Types of Rheological Response

Differences between solid-like and liquid-bke complex fluids show up in all three of the shearing measurements discussed thus far the shear start-up viscosity t), the steady-state viscosity rj(y), and the linear viscoelastic moduli G co) and G (o). The start-up stresses a = y/ +() , t) of prototypical liquid-like and solid-like complex fluids are depicted in Fig. 1-6. For the liquid-like fluid the viscosity instantaneously reaches a steady-state value after inception of shear, while for the solid-like fluid the stress grows linearly with strain up to a critical shear strain, above which the material yields, or flows, at constant shear stress. 

The storage and loss moduli G and G for our prototypical liquid-like and solid-like fluids are shown in Fig. 1-8. For the liquid-like fluid, the storage modulus is much lower than the loss modulus, and it scales with frequency as G x(o, the loss modulus is linear in frequency, G oc co. The low-frequency liquid-like region in which G and G obey these power laws is called the terminal region. For the solid-like fluid, G S G , and G is nearly frequency-independent. 

Note that at the temperature 150 C, the transition from liquid-like to solid-like behavior occurs at frequency 1 sec, as indicated by the crossover of G and G roughly at this frequency. For this melt, a corresponding crossover in the steady-state shear-viscosity curve from a liquid-like plateau in to a solid-like shear thinning region occurs at a shear rate roughly in the vicinity of 1 sec (see Fig. 1-9). Thus, the crossover shear rate yc in steady shearing is about equal numerically to the crossover frequency coc 

A handy, though very crude, rule of thumb is that the order of magnitude of the zero shear viscosity rjo is given by the product of the characteristic relaxation time and the characteristic modulus that is. 

This rule of thumb goes back to Maxwell (1867), who said that a viscous fluid with viscosity rjo can be thought of as a relaxing solid with modulus G that relaxes in a time period r hence, r]a Gx. Another handy rule is that the characteristic modulus of a liquid is roughly equal XovksT, where v is the number of structural units per unit volume. For a suspension of spheres, v is the number of spheres per unit volume, while for a small-molecule liquid, V is the number of molecules per unit volume thus, v = pNa/M, where p is the fluid density, M is the molecule s molecular weight, and is Avogadro s number. Hence, for a small-molecule liquid with density p = Ig/cm, Af = 100 g/mol, and T = 300 K, we estimate G 2.4 x 10 Pa = 24 MPa. 

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