Volume effect

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. 

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 (EXCLUDED-VOLUME EFFECTS... 

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... 

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... 

A variety of experimental techniques have been employed to research the material of this chapter, many of which we shall not even mention. For example, pressure as well as temperature has been used as an experimental variable to study volume effects. Dielectric constants, indices of refraction, and nuclear magnetic resonsance (NMR) spectra are used, as well as mechanical relaxations, to monitor the onset of the glassy state. X-ray, electron, and neutron diffraction are used to elucidate structure along with electron microscopy. It would take us too far afield to trace all these different techniques and the results obtained from each, so we restrict ourselves to discussing only a few types of experimental data. Our failure to mention all sources of data does not imply that these other techniques have not been employed to good advantage in the study of the topics contained herein. 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

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... 

Design concerns for drive cycle emission control include canister volume requirements, purge volume effects, and the use of return vs. retumless fuel systems. 

To use the volume effectively, the reader should first review the Glossary of Terms section immediately preceding the first chemical entry. This section contains precise definitions used for certain parameters where data have been obtained for each chemical. A review of these terms will help the reader interpret certain information. In addition, a list of abbreviations used throughout the volume is also provided in the front section of the handbook. 

J. S. Pedersen, M. Laso, P. Schurtenberger. Monte Carlo study of excluded volume effects in worm-like micelles and semi-flexible polymers. Phys Rev E 54 R5917-R5920, 1996. 

Suppose we have a physical system with small rigid particles immersed in an atomic solvent. We assume that the densities of the solvent and the colloid material are roughly equal. Then the particles will not settle to the bottom of their container due to gravity. As theorists, we have to model the interactions present in the system. The obvious interaction is the excluded-volume effect caused by the finite volume of the particles. Experimental realizations are suspensions of sterically stabilized PMMA particles, (Fig. 4). Formally, the interaction potential can be written as... 

Any real sample of a colloidal suspension has boundaries. These may stem from the walls of the container holding the suspension or from a free interface towards the surroundings. One is faced with surface effects that are small compared to volume effects. But there are also situations where surface effects are comparable to bulk effects because of strong confinement of the suspension. Examples are cylindrical pores (Fig. 8), porous media filled with suspension (Fig. 9), and thin colloidal films squeezed between parallel plates (Fig. 10). Confined systems show physical effects absent in the bulk behavior of the system and absent in the limit of extreme confinement, e.g., a onedimensional system is built up by shrinking the size of a cylindrical pore to the particle diameter. 

Calibration curves for PS and PMMA are shown in Figs. 15.3-15.5. The slight differences in courses of calibration curves for PS in THF, chloroform, and toluene, as well as the curve for PMMA in THF (Fig. 15.3), can be explained by the flow rate variations for different pumping systems and by the hydrodynamic volume effects, respectively. The calibration curves for PMMA in mixed eluents THF/toluene are shown in Fig. 15.4. Three percent of THF in toluene assured a reasonable SEC elution of PMMA. However, more chloroform was needed to obtain a good SEC elution of PMMA in mixed eluent chloroform/toluene... 

In the following paper, the possibility of equilibration of the primarily adsorbed portions of polymer was analyzed [20]. The surface coupling constant (k0) was introduced to characterize the polymer-surface interaction. The constant k0 includes an electrostatic interaction term, thus being k0 > 1 for polyelectrolytes and k0 1 for neutral polymers. It was found that, theoretically, the adsorption characteristics do not depend on the equilibration processes for k0 > 1. In contrast, for neutral polymers (k0 < 1), the difference between the equilibrium and non-equilibrium modes could be considerable. As more polymer is adsorbed, excluded-volume effects will swell out the loops of the adsorbate, so that the mutual reorientation of the polymer chains occurs. 

The value of the exponent a obtained in the above-mentioned experiments is in remarkable accord with predictions based on a consideration of excluded kinetic volume effects. Khokhlov51 proposed, that for a slow, chemically controlled, reaction between the ends of long chains a should be 0.16. The value of a was suggested to increase to 0,28 for chain end-mid chain reaction and to 0.43 for midchain-mid chain reaction. The latter provides one possible explanation for the greater exponent for higher acrylates (Table 5.11.32... 

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... 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

The Rouse and Zimm models are valid only under 0-conditions. To extend their range of applicability into good solvent conditions, several improvements have been proposed to include excluded volume effects. Dynamical scaling, however, provides probably the simplest approach to the problem [30],... 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

A series of theoretical studies of the SCV(C)P have been reported [38,40,70-74], which give valuable information on the kinetics, the molecular weights, the MWD, and the DB of the polymers obtained. Table 2 summarizes the calculated MWD and DB of hyperbranched polymers obtained by SCVP and SCVCP under various conditions. All calculations were conducted, assuming an ideal case, no cyclization (i.e., intramolecular reaction of the vinyl group with an active center), no excluded volume effects (i.e., rate constants are independent of the location of the active center or vinyl group in the macromolecule), and no side reactions (e.g., transfer or termination). 

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