Virial coefficients, renormalization

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

Second, the effective virial coefficient B characterizing the interaction of segments differs from the usual virial coefficient B of the solution of disconnected rods -connectivity of segments into long chains is the reason. The corresponding renormalization of the virial coefficient has been studied in detail 32 33). In the application to the semiflexible macromolecule under consideration the result is (B - B)/B 1/p < 1, i.e. for the long rigid rods (p > 1) the renormalization is unessential. 

The 6 temperature of the solution of semiflexibie macromolecules in the limit p > 1 practically coincides with the 6 temperature of the solution of disconnected rods with the same p value - the reason is the small degree of renormalization of the virial coefficients (for a more detailed discussion see52,33)). Thus, the result of Eq. (2.18) for eld also remains valid for this case. From the comparison of Eqs. (2.18) and (3.6) it can be concluded that the 0 point is always (independently of L/C) situated in the low temperature region of the phase diagram, well below the triple point. 

In this section we "semi-empirically" adapt some scaling ideas from the Group Renormalization theory (12, 15) of polymer solutions to obtain expressions for the osmotic virial coefficients of Equations 6 and 7 in terms of the degree of polymerization. In the following discussion we will occasionally omit the indices on the osmotic virial coefficients for the sake of simplicity. 

In field theory, one uses a parameter related to the behaviour of the renormalized four-leg vertex and incorrectly called the renormalized interaction . In a very similar way, in polymer theory, the second virial coefficient can be used to define the interaction between two polymers. We proceed as follows. 

Direct renormalization calculation of the second virial coefficient and of critical exponents to second-order in e... 

This is quite reasonable since h is the parameter which determines the effective strength of the three-body interaction and since this fact entails that the other virial coefficients must be given for all d by finite expansions in powers of h. Thus no renormalization is necessary to express g in terms of h and according to (14.6.36), we have... 

A priori, the renormalization of the terms porportional to z is not difficult, and these terms could be treated in the limit S/s0 - oo, by direct application of the methods which we used to study the swelling and the virial coefficients. In particular, we see that the term proportional to z appears in the form... 

Going beyond the limiting law it is found that the modified (or renormalized) virial eoefiieients in Mayer s theory of eleetrolytes are fimetions of the eoneentration through their dependence on k. The ionic second virial coefficient is given by [62]... 

Examination of the terms to O(k ) in the SL expansion for the free energy show that the convergence is extremely slow for a RPM 2-2 electrolyte in aqueous solution at room temperature. Nevertheless, the series can be summed using a Pade approximant similar to that for dipolar fluids which gives results that are comparable in accuracy to the MS approximation as shown in figure A2.3.19(a). However, unlike the DHLL + i 2 approximation, neither of these approximations produces the negative deviations in the osmotic and activity coefficients from the DHLL observed for higher valence electrolytes at low concentrations. This can be traced to the absence of the complete renormalized second virial coefficient in these theories it is present... 

In the high salt limit, UbCp -C Oion, the contribution of the translational entropy of mobile ions, disproportionated between the interior and the exterior of the corona, is equivalent to a renormalization of the second virial coefficient of monomer-monomer interactions, as va Veff = Va + [see (84)]. 

Some details of the renormalization group approach for polymers as done in Sec. 4.2.1 are given here [12]. We consider the problem of two interacting directed polymers and study the second virial coefficient. The second virial coefficient is related to the two-chain partition function with all the ends free. Dimensional regularization is to be used here. 

Figure 5.15. Renormalization of the interchain two-body interactions (cf. Figure 5.14, the second virial coefficient vanishes simultaneously with the renormalized (Oono and Freed, 1981a)) (Reprinted with permission from Y.Oono, K.F.FVeed. J. Chem. Phys. 75 (1981) 993-1008. Copyright 1981 American Inalitute of Physics]...

The terms in the new series are ordered differently from those in the original expansion and Mayer showed that the Debye-Huckel limiting law follows as the leading correction to the ideal behavior for ionic solutions. In principle, the theory enables systematic corrections to the limiting law to be obtained as the concentration of the electrolyte increases for any Hamiltonian which defines the short-range potential u j (r), not just the one which corresponds to the RPM. A modified (or renormalized) second virial coefficient was tabulated by Porrier (1953), while Meeron (1957) and Abe (1959) derived an expression for this in closed form. Extensions of the theory to non-pairwise additive solute potentials have been discussed by Friedman (1962). 

Mayer s paper was an important milestone in the development of electrolyte theory and the principle ideas behind this theory and the main results at the level of the renormalized second virial coefficient will be presented below. It follows from Eqs. (3) and (11) that the Mayer f-function for the solute pair potential can be written as the sum of terms ... 

We now present briefly more explicit calculations of the mutual virial coefficients obtained with the use of des Cloizeaux direct renormalization method for blends of linear flexible polymers in a common good solvent, a common 0-solvent and a selective solvent and for blends of rodlike polymers and flexible polymers in a 0-solvent (marginal behavior). These calculations enable one to find (universal) prefactors relating the mutual virial coefficient to the chain volume (in Eq. 7) in the asymptotic limit. Moreover they give the corrections to the scaling behavior which explicitly depend on the interactions between unlike monomers and are actually responsible for the phase separation of flexible polymer blends in a good solvent. 

The direct renormalization method also allows the determination of the mutual virial coefficient in a common 0-solvent (g = g = 0) and a selective solvent (g = 0, g = g ). In both cases, for symmetric polymers, when the radius ratio is equal to unity we find a hard sphere interaction characterized by a dimensionless virial coefficient... 

In a dilute solution in a common good solvent for both blocks, the interactions between. different copolymers may be studied using the same direct renormalization procedures as the interactions between two homopolymers A and B equivalent to the two blocks.As for blends, in the asymptotic limit of infinite molecular masses, the chemical difference between the two blocks is irrelevant and the dimensionless virial coefficient gc between block copolymers defined by Eq. (10) is equal to the same value g as for homopolymers. The interactions which may provoke the formation of mesophases are here again due to the corrections to the scaling behavior ... 

The interaction between flexible polymer chains of different chemical nature A and B is measured by their second virial coefficient GaB- We have calculated Gab using Descloizeaux direct renormalization method in a solvent which is either a good or a 0 solvent for the chains when the A-B interaction is repulsive. 

As mentioned above, an independent variable (the so-called renormalized coupling constant) g is used in the analysis of J and K. It is related to the second virial coefficient and so its designation as an interaction constant or coupling constant is not without merit. However, the designation renormalized is historical and a complete misnomer in the present context. The quantity gf is a well-defined and finite quantity in the two-parameter model and involves absolutely no renormalization. It is given by... 

A chain conformational renormalization group method has been developed by Miyaki and Freed " for star polymers. Explicit expressions for the distribution functions for intersegment distance vectors and their moments, b, the osmotic second virial coefficient and the related functions, etc., have been reported based on the 8-expansion approximation. A detailed comparison of these results with those of linear chains, scaling predictions and the experimental data is also reported in the literature. 

---