Semidilute polymer solutions

So far we have paid attention mostly to dilute solutions, c c, in which polymer chains are more or less separated from each other. Chapter 2 focused on thermodynamics, and Chapter 3 focused on dynamics. These solutions were mostly ideal. We also learned how the concentration c might change the thermodynamics and dynamics, as represented by the osmotic pressure and the diffusion coefficient, from those in the ideally dilute solutions. 

This chapter is about semidilute solutions, c c. We learn both thermodynamics and dynamics. The properties of semidilute solutions are drastically different from those of dilute solutions. With a mere tenfold increase in the concentration, the osmotic pressure can easily increase by a factor of several hundred. In the ideal solution, in contrast, the osmotic pressure is proportional to c. Furthermore, the overall chain motion is slow in semidilute solutions because the chains are entangled semidilute solutions of a high-molecular-weight polymer can barely flow. The solutions are highly viscous and may even behave like elastic rubber. 

The osmotic pressure and the time scale of motion depend heavily on concentration and molecular weight. The dependence is universal for a certain class of solutions each class, however, exhibits a characteristic dependence. For many years, we had not had a good understanding of those characteristics until the blob concept, the scaling theory, and the reptation model were introduced in 1970s. With simple ideas and simple mathematics, these concepts elegantly explained the observed complicated dependence. 

Semidilute solution, c c, is unique to polymer solutions. Because c is low, the semidilute regime extends to a low concentration (in terms of g/L). As we have 

The semidilute regime is often specified by c c c. With c at around 10 g/L for polystyrene of = 6 X 10 g/mol, for instance, and c at around 300 g/L, the double inequality may appear to impose a severe restriction on the accessibility by an ordinary polymer. In practice, however, solutions several times as concentrated as c already qualify as semidilute solutions. With ambiguity in the definition of c (Eqs. 1.108-1.110), it does not make sense to ask how high the concentration should be for the solution to be semidilute. As we will see in many experimental results, there is an easy way to find whether or not the concentration is sufficiently high. 

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For example, at MW = 4 X 10, c = 12 g/liter, and at MW = 5 X 10, c " = 62 g/liter. A polymer solution with concentration c > c is called a semidilute solution because mass concentration is low yet repulsive interactions between solutes are strong. Thermodynamics, viscoelasticity, and diffusion properties of semidilute polymer solutions have been studied extensively since the 1960s. 

Cukier [87], using an effective medium-type approach, analyzed the diffusion of Brownian spheres in two semidilute polymer solutions The first was composed of long... 

Cukier, RI, Diffusion of Brownian Spheres in Semidilute Polymer Solutions, Macromolecules 17, 252, 1984. 

Scaling in Semidilute Polymer Solutions A Monte Carlo Test. 

A surprising feature of the results in Table II is that the best solvent (1 1 MIBK/MeOH) yields a lower solution viscosity than the worst solvent (MeOH) despite greater aggregation in the better solvent. The viscosity of a semidilute polymer solution may be expressed as the product of a structure factor F and friction factor f... 

Saha, S., Heuer, D. M., and Archer, L. A. (2006). Electrophoretic mobility oflinear and star-branched DNA in semidilute polymer solutions. Electrophoresis 27, 3181—3194. 

The flow-strength criteria stated in equation (10.2) has been examined experimentally by Fuller and Leal [72] and Dunlap and Leal [149] using four- and two-roll mills, respectively. These devices allow one to systematically vary the flow type (the relative amount of pure extension to pure rotation). The birefringence was measured for dilute and semidilute polymer solutions as a function of both the magnitude and type of the flow. Simple molecular models of flexible polymer chains suggest that such data, when plotted as a... 

Cukier RI. Diffusion of Brownian spheres in semidilute polymer solutions. Macromolecules 1984 17 252-255. 

Consider a semidilute polymer solution of chains with A b monomers, volume fraction

excluded volume v. A trace amount of longer chemically identical chains with Aa monomers is added to the solution. What is the size Ra of these A-chains, if they are assumed not to overlap with each other and not to change the overall volume fraction (pi Derive Eqs (5.23) and (5.26) for good solvents with 0 < v < from Eqs (5.38) and (5.39) for athermal solvents. Hint Renormalize the monomer to the thermal blob. 

Consider a semidilute polymer solution at room temperature with Flory interaction parameter x = 0 4, having N— 10 Kuhn monomers of length... 

The diffusion coefficient in semidilute polymer solutions is determined from the fact that the chain diffuses a distance of order of its own size in its reptation time ... 

Thin liquid films made from the mixed solutions below CAC exhibit a stratification phenomenon, with a stratum thickness corresponding to the mesh size of the polymeric network, i.e. the distance between overlap points of two polymer chains. The oscillatory forces are particular to polyelectrolytes and disappear when the electrostatic forces are screened with salt. The study of freely suspended films gives new useful insights into the structure of semidilute polymer solutions which are presently the object of numerous speculations. 

Despite numerous efforts, the dynamic properties of semidilute polymer solutions in organic solvents at T < 0 are poorly understood. To our knowledge, only one DLS experiment has been performed so far in the poor solvent domain of supercritical polymer solutions [4]. The most important issue which is yet to be resolved is whether the viscosity of the solvent or that of the solution should be used to calculate the dynamic correlation length from the Stokes-Einstein Eq.l2 [31]. 

Corresponding-states studies of the viscosity-concentration behavior of dilute and semidilute polymer solutions with Robert Simha led one of the authors (JLZ) to their applications in turbulent flows, a phenomenon that is generally called drag reduction. About six decades ago, Mysels [Mysels, 1949 Agoston et al., 1954] and Toms [1949] discovered that small amounts of aluminum soaps and high polymers added to a fluid in turbulent flow could significantly reduce pressure losses. 

Despite the fact that here one has the typical composition of a microemulsion, i.e., surfactant-water-oil, one does not find a low viscosity microemulsion but instead a highly viscous system. The addition of water results in the formation of flexible cylindrical reverse micelles that form a transient network of entangled micelles and has been characterized by means of dynamic shear viscosity measurements [73,74]. Light scattering experiments on systems with cyclohexane as the oil have demonstrated that a water-induced micellar growth occurs and that these systems may be described analogously to semidilute polymer solutions [75-77]. 

The first theories that implemented a proper balance of intramolecular interactions and conformational elasticity of the branches were developed by Daoud and Cotton [21] and by Zhulina and Birshtein [22-24]. These theories use scaling concepts (the blob model), originally developed by de Gennes and Alexander to describe the structure of semidilute polymer solutions [64] and planar polymer brushes [65, 66]. Here, the monomer-monomer interactions were incorporated on the level of binary or ternary contacts (corresponding to good and theta-solvent conditions, respectively), and both dilute and semidilute solutions of star polymers were considered. Depending on the solvent quality and the intrinsic stiffness of the arms, the branches of a star could be locally swollen, or exhibit Gaussian statistics [22-24]. 

Theoretical analyses predicted the propagation of low-frequency shear waves in both semidilute polymer solutions [49] and dilute colloidal crystals [50]. Different experimental techniques were applied for their detection and for the determination of the shear modules of colloidal samples [51-54]. The dispersion equation of the transverse waves for the low-frequency regime (wavelengths much larger than the interparticle distance) [55]... 

Cabane, B., and Duplessix, R, Decoration of semidilute polymer solutions with surfactant micelles, J. Physique, 48, 651, 1987. 

In mixtures of low-molar-mass components, the structure of the components will not depend on concentration. However, the structure of the polymer depends on solvent concentration. At a certain polymer concentration (overlapping concentration c ), entanglements of the polymer chains will occur (semidiluted polymer solution). This effect is neglected in most thermodynamic equations. 

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