Viscoelastic model dynamics

The viscoelastic effects on the morphology and dynamics of microphase separation of diblock copolymers was simulated by Huo et al. [ 126] based on Tanaka s viscoelastic model [127] in the presence and absence of additional thermal noise. Their results indicate that for

bulk modulus of both blocks, the area fraction of the A-rich phase remains constant during the microphase separation process. For each block randomly oriented lamellae are preferred. 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

If data is needed for the more sophisticated viscoelastic models now being introduced then results from forced vibration dynamic tests (Chapter 9) or stress relaxation tests (Chapter 10), as appropriate, would be used. 

To make MCT calculations simpler, the solvent dynamic quantities are sometimes modeled in such a way that the self-consistency is avoided. This is where the viscoelastic models (VEMs) play an important role. The VEM is usually expressed as a Mori continued fraction where the frequencies are de-... 

Equation (210) when compared with the viscoelastic model of the dynamic structure factor we can identify the memory kernel in the viscoelastic model, which is written as... 

Thus we note that the memory kernel has a short-time and a long-time part. It is the long-time part which is not present in the viscoelastic model, becomes important in the supercooled-liquid-near-glass transition, and gives rise to the long-time tail of the dynamic structure factor. 

The linear viscoelastic models (LVE), which are widely used to describe the dynamic rheological response of polymer melts below the strain limit of the linear viscoelastic response of polymers. The results obtained are characteristic of and depend on the macromolecular structure. These are widely used as rheology-based structure characterization tools. 

The nonlinear viscoelastic models (VE), which utilize continuum mechanics arguments to cast constitutive equations in coordinate frame-invariant form, thus enabling them to describe all flows steady and dynamic shear as well as extensional. The objective of the polymer scientists researching these nonlinear VE empirical models is to develop constitutive equations that predict all the observed rheological phenomena. 

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. 

To obtain a closed solution it is necessary to assume a viscoelastic model. For example, if a standard solid model is adopted, the following value for the dynamic modulus is found ... 

Recent work (Altmann, 2002) has focussed on combining a dynamic Monte Carlo percolation-grid simulation for reaction kinetics and an enthalpy-based group-interaction viscoelastic model to develop a model for the chemorheological and network properties of reactive systems. More emphasis will be placed on this model in Chapter 6. 

Multiplicity, bifurcation, stability, and hysteresis in dynamic solutions, nonisothermal viscoelastic model Parameter estimation to characterize convective heat transfer... 

The validity of the viscoelastic model (5.32) has been tested against experimental and molecular dynamics simulation results [26, 27, 28]. The detailed comparison has established that the viscoelastic model works remarkably well for wavenumbers k km, where km denotes the first peak position of the static structure factor S k). However, it has also been found that the situation is not so satisfactory for smaller wavenumbers, where the viscoelastic model is shown in some circumstances to yield even qualitatively incorrect results. This failure was attributed to the fact that the single relaxation time model (5.31) cannot describe both the short-time behavior of the memory function, dominated by the so-called binary collisions, and in particular the intermediate and long-time behavior where in the liquid range additional slow processes play an important role (see the next subsection). It is obvious that these conclusions demand a more rigorous consideration of the memory function, which lead to the development of the modern version of the kinetic theory. Nevertheless, the viscoelastic model provides a rather satisfactory account of the main features of microscopic collective density fluctuations in simple liquids at relatively large wavenumbers, and its value should not be undervalued. 

Borg, T. and Paakkonen, E.J. (2010) Linear viscoelastic models Part IV. Prom molecular dynamics to temperature and viscoelastic relations using control theory. J. Non-Newtonian Fluid Medt., 165 (1-2), 24-31. 

For the generalized viscoelastic model given in Problem 4.12, show that real and imaginary parts of the complex dynamic modulus G oi) are given by... 

The blown film process is known to be difficult to operate, and a variety of instabilities have been observed on experimental and production film lines. We showed in the previous chapter (Figure 10.10) that even a simple viscoelastic model of film blowing can lead to multiple steady states that have very different bubble shapes for the same operating parameters. The dynamical response, both experimental and from blown film models, is even richer. The dynamics of solidification are undoubtedly an important factor in the transient response of the process, but the operating space exhibits a variety of response modes even with the conventional approach of fixing the location of solidification and requiring that the rate of change of the bubble radius vanish at that point. 

The non linear viscoelasticity of various particles filled rubber is addressed in range of studies. It is found that the carbon black filled-elastomer exhibit quasi-static and dynamic response of nonlinearity. Hartmann reported a state of stress which is the superposition of a time independent, long-term, response (hyperelastic) and a time dependent, short-term, response in carbon black filled-rubber when loaded with time-dependent external forces. The short term stresses were larger than the long term hyperelastic ones. The authors had done a comparative study for the non linear viscoelastic models undergoing relaxation, creep and hysteresis tests [20-22]. For reproducible and accurate viscoelastic parameters an experimental procedure is developed using an ad hoc nonlinear optimization algorithm. 

Abstract Phase separation in isotropic condensed matter has so far been believed to be classified into solid and fluid models. When there is a large difference in the characteristic rheological time between the components of a mixture, however, we need a model of phase separation, which we call viscoelastic model . This model is likely a general model that can describe all types of isotropic phase separation including solid and fluid model as special cases. We point out that this dynamic asymmetry between the components is quite common in complex fluids, one of whose components has large internal degrees of freedom. We also demonstrate that viscoelastic phase separation in such dynamically asymmetric mixtures can be characterized by the order-parameter switching phenomena. The primary order parameter switches from the... 

Here, we focus our attention on phase separation in complex fluids that are characterized by the large internal degrees of freedom. In all conventional theories of critical phenomena and phase separation, the same dynamics for the two components of a binary mixture, which we call dynamic symmetry between the components, has been implicitly assumed [1, 2]. However, this assumption is not always valid especially in complex fluids. Recently, we have found [3,4] that in mixtures having intrinsic dynamic asymmetry between its components (e.g. a polymer solution composed of long chain-like molecules and simple liquid molecules and a mixture composed of components whose glass-transition temperatures are quite different), critical concentration fluctuation is not necessarily only the slow mode of the system and, thus, we have to consider the interplay between critical dynamics and the slow dynamics of material itself In addition to a solid and a fluid model, we probably need a third general model for phase separation in condensed matter, which we call viscoelastic model . 

Viscoelastic model None Polymer solutions Viscoelasticity Dynamic asymmetry... 

In the viscoelastic model, the phase-separation mode can be switched between the fluid mode and elastic gel mode . The dynamic process of viscoelastic phase separation is schematically drawn in Fig. 1. It is characterized by the switching of phase-separation modes between fluidlike and elastic gel-like ones [4]. This switching is likely caused by the change in the coupling between stress fields and velocity fields, which is described by Eq. (6) According to Eq. (6), the two extreme cases, namely, (i) fluid model (xfj const.) and (ii) elastic gel model (G(t), K t) const.), correspond to and t[Pg.180]

They presented a theoretical approach to predict the behavior of silicone rubber under uniaxial stress. The model is based on the concept of the classical Maxwell treatment of viscoelasticity and stress relaxation behavior, and the Hookean spring component was replaced by an ideal elastomer component. From the test data, the substitution permits the new model estimation of the cross-link density of the silicone elastomer and allows a stress level to be predicted as a complex function of extension, cross-link density, absolute temperature, and relaxation time. Tock and co-workersh" ] found quite good agreementbetweenthe experimental behavior based on the new viscoelastic model. By using dynamic mechanical analysis (DMA), the authors would have been able to obtain similar information on the silicone elastomer. 

In experiments with marine sands (Nikolaevskll and Vllchinskaya [4]) It was found that every dynamic action created running waves with frequencies which could be Identified as dominant frequencies. This phenomenon appears to be common for any real geological medium which Is fragmented by crack systems and pores. Therefore, It Is possible to use viscoelastic models In the study of the dynamics of geomaterlals. However, It Is necessary to find representations of rheological coefficients which may not be a simple combination of elastic moduli and relaxation times. 

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