Isothermal Viscoelastic Model

The non-isothermal viscoelastic cell model was used to study foam growth in the continuous extrusion of low density foam sheet. Surface escape of blowing agent was successfully incorporated to describe the foaming efficiency. Reasonable agreement was obtained with experimental data for HCFC-22 blown LDPE foam in the sub-centimetre thickness domain. 11 refs. 

In the following analysis we first present the Newtonian isothermal model, which leads to an analytical solution. Then we discuss the Newtonian nonisothermal model, which gives insight into the complexities of the coupled heat and momentum transfer equations. PET, Nylon, and polysiloxanes are three typical polymers which are almost Newtonian at spinning conditions. Finally, we introduce the non-Newtonian isothermal model together with its associated difficulties. High-density polyethylene (HDPE), LDPE, polypropylene (PP), and polystyrene (PS) are all pseudoplastic and viscoelastic and fall into the latter category. 

C. Sollogoub, Y. Demay, and J.F. Agassant, Non-isothermal viscoelastic numerical model of the cast-film process. Journal of Non-Newtonian Fluid Mechanics, 138,76-86 (2006). 

Although all polymer processes involve complex phenomena that are non-isothermal, non-Newtonian and often viscoelastic, most of them can be simplified sufficiently to allow the construction of analytical models. These analytical models involve one or more of the simple flows derived in the previous chapter. These back of the envelope models allow us to predict pressures, velocity fields, temperature fields, melting and solidification times, cycle times, etc. The models that are derived will aid the student or engineer to better understand the process under consideration, allowing for optimization of processing conditions, and even geometries and part performance. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

Fig. 11 shows master curves extrapolated with the model. These have the same general features as master curves plotted by shifting data isotherms. The two differ slightly because different assumptions are involved. The conventional method of shifting data makes all isotherms congruent the same viscoelastic processes are assumed at all temperatures. The phenomenological model is not limited to this assumption. When the thermal spread varies with frequency, the model extrapolates to isotherms like fig. 11. Note... 

The above analysis of the viscoelastic behaviour for adsorption layers of a reorientable surfactant leads to important conclusions. It is seen that the most important prerequisite for a realistic prediction of the elastic properties is the adequacy of the theoretical model used to describe the equilibrium adsorption of the surfactant. For example, when we use the von Szyszkowski-Langmuir equation instead of the reorientation model to describe the interfacial tension isotherm, this rather minor difference drastically affects the elasticity modulus of the surface layer. The elasticity modulus, therefore, can be regarded to as a much more sensitive parameter to find the correct equation of state and adsorption isotherm, rather than the surface or interfacial tension. Therefore the study of viscoelastic properties can give much more insight into the nature of subtle phenomena, like reorientation, aggregation etc. 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

The theory of non-isothermal viscoelastic behavior as developed by Hopkins [2] and Haugh [3] may be based on the representation of linear viscoelastic behavior by mechanical models. The linear viscoelastic behavior of polymers in simple shear at constant temperature and prescribed stress history may be expressed in terms of the deformation of a generalized Kelvin model. Spring constants and dashpot viscosity constants of the model have to be appropriately chosen the choice depends on temperature. For the non-isothermal treatment, the elasticity of the springs and the viscosities of the dashpots have to be inserted as functions of temperature. Due to the prescribed temperature history, they become functions of time. 

We have reviewed here the simplest, isothermal version of CDLG models for two-phase fluid dynamics on the microscopic scale. Applications of these models for studying interfacial dynamics in liquid-vapor and liquid-liquid systems in microcapillaries were discussed. The main advantage of our approach is that it models the exphcit dependence of the interfadal structure and dynamics on molecular interactions, including surfactant effects. However, an off-lattice model of microscopic MF dynamics may be required for incorporating viscoelastic and chain-connectivity effects in complex fluids. Isothermal CDLG MF dynamics is based on the same local conservation laws for species and momenta that serve as a foundation for mechanics, hydrodynamics and irreversible thermodynamics. As in hydrodynamics and irreversible thermodynamics, the isothermal version of CDLG model ean be... 

Data on extensional stresses in fiber melt spinning, and their constitutive modeling for both isothermal and nonisothermal fiber spinning, can be found in the several books referred to earlier. Further modeling elforts are discussed by Denn.(66) However, it is fair to say that no totally acceptable rheological model has so far been found to quantitatively explain viscoelastic fiber-spinning results. The question of rheological constitutive equations is examined briefly in Section 7. 

In the following section a review of the constitutive equations used to model nonlinear viscoelastic solids tmdergoing isothermal deformation is provided. 

The Knauss-Emri Model. There have been several works in the literature in which volume- or ee-ooZume-dependent clocks were used to describe the nonlinear viscoelastic response of polymeric glasses. The chief success among these is the ICnauss-Emri model (163) in which the reduced time was defined in terms of a shift factor that depended on temperature, stress, and concentration of small molecules in such a way that the responses depended on the free volume induced by each of these parameters. For an isothermal single phase and homogeneous material, the equations are... 

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. 

Analyses of the Newtonian isothermal and nonisothermal models can be even further complicated by the introduction of the viscoelastic nature of the polymers in the melt state. The viscoelasticity of the polymer is important in cases where the relaxation time. A., is of the same order of magnitude or slower than the characteristic time constant of the process, which might be taken to be equal to vo/L. The ratio of these two time constants is called the Deborah number (Eq. 3.90), and it is equal to... 

The described models are mostly calibrated according to static or dynamic experiments. While static experiments result in common stress-strain diagrams for mostly isothermal load situations, dynamic experiments allow the determination of the temperature dependent modulus. These values are usually measured using a dynamic mechanical analysis (DMA), which operates in the range of linear viscoelasticity at low stress- and strain-amplitudes. 

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