Excluded-volume theories

Decreasing the degree of crosslinking will increase the water uptake for a mass of dry gel, though compromises in the efficiency will result. The effect of crosslinks on the separation of vitamin B-12, a nonionic solute of molecular weight 1355, is shown in Fig. 4 [16]. As the crosslink density decreases, the polymer chain length between crosslinks increases, yielding a looser structure which vitamin B-12 can more easily penetrate. The behavior fits well with the prediction from Flory excluded volume theory [16] ... 

Fig. 4. Separation efficiency vs degree of crosslinking. The solute is vitamin B-12 and the gel is hydrolyzed polyacrylamide. The solid line in the inset is the prediction from Flory excluded volume theory (Eq. 3)...

All excluded volume theories for branched chains suffer, however, from a principal deficiency since the assumption is tacitly made that all monomeric units in the molecule may have, in principle, the chance to interact with each other. This is, however, a too extensive assumption since for sterical reasons two remote segments can never form a contact. Hence, the excluded volume effect is highly overestimated for densely branched chains. In fact, highly branched polymers show the phenomenon of swelling but no detectable distortion of Gaussian chain behavior117,137,138,179. ... 

In summarizing the intrinsic viscosity relations presented in this section, it must be admitted that they represent nothing more than rather small semi-empirical refinements of the Flory excluded volume theory and the Flory-Fox viscosity theory. For a large fraction of the existing body of experimental data, they offer merely a slight improvement in curve-fitting. But for polymers in good solvents it is believed that a more transcendental result has been achieved. The new equations permit more reliable assessment of unperturbed chain dimensions in such cases, and in some instances (e. g., certain cellulose derivatives see Section III B) they offer possible explanations of heretofore paradoxical solution behavior. 

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. 

This nr1 dependence of the second virial coefficient has been already predicted in the Flory excluded volume theory [6] let a be the expansion factor of a chain molecule and Vs the molar volume of solvents. Following Flory, one has... 

Nb is the number of animals with b bonds), and making use of the aforementioned series expansion (Sect. 6), he could show that the exponent 0 collapses into the classical value, 0 = 5/2, at 8 dimensions, in support of the Lubensky field theory. The Parisi and Sourlas e = 8 -d expansion [34] gave almost identical results for 0, thus supporting dc= 8 as well. The Isaacsson-Lubensky mean-field theory [30] based on the Flory excluded volume theory and the screening concept is more intelligible, which reads... 

According to the Flory excluded volume theory and the de Gennes screening effect, the exponent of the molecular dimension is given by the Isaacson-Lubensky formula ... 

The effect of molecular weight is qualitatively explained by excluded volume theories, which are described later. 

Cm is a parameter depending on the properties of the polymer-solvent system and ocg is an expansion parameter. Both the terms are defined in the following section on Excluded Volume Theories. At T = 9, X becomes zero, F X) becomes equal to 1, and the Flory-Krigbaum theory reduces to the Flory-Huggins theory. 

An expression for the apparent excluded volume of a coil can be obtained by comparing the perturbation second virial coefficient [Eq. (3.117)] with the one from excluded-volume theory for compact molecules [Eq. (3.115)]. This... 

Presentation of light-scattering results according to excluded-volume theory, Eur. Polym. J., 15, 111, 1979. 

Z collision number Z z fraction z ionic charge z coordination number z dissymmetry (light scattering) z parameter in excluded volume theory z number of nearest neighbors... 

A change of some properties with change of size of the system provides us with the opportunity to recognize the entities of its structure. The great theoretical support has been provided in polymer science with excluded volume theory of Flory [1], It provides the following law that describes scaling of a polymer molecules radius of gyration Rq with number of its backbone bonds N. ... 

The excluded volume theories of Onsager and Flory show that depending on the axial ratio of the rod-like particles, there is a critical concentration above which a nematic phase is formed. This concentration does not depend on the temperature of the system as these theories are essentially athermal, i.e. the part of the free energy leading to the anisotropy is an entropy term. 

In carbon black-polymer composites, the value of Xc varies from about 40 vol.% in the case of spherical CB (e.g., N990) particles [51] to about 5 vol.% (e.g., for XC-72 [51] and Ketjenblack [52]) following the increase of the aspect ratio of the particles to the order of 10 [34]. The latter result is in excellent agreement with the excluded volume theory that predicts such a dependence of Xc on the aspect ratio (see Eq. (5.11)) if we identify the shell thickness d with a critical average distance, dc, that the tuimeling electrons should cross [29], This trend is continued when we consider the x values for CNTs [43], where x values on the order of 1 vol.% are observed for aspects ratios on the order of 10, and 0.1 vol.% for aspect ratios on the order of lO -lO. ... 

Ever since Hermann Staudinger developed the macromolecular hypothesis in the 1920s (41), polymer scientists have wondered about the spatial arrangement of polymer chains, both in dilute solution and in the bulk. The earliest models included both rods and bedspring-like coils. X-ray and mechanical studies led to the development of the random coil model. In this model the polymer chains are permitted to wander about in a space-fiUing way as long as they do not pass through themselves or another chain (excluded-volume theory). 

In all these studies, ACM allowed, with the massive data provided, the behavior of the polyelectrolyte conformational and interaction parameters to be followed at a level of detail previously unobtainable by manual gathering of individual data points. Correlations were made between experimental data and of electrostatic persistence length and electrostatic excluded volume theories, with no adjustable parameters. 

The hard sphere model is based on the excluded volume of spherical particles. An excluded volume theory has been developed to account for the orientational ordering of liquid crystal molecules, assuming them to be hard rods. This is the Onsager theory and its variants, outlined in Section 5.5.2. Excluded volume interactions influence the conformation of polymer chains. The conformation of an ideal chain is described by a random walk. However,... 

At intermediate and high salt concentration the complication arises from excluded volume effects. This question was investigated long ago by Nagasawa et al. [4, 272, 273] solely in terms of classical excluded volume theory, i.e., 1, = Ip f(cf), as the concept of the electrostatic persistence length was not known at the time. The conclusion was that experimentally determined electrostatic expansion coefficients do vary as... 

A useful qualitative way of looking at the question of what is, as well as addressing many other problems in semi-dilute systems, is in terms of bloos, a term introduced by de Gennes. Let us regard the bonded units, expected to be within distance of each other, as a blob. The number of monomers in a blob, g, is given by the distance-number relation of excluded volume theory, since these units are not shielded from excluded volume forces ... 

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