Dynamic modulus dilute solution

Fig. 25 Viscoelastic behavior of a semi-dilute solution of DNA. Elastic modulus (G, filled symbols) and viscous modulus (G", open symbols) are plotted, together with their ratio G"IG = tan 5 (solid curve), for a 93 mg/mL DNA solution subjected to a heating-cooling cycle. The entanglement of DNA helices and, at high temperature, of single strands causes the almost monotonous increase of the dynamic moduli. Reproduced with permission from [110]...

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

These linear viscoelastic dynamic moduli are functions of frequency. For a suspension or an emulsitm material at low frequency, elastic stresses relax and viscous stresses dominate with the result that the loss modulus, G", is higher than the storage modulus, G. For a dilute solution, G" is larger than G over the entire frequency range, but they approach each other at higher frequencies as shown in Fig. 3. 

It is the aim of this subsection to present the derivation of analytic viscoelastic expressions in the GGS framework. As before, we take the GGS structure to be embedded in the viscous mediiun for example, starting with a dilute polymer solution, the medium is given by the solvent. To measure the mechanical properties, one creates a macroscopic perturbation (strain) in the embedding medium. This external perturbation produces, in turn, a mesoscopic strain at the level of the polymer (GGS). A subsequent change of the polymer s configmation leads to the relaxation of the stress in the sample, which can be measmed, allowing us to determine, for instance, the dynamic modulus. 

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

Dilute polyelectrolyte solutions, such as solutions of tobacco mosaic virus (TMV) in water and other solvents, are known to exhibit interesting dynamic properties, such as a plateau in viscosity against concentration curve at very low concentration [196]. It also shows a shear thinning at a shear strain rate which is inverse of the relaxation time obtained from the Cole-Cole plot of frequency dependence of the shear modulus, G(co). 

The value of the powers, superscripts s and /, in Eqs. (6) and (7) are 1.3 and 1.8, respectively, by replacing p with the polymer concentration. There are many careful measurements reported on the modulus near the gel point. Tokita and Hikichi [187] measured the modulus of dilute agarose aqueous solution as a function of temperature and determined the value of / to be 1.9 and the critical temperature, = 80°C. The concentration dependence of the agarose aqueous solution at 20°C was also measured and obtained as = 0.0137 g/lOOmL, /=1.93. Gauthier-Manuel et al. [188] measured the dynamic viscoelasticity if = 10 Hz) of silica particle suspension as a function of reaction time and obtained / = 2. The experimental value is closer to the index 1.9 of the 3D percolation theory than is / = 3 of the Flory-Stockmayer... 

Let us look at typical behavior of these material functions. In Figure 3.3.5 we see that G versus o) looks similar to G versus 1/r from Figure 3.3.1. For rubber it becomes constant at low frequency (long times), and for concentrated polymeric liquids it shows the plateau modulus Ge and decreases with co in the limit of low frequency. The loss modulus is much lower than G for a crosslinked rubber and sometimes can show a local maximum. This maximum is more pronounced in polymeric liquids, especially for narrow molecular weight distribution. The same features are present in dilute suspensions of rodlike particles, but not for dilute random coil polymer solutions, as Figure 3.3.3b shows. These applications of the dynamic moduli to structural characterization are discussed in Chapters 10 and 11. 

We first consider the behavior of the dynamic storage modulus and the dynamic loss modulus. Colby, et a/. (15) report an extremely extensive series of measurements of the storage and loss moduli of a 925 kDa (M ) polybutadiene having a narrow molecular weight distribution (M /Mn < 1.1 M /Mw < 1-1). Solutions were made in the Theta solvent dioctylphthalate (DOP) at 12°C above the Theta temperature, and the good solvent phenyloctane (PO). Viscosities were reported at 15 volume fractions extending from extreme dilution 0.001) up to the melt full... 

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