Second virial coefficients excluded-volume

Here, the second virial coefficient (excluded-volume parameter), vb = [1 -2x T)]<0, is negative because water is a poor solvent for the hydrophobic block B. The third virial coefficient, Wb, is positive, and x= T- e /d is the relative deviation from the theta temperature. At small deviations from the theta point, r < 1, the surface tension y and the polymer volume fraction (p are related as y/k T = (p. However, at larger deviations from the theta point, (p becomes comparable to unity and the latter relationship breaks down. Because in a typical experimental situation (p = 1, we treat (p and y as two independent parameters. Note that in a general case, surface tension y and width A of the core-corona interface depend on both the polymer-solvent interaction parameter Xbs T) for the core-forming block and the incompatibility Xab between monomers of blocks A and B. That is, y could depend on the concentration of monomers of the coronal block A near the core surface. We, however, neglect this (weak) dependence and assume that the surface tension y is not affected by conformations of the coronal blocks in a micelle. 

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... 

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. 

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. 

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. 

The parameter a which we introduced in Sec. 1.11 to measure the expansion which arises from solvent being imbibed into the coil domain can also be used to describe the second virial coefficient and excluded volume. We shall see in Sec. 9.7 that the difference 1/2 - x is proportional to. When the fully... 

It is interesting to note that for a van der Waals gas, the second virial coefficient equals b - a/RT, and this equals zero at the Boyle temperature. This shows that the excluded volume (the van der Waals b term) and the intermolecular attractions (the a term) cancel out at the Boyle temperature. This kind of compensation is also typical of 0 conditions. 

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... 

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) ... 

N is the total number of monomers, (p the polymer volume fraction and Pi and Pi/2 the form factors of the total copolymer and of the single blocks respectively. 12=Vd=Vh is the excluded volume interaction parameter which relates to the second virial coefficient A2=vN/ 2Mc). 

The concentration effects for the oligomers and also for the excluded (high) polymer species, are usually small or even negligible, k values depend also on the thermodynamic quality of eluent [108] and the correlation was found between product A2M and k, where A2 is the second virial coefficient of the particular polymer-solvent system (Section 16.2.2) and M is the polymer molar mass [109]. Concentration effects may slightly contribute to the reduction of the band broadening effects in SEC the retention volumes for species with the higher molar masses are more reduced than those for the lower molar masses. 

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. 

It is illuminating to consider some representative examples of effects of biopolymer geometrical structure on the theoretical expression for the excluded volume term of the second virial coefficient on the molal scale (cm /mol). The simplest case is that of interacting solid spheres (Tan-ford, 1961) ... 

Let us now consider some actual numerical data for specific mixed biopolymer systems. Table 5.1 shows a set of examples comparing the values of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated theoretically on the basis of various simple excluded volume models using equations (5.32) to (5.35). For the purposes of this comparison, the experimental data were obtained under conditions of relatively high ionic strength (/ > 0.1 mol dm- ), i.e., under conditions where the contribution of the electrostatic term (A if1) is expected to be relatively insignificant. 

Table 5.1 Comparison of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated from theory on the basis of the excluded volume contribution only.

Here ps is the biopolymer immobilization density A2us = 2%D /3 is the second virial coefficient based on excluded volume for a biopolymer of equivalent diameter D (a sphere of equal volume) (Neal and Lenhoff, 1995) and = As/V0 is the chromatographic phase ratio. The surface area As accessible to the biopolymer in the mobile phase is available in the literature, especially for proteins (Tessier et al., 2002 Dumetz et al., 2008). 

Neal, B.L., Lenhoff, AM. (1995). Excluded-volume contribution to the osmotic second virial coefficient for proteins. AlChE Journal, 41, 1010-1014. 

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). 

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). 

The essence of this model for the second virial coefficient is that an excluded volume is defined by surface contact between solute molecules. As such, the model is more appropriate for molecules with a rigid structure than for those with more diffuse structures. For example, protein molecules are held in compact forms by disulfide bridges and intramolecular hydrogen bonds by contrast, a randomly coiled molecule has a constantly changing outline and imbibes solvent into the domain of the coil to give it a very soft surface. The present model, therefore, is much more appropriate for the globular protein than for the latter. Example 3.3 applies the excluded-volume interpretation of B to an aqueous protein solution. 

Outline the logic used in deriving expressions for the osmotic pressure and second virial coefficient due to excluded-volume interactions ... 

Molecular weight from osmotic pressure measurements Degree of polymerization and molecular weight distribution Excluded volume from osmotic pressure measurements Theta temperature from second virial coefficient data Evaluation of charges on macroions from osmotic pressures... 

As pointed out in Chapter III, Section 1 some specific diluent effects, or even remnants of the excluded volume effect on chain dimensions, may be present in swollen networks. Flory and Hoeve (88, 89) have stated never to have found such effects, but especially Rijke s experiments on highly swollen poly(methyl methacrylates) do point in this direction. Fig. 15 shows the relation between q0 in a series of diluents (Rijke assumed A = 1) and the second virial coefficient of the uncrosslinked polymer in those solvents. Apparently a relation, which could be interpreted as pointing to an excluded volume effect in q0, exists. A criticism which could be raised against Rijke s work lies in the fact that he determined % in a separate osmotic experiment on the polymer solutions. This introduces an uncertainty because % in the network may be different. More fundamentally incorrect is the use of the Flory-Huggins free enthalpy expression because it implies constant segment density in the swollen network. We have seen that this means that the reference dimensions excluded volume effect. 

Equation (1.3) is by no means the only power law found in the excluded volume limit. For the second virial coefficient Akf, for instance, we find... 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

This formula is the basis of the Zimm Plot which consists in plotting the inverse of the scattering intensity, S "1 (Q), vs Q2 which shows a linear variation at low Q and in dilute solutions. Extrapolated values (Q - 0, Cp - 0) of the intercept and the slope yield the degree of polymerization N and the excluded volume v (or second virial coefficient A2) respectively. Zimm s formula can describe scattering data accurately well into the semidilute concentration region. This region is defined for concentrations above an overlap concentration C which is defined in either of the two following ways ... 

In addition to the aforementioned practical difficulty of the method based anA M/fy], it must be realized that its theoretical basis is also not secure. The excluded volume problem of two entangled chains, which is a fundamental part of the theory of the second virial coefficient, has not been much advanced beyond the first-order perturbation stage, and as a result the function /(a) of Eq. (10) is only imperfectly known. For these... 

In dilute solution, all macromolecular chains undergo interactions with each other resulting in the so-called intermolecular excluded volume effect, corresponding to the intermolecular potential. This effect is also observed if one does not assume particular cohesive forces to occur between the macromolecular chains. Under these conditions, the second virial coefficient is calculated from the equation1,2) ... 

You may recall that the temperature where % 2is what Floiy called the theta tern--perature and can now be seen to describe the situation where the second virial coefficient becomes zero (Figure 12-10). This means that at this point pair-wise interactions cancel and the chain becomes nearly ideal, as we discussed in the section on dilute solutions (Chapter 11), where we referred to the Floiy excluded volume model in which the chain expansion factor is given by Equation 12-18 ... 

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