Third virial coefficient reduced

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. 

Reduced third virial coefficient 4th virial coefficient, density expansion... 

Data for third virial coefficients are often lacking, but generalized correlations are available. Equation (4-68) may be rewritten in reduced form as... 

For the most satisfactory correlation of third virial coefficients it is best to turn to empirical methods of which the method of Orbey and Vera [83-orb/ver] is particularly simple and effective. The third virial coefficient in reduced form is given by... 

X is the wavelength in vacuo, N is Avogadro s number, while A and A are the second and third virial coefficients. The term P(9 is the form factor which is a function of the size and shape of the macromolecule in solution and represents the modulation of the intensity of scattered radiation due to the finite size of the molecule and to its deviation from sphericity. The term dn/dc is the specific refractive index increment and represents the change in solution refractive index as a function of solute concentration. If experiments are conducted in the limit of zero scattering angle where P(9) = 1 as well as at sufficiently low concentrations where only the second virial coefficient need be considered, then eq. (1) reduces to... 

State (12.2.15) [9]. At 76 K, B22 = -12 cc/mol [10], so unlike attractions between solute and solvent are stronger than those between solvent molecules. As pressure increases, the effect of this attraction on overcomes the slight tendency of CH4 to volatilize, so solubility increases. At higher pressures, where the third virial coefficient C becomes important, this decrease in q> with increasing P is reduced, because C is positive while B is negative. 

Figure 1 Reduced second and third virial coefficients vs. reduced temperature for the Lennard-Jones 6—12 potential...

Figure 1 shows reduced virial coefficients calculated for a more realistic two-parameter potential, the Lennard-Jones 6—12 potential. The only significant difference between the Lennard-Jones and square-well second virial coefficients occurs at high reduced temperatures, where experimentally B is observed to have a maximum value. This maximum is related to the softness of the repulsive portion of the potential it therefore cannot be reproduced by the infinitely steep square well. The third virial coefficients calculated from the square well are qualitatively correct but show significant departures from the Lennard-Jones values. 

A semi-empirical method for calculating the additive portion of the interaction third virial coefficient from a knowledge of the reduced third virial coefficient of a pure substance was proposed by Rowlinson, Sumner, and Sutton. Their equation can be written in the form... 

In discussing A3 we often use a reduced third virial coefficient g defined by... 

Fig. 2-14. Molecular weight dependence of the reduced third virial coefficient g for PS in benzene [91]. 

Cpy = third virial coefficient for "simple fluids" reduced by Vj,... 

C = reduced third virial coefficient from L-J potential C = see Equation (9a)... 

Note that these results for the theta chains are different from the chain dimensions of ideal chains in the slit and the tube. The confined ideal chains have the dimension of in both confining geometries. The difference between the ideal chain and the theta chain shows up because of the third virial coefficient A3. Compensation of the excluded volume effect by the attractive polymer-polymer interaction allows the theta chain to have the same dimension as that of the ideal chain, but it is valid only in the three-dimensional free solution. In spaces of a reduced dimensionality, the same attractive interaction cannot mask the excluded volume. 

Figure 3.2 The dimensionless third virial coefficient C(p /RT ) of argon as a function of reduced temperature T/T, where p is the critical pressure and T is the critical temperature. Values computed from the equation of state of Tegeler et alf...

It is well known that dimensions of charged polymers and [ti] are affected by intramolecular electrostatic interactions [43]. In this context, the use of the ACM allowed detailed measurements of electrostatic effects in the copolyelectrolyte solutions to be made while varying ionic strength and polyelectrolyte concentration [44-47]. The evolution of second and third virial coefficients, A and A3, the angular scattering envelope, and the reduced viscosity were determined simultaneously. 

In the limit of vanishing Z3, equations (32), (35) and (36) all reduce to the results of the two-parameter theory as discussed, for example, in Yamakawa s book (see also below. Section 3.2.4.3). An assessment of Z3 can be made from measurements of the third virial coefficient (Section 3.3.2.2) under conditions with A2 = 0. It is indeed found to be small, of the order of 10 to 10" for the few systems studied. 

The third virial coefficients for PS in toluene and benzene and for PIB in cyclohexane, all good solvent systems, were fotmd to vary in proportion to unless M < 10 . This exponent 0.6 ought to be observed if the reduced third virial coefficient g defined by... 

Figm 15 Reduced third virial coefficients for PS in benzene at 25 °C (unfilled circles) and PIB in cyclohexane at 25 °C (filled circles), compared with theoretical cunres. Solid line, for 26=0.25 and 2 =0.025 dashed line, dotted-dashed line, coil limit of P2 ( a. two-parameter theory). 

The acentric factor, CO, was the third parameter used (20) in an equation based on the second virial coefficient. This equation was further modified and is suitable for reduced temperatures above 0.5. 

For non-polar molecules, Tsonopoulos [74-tso] modified the Pitzer and Curl [57-pit/cra] relationship for the reduced second virial coefficient in terms of T the temperature reduced by the critical temperature, and a third parameter, the acentric factor (O, which had been introduced to extend the application to non-spherical non-polar molecules, to give ... 

Thus, the two sets of virial coefficients are related by D = 6n(wo)" Finally, a third set of coefficients is that of the reduced values,... 

Practical correlations of virial coefficients employ as scaling parameters the critical temperature and the characteristic molar volume RTJpc, where pc is the critical pressure, and seek to represent Bip RT and C(pJRTcf as universal functions of the new reduced temperature Ti=TITc. Although the principle, as stated above, applies to only a small number of simple fluids, Pitzer showed that many different kinds of molecular complexity may be accounted for by the inclusion of a third parameter to which he called the acentric factor. This parameter is defined in terms of the vapour pressure psm, by the equation... 

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