Osmotic pressure of semi-dilute solutions

However, let note, that the assumption about independence of the osmotic pressure of semi-diluted solutions on the length of a chain is not physically definitely well-founded per se it is equivalent to position that the system of strongly intertwined chains is thermodynamically equivalent to the system of gaped monomeric links of the same concentration. Therefore, both Flory-Huggins method and Scaling method do not take into account the conformation constituent of free energy of polymeric chains. 

In that way, the thermodynamic approach with the use of conformational term of chemical potential of macromolecules permitted to obtain the expressions for osmotic pressure of semi-diluted and concentrated solutions in more general form than proposed ones in the methods of self-consistent field and scaling. It was shown, that only the osmotic pressure of semi-diluted solutions does not depend on free energy of the macromolecules conformation whereas the contribution of the last one into the osmotic pressure of semi-diluted and concentrated solutions is prelevant. 

Fx Universal constant concerning the osmotic pressure of semi-dilute solutions in good solvent (for x > 1, F(x) a F, x 1 [Pg.916]

Analysis of osmotic pressure of semi-diluted polymeric solutions by Scaling method is based [3] on two positions. Accordingly to the first one it is assumed that the polymeric chain is in good solvent for which % < 1 / 2. This position is necessary in order to index v in the expression (2) will be determined by the ratio... 

The following expression is initial for the determination of osmotic pressure of semi-diluted polymeric solutions accordingly to Scaling method ... 

Near the theta point, the osmotic pressure of the dilute solution is closely related to the chain length of polymers, as demonstrated in (4.43). In semi-dilute solutions, however, the osmotic pressure is related to the degree of interpenetration of polymer coils, no longer related to the chain length. Using the blob model, we have... 

On the contrary, the measurement of the osmotic pressure for different overlap ratios provides essential information on the structure of semi-dilute solutions. Data have been available since the time when this technique was first used. However, this approach required a theoretical basis and we have a good theory only since 1975. 

Reverse osmosis is the inverse of natural osmosis under pressure. Osmosis is the solvent flow from a more dilute solution toward a more concentrated one through a semi-permeable membrane. The process is driven by the osmotic pressure of the concentrated solution, which depends on the chemical identities of the solvent and the dissolved substance, as well as on their ratio. If the external pressure acting on the more concentrated solution is higher than its osmotic pressure, reverse osmosis, i.e. a solvent flow toward the more diluted solution happens through the membrane. Reverse osmosis is most commonly known for its use in drinking water purification from seawater, removing the salt and other effluent materials from the water molecules. 

How do we predict the thermodynamic properties of semi-dilute solutions The basic notion is a scaling law for the osmotic pressure, proved by des Cloiseaux for one specific example. This scaling law is a natural generalization of eq. (III.20) and reads... 

The second position assumes that in semi-diluted solutions the polymeric chains are as much strong intertwined that the all thermodynamic values, in particular the osmotic pressure, achieve the limit (at N —>oc) depending only on the concentration of monomeric links, but not on the chain length. 

That fact the scaling method and presented thermodynamic approach from seeming opposite positions lead to practically the same result in the form (8) and (35) can be named as mysterious incident if it were not two circumstances. First is exactly free energy of the conformation makes the main contribution into the osmotic pressure of the semi-diluted and concentrated solutions. The second is the peculiarity of the point c = c. ... 

As we have already shown that the osmotic pressure of dissolved substances m dilute solution obeys the gas laws, one can infer that the thermodynamic deduction, already given, of the mass law for the gaseous state can at once be transferred to the state of solution An analogous cycle could be earned out with two equilibnum boxes immersed in a reservoir of solvent of infinite size, a series of semi-permeable pistons... 

For relatively small values of C (very dilute solutions), the first term is dominant and, of course, one gets the perfect gas law IIfi = C. For relatively large values of C (semi-dilute solutions), the osmotic pressure now depends only on C = A C. 

Osmotic pressure of dilute and semi-dilute solutions in moderately good solvent one-loop approximation (b 0, c = 0)... 

Thus, we find the law which gives the osmotic pressure as a function of concentration for semi-dilute solutions. 

Osmotic pressure of a semi-dilute solution in the tricritical domain... 

When the condition tp < 1 is no longer satisfied, the osmotic pressure ceases from being a universal function of CX3 (which represents the rate of overlap in semi-dilute solutions). Then, in such conditions, the volume fraction (p becomes the essential parameter which determines the pressure. In fact, this is just the property that Noda et al.21 showed experimentally. 

In fact, as he illustrated, reported osmotic pressure data on many polymer + solvent systems can be fitted precisely with eq 1.10 over a range including both dilute and semi-dilute solutions. Thus, Schafer s equation should be quite useful for estimating 11 of polymer solutions over a relatively wide concentration range. 

The repulsive energy between the blobs should be comparable with the energy level of kT for thermal fluctuations. Accordingly, the osmotic pressure of the semi-dilute solution at the theta point becomes... 

In practice, from the dilute solutions to semi-dilute solutions, the scaling law of the osmotic pressure changes rather gradually, as roughly given by... 

Schafer and Witten" have applied the RG to excluded volume, and established scaling laws , for example for the osmotic pressure. One of the objects of the RG method is to establish such scaling laws, and to demonstrate scale invariance . Then experimentally observable correlation functions can be shown to obey particular scaling behaviour, and the critical exponent calculated may be compared with that obtained by experiment. Critical exponents calculated by the RG will generally differ from that obtained by classical mean field e.g. SCF approaches - Mackenzie " in a recent review has pointed out that discrimination between the two lies with experiment. For example, Le Guillou and Zinn-Justin have calculated v in equation (7) to be 0.588 (c/. the SCF-fifth-power law value of 0.60). However, to discriminate between these values is beyond the capability of current experimental techniques. Moore has used the RG to explore the asymptotic limit, and recently demonstrated that when the ternary cluster integral vanishes, an expression for the osmotic pressure may be derived which holds for both poor and good solvents, in semi-dilute solutions. 

---