Second Virial Coefficients of Polymer

Many presentations of the second virial coefficient of polymer solutions contain different expressions for the quantities we have discussed. The difference lies in the fact that the factor p( - 0/T) appears in place of 1/2 - x-There are several attitudes we can take toward this difference. For one thing, we can regard the discrepancy as nothing more than different notation ... 

This is made possible by considering the second virial coefficients of polymers in the various solvents after different solution histories. The second virial coefficient,... 

Casassa E.F. (1962). Effect of heterogeneity and molecular weight on the second virial coefficient of polymers in good solvent. Polymer, 3, 625-638. 

The fact that the second virial coefficient of polymer chains becomes negative in worse than 0-solvents is indicative of mutual segmental attraction (Flory, 1953). Moreover, as discussed in Chapter 8, when sterically stabilized particles coagulate under the action of van der Waals forces, an entirely different pattern of flocculation behaviour ensues. Specifically, flocculation occurs in dispersion media that are better solvents than 0-solvents, i.e. when the segments are undoubtedly mutually repulsive. 

The second virial coefficient of polymer solutions has been the subject of theoretical study for decades. Nonetheless, no persuasive explanation is as yet available concerning its molecular weight dependnece in good solvent systems. To be important, the defect of the existing theories of A2 is not limited to this point. They reveal another discrepancy with experiment when we look at the behavior of A2 in poor solvents below the 0 temperature. 

Some implicit databases are provided within the Polymer Handbook by Schuld and Wolf or by Orwoll and in two papers prepared earlier by Orwoll. These four sources list tables of Flory s %-function and tables where enthalpy, entropy or volume changes, respectively, are given in the literature for a large number of polymer solutions. The tables of second virial coefficients of polymers in solution, which were prepared by Lechner and coworkers (also provided in die Polymer Handbook), are a valuable source for estimating the solvent activity in the dilute polymer solution. Bonner reviewed vapor-liquid equilibria in concentrated polymer solutions and listed tables containing temperature and concentration ranges of a certain number of polymer solutions." Two CRC-handbooks prepared by Barton list a larger number of fliermodynamic data of polymer solutions in form of polymer-solvent interaction or solubility parameters." ... 

KUW Kuwahara, N. and Miyake, Y., Second virial coefficient of polymer solutions, Kobunshi Kagaku, 18, 153, 1961. 

Sedimentation Coefficients, Diffusion Coefficients, Partial Specific Volumes, Frictional Ratios, and Second Virial Coefficients of Polymers in Solution, ... 

Second Virial Coefficients of Poly(alkenes) and Poly(allcynes) VII / 165 D. TABLE OF SECOND VIRIAL COEFFICIENTS OF POLYMERS IN SOLUTION... 

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. 

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. 

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

It is important to note that the values of the second virial coefficients determined by light scattering are weight-average quantities. The general expression for the second virial coefficient of a polydisperse polymer is as follows (Casassa, 1962) ... 

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. 

Kobayashi, H. Molecular weight dependence of intrinsic viscosity, diffusion constant, and second virial coefficient of polyacrylonitrile. J. Polymer Sci. 39, 369-388 (1959). 

The free energy change which takes place when polymer chains interpenetrate is influenced by factors such as temperature, pressure and solvent composition. The point at which this free energy change is equal to zero is known as the 0 (theta)-point and such a solvent is called a 0-solvent. More formally, a 0-point is defined as one where the second virial coefficient of the polymer chains is equal to zero. It can be determined by light scattering and by osmometry. 

This fact can be demonstrated as follows. Let us determine the value of the well-known Flory parameter x, which corresponds to the 6 point (i.e. to the point of inversion of the second virial coefficient of the solution of rods) in the Flory theory of Ref.9). This can be done by expanding the chemical potential of the solvent in the isotropic phase (Eq. (16) of Ref.9 ) into powers of the polymer volume fraction in the solution, and by equating the coefficient at the quadratic term of this expansion to zero this procedure gives Xe = 1/2 independently of p. On the other hand, it is well known26,27) that the value of x decreases with increasing p and that X < 1 at p > 1. The contradiction obtained shows that the expressions for the thermodynamic functions used in Ref.9) are not always correct... 

In polymer science, the ideal form of the thermodynamic equations is preserved and the nonideality of polymer solutions is incorporated in the virial coefficients. At low concentrations, the effects of the cl terms in any of the equations will be very small, and the data are expected to be linear with intercepts which yield values of and slopes which arc measures of the second virial coefficient of the polymer solution. Theories of poly mer solutions can be judged by their success in predicting nonideality. This means predictions of second virial coefficients in practice, because this is the coefficient that can be measured most accurately. Note in this connection that the intercept of a straight line can usually be determined with more accuracy than the slope. Thus many experiments which are accurate enough for reasonable average molecular weights do not yield reliable virial coefficients. Many more data points and much more care is needed if the experiment is intended to produce a reliable slope and consequent measure of the second virial coefficient. 

It is to be expected that measurements of the osmotic pressures of I he same polymer in different solvent should yield a common intercept. The slopes will differ (Fig. 3-1 a), however, since the second virial coefficient reflects polymer-solvent interactions and can be related, for example, to the Flory-Huggins interaction parameter x (Chapter 12) by... 

Such a product assumption has been used successfully in the evaluation of the second virial coefficient of chain polymers 48). With respect to the singlect distribution function the reader is refered to Reference (47) and also Reference (49). 

Determine the number-average molar mass of the polymer and the second virial coefficient of the solution Aj. 

At low concentrations, the effects of the terms in any of the above virial equations, Eqs. (4.52) to (4.54), will be very small, and the data of n/c versus c are expected to be linear with intercepts at c = 0 yielding values of M and slopes that are measures of the second virial coefficient of the polymer solution. 

The second virial coefficient of polyelectrolytes is treated theoretically either by applying the theory for charged spherical colloids with a correction for the chain character of the polyion or as an extension of the theory of the second virial coefficient for nonionic linear polymers. An example of an extension of the theory of the second virial coefficient for nonionic linear polymers to polyelectrolytes is the above-mentioned Yamakawa approach of using perturbation theory of excluded volume to calculate A2 [42],... 

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. 

Fig. 13.1. The second virial coefficient of a polymer solution is directly proportional to the ratio g(z)/g whose dependence with respect to the parameter z is represented above. The value of g corresponding to this curve is g = 0.233.

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