Marrucci Model

As the title indicates, Petrie s chapter also addresses work through 1983 on blow molding and thermoforming. The Newtonian and Maxwell fluid results shown here are from the following, which also includes results for still another Maxwell-type equation with bounded extensional stresses (the Marrucci model) ... 

In Eq. 11.24, the evolution of the orientation tensor S is given as a differential equation, analogous to the differential Eq. 11.21 of Eikhtman etal., rather than the history integral equation in Eq. 11.15. While differential equations are much easier to handle numerically than history integral equations, many of these equations, including that of likhtman, et al., lead to a zero second normal stress difference in simple shear. However, in Eq. 11.24 of the lanniruberto and Marrucci model, the use of the square of the orientation tensor leads to a ratio of Nj/Nj of -0.25 at low shear rates, which is close to the values typically observed in experiments on entangled solutions and melts [41,42]. 

A purely viscous non-Newtonian approach was followed by Han and Park (1975b). They used the power-law model and the energy equation, assuming that the effects of crystallization were insignificant. The agreement of this model with experimental data in terms of the bubble radius and thickness as a function of the axial distance for LDPE and HDPE was reported to be reasonable. In terms of viscoelastic models, Luo and Tanner (1985) considered the Leonov model, and Cain and Denn (1988) considered the upper convected Maxwell and Marrucci models in nonisothermal cases of film blowing. In some of the cases analyzed, multiple steady-state solutions were present (see also Problem 9C.2). 

To evaluate the effect of holdup on bubble velocity, Marrucci (M3) used a spherical cell model of radius b such that... 

Optical measurements often have a greater sensitivity compared with mechanical measurements. Semidilute polymers, for example, may not be sufficiently viscous to permit reliable transient stress measurements or steady state normal stress measurements. Chow and coworkers [113] used two-color flow birefringence to study semidilute solutions of the semirigid biopolymer, collagen, and used the results to test the Doi and Edwards model discussed in section 7.1.6.4. That work concluded that the model could successfully account for the observed birefringence and orientation angles if modifications to the model proposed by Marrucci and Grizzuti [114] that account for polydispersity, were used. 

D. Aciemo, F. P. La Mantia, G. Marrucci, and G. Titomanlio, A Non-linear Viscoelastic Model with Structure-dependent Relaxation Times I. Basic Formulation, J. Non-Newt. Fluid Meek, 1, 125-146 (1976). 

G Marrucci. Microrheological modelling. In JA Covas, JF Agassant, AC Diogo, J Vlachopoulos, K Walters, eds. Rheological Fundamentals of Polymer Processing . NATO ASI Ser. London Kluwer Academic Publishers, 1994,p 37... 

Two theories of viscoelasticity in reptating chain systems have appeared since the original Doi-Edwards publications, the theory of Marrucci and coworkers and the theory of Curtiss and Bird ° They differ in various ways from the Doi-Edwards tube model and from the model suggested in Part II. It is difficult and probably premature to provide detmled criticisms at the present time, but it is perhaps worthwhile to point out at least a few of the differences. 

Concerning the mathematical modeling of LCPs, most attempts have concentrated on lyotropic systems. Doi [103] has developed a defect free model for lyotropic rodlike polymers in the isotropic and anisotropic states and Larson and Mead [ 104] have proposed a polydomain phenomenological description of lyotropic systems. Marrucci and Maffettone [105] solved a diffusion... 

The accumulation of CR-jumps results in the chain motion in the direction lateral to the tube axis over a distance well above the tube diameter a (oc M buik)/ as schematically shown at the bottom of part (b-3) of Figure 3.6. Thus, the CR mechanism d5mamically dilates an effective tube diameter defined in a coarse-grained time scale, and the chain is regarded to be constrained in the dilated tube (supertube) in that time scale, as first pointed out by Marrucci (1985). A model based on this molecular picture was first proposed by Marrucci (1985) and later refined by Ball and McLeish (1989) and by Milner and McLeish (1998). Because this coarse-graining molecular picture... 

The time evolution (decay) of ( /(f) has been calculated in several full-DTD models, and the corresponding G(t) and complex modulus G (o)) = G ((o) + iG"(o>) have been compared with the data (see Ball and McLeish, 1989 Marrucci, 1985 Milner and McLeish, 1998). Agreement of the model prediction and data was reported for monodisperse linear and star-branched polymer. 

Marrucci, G. 1985. Relaxation by reptation and tube enlargement— A model for poly-disperse polymers. /. Polym. Sci. Polym. Phys. Ed. 23 159-177. 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

Fig. 7.12 Illustration of extensional viscosity versus the extensional rate curve predicted by the molecular theory based on the standard tube model for the stable extensional flow of linear polymers. Starting from the low extensional rate, the viscosity first keeps in 3%, then decays, after deformation begins to increase, till to saturation (Marrucci and lannirubertok 2004) (Readapted with permission)...

The strain measure which follows from the model by Marrucci is indistinguishable from curve 1 in Fig. 2. It is in good agreement with expe-... 

On the other hand, Marrucci et al (Marrucci G. and Maffettone 1989) have solved a two dimensional version of the Doi model for nematics (Doi M. and Edwards S.. F. 1986), in which the molecules are assumed to lie in the plane perpendicular to the vorticity axis, that is, in the plane parallel to both, the direction of the velocity and the direction of the velocity gradient. Despite this simplification, the predicted range of shear rates over which Ni is negative, is in excellent agreement with observations. This result opens up the possibility that negative first normal stress differences may be predicted in a two dimensional flow. 

Marrucci, G. (1996) Dynamics of entanglements a nonlinear model consistent with the Cox-Merz rule. /. Non-Newtonian FluidMech., 62, 279. 

A more original approach has been proposed by Marrucci and Viovy and, more recently, put into a concise sketch by des Cloizeaux. In the framework of the reptation model, noting that a topological constraint between two chains, A and B, can be relaxed either (and independently) by the motion of A or B, des Cloizeaux suggests that the relaxation modulus should in fact be proportional to M t), with M(t) obeying normal blending laws M(t) = (0 is the fraction of unrelaxed chains... 

M. Rubinstein (Eastman Kodak Company) In the des Cloizeaux double reptation model which is similar to the Marrucci Viovy model, it is assumed that a release of constraint chain A imposes on chain B when chain A reptates away completely relaxes the stress in that region for both chains. This would imply that for a homopolymer binary blend of long and short chains would be completely relaxed after each of these K entanglements is released only once. But if an entanglement is released, another one is formed nearby. I believe that to completely relax this section one needs disentanglement events and that the Verdier-Stockmayer flip-bond model or the Rouse model is needed to describe the motion and relaxation of the primitive path due to the constraint release process, as was proposed by Prof, de Gennes, J. Klein, Daoud, G. de Bennes and Graessley and used recently by many other scientists. The fact that double reptation is an oversimplification of the constraint release process has been confirmed by experiments. 

Macosko CW (1994) Rheology principles, measuremaits and applications. Wiley, New York Maier D, Eckstein A, Friedrich C, Honerkamp JL (1998) Evaluation of models combining rheological data with molecular weight distribution. J Rheol 42 1153-1173 Majumder KK, Hobbs G, Bhattacharya SN (2007) Molecular, ifaeological, and crystalline properties of low-density polyethylene in blown film extrusion. Polym Eng Sci 47 1983-1991 Marrucci G (1991) Liquid crystallinity in polymers principles and fundamental properties. VCH, New York... 

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