Graessley’s model

This characteristic shear rate is equal to the reciprocal average entanglement time Tn at equilibrium. The solid curve is from Graessley s model, which reads... 

For the special case = 1, Eq. 183 recovers Graessley s result, Eq. 172. For quite long chains, 1, the maximal relaxation time tmax goes as ri to, which leads to a considerable broadening of the relaxation spectrum in the domain of short times, a fact due to the inclusion of the chains between the branching points. Thus, for tree-like networks the relaxation spectrum turns out to be broader than in Graessley s model of the previous section. 

The MWBD method also requires an independent measure of the branching structure factor e. For our analysfs of polyvinyl acetate, it was obtained by comparing M and Bf values calculated from SEC data, analyz d using the MWBD method and various epsilons, and the Mfj and Bj values predicted by Graessley s (21) kinetic model. An epsilon value of 1.0 was found to fit best. 

Due to difficulties in measuring the zero-shear viscosity of such high molecular weight polymers, and thus deducing the monomeric friction coefficient from Graessley s uncorrelated drag model [43], the following equation adapted from the modified Rouse theory has been applied [8]. 

During the years 1992-1998, numerous publications emerged from Prof. Graessley s laboratory. The model PEs with different stmctures (see Fig. 18.10) were commercial (e.g., HDPE or PP) or from laboratory (e.g., hydrogenation/deuteration of diolefins, anionic reaction for PIB, Z-N catalysis using a V-based catalyst in Cg for poly (ethylene-co-a-olefin) or later a metallocene catalyst. The thermodynamic properties of numerous PO blends were extracted from the pressuie-volume-temperature (PVT) data (Walsh et al. 1992 Krishnamoorti et al. 1996) or from SANS results (Krishnamoorti et al. 1994, 1995 Graessley et al. 1994, 1995 Reichart et al. 1997 Alamo et al. 1997). 

Because the ansatz does not invoke a detailed molecular picture, its predictive powers are restricted. The approach yields the functional dependence of various viscoelastic parameters on frequency or shear rate. The price exacted for the simplicity of derivation is that the ansatz is fundamentally unable to predict numerical values for functional parameters, let alone predict the dependence of those parameters on solvent quality, polymer concentration, or molecular weight. The ansatz is thus substantially noncommunicating with treatments of polymer viscoelasticity that invoke detailed microscopic models of polymer dynamics, such as the models of Bird, etal.(4,5), Graessley(6,7), or Raspaud, etal. S). 

Considerable progress has been made in recent years in theoretical as well as experimental studies of the relation between the elastic behaviour and the structure of polymer networks. Theoretical studies were published by Flory " Ronca and Allegra, Edwards Freed a.o., while Rempp c.s. published an excellent survey of the synthesis and properties of Model Networks . For earlier surveys of the field we refer to DuSek and Prins and Graessley ... 

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