Excluded volume concept

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. 

Bose, A. Aqueous Size Exclusion Chromatography of Non-Derivatized Cellulose Application of Excluded Volume Concepts to Calibration Ph.D. Thesis, Purdue Univ., LaFayette, IN, 1980. 

This is a landmark paper on the nature of the vitreous body, describing the mechanochemical (or double-network ) model. This model explains satisfactorily the correlations between some properties of the vitreous (composition, rheology, volume, cell population, transparency) and the physicochemical principles governing its stability (frictional interaction, expansion/contraction, the excluded-volume concept, and the molecular-sieve effect). 

For the non-adsorbing polymer case, the depletion force w ill be generated. The depletion mechanism was first theoretically addressed in ref. [23] using the excluded volume concept. Other approaches such as the density ftmctional theory [24] and the virial expansion [25] were developed for deriving the exact expression for the depletion force. Simply, the interaction potential due to the depletion force can be expressed [15]... 

This "excluded volume" concept was further illustrated by Winey and her coworkers who prepared SWCNT/PS nanocomposites by homogeneously coating SWCNTs (exfoliated in aqueous solutions without using surfactant) on the surface of softened flakes or pellets of PS, maintained above the glass transition of the polymer. After processing of the coated PS particles by compression molding, it was shown that SWCNTs were predominantly present in the interfacial volume between the pellets and formed a continuous three-dimensional cellular network. The nanocomposites obtained had conductivity values of the order of 10 S/m for 1 wt% of SWCNTs and a percolation threshold of about 0.2-0.3 wt%, i.e., half of the value of one of the reference samples for which SWCNTs of the same batch were homogeneously dispersed into the same PS matrix by an alternative method. ... 

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. 

In general, each molecule in a very dilute solution in a good solvent (low xi) will tend to exclude all others from the volume which it occupies. This leads to the concept of an excluded volume from which a given... 

Before concluding this discussion of the excluded volume, it is desirable to introduce the concept of an equivalent impenetrable sphere having a size chosen to give an excluded volume equal to that of the actual polymer molecule. Two such hard spheres can be brought no closer together than the distance at which their centers are separated by the sphere diameter de. At all greater distances the interaction is considered to be zero. Hence / = for a dey and fa = 0 for a[Pg.529]

The relativistic EOS of nuclear matter for supernova explosions was investigated recently [11], To include bound states such as a-particlcs, medium modifications of the few-body states have to be taken into account. Simple concepts used there such as the excluded volume should be replaced by more rigorous treatments based on a systematic many-particle approach. We will report on results including two-particle correlations into the nuclear matter EOS. New results are presented calculating the effects of three and four-particle correlations. 

In [11] the alpha-particles were included into the EOS, and detailed comparisons of the outcome with respect to the alpha-particle contribution has been made. We will elaborate this item further, first by using a systematic quantum statistical treatment instead of the simplifying concept of excluded volume, second by including also other (two- and three-particle) correlations. 

In earlier experiments the effect of branching on the second virial coefficient was not seriously considered because the accuracy of measurements were not sufficient at that time. With the refinements of modern instruments a much higher precision has now been achieved. Thus A2 can also now be measured with good accuracy and compared with theoretical expectations. The second virial coefficient results from the total volume exclusion of two macromolecules in contact [3,81]. Furthermore, this total excluded volume of a macromolecule can be expressed in terms of the excluded volume of the individual monomeric units. In the limit of good solvent behavior this concept leads to the expression [6,27] as shown in Eq. (24) ... 

In Section 3.4a we examine a model for the second virial coefficient that is based on the concept of the excluded volume of the solute particles. A solute-solute interaction arising from the spatial extension of particles is the premise of this model. Therefore the potential exists for learning something about this extension (i.e., particle dimension) for systems for which the model is applicable. In Section 3.4b we consider a model that considers the second virial coefficient in terms of solute-solvent interaction. This approach offers a quantitative measure of such interactions through B. In both instances we only outline the pertinent statistical thermodynamics a somewhat fuller development of these ideas is given in Flory (1953). Finally, we should note that some of the ideas of this section are going to reappear in Chapter 13 in our discussions of polymer-induced forces in colloidal dispersions and of coagulation or steric stabilization (Sections 13.6 and 13.7). 

Before considering how the excluded volume affects the second virial coefficient, let us first review what we mean by excluded volume. We alluded to this concept in our model for size-exclusion chromatography in Section 1.6b.2b. The development of Equation (1.27) is based on the idea that the center of a spherical particle cannot approach the walls of a pore any closer than a distance equal to its radius. A zone of this thickness adjacent to the pore walls is a volume from which the particles —described in terms of their centers —are denied entry because of their own spatial extension. The volume of this zone is what we call the excluded volume for such a model. The van der Waals constant b in Equation (28) measures the excluded volume of gas molecules for spherical molecules it equals four times the actual volume of the sphere, as discussed in Section 10.4b, Equation (10.38). 

Consider two particles with adsorbed layers approaching each other. The adsorbed layers on the core particles first begin to overlap at the outermost extreme of the fringe, at which the surface exerts the least influence. As a first approximation, then, the initial encounter between two approaching core particles is comparable to the approach of two polymer coils in solution. In Chapter 3, Section 3.4a, we saw that the concept of excluded volume could be... 

In ref. 63 a lower numerical factor than that of eq. (5.3) is given. This is due to the fact that the original Flory-Fox parameter, viz. 2.1 X 1023, has been used there (98). The present numerical factor is preferred as it is consistent with the concept of Gaussian coils in the absence of excluded volume. A comparison between experimental results of ref. (69)... 

In Chap. 6 we learned that in the excluded volume limit ftc > 0,n —> oo, the cluster expansion breaks down, simply because it orders according to powers of z = j3enef2 —> oo. To proceed, we need a new idea, going beyond perturbation theory. The new concept is known as the Renormalization Group (RG), which postulates, proves, and exploits the fascinating scale invariance property of the theory. 

Summary The classical treatment of the physicochemical behavior of polymers is presented in such a way that the chapter will meet the requirements of a beginner in the study of polymeric systems in solution. This chapter is an introduction to the classical conformational and thermodynamic analysis of polymeric solutions where the different theories that describe these behaviors of polymers are analyzed. Owing to the importance of the basic knowledge of the solution properties of polymers, the description of the conformational and thermodynamic behavior of polymers is presented in a classical way. The basic concepts like theta condition, excluded volume, good and poor solvents, critical phenomena, concentration regime, cosolvent effect of polymers in binary solvents, preferential adsorption are analyzed in an intelligible way. The thermodynamic theory of association equilibria which is capable to describe quantitatively the preferential adsorption of polymers by polar binary solvents is also analyzed. 

The slope of the potential energy curve for repulsive forces is negative, indicating that the force is in the positive direction (i.e., tending to increase the distance between molecules). Repulsive forces are very short range. They only become important when molecules are very close to each other, but they rise quickly to very large values over a very short distance. Because of this, van der Waals treated the repulsive forces using the concept of an excluded volume, [i.e.,... 

The origins of the present three-dimensional molecular-level branching concepts can be traced back to the initial introduction of infinite network theory by Flory [62-65] and Stockmayer [66, 67], In 1943, Flory introduced the term network cell, which he defined as the most fundamental unit in a molecular network structure [68]. To paraphrase the original definition, it is the recurring branch juncture in a network system as well as the excluded volume associated with this branch juncture. Graessley [69, 70] took the notion one step further by describing... 

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