Osmotic pressure of polymer solutions

Osmotic pressure of polymer solutions was discussed in detail in Section 10.3.3 because it is important to steric stabilization. The final result of that development presented here for completeness. The osmotic pressure of a polymer solution determined fiiom the Floiy-Huggins theory [22] and is given by 

The i gT /)2/(jcVi) (or RgTc2/M 2) term in this equation is the classical van t Hoff infinite 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 

3 Osmotic Pressure of the Double Layer in a Colloidal Suspension 

An aqueous colloidal suspension also has an osmotic pressure associated with both the double layer of the particles in solution and the structure of the particles. The osmotic pressure term for the structure is given in Section 11.6 for both ordered and random close packing. The osmotic pressure associated with the double layer surrounding the ceramic particles in aqueous solution is discussed here. 

If we consider a spherical cell containing at its center a spherical particle of radius, a, and a shell of fluid of radius, /B, as a one-partide example of the suspension with a volume fraction d [= (a//3f], then the osmotic pressure of the suspension is given by [23] 

In the limit of dilute polymer solutions the osmotic pressure is given by the ideal Van t Hoff law Pid = rnkT (see Sect. 2.2). For the osmotic pressure of non-ideal polymers in solution one can write down a general virial series 

Perturbation expansions in terms of the excluded volume in principle yield the second and higher order osmotic virial coefficients [38, 50]. This procedure becomes rather cumbersome for A4 and higher-order coefficients and established scaling exponents [40] for the semi-dilute polymer concentration regime cannot be reproduced in a virial expansion. 

In fact, in the semi-dilute case the picture is simple the chains overlap to such a degree that the characteristic length scale is determined by the correlation length rather than the coil size set by the chain length M. The corresponding volume is denoted as a blob. The osmotic pressure can then be viewed upon as an ideal gas of blobs, so Psd with the number of blobs Therefore the scaling result becomes P d/kT cfip.  

A convenient expression that enables to describe both the dilute and semi-dilute polymer concentration regimes follows from a simple additivity rule P = Fid -b Pid- This additivity follows from the Flory-Huggins theory [37] for a -solvent but appears to be an excellent approximation for good solvents as well [41]. This leads to the following expression for the ratio P/P[d 

The numerical coefficient ( follows from Flory-Huggins theory for a -solvent and from RGT for a good solvent. Equation (4.28) turns out to be extremely accurate in comparison with experimental and computer simulation data [41]. For a good solvent the result is plotted in Fig. 4.9. Under -solvent conditions Flory-Huggins theory reproduces (4.27). 

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Maron,S.H., Nakajima,N. A theory of the thermodynamic behavior of nonelectrolyte solutions. III. The osmotic pressure of polymer solutions. J. Polymer Sci. 42, 327-340 (1966). 

For dilute polymer solutions, the osmotic pressure can be approximated by the limiting van t Hoff expression. A more general expression for the osmotic pressure of polymer solutions is given by (17)... 

Recall the mean-field virial expansion for the osmotic pressure of polymer solutions discussed in Section 4.5.1 [Eq. (4.67)]. 

Fig. 18. Osmotic pressure of polymer solution in dependence on volume fraction

Although this expression is the basis for the interpretation of osmotic pressures of polymer solutions to obtain relative molecular masses (A/2) of polymers, this is not why it is reproduced here. The right-hand side of equation (5.2.8) has two components, the first term (RT/M2) is that expected for an ideal solution at infinite dilution. The second component is an excess term arising from two-body interactions generally represented by the second virial coefficient ... 

Maron SH, Nakajima N (1959) A theory of the thermodynamic behavior of non-electrolyte solutions. II. Application to the system mbber-benzene. J Polym Sci 40 59-71 Maron SH, Nakajima N (1960) A theory of the thermodynamic behavior of non-electrolyte solutions. III. The osmotic pressure of polymer solutions. J Polym Sci 42 327-340 Orwall RA, Flory PJ (1967) Equation-of-state parameters for normal alkanes. Correlation with chain length. J Am Chem Soc 89 6814—6822 Prigogine I (1957a) The molecular theory of solution. North-Holland, Amsterdam Prigogine I (1957b) Molecular theory of solutions. Chapter 16. North-Holland, Amsterdam Rowlinson JS (1970) Structure and properties of simple liquids and solutions. Faraday Disc Chem Soc 49 30-42... 

Osmotic pressure Fedors (35) used the osmotic pressure of polymer solutions to determine solubility parameters. 

Cherutich, C. K., Osmotic Pressure of Polymer Solutions in the Presence of Fluid Deformation, M.S. thesis. Chemical Engineering, State University of New York, Buffalo, 1990. 

In this section we briefly consider the osmotic pressure of polymers which carry an electric charge in solution. These include synthetic polymers with ionizable functional groups such as -NH2 and -COOH, as well as biopolymers such as proteins and nucleic acids. In this discussion we shall restrict our consideration... 

As Morawetz puts the matter, an acceptance of the validity of the laws governing colligative properties (i.e., properties such as osmotic pressure) for polymer solutions had no bearing on the question whether the osmotically active particle is a molecule or a molecular aggregate . The colloid chemists, as we have seen, in regard to polymer solutions came to favour the second alternative, and hence created the standoff with the proponents of macromolecular status outlined above. 

The partitioning principle is different at high concentrations c > c . Strong repulsions between solvated polymer chains increase the osmotic pressure of the solution to a level much higher when compared to an ideal solution of the same concentration (5). The high osmotic pressure of the solution exterior to the pore drives polymer chains into the pore channels at a higher proportion (4,9). Thus K increases as c increases. For a solution of monodisperse polymer, K approaches unity at sufficiently high concentrations, but never exceeds unity. 

The van t Ho ff equation is used to determine the molar mass of a solute from osmotic pressure measurements. This technique, which is called osmometry, involves the determination of the osmotic pressure of a solution prepared by making up a known volume of solution of a known mass of solute with an unknown molar mass. Osmometry is very sensitive, even at low concentrations, and is commonly used to determine very large molar masses, such as those of polymers. 

Acrylic resins are polymeric materials used to make warm yet lightweight garments. The osmotic pressure of a solution prepared by dissolving 47.7 g of an acrylic resin in enough water to make 500. mL of solution is 0.325 atm at 25°C. (a) What is the average molar mass of the polymer ... 

The interpretation of A becomes clearer when two plates, originally at very small distance from each other, are separated. At a certain separation, equal to 2A, polymer penetrates into the gap. In dilute solutions, where the chains behave as individual coils, A is expected to be of the order or r, the radius of gyration. However, at concentrations where t e coils overlap, the osmotic pressure of the solution becomes so high that narrower gaps can be entered, and A becomes smaller than Tg. 

Further development of the Flory-Huggins method in direction of taking into account the effects of far interaction, swelling of polymeric ball in good solvents [4, 5], difference of free volumes of polymer and solvent [6, 7] leaded to complication of expression for virial coefficient A and to growth of number of parameters needed for its numerical estimation, but weakly reflected on the possibility of equation (1) to describe the osmotic pressure of polymeric solutions in a wide range of concentrations. 

An alternative approach is based on the theoretical foundation described earlier for the colligative properties. If the solution is isotonic with blood, its osmotic pressure, vapor pressure, boiling-point elevation, and freezing-point depression should also be identical to those of blood. Thus, to measure isotonicity, one has to measure the osmotic pressure of the solution and compare it with the known value for blood. However, the accurate measurement of osmotic pressure is difficult and cumbersome. If a solution is separated from blood by a true semipermeable membrane, the resulting pressure due to solvent flow (the head) is accurately measurable, but the solvent flow dilutes the solution, thus not allowing one to know the concentration of the dissolved solute. An alternative is to apply pressure to the solution side of the membrane to prevent osmotic solvent flow. In 1877, Pfeffer used this method to measure osmotic pressure of sugar solutions. With the advances in the technology, sensitive pressure transducers, and synthetic polymer membranes, this method can be improved. However, results of the search for a true semipermeable membrane are still... 

It is well known that the osmotic pressure of a solution of one polymer can be scaled with a single dimensionless variable "S" which is proportional to polymer concentration at least for the case of mixtures with good solvents in the dilute to semidilute regime (12, 17). This implies that the osmotic compressibility factor (ti/cRT) can be expressed as some function of "S" only as shown in Equation 11. 

It is to be expected from Eq. (4.52). that measurements of the osmotic pressures of the solutions of the same polymer in different solvents should yield plots with a common intercept (at c = 0) but different slopes (see Fig. 4.6), since the second virial coefficient, which reflects polymer-solvent interactions, will be different in solvents of differing solvent power. For example, the second virial coefficient can be related to the Flory-Huggins interaction parameter x (see p. 162) by... 

Moreover, we are going to show that the osmotic pressure of a solution of monodisperse polymers obeys an elementary scaling law. Let us start from (10.4.53)... 

Noda et a/.21 pursued this experimental study, but at higher concentrations. They measured the osmotic pressure of polymer chains in solution, for T> TF and > c so as to remain always in the poor solvent state with a strong chain overlap (see Fig. 13.26, p. 642). In this physical situation, the volume fractions of polymer are high for instance, it may occur that

0.3. Thus, one gets out of the theoretical framework fixed in Chapters 13 and 14. To interpret the pressure measured by the authors quoted above, a theory for the liquid polymer state is needed. However, we present here their results without referring to any theory of this sort because, per se, these results manifest properties which are those of solutions of overlapping chains with (p < 1 (they were studied in Chapter 15 and at the beginning of this section). 

In a study designed to prepare new gasoline-resistant coatings, a polymer chemist dissolves 6.053 g of poly(vinyl alcohol) in enough water to make 100.0 mL of solution. At 25°C, the osmotic pressure of this solution is 0.272 atm. What is the molar mass of the polymer sample ... 

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