Second virial coefficient molecular weight dependence

THE DEPENDENCE OF THE SECOND VIRIAL COEFFICIENTS ON MOLECULAR WEIGHT THREE MODELS... 

To overcome the problem of non-ideality the work be carried out at the Q temperature because in nonideal solutions the apparent Molecular weight is a linear function of concentration at temperatures near Q and the slope depending primarily on the second virial coefficient. 

The properties of solutions of macromolecular substances depend on the solvent, the temperature, and the molecular weight of the chain molecules. Hence, the (average) molecular weight of polymers can be determined by measuring the solution properties such as the viscosity of dilute solutions. However, prior to this, some details have to be known about the solubility of the polymer to be analyzed. When the solubility of a polymer has to be determined, it is important to realize that macromolecules often show behavioral extremes they may be either infinitely soluble in a solvent, completely insoluble, or only swellable to a well-defined extent. Saturated solutions in contact with a nonswollen solid phase, as is normally observed with low-molecular-weight compounds, do not occur in the case of polymeric materials. The suitability of a solvent for a specific polymer, therefore, cannot be quantified in terms of a classic saturated solution. It is much better expressed in terms of the amount of a precipitant that must be added to the polymer solution to initiate precipitation (cloud point). A more exact measure for the quality of a solvent is the second virial coefficient of the osmotic pressure determined for the corresponding solution, or the viscosity numbers in different solvents. 

Kobayashi, H. Molecular weight dependence of intrinsic viscosity, diffusion constant, and second virial coefficient of polyacrylonitrile. J. Polymer Sci. 39, 369-388 (1959). 

Tables 8.2.2 and 8.2.3 give typical results for two series of polydisperse lignin fractions obtained from acidic organosolv delignification of black cottonwood (Pla et al. 1986) and from alkaline delignification of western hemlock (Dolk et al. 1986). In both cases, LALLS allows accurate determination of low molecular weight values. The nearly identical dn/dc values for a given series of lignin fractions indicate the good reproducibility and accuracy of the technique. However, the second virial coefficients, A2, vary considerably depending upon the fraction measured.

The second virial coefficients of pol5uner solutions depend on the molecular weight of the polsmier molecules. This dependece is empirically given by ... 

This investigation has enhanced our understanding of the factors which contribute to the molecular weight dependence of protein partitioning. The molecular weight dependence of the protein partition coefficient results from a competition between two terms in the partition coefficient expansion, namely the crossed second virial coefficient and the differences between the polymer concentrations in the top and bottom phases. While the trend in binodal concentrations tends (in part) to favor the trends observed experimentally, the trend in the second virial coefficient tends to oppose the experimental trends. 

Under conditions of partly screened interactions in dilute solutions (high added salt concentration cs and low polymer concentration c), the solution osmotic pressure can be expressed via a virial expansion (Eq. 24). Then light scattering becomes a useful tool to obtain values of second virial coefficients characterizing interactions in solution. The second virial coefficient can be calculated from the slope of the dependence given by Eq. 25. The relation between the true and the apparent second virial coefficient is similar to the relation between the true and the apparent molecular weight (see the previous section for more details and the meaning of the symbols) ... 

The dependence of the second virial coefficient on basic quantities such as the solution ionic strength or the polymer molecular weight is usually of greater scientific interest than the knowledge of true vs. apparent values. Therefore the time-consuming dialysis step leading to true rather than apparent values is usually omitted. Hence measured second virial coefficients are usually only apparent values. 

The dependence of the second virial coefficient of coil-shaped molecules is difficult to calculate since the excluded volume is a complicated function of the molecular weight (see Section 4.5.2). The usual method is to replace the excluded volume of the molecule u in equation (6-64) by the excluded volume of the chain segment Ihe molecular weight M2 of the molecule is also replaced by the formula molecular weight M of the chain segment. The molecular weight dependence of the excluded volume is expressed in terms of a function h z) whose coefficients have been evaluated theoretically ... 

The hydrodynamic virial coefficient, Iq, is defined as the concentration coefficient in D=Dq(1+Jcj)C+...) Dq is the value of D at infinite dilution kj)= 2A2M - kf-V2 where M is the polymer molecular weight and kf describes the concentration dependence of the friction coefficient, f where f=fQ(l+k ) and V2 is the partial specific volume, kj) is thus the sum of a static factor, proportional to the second virial coefficient, A2, and the concentration dependence of the friction coefficient. Expressions for kf have been... 

---