Linear viscoelasticity dilute solutions

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

Rouse, P.E. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21,1272-1280 (1953). 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

New heterocyclic polymers designed especially for service at elevated temperatures have intriguing properties, some of which are in contrast to properties usually associated with linear noncrystalline polymers. These polymers have sometimes been described as stiff chains because of the long inflexible repeat units of which they are comprised. Relatively few quantitative studies have yet appeared in the dilute solution properties or the viscoelastic behavior of the new heterocyclic polymers—partly because of the difficulties inherent in working with the poorly soluble materials. Some studies on the polyimide with the (idealized) structure ... 

These predictions of the Zimm model are compared with experimental data on dilute polystyrene solutions in two -solvents in Fig. 8.7. The Zimm model gives an excellent description of the viscoelasticity of dilute solutions of linear polymers. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Shear Wave Propagation. A pulse shearometer (Rank Bros.) was used to measure the propagation velocity of a shear wave through the weak gels formed by the solutions of HMHEC in dilute NaCl. The polymer concentration range studied was 0.5-2.0%. With this apparatus, the frequency of the shear wave is approximately 1200 rad s" and the strain is <10 . At this strain, n pst systems behave in a linear viscoelastic fashion, and the wave-rigidity modulus, G is... 

Shankar V, Pasquali M, Morse DC (2002) Theory of linear viscoelasticity of semiflexible rods in dilute solution. J Rheol 46 1111... 

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. 

Rouse, P.E. Jr. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. 

The molecular theory of polymer viscoelasticity rests on the work of Bueche (101-104), Rouse (105), and Zimm (106,107), investigating behavior of diluted solutions of linear polymers. The molecular theory of viscoelasticity has not been very successful in describing viscoelasticity of solid polymers over the whole temperature interval, and thus modified theory of rubber elasticity has to be used... 

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