Virial regime

Some authors [59] call the region c < c the virial regime. Probably they consider that the virial expansion for osmotic pressure would diverge as c approaches c. ... 

In very dilute solutions we have considered mainly single chain motions, and we have studied dynamic quantities in the limit of zero concentration. When the concentration is non-zero, one must take into account interactions between different chains, both static interactions and hydrodynamic interactions. If the concentration is smaller than the overlap concentration c, in the so-called virial regime, these effects are small and can be treated as perturbations. This is not the case, however, in a semi-dilute regime, where the motions become collective and thus renormalize the local monomer friction C which becomes concentration dependent. 

In the opposite limit of a negative second virial coefficient, V2 < 0, corresponding to the bad or poor solvent regime, the polymer coil will be collapsed due to attraction between monomers. In this case, the attraction term in the free energy is balanced by the third-virial term in a low-density expansion (where we assume that V3 > 0),... 

Larson s results [154] are divided into the three shear rate regimes - tumbling, wagging, and steady-state - as explained below. He chose the strength of mean-field potential 2L2dc in Eq. (41) to be 10.67, which corresponds to the concentration cA of the nematic phase coexisting with the isotropic phase (in the second virial approximation), and expressed the shear rate in terms of T defined by... 

This series is known as the virial expansion, where Bn are the virial coefficients. In principle, these last can be calculated if the potential is known. In practice, however, the calculation is feasible only for the first few coefficients. In this framework, Eq. (6) is only applicable to low density regimes. Obviouly, a complete theory is expected to confidently calculate the highest possible number of known virial coefficients. Such a calculation is one of the benchmarks of its overall accuracy. 

The parameter r can be optimized for thermodynamic consistency between the virial and compressibility routes to the free energy of the system. As discussed in Section 5.3.2, the KHM closure (7) brings about an additional long-range term in the direct correlation functions in the critical regime. 

The interval z p z, also depends on the nature of the observable. Figure 15.11 shows, for instance, that the asymptotic behaviour of the osmotic parameter g(z) is practically reached when z P 10. For the end-to-end swelling, it is necessary to wait until zp 30 (see Fig. 15.12). We shall now examine to what extent we can observe the asymptotic regimes of the gyration swelling, of the second virial coefficient (for 2 P 1) and of the form function (for z p 1 and qRp l). 

Better experimental values of the exponent v are obtained from measurements of the radius of gyration of polymers and of the second virial coefficient, in dilute solutions. Indeed, when the number of links increases, these quantities reach the asymptotic regime much more rapidly. However, the realization of a true asymptotic state for these quantities remains the main difficulty to overcome. Some progress has recently been made thanks to the elaboration of samples of high molecular mass Mw = 5 x 107. 

It is necessary to consider different regimes dependent on the dimensions of polymer chains. A long chain collapses into a globule consisting of close-packed blobs of size The blobs are either Gaussian or swollen depending on the second virial coefficient value. In a 0 solvent, the chain radius in the collapsed state is given by ... 

We, therefore, find that the PE star size, R, in the salt-dominated regime decreases as upon an increase in salt concentration. Note that the same scaling dependence is obtained when the electrostatic repulsions between the charged monomers are accounted for through the screened Coulomb binary interaction potential, u r)/k T = l r exp -r/rj)). A further increase in Oion leads to an additional decrease in When Uett becomes on the order of the bare virial coefficient u, the star size R approaches that of a neutral star, (4). Therefore, the size 7 of a quenched PE star demonstrates a plateau at low salt concentrations (in the osmotic regime), decreases in the salt-dominated regime as 7 und ap-... 

Actually, the entire present development stays under the valence or long-range regime of electrons in atoms and molecules in various forms and approximations of conceptual DFT. The minus sign in Eq. (4.203) agrees with the opposite phenomenological behavior in density and potential variation, as provided by Poisson equation—for instance (Putz et al., 2005), and is in accordance with alternative derivation based on ehemical action principle and virial theorem (Putz, 2009a). 

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