Volume, excluded

Suppose that chains A and B each have N monomers. Because the radius of a random-flight chain molecule is proportional to (Equation (32.7)), the 

for typical polymers, N = 10 -10, the interior of a random-flight chain is nearly pure solvent. 

If N is large, Psuccoss 0. Even though the interior of a random-flight poly mer chain is mostly solvent, two polyaner chains with N 1 ha e a very strong steric tendency not to interpenetrate each other. This is called excluded volume. 

Excluded volume is different in different dimensionalities. To see this, look at Equation (33.1). Let d represent the dimensionality d = 3 for polymers in three-dimensional space, or d = 2 for polymers on two-dimensional surfaces, etc. You can express the concentration (N/volume) of polymer segments in any dimensionality d as /V/R where R = Generalizing Equation (33.4) 

Steric repulsions also cause random-flight polyaners to repel rigid impenetrable surfaces. 

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A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

Flere b corresponds to the repulsive part of the potential, which is equivalent to the excluded volume due to the finite atomic size, and a/v corresponds to the attractive part of the potential. The van der Waals equation... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

The major deficiency of the equation as written is that there is no excluded volume, a deficiency DFl could rectify for the central ion, but not for all ions around the central ion. This deficiency has been addressed within the DFl framework by Outhwaite [9]. 

We have seen that the DFI theory in the limiting case neglects excluded volume effects in fact the excluded volume of the centra ion can be introduced into the theory as explained after A2.4.48. If the radius of the ions is taken as a for all ions, we have, in first order. 

The simplest extension to the DH equation that does at least allow the qualitative trends at higher concentrations to be examined is to treat the excluded volume rationally. This model, in which the ion of charge z-Cq is given an ionic radius d- is temied the primitive model. If we assume an essentially spherical equation for the u. . 

Schafer L 1999 Excluded Volume Effects in Polymer Solutions (Berlin Springer)... 

SO tliat tire characteristic ratio can be evaluated from tire plateau value of i ([Pg.2518]

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

In tire simple version of tire lattice representation of proteins tire polypeptide chain is modelled as a sequence of connected beads. The beads are confined to tire sites of a suitable lattice. Most of tire studies have used tire cubic lattice. To satisfy tire excluded volume condition only one bead is allowed to occupy a lattice site. If all tire beads are identical we have a homopolymer model the characteristics of which on lattices have been extensively studied. 

Chan H S and Dill K A 1989 Intrachain loops in polymers effects of excluded volume J. Chem. Phys. 90 493-509... 

Solvent-excluded surfaces correlate with the molecular or Connolly surfaces (there is some confusion in the literature). The definition simply proceeds from another point of view. In this c ase, one assumes to be inside a molecaile and examines how the molecule secs the surrounding solvent molecules. The surface where the probe sphere does not intersect the molecular volume is determined. Thus, the SES embodies the solvent-excluded volume, which is the sum of the van der Waals volume and the interstitial (re-entrant) volume (Figures 2-119. 2-120). 

Richmond T J 1984. Solvent Accessible Surface Area and Excluded Volume in Proteins. Journal oj Molecular Biology 178 63-88. 

The solvent-excluded volume is a molecular volume calculation that finds the volume of space which a given solvent cannot reach. This is done by determining the surface created by running a spherical probe over a hard sphere model of molecule. The size of the probe sphere is based on the size of the solvent molecule. 

That state of affairs in which the poorness of the solvent exactly compensates for the excluded volume effect is called a 0 condition or Flory condition, after... 

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. 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

Regardless of the particle geometry, the excluded volume exceeds the actual volume of the molecules by a factor which depends on the shape of the particles. 

Figure 8.10 Excluded volume for two spheres (dotted surface) as determined by the distance of closest approach.

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