Constitutive equations Molecular models

Polymeric fluids are the most studied of all complex fluids. Their rich rheological behavior is deservedly the topic of numerous books and is much too vast a subject to be covered in detail here. We must therefore limit ourselves to an overview. The interested reader can obtain more thorough presentations in the following references a book by Ferry (1980), which concentrates on the linear viscoelasticity of polymeric fluids, a pair of books by Bird et al. (1987a,b), which cover polymer constitutive equations, molecular models, and elementary fluid mechanics, books by Tanner (1985), by Dealy and Wissbrun (1990), and by Baird and Dimitris (1995), which emphasize kinematics and polymer processing flows, a book by Macosko (1994) focusing on measurement methods and a book by Larson (1988) on polymer constitutive equations. Parts of this present chapter are condensed versions of material from Larson (1988). The static properties of flexible polymer molecules are discussed in Section 2.2.3 their chemistry is described in Flory (1953). 

So in the world of polymer melts, the intermediary of a constitutive equation, or equation set, albeit molecularly derived, would seem unavoidable at present. The success of the branched-polymer model discussed above would suggest that... 

In network models the molecular arguments supply a form for the constitutive equation, but do not provide the detailed connections to molecular structure. As such, they provide a bridge between molecular theories which incorporate specific structural information in rather specific flow situations and continuum models which can generalize such information to arbitrary flows. 

The proposed mechanisms of models to explain the drag reduction phenomenon are based on either a molecular approach or fluid dynamical continuum considerations, but these models are mainly empirical or semi-empirical in nature. Models constructed from the equations of motion (or energy) and from the constitutive equations of the dilute polymer solutions are generally not suitable for use in engineering applications due to the difficulty of placing numerical values on all the parameters. In the absence of a more generally accurate model, semi-empirical ones remain the most useful for applications. 

Fig. 3.7 (a) Uniaxial, (b) equibiaxial, and (c) planar extensional viscosities for a LDPE melt. [Data fromP. Hachmann, Ph.D. Dissertation, ETH, Zorich (1996).] Solid lines are predictions of the molecular stress function model constitutive equation by Wagner et al, (65,66) to be discussed in Section 3.4. 

We have tried to give a quick glimpse of the interrelationships among some commonly used constitutive equations for polymer melts and solutions. None predicts quantitatively the entire spectrum of the rheological behavior of these materials. Some are better than others, becoming more powerful by utilizing more detailed and realistic molecular models. These, however, are more complex to use in connection with the equation of motion. Table 3.1 summarizes the predictive abilities of some of the foregoing, as well as other constitutive equations. 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

For example, when we consider the design of specialty chemical, polymer, biological, electronic materials, etc. processes, the separation units are usually described by transport-limited models, rather than the thermodynamically limited models encountered in petrochemical processes (flash drums, plate distillations, plate absorbers, extractions, etc.). Thus, from a design perspective, we need to estimate vapor-liquid-solid equilibria, as well as transport coefficients. Similarly, we need to estimate reaction kinetic models for all kinds of reactors, for example, chemical, polymer, biological, and electronic materials reactors, as well as crystallization kinetics, based on the molecular structures of the components present. Furthermore, it will be necessary to estimate constitutive equations for the complex materials we will encounter in new processes. 

Nevertheless, each author remains responsible for his own contribution. Papers have been gathered in sections covering successively "Molecular dynamics", "Constitutive equations and numerical modelling", "Simple and complex flows". However, each paper can be read independently. 

Time constants are related to the relaxation times and can be found in equations based on mechanical models (phenomenological approaches), in constitutive equations (empirical or semiempirical) for viscoelastic fluids that are based on either molecular theories or continuum mechanics. Equations based on mechanical models are covered in later sections, particularly in the treatment of creep-compliance studies while the Bird-Leider relationship is an example of an empirical relationship for viscoelastic fluids. 

The prediction of the above equation is the closest of any molecular theory to experimentally observed 3.4 power. A constitutive equation based on the reptation model can be written as ... 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

These buzz words include anything that connects one scale to the next, starting from a straight calculation of a constitutive equation parameter from molecular modelling to some more sophisticated schemes of mapping and reverse mapping of scales. The basic idea and attraction is depicted in Figure 2 ... 

Here the situation is very similar to that encountered in connection with the need for continuum (constitutive) models for the molecular transport processes in that a derivation of appropriate boundary conditions from the more fundamental, molecular description has not been accomphshed to date. In both cases, the knowledge that we have of constitutive models and boundary conditions that are appropriate for the continuum-level description is largely empirical in nature. In effect, we make an educated guess for both constitutive equations and boundary conditions and then normally judge the success of our choices by the resulting comparison between predicted and experimentally measured continuum velocity or temperature fields. Models derived from molecular theories, with the exception of kinetic theory for gases, are generally not available for comparison with the empirically proposed models. We discuss some of these matters in more detail later in this chapter, where specific choices will be proposed for both the constitutive equations and boundary conditions. 

Although this simplified version of Fourier s heat conduction law is well known to be an accurate constitutive model for many real gases, liquids, and solids, it is important to keep in mind that, in the absence of empirical data, it is no more than an educated guess, based on a series of assumptions about material behavior that one cannot guarantee ahead of time to be satisfied by any real material. This status is typical of all constitutive equations in continuum mechanics, except for the relatively few that have been derived by means of a molecular theory. 

Spin densities (p) are theoretical quantities, defined as the sum of the squared atomic orbital coefficients in the nonbonding semi-occupied molecular orbital (SOMO) of the radical species (Hiickel theory). For monoradical species, the spin density is connected to the experimental EPR hyperfine coupling constant a through the McConnell equation [38]. This relation provides the opportunity to test the spin density dependence of the D parameter [Eq. (8)] for the cyclopentane-1,3-diyl triplet diradicals 10 by comparing them with the known experimental hyperfine coupling constants (ap) of the corresponding substituted cumyl radicals 14 [39]. The good semiquadratic correlation (Fig. 9) between these two EPR spectral quantities demonstrates unequivocally that the localized triplet 1,3-diradicals 9-11 constitute an excellent model system to assess electronic substituent effects on the spin density in cumyl-type monoradicals. 

The constitutive equations to be adopted are defined empirically, though the coefficients in these equations (viscosity coefficient, heat conduction coefficient, etc.) may be determined at the molecular level. Often, however, these coefficients are determined empirically from the phenomena themselves, though the molecular picture may provide a basis for the interpretation of the data. It is for this reason that the description of the fluid state based on a continuum model and concepts is termed a phenomenological description or model. 

The theoretical treatment of such non-Newtonian behavior has been through the development of constitutive equations to replace the Newtonian relationship between stress and strain rate, and to be used in continuum mechanical treatments of real flows. The constitutive equation has often been largely empirical in origin (like, for example, power law fluids), but increasingly has been derived from molecular theories, where macromolecules are modeled as bead-spring, bead-rod, or finitely extendable dumbbells. 

An important concept in continuum mechanics is the objectivity, or admissibility, of the constitutive equation. There are the covariant and contravariant ways of achieving objectivity. The molecular theories the elastic dumbbell model of this chapter, the Rouse model to be studied in the next chapter, and the Zimm model which includes the preaveraged hydrodynamic interaction, all give the result equivalent to the contravariant way. In this appendix, we limit our discussion of continuum mechanics to what is needed for the molecular theories studied in Chapters 6 and 7. More detailed discussions of the subject, particularly about the convected coordinates, can be found in Refs. 5 and 6. 

Concrete definition of equilibrium state must be performed for each constitutive model (characterized by the observer s scales of Sect. 1.1 and mainly Sect. 2.3) by time fixing of some quantities from those determining their states (see Rem. 6). Time persistency is usually difficult to achieve (because of molecular fluctuation) and therefore to describe real materials by such constitutive models we must add to constitutive equations (as their regularity) the conditions of stability by which the time permanence of equilibrium state S4 is assured. For details see Sects. 2.1-2.4, 3.8, 4.7 and Rems. 7,9, 11 in Chap. 2. Although one eqrulibrium state would suffice, typically there are more equilibrium states often forming the equilibrium process as their time sequence, see Rem. 12. 

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