Second virial coefficient-molecular weight

Measurements of viscosity and light scattering have proven that maximum coil dimensions of PAAm/AAcNa in salt solutions are achieved at about 67 mole ) AAcNa. The maximum behaviour was researched by measuring the radius of gyration, second virial coefficient, molecular weight, preferential solvation, viscosity and absorption bands of the copolymer series. Thus, an increase in the possible ways of arranging AAcNa-AAm-AAcNa units (via H-bonds) will lead to extended coil dimensions and therefore influence the viscosity level. In addition, it can... 

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

In the next section we shall describe the use of Eq. (8.83) to determine the number average molecular weight of a polymer, and in subsequent sections we shall examine models which offer interpretations of the second virial coefficient. 

Fig Second virial coefficients as a function of Molecular weight for a solution of poly (methyl methacrylate) in acetone. 

The weight-average Molecular weightin the inverse of the intercept at c = 0 and q = 0 in the Zimm plot. The second virial coefficient can be calculated from the slope of the lines at constant angle by eq. 8.34, and to a first approximation is independent of angle. The radius of gyration is... 

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. 

Archibald (1947) showed that measurement of c and dc/dr at the cell boundaries permit Molecular weight determination at any stage in the equilibrium process. However, when measurements are made early enough, before the molecular species have time to redistribute in the cell, the weight-average Molecular weight and second virial coefficient can be evaluated. In practice, measurements can be made... 

Of the preponderance of small ions, the colligative properties of polyelectrolytes in ionising solvents measure counterion activities rather than Molecular weight. In the presence of added salt, however, correct Molecular weights of polyelectrolytes can be measured by membrane osmometry, since the small ions can move across the membrane. The second virial coefficient differs from that previously defined, since it is determined by both ionic and non-ionic polymer-solvent interactions. 

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

If the average molecular weight of the sample is known its second virial coefficient can be predicted using the Kok-Rudin method (6). Input parameters for this calculation are M,... 

The existence of aggregates is evidenced by virial coefficients wh ch are lower than the theoretical values, for the measured M,. This is because the second virial coefficient decreases with increasing molecular weight. Supermolecular aggregates appear to have very high effective molecular weights. 

The MOLWT-II program calculates the molecular weight of species in retention volume v(M(v)), where v is one of 256 equivalent volumes defined by a convenient data acquisition time which spans elution of the sample. I oment of the molecular weight distribution (e.g., Mz. Mw. Mn ) are calculated from summation across the chromatogram. Along with injected mass and chromatographic data, such as the flow rate and LALLS instruments constants, one needs to supply a value for the optical constant K (Equation la), and second virial coefficient Ag (Equation 1). The value of K was calculated for each of the samples after determination of the specific refractive index increment (dn/dc) for the sample in the appropriate solvent. Values of Ag were derived from off-line (static) determinations of Mw. 

Selected entries from Methods in Enzymology [vol, page(s)] Association constant determination, 259, 444-445 buoyant mass determination, 259, 432-433, 438, 441, 443, 444 cell handling, 259, 436-437 centerpiece selection, 259, 433-434, 436 centrifuge operation, 259, 437-438 concentration distribution, 259, 431 equilibration time, estimation, 259, 438-439 molecular weight calculation, 259, 431-432, 444 nonlinear least-squares analysis of primary data, 259, 449-451 oligomerization state of proteins [determination, 259, 439-441, 443 heterogeneous association, 259, 447-448 reversibility of association, 259, 445-447] optical systems, 259, 434-435 protein denaturants, 259, 439-440 retroviral protease, analysis, 241, 123-124 sample preparation, 259, 435-436 second virial coefficient [determination, 259, 443, 448-449 nonideality contribution, 259, 448-449] sensitivity, 259, 427 stoichiometry of reaction, determination, 259, 444-445 terms and symbols, 259, 429-431 thermodynamic parameter determination, 259, 427, 443-444, 449-451. 

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. 

Where Mw 1 the weight average molecular weight or the apparent molecular weight in the cape of copolyraerp, is the second virial coefficient, and K is an optical constant defined as... 

In Flory s theory (/< ), a polymer-solvent system is characterized by a temperature 0 at which (i) excluded-volume effects are just balanced by polymer-solvent interactions, so that os=l, (ii) the second virial coefficient is zero, irrespective of the MW of the polymer, and (iii) the polymer, of infinite molecular weight, is just completely miscible with the solvent The fundamental definition of the temperature is a macroscopic one, namely that for T near 0 the excess chemical potential of the solvent in a solution of polymer volume fraction v2 is of the form (18) ... 

In practice, from a knowledge of measured values of the osmotic second virial coefficients it is rather easy to calculate the spinodal curve. It is worthy of note here to observe that reciprocal values of m, for biopolymers of rather high molecular weight (> 104 g/mol) are often comparable with the magnitude of A 24. This requires that, as well as values of the osmotic second virial coefficients, the molecular weight should also be taken into account in the prediction of the boundary conditions relating to phase separation. 

Casassa E.F. (1962). Effect of heterogeneity and molecular weight on the second virial coefficient of polymers in good solvent. Polymer, 3, 625-638. 

Figure 6.10 Effect of CITREM on the molecular and thermodynamic parameters of maltodextrin SA-2 (DE = 2) in aqueous medium (phosphate buffer, pH = 7.2, ionic strength = 0.05 M 20 °C) (a) weight average molar mass, Mw (b) radius of gyration, Ra (c) structure sensitive parameter, p, characterizing die architecture of maltodextrin associates (d) second virial coefficient, A2 or A2, on the basis of the weight ( ) and molal (A) scales, respectively. The parameter R is defined as the molar ratio of surfactant to glucose monomer units in the polysaccharide. The indicated cmc value refers to the cmc of the pure CITREM solution. Reproduced from Anokhina et al. (2007) with permission.

Some of the experimental details of osmometry and the use of osmometry for determining molecular weights and second virial coefficients are discussed in Section 3.3. 

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