Semidilute solutions concentration dependence

As discussed in section 7.1.6.4, semidilute solutions of rodlike polymers can be expected to follow the stress-optical rule as long as the concentration is sufficiently below the onset of the isotropic to nematic transition. Certainly, once such a system becomes nematic and anisotropic, the stress-optical rule cannot be expected to apply. This problem was studied in detail using an instrument capable of combined stress and birefringence measurements by Mead and Larson [109] on solutions of poly(y benzyl L-glutamate) in m-cresol. A pretransitional increase in the stress-optical coefficient was observed as the concentration approached the transition to a nematic state, in agreement of calculations based on the Doi model of polymer liquid crystals [63]. In addition to a dependence on concentration, the stress-optical coefficient was also seen to be dependent on shear rate, and on time for transient shear flows. 

It is expected that the same picture that gives a good account of the linear viscoelastic behavior of polymer melts should also hold for semidilute and concentrated solutions. In the case of semidilute solutions some conclusions can be drawn from sealing arguments (19,3, p. 235). In this way, concentration dependence of the maximum relaxation time tmax the zero shear rate viscosity r Q, and the plateau modulus G% can be obtained, where t is the viscosity of the solvent. The relevant parameters needed to obtain Xmax as a function of concentration are b, c, N, kgT, and Dimensional analysis shows that... 

The vast majority of polymer micelles have spherical shape in dilute or semidilute solutions, whereas low-molecular-weight surfactants form structures that are strongly concentration dependent lamellae, sheets, rods, and spheres. 

The concentration dependence of polymer size in semidilute solution is determined by substituting the expressions for the concentration dependence of the correlation length [Eq. (5.23)] and the number of monomers in a correlation blob [Eq. (5.24)] into Eq. (5.25) ... 

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. 

The concentration dependence of the correlation length in semidilute 6>-solutions is stronger than that in good solvent ... 

Recall from Chapter 5 that the crossover concentration (p Ki jb [Eq. (5.36)] denotes the boundary between semidilute and concentrated solutions. For 0 > 0 chains are nearly ideal in concentrated solutions, whereas for 0 < 0 chains are swollen on intermediate scales. Network modulus and equilibrium swelling depend on the relative value of preparation and fully swollen concentrations (0o and l/Q) with respect to the crossover concentration 0. Since the swollen concentration is always lower than the preparation concentration (l/Q < 0o) there are three... 

From Eqs (8.76) and (8.77), the concentration dependence of the relaxa-tion time of the chain in semidilute solution is obtained ... 

The length scales a, and R are plotted as a function of concentration for a "typical good solvent in Fig. 9.7. All three length scales change their con-centration dependences from athermal to ideal at the concentration separating semidilute and concentrated solutions. 

Concentration dependence of viscosity in semidilute solutions of polystyrene at 35 °C. (a) Solutions in the good solvent toluene have (pj(p reduce data for different molar masses to a universal curve, using data from M. Adam and... 

To proceed further (i.e., to extend the treatment to encompass temperature dependence and the crossover from semidilute to concentrated solution regime, it is necessary to incorporate information regarding the free-volume characteristics of polymer and solvent. Utracki and Simha [1981] find that a scaling equation of the form... 

Theoretical discussion of the concentration and molecular weight dependence of the viscosity of semidilute solutions of rigid-rod macromolecules has been reported. Doi and Edwards [1978] derived the well-known result... 

Furthermore, the concentration dependence of the apparent diffusion coefficient has been investigated. The determination of the hydrodynamic radius from the diffusion coefficient is valid only when pure self-diffusion is measured, that is, when the experiment is performed in the dilute concentration range. In the semidilute or concentrated regime, interactions between individual solute molecules have to be considered. Whether interactions... 

Figure 1. Polymer concentration dependences of 7 sp in the absence and presence of added-salt. The triangles, squares and circles denote the data for MVPK-11, MVPK-12 and MVPK-13, respectively. The upward, rightward and downward pips indicate the data in 0.01, 0.1 and 0.5M NaCl solutions, respectively. The symbols without pip denote the data in the absence of added-salt. The dotted and solid lines were drawn to connect smoothly the data in dilute and semidilute regions, respectively. (Reproduced from ref. 10).

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