Polymer solutions excluded volume

A solution that contains a polymer at a volume fraction cf> is in contact with a pore of diameter d. The pore walls are neutral to the polymer, that is, the pore does not have an attractive or repulsive interaction other than the excluded volume (polymers cannot intersect the pore wall). At equilibrium, the polymer volume fraction is 4>i in the pore. We allow ct>i and < e to be anywhere from dilute to semidilute regimes. Except near the pore-exterior boundary, is uniform along the pore (Fig. 4.22). 

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

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... 

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. 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

One thing that is apparent at the outset is that polymer molecules in solution are very different species from the rigid spheres upon which the Einstein theory is based. On the other hand, we saw in the last chapter that the random coil contributes an excluded volume to the second virial coefficient that is at least... 

P. G. De Gennes. Exponents for the excluded volume problem as derived by the Wilson method. Phys Lett 38A 339, 1972 J. des Cloiseaux. The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys 26 281-291, 1975. 

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 analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

The conformations adopted by polyelectrolytes under different conditions in aqueous solution have been the subject of much study. It is known, for example, that at low charge densities or at high ionic strengths polyelectrolytes have more or less randomly coiled conformations. As neutralization proceeds, with concomitant increase in charge density, so the polyelectrolyte chain uncoils due to electrostatic repulsion. Eventually at full neutralization such molecules have conformations that are essentially rod-like (Kitano et al., 1980). This rod-like conformation for poly(acrylic acid) neutralized with sodium hydroxide in aqueous solution is not due to an increase in stiffness of the polymer, but to an increase in the so-called excluded volume, i.e. that region around an individual polymer molecule that cannot be entered by another molecule. The excluded volume itself increases due to an increase in electrostatic charge density (Kitano et al., 1980). 

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. 

Cifra, P. and Bleha, T. Conformer populations and the excluded volume effect in lattice simulations of flexible chains in solutions, Polymer, 34 (1993). 3716-3722... 

It is now useful to consider how we may define the concentration of polymers in solution. Dilute solutions are ones in which the polymer molecules have space to move independently of each other, i.e. the volume available to a molecule is in excess of the excluded volume of that molecule. Once the concentration reaches a value where there are too many molecules in solution for this to be possible, the solution is considered to be semi-dilute.8 The concentration at which this occurs is denoted by c. For convenience c is defined as the concentration when the volumes of the coils just occupy the total volume of the system, i.e. 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

The size of a polymer molecule in solution is influenced by both the excluded volume effect and thermodynamic interactions between polymer segments and the solvent, so that in general =t= . The Flory (/S) expansion factor a is introduced to express this effect, by writing ... 

The size of molecules in solution, measured by , is important both in its own right and because of the effects of changes in on interactions between molecules in solution. For linear polymers, the expansion factor a of Eq. (3.2) can be expressed as a power series in a dimensionless parameter z which is related to the excluded volume per segment pair, p, and the molecular size ... 

In Flory s theory (/< ), a polymer-solvent system is characterized by a temperature 0 at which (i) excluded-volume effects are just balanced by polymer-solvent interactions, so that os=l, (ii) the second virial coefficient is zero, irrespective of the MW of the polymer, and (iii) the polymer, of infinite molecular weight, is just completely miscible with the solvent The fundamental definition of the temperature is a macroscopic one, namely that for T near 0 the excess chemical potential of the solvent in a solution of polymer volume fraction v2 is of the form (18) ... 

Under the conditions of screening of electrostatic interactions between polyions, as occurs at high ionic strength (say, / > 0.1 mol dm- ), or in solutions containing neutral (non-ionic) polymers, the excluded volume term is the leading term in the theoretical equation for the second virial coefficient. In this latter type of situation, the sizes and conformation/ architecture of the biopolymer molecules/particles become of substantial importance. 

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

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). 

What is theta temperature (or, the Flory temperature) What are the relative magnitudes of the excluded-volume interactions and the energetic interactions in a dilute polymer solution at its theta temperature ... 

In the final section, we build on the thermodynamic theories of polymer solutions developed in Chapter 3, Section 3.4, to provide an illustration of how a thermodynamic picture of steric stabilization can be built when excluded-volume and elastic contributions determine the interaction between polymer layers. 

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... 

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