Second virial coefficient Flory-Huggins

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. 

To use the Flory-Huggins theory as a source for understanding the second virial coefficient, we return to Eq. (8.53), which gives an expression for jui -jui°. Combining this result with Eq. (8.79) gives... 

Another important application of experimentally determined values of the osmotic second virial coefficient is in the estimation of the corresponding values of the Flory-Huggins interaction parameters x 12, X14 and X24. In practice, these parameters are commonly used within the framework of the Flory-Huggins lattice model approach to the thermodynamic description of solutions of polymer + solvent or polymer] + polymer2 + solvent (Flory, 1942 Huggins, 1942 Tanford, 1961 Zeman and Patterson, 1972 Hsu and Prausnitz, 1974 Johansson et al., 2000) ... 

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. 

All that remains to be done to complete our derivation of the second virial coefficient in terms of the Flory-Huggins theory is convert volume fractions into practical concentration units. First, we can express the volume fraction of the solute in terms of partial molar volumes ... 

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. 

It should also be mentioned that polymer-solvent interactions can be characterized by the second virial coefficients that appear in equations (8) and (13) and by the free energy of interaction parameter Z1 that appears in the Flory-Huggins theory of polymer solution thermodynamics.1,61... 

Once the second virial coefficient has been obtained for a given polymer - solvent system one can calculate the corresponding Flory - Huggins interaction parameter, X, from the equation ... 

Calculate the for solutions in chlorobenzene of polystyrenes with molar weights of 2 x 105,5 x 105 and 106 g/mol the intrinsic viscosity, the critical concentration, the swelling factor, the hydrodynamic swollen volume, the second virial coefficient and the Flory-Huggins interaction parameter. 

Interaction parameter in Flory-Huggins treatment of polymer mixtures after normalization on a per monomer basis this becomes Xay also susceptibilities in discussion of density functional theories. Applied external field acting on species a in conformation X". Applied external field as a function of position acting on species a. Contribution of attractive interactions to the second virial coefficient for species pair ay also van der Waals 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... 

It can be seen that for small values of z, the perturbation second virial coefficient given by Eq. (3.117) is equal to the factor in parenthesis. Since N is proportional to M2, the virial coefficient should be independent of molecular weight in the limit of small z. This is the same result we have derived earlier from the Flory-Huggins theory. In the limit of small z, Eq. (3.117) is frequently combined with Eq. (3.86) to yield... 

The second virial coefficient B2 can be related to the interaction parameter Xi via the Flory-Huggins relationship... 

On replacement of the second virial coefficient A2 with the result of the Flory-Huggins equation (Chapter 7) one obtains ... 

The parameter xi is a measure of the interaction enthalpy per solvent molecule. It is called the Flory-Huggins interaction parameter, or simply interaction parameter, or sometimes chi parameter. Since the second virial coefficient is given by... 

The concentration dependence of the osmotic pressure is another important test of the Flory-Huggins theory. We will first derive an expression relating osmotic pressure and polymer concentration and then define the second virial coefficient (A2). 

Solution activity data obtained by osmometry on dilute solutions showed that the second virial coefficient is dependent on molar mass, contradicting the Flory-Huggins theory. These problems arise from the mean-field assumption used to place the segments in the lattice. In dilute solutions, the polymer molecules are well separated and the concentration of segments is highly non-uniform. Several scaling laws were therefore developed for dilute (c < c is the polymer concentration in the solution, c is the threshold concentration for molecular overlap) and semi-dilute (c > c ) solutions. In a good solvent the threshold concentration is related to molar mass as follows ... 

Now, according to the Flory-Huggins theory, the second virial coefficients are related to the interaction parameters as follows ... 

For neutron scattering experiments, the Flory-Huggins interaction parameter can be obtained from the second virial coefficient from the expression ... 

The slope of the initial linear region of the plot can be used with Equation (3.100) to evaluate the second virial coefficient A2 from which it is possible to calculate the Flory-Huggins polymer-solvent interaction parameter preferably by use of Equation (3.103) rather than Equation (3.102). In good solvents A3 is significant and the plots (see Fig. 3.11) show distinct upward curvature at higher c due to the term in Equation... 

The model of excluded volume thus predicts that the second virial coefficient A2 = 2 decreases as the molar mass of the polymer increases, in contrast to the regular Flory-Huggins model. 

The Flory-Krigbaum model leads to an expression for the second virial coefficient (A2) that differs by the factor F(Y) from relation (4.27), given by the Flory-Huggins model ... 

For good solvents (x < 0), the Flory-Huggins theory predicts that A2 is proportional to w and is independent of chain molecular weight. However the experimental data show that A2 Attempts to explain this molecular weight dependence of the second virial coefficient have been made by Flory and Krigbaum and others. " A complete theory will not involve a lattice model and treat the averages more carefully. It is still an active subject. 

The different crosslinking behaviours of photopolymer films formed from different solvents can be explained with the help of different polymer conformations in these solvents. Studies were performed with one selected photopolymer (27). The second virial coefficients (A2), determined by light scattering, revealed the most coiled photopolymer structure in THF solution. This is to be expected since the magnitude A2 (found to be lowest in THF solution) is related to the Flory-Huggins constant Xi widely used in fundamental polymer studies as measure for the polymer-solvent interactions. If A2>0, the interactions between polymer molecules and solvent molecules are more attractive than the solvent-solvent interactions. Such solvents are considered to be thermodynamically good for the polymer. Thus, the low value for THF means that this solvent can be considered to be poor , leading to a more... 

Flory-Huggins interaction parameter Circular frequency Interaction parameter Second virial coefficient Concentration... 

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