A2 Second virial coefficient

Abbreviations A H Huggins coefficient M, molar mass R, radius of gyration RD, core radius p, association number AmcH°, standard enthalpy of micellization, AmlcG°, standard Gibbs energy of micellization A2, second virial coefficient. Ru, hydrodynamic radius. 

One sees that, if i 4> n which is equivalent, in the experimentalist notations, to A2CM,(A2 second virial coefficient, c concentration and M molecular weight) is large, one recovers Eq.(l8). If one works in dilute solution, the scattering intensity depends only on J q) and the copolymer structure has no effect. One has to realize that in bulk as well as in solution, it will be difficult from an experimental point of view to use this last approximation for the determination of J(q). If the labeled central part of the copolymer is small compared to the wavelength and to the other part of the copolymer, the contrast will be low and the experiment difficult, but, with the accuracy of the present neutron scattering equipment, it does not seem to be impossible to measure J(q) on such systems. 

Figure A2.1.7. The second virial coefficient 5 as a fiinction of temperature T/T. (Calculated for a gas satisfying the Leimard-Jones potential [8].)...

Figure A2.3.1 Second virial coefFicient BJT) of several gases as a fimction of temperature T. (From [10]).

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

Figure A2.3.6 (a) Reduced second virial coefFicient fimction of and (b) In versus 1/Jj for...

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

At least two runs were carried out on each of the seven narrow MWD PS standards and data were processed with MOLWT using the second virial coefficient (A2) relationship (14). 

The true second virial coefficient can be calculated via A2 — A2 aPP(Mapp/Ma,). 

For accuracy in light-scattering measurement the proper choice of solvent is necessary. The difference in refractive index between polymer and solvent should be as large as possible. Moreover, the solvent should itself have relatively low scattering and the polymer-solvent system must not have too high a second virial coefficient as the extrapolation to zero polymer concentration becomes less certain for high A2. Mixed solvent should be avoided unless both components have the same refractive index. 

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. 

As the second virial coefficient of non-ideality, A2, is generally finite, the molecular weight is given by... 

Essentially, separate experiments on each polymer in the same solvent yield vA, Ma and the second virial coefficient (A2)a as well as the corresponding quantities for polymer B. When a mixture of the two polymers in which the composition of the polymers are WA and WB is dissolved in the same solvent, there are two approaches. 

The osmotic second virial coefficient A2 is another interesting solution property, whose value should be zero at the theta point. It can be directly related with the molecular second virial coefficient, expressed as B2=A2M /N2 (in volume units). For an EV chain in a good solvent, the second virial coefficient should be proportional to the chain volume and therefore scales proportionally to the cube of the mean size [ 16]. It can, therefore, be expressed in terms of a dimensionless interpenetration factor that is defined as... 

The next step consists of the determination of the size of the macromolecules in space. Two equivalent sphere radii can be measured directly by means of static and dynamic LS. Another one can be determined from a combination of the molar mass and the second virial coefficient A2. Similarly, an equivalent sphere radius is obtained from a combination of the molar mass with the intrinsic viscosity. This is outlined in the following sections. 

---