Semidilute solution osmotic pressure

With respect to the osmotic pressure a semidilute solution behaves like an ideal gas of rc -blobs. 

The mean-field theory correctly predicts that the osmotic pressure is independent of molar mass in semidilute solution. However, the mean-field theory does not take into account the correlations between monomers along the chain, but instead assumes that they are distributed uniformly as in a solution of monomers. This is the reason why the two-body interac-Tioiis [Eq. (5.41)] become important at volume fraction --------------------... 

The final relation was obtained using Eq. (5.27) for 0. The osmotic pressure in semidilute solution is independent of chain length. Therefore the exponent of Ain Eq. (5.46) must be zero... 

In the final expression, Eq. (5.23) was used for the correlation length The scaling prediction for osmotic pressure is significantly different from the mean-field prediction because the exponents for the concentration dependence differ (2.3 instead of Z). t he scaling prediction is in excellenT agreement with experiments, as demonstrated in Fig. 5.7 (the high concentrations are described by IT/c c ). Equation (5,49) demonstrates that the osmotic pressure provides a direct measure of the correlation length in semidilute solutions. 

In dilute solutions 0 < 0, this scaling function approaches unity h (j>l4> ) 1 and the osmotic pressure obeys the van t Hoff law. In semidilute 0-solutions, h((f>j(f) ) is again assumed to be a power law ... 

The exponent y is determined from the condition that the osmotic pressure in semidilute solutions is independent of chain length, II N ... 

The scaling argument leads to the same prediction in semidilute 0-solutions as the mean-field theory [Eq. (5.57)]. The osmotic pressure in semidilute solutions is again of the order of the thermal energy kT per correlation volume ... 

This equation holds for theta, good, and athermal solvents. Hence, osmotic pressure or osmotic compressibility measurements provide a con-venient means ot measuring the correlation length in semidilute solutions. 

At swelling equilibrium, the elasticity is balanced by the osmotic pressure n of a semidilute solution of uncrosslinked chains at the same con-centration. Since the modulus is proportional to the elastic free energy per unit volume, any gel swells until the modulus and osmotic pressure are balanced. The equilibrium swelling ratio Q is the ratio of the volume in the fully swollen state and the volume in the dry state ... 

It is important to emphasize the fact that the osmotic pressure in Eq. (7.72) is the osmotic pressure of a semidilute solution of linear chains at the same volume fraction as the gel. This is not to be confused with the osmotic pressure of the gel calculated from its definition in Eq. (4.62), which includes effects from the elasticity of the gel. 

What is the physical significance of the fact that the value of the stress relaxation modulus at the relaxation time of a correlation blob G(r ) is proportional to the osmotic pressure in semidilute solutions ... 

Tnteractions are not important. The dynamics on these intermediate scales (for r < t< Te) are described by the Rouse model with stress relaxation modulus similar to the Rouse result for unentangled solutions [Eq. (8.90) with the long time limit the Rouse time of an entanglement strand Tg]. At Te, the stress relaxation modulus has decayed to the plateau modulus Gg[kT per entanglement strand, Eq. [(9.37), see Fig. 9.9)]. The ratio of osmotic pressure and plateau modulus at any concentration in semidilute solution -in athermal solvents is proportional to the number of Kuhn monomers in ... 

The star architecture effects are more important for I q 0) than for Dc because the ratio of the corresponding correction terms, k / k — k, is large when k k. Nevertheless, the experimental Dc c/c reveals a stronger speed-up of Dc with concentration in multiarm stars compared to the semidilute linear polymer solutions. The hard core contribution to the osmotic pressure is essentially hidden in the inhomogeneous density profile and the thermodynamic properties of the star solutions are primarily determined by their polymeric character. 

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. 

For semidilute solutions the segment of a blob moves as a unit. The osmotic pressure for this solution depends on the number of blobs per unit volume (1/f ). 

This relation indicates that in a semidilute regime it is the same for solutions of polymers of different chain length or molecular weight (Fig. 25.8) thus, at p > p the osmotic pressure is independent of M, contrasting with the behavior of a dilute solution, in which k is strongly dependent on M. 

In the scaling theory, introduced by de Gennes [15], the osmotic pressure behaves like some powers (m) of concentration and becomes independent of the degree of polymerization N). For semidilute solutions (large x). 

There are three main modes of interaction between a polymer solution and a solid surface. The first interaction mode is depletion [2,3]. If the monomers are repelled by the surface (or in other words if the attractive interaction between the solvent molecules and the surface is larger than the interaction between the monomers and the surface), the polymer concentration in solution decreases as the surface is approached and a region depleted in polymer exists in the vicinity of the surface. The size of this region is the size of the polymer chain if the solution is dilute and the size of the correlation length of the solution if the solution is semidilute (if the polymer chains overlap). When two surfaces are brought in close contact, the density in the gap between the surfaces is smaller than the bulk concentration and the osmotic pressure in the gap is smaller than the bulk osmotic pressure. This osmotic pressure difference induces an attraction between the surfaces. The depletion interaction is not specific to polymers and exists with any particle with a size in the colloidal range [4]. It has sometimes been used to induce adhesion between particles of mesoscopic size such as red blood cells. The only limitation to this qualitative description of the depletion force is that at equilibrium the polymer chains (or any other particles) must leave the gap as the surfaces get closer. There is no attractive depletion force if they remain trapped in the gap. We will not consider further the depletion interaction. 

We have considered in this expression the polymer inside the layers as a semidilute solution and used the sealing expression for the osmotic pressure [18]. Note that we ignore here the variation of the monomer eoneentration at the edge of the grafted layer. Equation (2) thus overestimates the interaetion between the two grafted layers when they are close to overlapping, h 2H. 

In solution, we have considered the scaling behavior of a single PE (Sect. 2.7.3.1). The importance of the electrostatic persistence length was stressed. The Manning condensation of counterions leads to a reduction of the effective linear charge density (Sect. 2.7.3.1.1). Excluded volume effects are typically less important than for neutral polymers (Sect. 2.7.3.1.2). Dilute PE solutions are typically dominated by the behavior of the counterions. So is the large osmotic pressure of dilute PE solutions due to the entropic contribution of the counterions (Sect. 2.7.3.2). Semidilute PE solutions can be described by the RPA, which in particular yields the characteristic peak of the structure factor. 

Renormalization group theory (see, e.g., [35]) lies at the heart of this theory, justifying the use of scaling laws in the asymptotic limit, i.e., for infinitely long polymer chains and for dilute solutions. For semidilute solutions, however, this criterion is not so crucial because the polymer chains are overlapping and many properties, e.g., osmotic pressure, are independent of the chain length. 

Ohta and Oono (1982) applied the method of conformational space renormalization to calculate the osmotic pressure of solutions and the sizes of macromolecules in. semidilute solutions of polymers. 

The van t Hoff Law for osmotic pressure (Equation 5.19) depends explicitly on the molecular weight of the solute. One of the most remarkable properties of semidilute solutions is that the osmotic pressure is observed to be independent of the molecular weight of the macromolecules. Another property fliat is observed to be independent of molecular weight in semidilute solution is the mutual-diffusion coefficient, DJ c). These phenomena are discussed in Section 6.2 and explained in more detail in Section 6.3. 

The osmotic pressure of a semidilute polymer solution can be described by a power law in the concentration. It is convenient to express this relationship as ... 

Application of the model of a semidilute solution as a collection of regions of size can be made to both the osmotic pressure and the mutual-diffusion coefficient. It has been proposed by deGermes that the osmotic pressure should scale as the number density of screening regions ... 

The osmotic pressure in semidilute solution is given by the des Cloizeaux law ... 

In addition to the ionic contribution, polyelectrolyte solutions have the polymeric contribution to their osmotic pressure. In semidilute polymer solutions, the polymeric contribution is essentially k T per correlation volume ... 

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