Excluded-Volume Effects

Excluded Volume Effects.—Because of intramolecular segment-segment interactions, a polymer chain in a thermodynamically good solvent expands by a factor defined by 

For the excluded volume behaviour of chains at large z (the asymptotic regime), various methods have been used. For example Edwards and others have exploited the self-consistent field (SCF), while des Cloizeaux and de Gennes have pioneered 

Experimental data on the increase of polymer dimensions with increasing goodness of solvent has been obtained by Einaga and co-workers in a series of papers.They have examined a range of polystyrenes with A/ , between 4 x 10 and 6 x 10 , and measured l /l, and As part of this work they have 

Stanley, Introduction to Phase Transitions and Critical Phenomena . Clarendon Press, Oxford. 

Tanaka et and by Weill and co-workers. In the recent modified blob theory, the change from Gaussian to excluded volume statistics is not as sudden as was at first surmized, and an estimate of the width of the cross-over region can be made and compared with experimental data. 

Excluded Volume Effects.—In a thermodynamically good solvent, the dimensions of a polymer molecule are greater than in the unperturbed state because of excluded volume effects. This effect is usually characterized by an expansion factor a, with 

Similar expansion factors exist for e.g. the ratio of intrinsic viscosities, a. At temperatures just above the 0-temperature, we have the regime of the perturbation treatment. Its extension, the two-parameter theory, relates the expanded chain dimensions to the parameter with P the binary 

In an area dominated by Japanese workers, Norisuye and Fujita have evaluated the expansion factor of non-draining linear chains and rings for small z, using Kirkwood-Riseman theory. 

The negative coefficient in this expansion is contrary to that obtained by Tanaka and Yamakawa using the Zimm-Hearst theory. Minato and Hatano have evaluated the effect of excluded volume on the three separate principal components (along the principal axis of inertia) of a polymer chain. Solc has shown that the shape of a random flight chain is far from spherical, in fact the ratio of the lengths of the principal axes is, on average, in the ratio 

Now the perturbation theory has been applied to each of these components up to terms in z . The effect of excluded volume is again highly anisotropic, in fact for large z it is shown that Aj/w obeys the fifth power law, whereas XJn and XJn converge, implying the adoption of an extended ribbon configuration. 

There exists a mushrooming interest in the mesomorphic behavior of macromolecules in solution. The existence of macroscopic anisotropy requires (at least partial) chain rigidity. This may arise from next nearest neighbor steric restrictions for polymers constructed from bulky backbone monomers as in the polyamides 

The current s3nnposium, to a great extent, has focused on comb-like systems -ii wherein mesogenic side groups are attached to a flexible backbone, as 

Supported in part by the National Science Foundation and the Office of Naval Research. 

If the solvent is good, the only important intermolecular interactions are then steric short range excluded volume repulsions. 

The mesomorphic behavior of a solution of rigid rods has been studied by several authors.i-iZ-Ll In particular, for long rods of length b and radius a, OnsagerL has shown that the effective excluded volume per rod in the isotropic phase is of order b a. The actual volume occupied per rod is ira b. A lyotropic transition to a nematic phase occurs at a rod concentration O given by 

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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... 

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... 

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... 

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. 

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],... 

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). 

Arai, T, Sawatari, N., Yoshizaki, T., Einaga, Y. and Yamakawa, H. (1996) Excluded-volume effects on the hydrodynamic radius of atactic and isotactic oligo- and poly(methylmethacrylate)s in dilute solution. Macromolecules, 29, 2309-2314. 

The GvdW theory has been applied also to mixtures of Leonard-Jones fluids [15,16], The extension to mixtures is straightforward with respect to the binding energy and interaction with an external field but not quite so straightforward with respect to the excluded volume effect. The GvdW(S) theory produces for a mixture of c components an equation of state of the form... 

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

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. 

A similar effect may exist for hydrophobic interaction between solute and stationary phase, as one solute may adsorb more strongly to the stationary phase than another. It has also been remarked that a flexible polymer confined to a pore should be at a lower entropy than one in bulk solution, leading to exclusion in excess of that expected for a simple geometric solid.23 Even in the absence of interactions, a high concentration of a small component can lead to an excluded volume effect, since the effective volume inside the pore is reduced. 

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

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