Second virial coefficient of osmotic pressure

The second virial coefficient of osmotic pressure, A2, for a linear polymer of MW M is shown to be given by ... 

If solvents are used which do not possess a high dissolving power for both kinds of blocks (high second virial coefficients of osmotic pressure), phase separation occurs at considerably lower concentrations, and the solvent content of the aggregates is lower than that of the matrix. 

However, the structure and composition of microdroplets in which the reaction takes place are not the only parameters controlling polymerisation. We shall see later that particles in the initial microemulsion cannot be considered as independent reactors since interparticle interactions play an important role. Small angle light and neutron scattering experiments have shown that these interactions are attractive [6.20]. There is a clear increase in these attractive interparticle forces as the proportion of acrylamide is raised. In particular, this has two consequences the second virial coefficient of osmotic pressure takes negative values the peak in the structure factor which characterises a hard sphere system is no longer present. 

Figure 4. Adnesion energy (erg/cm ) for SUPC biiayers in 0.1 M salt (PBS) plus dextran polymers. Number average polymer indices - Np (number of glucose monomers). Solid and dasned curves - predictions from mean field theory with first and second virial coefficients from osmotic pressure measurements (14).

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

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. 

We have already seen that the second virial coefficient may be determined experimentally from a plot of the reduced osmotic pressure versus concentration. Since all other quantities in Equation (99) are measurable, the charge of a macroion may be determined from the second virial coefficient of a solution with a known amount of salt. As an illustration of the use of Equation (99), we consider the data of Figure 3.6 in Example 3.5. 

Theta conditions are identified experimentally as the situation in which the second virial coefficient of the osmotic pressure is zero. 

Thus, the osmotic pressure is first measured at different polymer concentrations, no c is then plotted vs. c, the values are linearly extrapolated to c 0, and the value of is determined from the y axis intercept. A2 is the second virial coefficient of the osmotic pressure. Solvents where A2 = 0 are called ideal or d solvents. 

The effect of polymer molecular weight on the viscosity when added at T < is shown in Figure 5. The viscosities were measured at a shear rate of 100 s and T = 25 C. The viscosity ratio is the viscosity of the dispersion plus polymer to the viscosity of the dispersion. The viscosity of the lamellar dispersion is reduced by the addition of the nonionic polymers of all molecular weights and the ranking of effectiveness (i.e., 20K > PVA = 3350 > 600) is consistent with the expected osmotic pressure due to the second virial coefficient of the polymers. At an equal molar concentration the second virial coefficient and the osmotic pressure due to the concentration gradient increases with molecular weight. 

The measure of the intermolecular pair interactions is the second virial coefficient A2 in the virial expansion of osmotic pressure (see Equations 1.3-37, 52-54, 146-148). 

As the second virial coefficient of the osmotic pressure expansion, the quantity (1/2 — y) is accepted here, while the difference between the characteristic parameter of the ternary interactions w and the trivial factor 1/6 acts as the third virial coefficient (sec... 

The reference temperature is not the true theta temperature at which the second virial coefficient of the osmotic pressure vanishes. The latter lies far below due to H-bonding and hydrophobic interaction in addition to the van der Waals interaction in the background. The parameters related to the strength of hydration, such as no, yn, were taken from Section 6.4 for PEO, and Section 6.5 for PNIPAM. 

In the previous four sections we have dealt with some aspects of very dilute aqueous solutions. From the formal point of view, it is sufficient to study the solvation properties of one solute j in a pure solvent. We now proceed to the next step and study a pure solvent with two solutes. In the absence of a solvent, two-particles-in-a-system determines the second virial coefficient in the density expansion of the pressure (section 1.8). Likewise, two-solute-in-a-solvent determines the second virial coefficient of the osmotic pressure (section 6.11). This quantity is expressed in terms of the pair correlation function by... 

As already noted, estimation of the value of 6 is difficult the prediction of conditions under which B shall precisely vanish would be even more precarious. However, the "Theta point," so-called, at which this condition is met is readily identified with high accuracy by any of several experimental procedures. An excluded volume of zero connotes a second virial coefficient of zero, and hence conformance of the osmotic pressure to the celebrated law of J. H. van t Hoff. The Theta point may be located directly from osmotic pressure determinations, from light scattering measured as a function of concentration, or from determination of the precipitation point as a function of molecular welfjht. ... 

All the experimental results obtained in the semidilute 0-solutions are consistent with the idea of de Gennes and Brochardthat at the theta temperature, where the second virial coefficient of the osmotic pressure vanishes, the mesh size of the transient gel is not proportional to the correlation length of concentration fluctuations as it is in the case of a good solvent. ... 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

The alternative value, which describes the polymer-solvent interaction is the second virial coefficient, A2 from the power series expressing the colligative properties of polymer solutions such as vapor pressure, conventional light scattering, osmotic pressure, etc. The second virial coefficient in [mL moH] assumes the small positive values for coiled macromolecules dissolved in the thermodynamically good solvents. Similar to %, also the tabulated A2 values for the same polymer-solvent systems are often rather different [37]. There exists a direct dependence between A2 and % values [37]. 

Following on from equation (3.5), we note that it is the value of the second virial coefficient A2 that determines the osmotic pressure of the biopolymer solution ... 

Here, the quantities jn ° and ji are, respectively, the chemical potentials of pure solvent and of the solvent at a certain biopolymer concentration V is the molar volume of the solvent and n is the biopolymer number density, defined as n C/M, where C is the biopolymer concentration (% wt/wt) and M is the number-averaged molar weight of the biopolymer. The second virial coefficient has (weight-scale) units of cm mol g. Hence, the more positive the second virial coefficient, the larger is the osmotic pressure in the bulk of the biopolymer solution. This has consequences for the fluctuations in the biopolymer concentration in solution, which affects the solubility of the biopolymer in the solvent, and also the stability of colloidal systems, as will be discussed later on in this chapter. 

In principle, the expressions for pair potentials, osmotic pressure and second virial coefficients could be used as input parameters in computer simulations. The objective of performing such simulations is to clarify physical mechanisms and to provide a deeper insight into phenomena of interest, especially under those conditions where structural or thermodynamic parameters of the studied system cannot be accessed easily by experiment. The nature of the intermolecular forces responsible for protein self-assembly and phase behaviour under variation of solution conditions, including temperature, pH and ionic strength, has been explored using this kind of modelling approach (Dickinson and Krishna, 2001 Rosch and Errington, 2007 Blanch et al., 2002). 

At small solute concentrations the second virial coefficient is the main contributor to the value of n, and so in practice the general equation (5.16) is usually restricted to just the term containing the second virial coefficient. At this level of approximation, the osmotic pressure of a ternary solution (biopolymer, + biopolymer, + solvent) may be expressed in the following simple form using the molal scale (Edmond and Ogston, 1968) ... 

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