Milner-McLeish model

Figure 9.13 Comparison of theory with data for the loss nrKxJuli of binary blends of nearly monodisperse, linear 1,4-polybutadiene (MW = 105,000) and three-arm star 1,4-polybutadiene (MW = 127,000) at r=25 °C.The star volume fractions, from right to left, are 0,0.2,0.5,0.75, and 1. The data are from Struglinski etal. [35]. The dashed lines are the Milner-McLeish model predictions, while the solid lines were obtained from the hierarchical model (see Section 9.5.2) both using a = 4/3.The parameter values are the same as in Rg. 9.6. From Park and Larson [49].

The advanced molecular models described in this chapter, namely the Milner-McLeish model and the hierarchical model, involve combinations of multiple relaxation mechanisms reptation, primitive path fluctuations, and constraint release described by both constraint release Rouse motion and dynamic dilution. However, all these mechanisms can be captured in algorithms in which entanglements are viewed as slip links between two chains see for example Fig. 9.22. 

The constraint-release models discussed above have been tested by comparing their predictions to experimental data, as shown in Figures 7.9 and 7.10. For linear polymers for which the molecular weight distribution is unimodal, and not too broad, dynamic dilution is not very important, and theories that account for constraint release without assuming any tube dilation are adequate. Such is the case with the version of the Milner-McLeish theory for linear polymers used to make the predictions shown in Fig. 6.13. The double reptation theory also neglects tube dilation. The dual constraint theory mentioned in Chapter 6 does include dynamic dilution, although its effect is not very important for narrowly dispersed linear polymers. As described above, dynamic dilution becomes important for some bimodal blends, and is certainly extremely important for branched polymers, as discussed in Chapter 9. 

Models such as the Milner-McLeish or dual constraint model appear to give good agreement with experimental star data for 1,4-polybutadiene, 1,4-polyisoprene, and polystyrene [8]. 

Graham and Olmsted [166,167] used coarse-grained kinetic Monte Carlo simulations to simulate anisotropic nucleation based on the chain configurations obtained from a molecular flow model, the Graham-Likhtman and Milner-McLeish (GLaMM) model [168,169].These simulations confirm the power law with exponent 4 up to reasonably high shear rates (molecular stretch up to 3 to 4). They actually found an exponential dependence on the square of the molecular stretch. A practical problem of such an expression is that it contains an extra parameter besides the prefactor for the stretch, there is a prefactor for the exponential function as a whole, which gives the quiescent sporadic creation rate. Since quiescent nucleation is predominantly athermal, this parameter cannot be determined for common melts. 

The current version of the tube model combines all the above mechanisms to describe the linear viscoelastic data of linear and star-branched chains. For example, the Milner-McLeish (MM) model, " adopting the DTD molecular picture for the CR mechanism, considerably well describes the data, as shown with the solid curves in Figures 9 and 10. Equally good description can be obtained for some other models combining those mechanisms in a way different from that in the MM modd. - - ... 

The model described by equations (3.42)-(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the particles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) considered mechanism of reptation motion of branched macromolecules of different architecture. 

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. 

In general, there are multiple relaxation processes in polymers, many of which are much too complex to be described by simple rheological theories (such as the double reptation model presented below), and it is not our objective to describe all such processes in detail. The interested reader can find the details in the book by Doi and Edwards [ 1 ], and in the review article by Watanabe [2]. Nevertheless, in Chapter 9 we will present some advanced theories for polymer melts, including theories of McLeish, Milner, and coworkers, that include all the known important mechanisms of polymer relaxation, and in Chapter 11, we will combine... 

Figure 9.8 Comparison of predictions of theory of Milner and McLeish [13] with measurements of G (filled symbols) and G" (unfilled symbols) for nearly monodlsperse 1,4-poly butadiene four-arm stars with total molecular weights, from right to left, of 45,200,121,000, and 162,000 at T = 27 °C. (The arm molecular weight are one quarter of the total molecular weight.) The symbols are experimental data for samples F, D, and B of Roovers [301.The solid lines are model predictions using a= l,the dashed lines using a = 4/3.The parameter values are the same as In Fig. 9.6, with a small shift in to account forthe change in temperature (Table 7.1). From Park and Larson [27].

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