Polymeric liquids modeling

Fig. 27 a and b. Schematic representation of the molecular structure of a side chain polymeric liquid crystals b polymer model membranes studied by 2H NMR... 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

The step-strain experiments discussed above furnish the simplest example of a strong flow. Many other flows are of experimental importance transient and steady shear, transient extensional flow and reversing step strains, to give a few examples. Indeed the development of phenomenological constitutive equations to systematise the wealth of behaviour of polymeric liquids in general flows has been something of an industry over the past 40 years [9]. It is important to note that it is not possible to derive a constitutive equation from the tube model in... 

Kulicke W-M, Kehler H, Bouldin M A consideration of the state of solution in the molecular modeling of the zero-shear viscosity for polymeric liquids Colloid Polym Sci (submitted)... 

Now we shall discuss the method used to calculate the "cup"-averaged MWD-H, in which all portions of a polymerized liquid are mixed and averaged in a "cup" (vessel) positioned after the reactor. In this analysis, recourse was made to the so-called "suspension" model of a tubular reactor. According to this model, the reaction mass is regarded as an assemblage of immiscible microvolume batch reactors. Each of these microreactors moves along its own flow line. The most important point is that the duration of the reaction is different in each microreactor, as the residence time of each microvolume depends on its position at any given time, i.e., on its distance from the reactor axis. 

In a complex, polymeric liquid, normal stresses as well as the shear stress can be present, and these contributions will influence the shape of the structure factor. The simplest rheological constitutive model that can account for normal stresses is the second-order fluid model [64], where the first and second normal stress differences are quadratic functions of the shear rate. Calculations using this model [92,93,94,90,60], indicate that the appearance of normal stresses can rotate the structure factor towards the direction of flow in the case of simple shear flow and can induce a four-fold symmetry in the case of exten-sional flow. 

The three constant Oldroyd model is a nonlinear constitutive equation of the differential corrotational type, such as the Zaremba-Fromm-Dewitt (ZFD) fluid (Eq. 3.3-11). [For details, see R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1, Wiley, New York, 1987, Table 7.3-2.]... 

Vergnaud J. M., 1991. Liquid Transport Processes in Polymeric Materials. Modeling and industrial applications. Prentice Hall, Englewood Cliffs, New Jersey. 

In the first place, the averaged model equations are highly nonlinear and require sophisticated numerical analysis for solution. For example, the attempt to obtain numerical solutions for motions of polymeric liquids, based upon simple continuum, constitutive equations, is still not entirely successful after more than 10 years of intensive effort by a number of research groups worldwide [27]. It is possible, and one may certainly hope, that model equations derived from a sound description of the underlying microscale physics will behave better mathematically and be easier to solve, but one should not underestimate the difficulty of obtaining numerical solutions in the absence of a clear qualitative understanding of the behavior of the materials. 

The two-way arrow between polymer rheology and fluid mechanics has not always been appreciated. Traditionally we look at polymer rheology as input to fluid mechanics, as a way to supply constitutive equations. Gary Leal pointed out the use of fluid mechanics to provide feedback to tell us whether the constitutive equation is satisfactory. In the past, we tested constitutive models by examining polymeric liquids with very simple kinematics, homogeneous flows as a rule, either simple shear or simple shear-free types of flows. We don t actually use polymers in such simple flows, and it s essential to understand whether or not these constitutive equations actually interpolate properly between those simple types of kinematics. So there s a two-way arrow that we have to pay more attention to in the future. 

I would also like to list some of the challenges that will provide the foundation for where the profession has to go (Fig. 2). This is not meant to be comprehensive, but to suggest some of what we should be doing. This wish list derives from work Bob Brown and I have done on modeling flows of polymer fluids. The first item has to do with the need to understand the effects of polymer structure and rheology on flow transitions in polymeric liquids and on polymer processing operations. In the past, we ve studied extensively the behavior of Newtonian fluids and how Newtonian flows evolve as, say, the Reynolds number is varied. We have tools available to... 

As a result of molecular orientation, it would appear less appropriate to use the cube root of the molar volume as a measure of the thickness of the monomolecular layer at the vapor-liquid interface. An accurate calculation of the monomolecular layer thickness requires a precise theoretical model of the structure of a liquid or a polymer which is beyond the scope of this paper. An example of this kind of approach is given by Roe (28) in his paper on polymeric liquids. 

Studies of the mechanism and kinetics of homogeneous polymerizations in the liquid phase are simpler than in the other cases. Therefore they are preferred when possible, or at least the partial problems of inhomogeneous polymerizations are modelled in the homogeneous phase. 

Another technique which uses microscopy is based on the miscibility of compounds with identical mesophases and was developed by the Halle liquid crystal group for model liquid crystals. Noel has applied this method to mixtures composed of well-known model liquid crystals with polymeric liquid crystals 3 - ). Assuming that the method is applicable to mixtures of polymers and low molecular weight compounds, the type of mesophase can be positively identified if the polymer and model are miscible. 

Figure 3.10 Predictions of the temporary network model [Eq. (3-24)] (lines) compared to experimental data (symbols) for start-up of uniaxial extension of Melt 1, a long-chain branched polyethylene, using a relaxation spectrum fit to linear viscoelastic data for this melt. (From Bird et al. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

During the last ten years the interest in polymeric liquid crystals (PLCs) has been growing rapidly. Nevertheless our fundamental understanding of their flow behaviour is still rather limited. This is due to the fact that PLC rheology is much more complicated than that of ordinary isotropic polymeric fluids (1). Systematic and reliable data are lacking so far although this is the kind of information needed for the development and assessment of theoretical models for these unusual fluids. 

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