Osmotic pressure second virial coefficient

N = 14 (Martin and Watts 1971)4 Second virial coefficient (osmotic pressure)... 

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... 

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. 

Since dilute solutions are considered we can expand the osmotic pressure in a virial series that is truncated at the second virial coefficients... 

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. 

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]. 

Most liquid-crystalline polymer solutions have a large second virial coefficient ( > 10 4 cm 3mol/g2) [41], which means that it is rather difficult to find poor or theta solvents for these polymers and that liquid-crystalline polymers in solution interact repulsively. This fact is essential in formulating their static solution properties (osmotic pressure, phase separation, etc.). 

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

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) ... 

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. 

Outline the logic used in deriving expressions for the osmotic pressure and second virial coefficient due to excluded-volume interactions ... 

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

Molecular weight from osmotic pressure measurements Degree of polymerization and molecular weight distribution Excluded volume from osmotic pressure measurements Theta temperature from second virial coefficient data Evaluation of charges on macroions from osmotic pressures... 

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. 

Table 1.2 Reduced osmotic pressures (tt/c)c=o, number average molecular weights Mn and osmotic second virial coefficient A2 for poly(pentachlorophenyl methacrylate) fractions in toluene at 25°C and benzene at 40°C (tt in cm of benzene or toluene) (c in g dl-1). (From ref. [44])...

The second virial coefficient A2 in the osmotic pressure equations can also be used to determine random coil dimensions (see Chap. 10). 

Rudin s aim was to predict the size of dissolved polymer molecules and the colligative properties of polymer solutions (hydrodynamic volume, second virial coefficient, interaction parameter, osmotic pressure, etc) from viscometric data (average molar mass, intrinsic viscosity, etc.). 

The i gTinfinite dilution expression for the osmotic pressure, and the second term can be referred to as the second virial term with a second virial coefficient B2[= (1/2 - Xi)< / i]- The second virial coefficient becomes 0 when xi = 1/2. This point is called the Flory point or the... 

It is to be expected that measurements of the osmotic pressures of I he same polymer in different solvent should yield a common intercept. The slopes will differ (Fig. 3-1 a), however, since the second virial coefficient reflects polymer-solvent interactions and can be related, for example, to the Flory-Huggins interaction parameter x (Chapter 12) by... 

More recently Lapanje and Tanford (59) have reported osmotic pressure measurements for reduced protein polypeptide chains in 6M guanidine hydrochloride. Second virial coefficient data and intrinsic viscosity data are combined by these authors to yield unperturbed dimensions of randomly coiled proteins. The result is assentially identical with that obtained earlier from viscosity data alone. 

For more than 60 years it is known that B2, the osmotic second virial coefficient (OSVC), constitutes an important thermodynamic characteristic of protein—protein interactions in protein solutions. " In a binary mixture solvent (1)—protein (2), B2 is connected to the osmotic pressure (n) via the virial equation... 

As a first application we calculate the second virial coefficient A2 of the osmotic pressure for a monodisperse solution According to Eq. (Id)... 

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