Second virial coefficient of osmotic

The second virial coefficient of osmotic pressure, A2, for a linear polymer of MW M is shown to be given by ... 

If solvents are used which do not possess a high dissolving power for both kinds of blocks (high second virial coefficients of osmotic pressure), phase separation occurs at considerably lower concentrations, and the solvent content of the aggregates is lower than that of the matrix. 

However, the structure and composition of microdroplets in which the reaction takes place are not the only parameters controlling polymerisation. We shall see later that particles in the initial microemulsion cannot be considered as independent reactors since interparticle interactions play an important role. Small angle light and neutron scattering experiments have shown that these interactions are attractive [6.20]. There is a clear increase in these attractive interparticle forces as the proportion of acrylamide is raised. In particular, this has two consequences the second virial coefficient of osmotic pressure takes negative values the peak in the structure factor which characterises a hard sphere system is no longer present. 

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 properties of solutions of macromolecular substances depend on the solvent, the temperature, and the molecular weight of the chain molecules. Hence, the (average) molecular weight of polymers can be determined by measuring the solution properties such as the viscosity of dilute solutions. However, prior to this, some details have to be known about the solubility of the polymer to be analyzed. When the solubility of a polymer has to be determined, it is important to realize that macromolecules often show behavioral extremes they may be either infinitely soluble in a solvent, completely insoluble, or only swellable to a well-defined extent. Saturated solutions in contact with a nonswollen solid phase, as is normally observed with low-molecular-weight compounds, do not occur in the case of polymeric materials. The suitability of a solvent for a specific polymer, therefore, cannot be quantified in terms of a classic saturated solution. It is much better expressed in terms of the amount of a precipitant that must be added to the polymer solution to initiate precipitation (cloud point). A more exact measure for the quality of a solvent is the second virial coefficient of the osmotic pressure determined for the corresponding solution, or the viscosity numbers in different solvents. 

We have already seen that the second virial coefficient may be determined experimentally from a plot of the reduced osmotic pressure versus concentration. Since all other quantities in Equation (99) are measurable, the charge of a macroion may be determined from the second virial coefficient of a solution with a known amount of salt. As an illustration of the use of Equation (99), we consider the data of Figure 3.6 in Example 3.5. 

Theta conditions are identified experimentally as the situation in which the second virial coefficient of the osmotic pressure is zero. 

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. 

Figure 4. Adnesion energy (erg/cm ) for SUPC biiayers in 0.1 M salt (PBS) plus dextran polymers. Number average polymer indices - Np (number of glucose monomers). Solid and dasned curves - predictions from mean field theory with first and second virial coefficients from osmotic pressure measurements (14).

Thus, the osmotic pressure is first measured at different polymer concentrations, no c is then plotted vs. c, the values are linearly extrapolated to c 0, and the value of is determined from the y axis intercept. A2 is the second virial coefficient of the osmotic pressure. Solvents where A2 = 0 are called ideal or d solvents. 

Another quantity besides the (volume-corrected) preferential solvation parameters that can be extracted from the KBIs is the molar second virial coefficient of a component, say A, in the mixture A + B. The limiting value of the G a as its concentration tends to zero, GXa. is minus twice the molar osmotic pairwise virial coefficient of this component (Matteoli and Lepori 1984). This quantity is related to the self-association of the component A as a solute at infinite dilution in component B as a solvent. When GXa is positive then self-association of A is appreciable, in spite of solvation by component B, and is the more pronounced as it becomes larger. Contrarily, if G a is negative then its solvation by component B (the solvent) dominates over its self-association. 

The effect of polymer molecular weight on the viscosity when added at T < is shown in Figure 5. The viscosities were measured at a shear rate of 100 s and T = 25 C. The viscosity ratio is the viscosity of the dispersion plus polymer to the viscosity of the dispersion. The viscosity of the lamellar dispersion is reduced by the addition of the nonionic polymers of all molecular weights and the ranking of effectiveness (i.e., 20K > PVA = 3350 > 600) is consistent with the expected osmotic pressure due to the second virial coefficient of the polymers. At an equal molar concentration the second virial coefficient and the osmotic pressure due to the concentration gradient increases with molecular weight. 

As the second virial coefficient of the osmotic pressure expansion, the quantity (1/2 — y) is accepted here, while the difference between the characteristic parameter of the ternary interactions w and the trivial factor 1/6 acts as the third virial coefficient (sec... 

The reference temperature is not the true theta temperature at which the second virial coefficient of the osmotic pressure vanishes. The latter lies far below due to H-bonding and hydrophobic interaction in addition to the van der Waals interaction in the background. The parameters related to the strength of hydration, such as no, yn, were taken from Section 6.4 for PEO, and Section 6.5 for PNIPAM. 

This approximation takes only into account the singularity as r/ goes to 1, but gives neither the correct low density nor at high density limits of the osmotic coefficient. For the low volume fractions typical of ionic solutions, identifying the second virial coefficients of the one component and that of the multicomponent models, one finds for the average diameter... 

A theoretic equation was derived by Kurata and Yamakawa for the osmotic second virial coefficient of a flexible polymer chain which takes into account not only the intermolecular interaction but also the intramolecular interaction of segments (such as the excluded volume effect). The equation given by... 

In the previous four sections we have dealt with some aspects of very dilute aqueous solutions. From the formal point of view, it is sufficient to study the solvation properties of one solute j in a pure solvent. We now proceed to the next step and study a pure solvent with two solutes. In the absence of a solvent, two-particles-in-a-system determines the second virial coefficient in the density expansion of the pressure (section 1.8). Likewise, two-solute-in-a-solvent determines the second virial coefficient of the osmotic pressure (section 6.11). This quantity is expressed in terms of the pair correlation function by... 

As already noted, estimation of the value of 6 is difficult the prediction of conditions under which B shall precisely vanish would be even more precarious. However, the "Theta point," so-called, at which this condition is met is readily identified with high accuracy by any of several experimental procedures. An excluded volume of zero connotes a second virial coefficient of zero, and hence conformance of the osmotic pressure to the celebrated law of J. H. van t Hoff. The Theta point may be located directly from osmotic pressure determinations, from light scattering measured as a function of concentration, or from determination of the precipitation point as a function of molecular welfjht. ... 

All the experimental results obtained in the semidilute 0-solutions are consistent with the idea of de Gennes and Brochardthat at the theta temperature, where the second virial coefficient of the osmotic pressure vanishes, the mesh size of the transient gel is not proportional to the correlation length of concentration fluctuations as it is in the case of a good solvent. ... 

The intermicellar potential U reflects the superposition of an osmotic repulsion and the exchange attraction. To obtain the second virial coefficient of micelle-micelle interactions, B, we approximate f/ as a square-well potential of width out and depth AFejchange having an infinite wall at L - Accordingly, 5 = J [1 exp(-[//jfeT)]r2dris... 

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