect 30. The P/(0) function depends on the distribution of mass within the particle and this, in turn, is changed if the volume effect is operative.

If the backbone as well as the side chains consist of flexible units, the molecular conformation arises out of the competition of the entropic elasticity of the confined side chains and the backbone [ 153 -155]. In this case, coiling of the side chains can occur only at the expense of the stretching of the backbone. In addition to the excluded volume effects, short range enthalpic interactions may become important. This is particularly the case for densely substituted monoden-dron jacketed polymers, where the molecular conformation can be controlled by the optimum assembly of the dendrons [22-26,156]. If the brush contains io-nizable groups, the conformation and flexibility may be additionally affected by Coulomb forces depending on the ionic strength of the solvent [79,80]. 

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