Modeling of Rheological Behavior

The study of the rheological nature and behavior involves phenomena that occur during the deformation and flow of materials. The responses of solids and Uquids to mechanical deformation are governed by two fundamental functional relationships (i) the stress (o)-strain ( ) curve and (ii) the flow curve-that is, the relationship between shear stress (i) and velodty gradient (D = dv/dy). The two Umiting equations are  

In ideally elastic solids, the deformation energy will be stored within the solid without any losses, whereas in ideally viscous liquids the deformation energy will be completely transformed into frictional heat In the real world however, the elastic constants connecting the tensorial components of stress and strain, including the modulus of elasticity E, the shear modulus G, the Poisson number v, the compressibility k, and other parameters, are not time-invariant. Consequently, even after releasing very small stresses, both residual and time-dependent deformations remain in engineering solids that require a certain relaxation time to restore the equilibrium state of the system (e.g., see Meyers and Chawla, 2009). 

To account for the nonideal nature of real soUds and liquids, the theory of Unear viscoelasticity provides a generaUzation of the two classical approaches to the mechanics of the continuum-that is, the theory of elasticity and the theory of hydromechanics of viscous Uquids. Simulation of the ideal boundary properties elastic and viscous requires mechanical models that contain a combination of the ideal element spring to describe the elastic behavior as expressed by Hooke s law, and the ideal element dash pot (damper) to simulate the viscosity of an ideal Newton Uquid, as expressed by the law of internal friction of a liquid. The former foUows the equation F = D -x (where F = force, x = extension, and D = directional force or spring constant). As D is time-invariant, the spring element stores mechanical energy without losses. The force F then corresponds to the stress a, while the extension x corresponds to the strain e to yield a = E - e. 

Coupling in series. The partial shear stresses Ts of the spring elements and Tp of the damping element equal the total shear stress T= Ts = Td, analogous to the total amount of electricity transported through a series of capacitors that is Q= Qi = Qj = 

Coupling in paraiiel. The partial shear stresses add up to the total shear stress, r=Ts + Td, as do the capacities C = Ct + C2 +. .. of an electrical circuit consisting 

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Chauveau C, Maillols H, Delonca H. Natrosol 250 part 1 characterization and modeling of rheological behavior [in French]. Pharm Acta Helv 1986 61 292-297. 

Nonlinear models of rheological behavior can be approximated by step functions, whereby the existence of a finite yield stress G plays a dominant role. Three typical nonlinear models include the Saint-Venant model of ideal plastic behavior, the Prandtl-Reuss model of an elastoplastic material, and the Bingham model of viscoelastic behavior. The first model can be mechanically approximated by a sliding block, the second by a Maxwell element and a sliding block in series, and the third by a dash pot damping element and a sliding block in parallel (Figure 2.14). 

The second normal stress difference has been found to be negative with a magnitude less than that of Wj. It is quite sensitive to assumptions used in deriving a tube model of rheological behavior. A useful material function is the normal stress ratio it,f) defined as follows ... 

The existence of a microdomain interphase due to the diffuse concentration gradient across the boundary has been predicted by statistical thermodynamic theories based on the mean-field approach. The results of several experimental works (e.g., the systematic deviation of SAXS intensity profiles from the behavior of sharp-boundary systems described by Porod s law and the modeling of rheological behavior measured by DMA ) support the view of a segmentally mixed diffuse interphase. However, various other models, such as a coarse interface with a sharp boundary, may also account for some of the observed results from SAXS data. The ambiguity... 

Lest one ignore the important role of rheological behavior and properties of fluid foods in handling and processing foods, they are covered in Chapter 8. Here, the topics covered include applications under isothermal conditions (pressure drop and mbcing) and under non-isothermal conditions (heat transfer pasteurization and sterilization). In particular, the isothermal rheological and nonisothermal thermorheological models discussed in Chapters 3 and 4 are applied in Chapter 8. 

It is well known [38, 118, 125, 280, 379] that for foam there exists a yield stress ro that classifies the types of rheological behavior of foam as follows for r < to, the foam is a solid-shaped substance, and for t > to, it is fluid-shaped. For this reason, mechanical models of foam must include the Saint-Venant body. One of the simplest macrorheological models of the foam body is shown in Figure 7.3. 

Dispersions with uniform particle size have been used to model the rheological behavior of dispersions of noninteracting hard spheres [3.86]. Liquid dispersions with the appropriate partiele size distribution have been made with a very high volume fraction [3.87]. 

In spite of the complexity of the situation, there have been attempts to create theories of rheological behavior of PLCs. Wissbrun [79] represents a PLC material as a space-filling system of domains. At rest, the minimum energy arrangement is achieved when the directors in the planes of contact are parallel. Under shear, the domains slide over each other. The model predicts shear sensitiveness, a phenomenon observed experimentally the curves of viscosity as a function of the shear rate are horizontal for low shear rates and then go down. In fact, for instance the results for PCarb - - PET/ 0.6PHB blends—if we employ such coordinates rather than those in Fig. 41.12—exhibit shear sensitiveness [76]. 

The applied shear stress causes the deformation, or strain, of the cube, that is, a shift of its upper face with respect to the lower face by an amount y. This shift is numerically equal to the tangent of the tilt experienced by the side face, that is, to the relative shear strain, y. When the strains are low, one can write that tan (y) y. The relationship between the stress t, the strain y, and their change as a function of time represents the mechanical behavior, which is the main subject of rheology. Let us start by reviewing three basic models of mechanical behavior elastic, viscous, and plastic. 

The recent developments mentioned above open the door to the development of quantitative models relating molecular structure to rheological behavior. The two direct applications of these models are the prediction of rheological behavior when the molecular structure is well known and the determination of key aspects of molecular structure through rheological measurements. Going beyond the scope of the present book, the relationship between melt... 

Like Leonardi et al [52] Van Ruymbeke et al [53] accounted for non-reptational mechanisms in their method, but they used different models for the relaxation processes. They inverted a model that they had previously proposed [54] for the calculation of rheological behavior from the molecular weight distribution. For the Rouse modes they used a modified version of an expression proposed by Pattamaprom et al [55]. Their modified equation is shown below. 

For the simulation of more complex flows, one needs a constitutive equation or a rheological equation of state. Nearly all of the many equations that have been proposed over the past fifty years are basically empirical in nature, and only in the last twenty-five years have such models been developed on the basis of mean field molecular theories, e.g., tube models. Although the early models were often developed with a molecular viewpoint in mind, it is best to think of them as continuum models or semi-empirical models. The relaxation mechanisms invoked were crude, involving concepts such as network rupture or anisotropic friction without the molecular detail required to predict a priori the dependence of viscoelastic behavior on molecular structure. While these lack a firm molecular basis and thus do not have universal validity or predictive capability, they have been useful in the interpretation of experimental data. In more recent times, constitutive equations have been derived from mean field models of molecular behavior, and these are described in Chapter 11. We describe in this section a few constitutive equations that have proven useful in one or another way. More complete treatments of this subject are given by Larson [7] and by Bird et al. [8]. 

Although aH these models provide a description of the rheological behavior of very dry foams, they do not adequately describe the behavior of foams that have more fluid in them. The shear modulus of wet foams must ultimately go to zero as the volume fraction of the bubbles decreases. The foam only attains a solid-like behavior when the bubbles are packed at a sufficiently large volume fraction that they begin to deform. In fact, it is the additional energy of the bubbles caused by their deformation that must lead to the development of a shear modulus. However, exactly how this modulus develops, and its dependence on the volume fraction of gas, is not fuHy understood. 

Flow Models. Many flow models have been proposed (10,12), which are useful for the treatment of experimental data or for describing flow behavior (Table 1). However, it is likely that no given model fits the rheological behavior of a material over an extended shear rate range. Nevertheless, these models are useful for summarizing rheological data and are frequently encountered in the Hterature. 

Conway, M.W. Almond, S.W. Briscoe, J.E. Harris, L.E. "Chemical Model for the Rheological Behavior of Crosslinked Fluid Systems," SPE Paper 9334, 1980 SPE Annual Technical Conference and Exhibition, Dallas, September 21-24. 

PTT exhibits melt rheological behavior similar to that of PET. At low shear rates the melt is nearly Newtonian. It shear-thins when the shear rate is >1000s 1 (Figure 11.10) [68], At the melt processing temperatures of PET, 290°C, and of PTT, 260°C, both polymers have similar viscosities of about 200Pas. However, PTT has a lower non-Newtonian index than PET at high shear rates. The flow behavior can be modeled by the Bueche equation, as follows ... 

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