Apparent second virial coefficient

The scattering from the solution of fully hydrogenous PS in THF-ds was used to scale the scattered intensity to absolute values. In order to determine molecular weight, corrections must be made for the effects of the second virial coefficient Aj. The literature value of Az was used for the PS/THF-ds solutions. The previously discussed light scattering experiments show that the apparent second virial coefficient for SPS/THF solutions is zero under these conditions. 

Under conditions of partly screened interactions in dilute solutions (high added salt concentration cs and low polymer concentration c), the solution osmotic pressure can be expressed via a virial expansion (Eq. 24). Then light scattering becomes a useful tool to obtain values of second virial coefficients characterizing interactions in solution. The second virial coefficient can be calculated from the slope of the dependence given by Eq. 25. The relation between the true and the apparent second virial coefficient is similar to the relation between the true and the apparent molecular weight (see the previous section for more details and the meaning of the symbols) ... 

Approach using the Apparent Second Virial Coefficient... 

TABLE 6.11. Values of (Apparent), Second Virial Coefficient A2, and Radius of Inertia of the Macromolecule / for Polymer XXXm in Different Solvents at Different Temperatures [67]... 

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. 

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... 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

To overcome the problem of non-ideality the work be carried out at the Q temperature because in nonideal solutions the apparent Molecular weight is a linear function of concentration at temperatures near Q and the slope depending primarily on the second virial coefficient. 

Other dilute solution properties depend also on LCB. For example, the second virial coefficient (A2) is reduced due to LCB. However, near the Flory 0 temperature, where A2 = 0 for linear polymers, branched polymers are observed to have apparent positive values of A2 [35]. This is now understood to be due to a more important contribution of the third virial coefficient near the 0 point in branched than in linear polymers. As a consequence, the experimental 0 temperature, defined as the temperature where A2 = 0 is lower in branched than in linear polymers [36, 37]. Branched polymers have also been found to have a wider miscibility range than linear polymers [38], As a consequence, high MW highly branched polymers will tend to coprecipitate with lower MW more lightly branched or linear polymers in solvent/non-solvent fractionation experiments. This makes fractionation according to the extent of branching less effective. 

The intrinsic viscosity of PVB is shown as a function of solvent composition for various MIBK/MeOH mixtures in Figure 6. Since [ij] increases with a (see Equation 8), the higher [ly] the better the solvent. Apparently, most mixtures of MIBK and MeOH are better solvents for PVB than either pure solvent. Based on Figure 6, PVB should have a weak selective adsorption of MIBK in a 1 1 solvent mixture and weak adsorption of MeOH in a 3 1 MIBK/MeOH solvent mix. These predictions are in accord with light scattering data discussed previously. The intrinsic viscosity data is also consistent with the second virial coefficient data in Table II in indicating that the 1 1 and 3 1 MIBK/MeOH mixtures are nearly equally good solvents for PVB, the 9 1 mix is a worse solvent, but still better than pure MeOH. 

Where Mw 1 the weight average molecular weight or the apparent molecular weight in the cape of copolyraerp, is the second virial coefficient, and K is an optical constant defined as... 

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. 

An expression for the apparent excluded volume of a coil can be obtained by comparing the perturbation second virial coefficient [Eq. (3.117)] with the one from excluded-volume theory for compact molecules [Eq. (3.115)]. This... 

The dependence of the second virial coefficient on basic quantities such as the solution ionic strength or the polymer molecular weight is usually of greater scientific interest than the knowledge of true vs. apparent values. Therefore the time-consuming dialysis step leading to true rather than apparent values is usually omitted. Hence measured second virial coefficients are usually only apparent values. 

This result is interesting because g is a physical quantity which defines the second virial coefficient of a polymer solution in good solvent and for very long chains. In other terms, g defines the second virial coefficient of a solution of Kuhnian chains. For d = 3 (e = 1), the preceding formula gives g = 0.266, a result which, apparently, is not very very precise, because the second term in (12.3.102) is not small with respect to the first one. This question is discussed, in more detail, in Chapter 13. 

The simplification introduced by restricting the rods to three discrete orientations will now be apparent. The rods must intersect only in those three directions, so that the cluster integrals in three dimensions can be factorized into products of cluster integrals in one dimension. Consider for example the second virial coefficient for parallel rods. By definition... 

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