Viscoelastic response of polymeric fluids

Shear thinning or pseudoplastic behavior is an important property that must be taken into account in the design of polymer processes. However, it is not the only property, and in Chapter 3 models that describe the viscoelastic response of polymeric fluids will be discussed. However, first we would like to solve some basic one-dimensional isothermal flow problems using the shell momentum balance and the empiricisms for viscosity described in this section. 

VISCOELASTIC RESPONSE OF POLYMERIC FLUIDS AND FIBER SUSPENSIONS... 

Another often useful material parameter is Poissons ratio, vp, which is defined as the ratio between the perpendicular (transverse) deformation (ALj in Figure 1.12) and the longitudinal deformation (AL) vp = -ALx/AL. In the general, viscoelastic case, similar considerations as in the case of shear flow hold for the temporal response of polymeric fluids at molecular scale, and the resulting macroscopic response. 

Behavior of Entangled Polymer Melts and Solutions Transient Response. While the steady-state response of polymers in shear and elongational flows is of much interest, there are also many instances in which the transient response is important because not all processes attain steady state. There are two important transient responses in the nonlinear regime of behavior. These are the stress relaxation response in which the deformation is held constant and the stress evolution with time is followed. This was discussed above for the linear viscoelastic case. In addition, the response to a constant rate of deformation can be an important transient response to study. Also note that creep experiments are sometimes used to characterize the nonlinear response of polymeric fluids and these will also be discussed briefly. 

The relaxation time (A), which describes the time required for the polymer coil to relax from a deformed state back to its equilibrium configuration, is a key parameter for characterizing a viscoelastic fluid. For a fluid with large A, the stresses relax slowly and the elastic effects can be observed even at low deformation rates. A fluid with small A can also exhibit significant elastic effects provided that the deformation rate is high. Clearly, both the fluid characteristic time (the relaxation time) and the flow characteristic time (e.g., the inverse of the deformation rate) are crucial in determining the viscoelastic response of a viscoelastic liquid. For many polymeric liquids, X lies between 10 s for dilute solutions and 10 s for concentrated solutions. 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

In a dynamic experiment, a small-amplitude oscillatory shear is imposed to a molten polymer confined in the rheometer. The shear stress response of the polymeric system can be expressed as in Equation 22.14. In this equation, G and G" are dynamic moduli related to the elastic storage energy and dissipated energy of the system, respectively. For a viscoelastic fluid, two independent normal stress differences, namely, first and second normal stress differences can be defined. These quantities are calculated in terms of the differences of the components of the stress tensor, as indicated in Equation 22.15a and 22.15b, and can be obtained, for instance, from the radial pressure distribution in a cone-and-plate rheometer [5]. Some other experiments used in the determination of the normal stress differences can be found elsewhere [9, 22] ... 

Viscoelastic material such as polymers combine the characteristics of both elastic and viscous materials. They often exhibit elements of both Hookean elastic solid and pure viscous flow depending on the experimental time scale. Application of stresses of relatively long duration may cause some flow and irrecoverable (permanent) deformation, while a rapid shearing will induce elastic response in some polymeric fluids. Other examples of viscoelastic response include creep and stress relaxation, as described previously. 

Qualitatively, the stress created by imposing oscillatory strains will be a function of the amplitude of strain, its frequency, and the properties of the polymeric solution. By applying sufficiently small strains, we can minimize the strain dependency from the material response. Thus, the amplitude of strain is no longer an important consideration, resulting in a linear viscoelastic response. Small strain measurements can be important for certain application as they do not adversely affect the structure of the fluid. 

The term rheology dates back to 1929 (Tanner and Walters 1998) and is used to describe the mechanical response of materials. Polymeric materials generally show a more complex response than classical Newtonian fluids or linear viscoelastic bodies. Nevertheless, the kinematics and the conservation laws are the same for all bodies. The presentation here is condensed one may consult other books for amplification (Bird et al. 1987a Huilgol and Phan-Thien 1997 Tanner 2000). We begin with kinematics. 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

Polymer motions, such as bond rotations among different configurations, reptations through space, intermolecular overlap interactions, etc., lead to viscoelastic responses in polymer solutions that can span many orders of magnitude in time, or in its conjugate variable, the frequency at which stress is applied. Liquids whose viscosity does not change with frequency of stress are termed Newtonian fluids. Most polymeric fluids deviate from Newtonian behavior under certain conditions, and are termed non-Newtonian. ... 

Transient Response Creep. The creep behavior of the polymeric fluid in the nonlinear viscoelastic regime has some different features from what were foimd with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation behavior (or vice versa). For most cases, the material properties that appear in the constitutive equations are written in terms of the relaxation response. We discuss this subsequently in the context of the K-BKZ model. 

Because of their complex structure the mechanical behavior of polymeric materials is not well described by the classical constitutive equations Hooke s law (for elastic solids) or Newton s law (for viscous liquids). Polymeric materials are said to be viscoelastic inasmuch as they exhibit both viscous and elastic responses. This viscoelastic behavior has played a key role in the development of the understanding of polymer structure. Viscoelasticity is also important in the understanding of various measuring devices needed for rheometric measurements. In the fluid dynamics of polymeric liquids, viscoelasticity also plays a crucial role. " Also in the polymer-processing industry it is necessary to include the role of viscoelastic behavior in careful analysis and design. Finally there are important connections between viscoelasticity and flow birefringence. ... 

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