Reptation model parameters

The detailed equations for D, r, Gn, and can be expressed in terms of a single reptation model parameter and the monomeric friction < efficient The latter can be estimated independently (Ref. 12, Chap. 12), and predictions of relationships among properties can be obtained by expressing the reptation model parameter in terms of one experimental property such as Gn-... 

Moreover, from Fig. 3.18 it is apparent that the model of des Cloizeaux also suffers from an incorrect Q-dependence of S(Q,f) in the plateau region, which is most apparent at the highest Q measured. It is important to note that the fits with the reptation model were done with only one free parameter, the entanglement distance d. The Rouse rate was determined earlier through NSE data taken for Kr. With this one free parameter, quantitative agreement over the whole range of Q and t using the reptation model with d=46.0 1.0 A was found. 

The tube model avoids dealing with specification of an average molecular weight between entanglement points (Mg), but the diameter of the tube d) is an equivalent parameter. Further, the ratio of the contour length to diameter L/d) is taken to be approximately the magnitude of the entanglements per molecule (M/Me). Some of the important relationships predicted from the reptation model include (Tirrell, 1994) ... 

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). 

The reptation model was the first to bring the large parameter N into play. As a result, a molecular theory of fluid polymer dynamics has been developed. All the previous theories of the dynamics of polymeric liquids were basically phenomenological. 

In panels (a)-(c), comparison of the blend moduli calculated from the model (curves Equations 3.69 through 3.73) with the moduli data (symbols) for various high-M polyisoprene/ poly(p-tert butyl styrene) (PI/PtBS) blends as indicated. The sample code nmnbers of the blends indicate 10 M of the components. The model considers the cooperative Rouse equilibration and successive constraint release (CR)/reptation relaxation of the component chains, and the model parameters summarized in Table 3.1 were determined experimentally. (Redrawn, with permission, from Watanabe, H., Q. Chen, Y. Kawasaki, Y. Matsumiya, T. Inoue, and O. Urakawa. 2011. Entanglement dynamics in miscible polyisoprene/poly(p-fert-butylstyrene) blends. Macromolecules 44 1570-1584). 

Tj represents some relaxation time, of component i, which in terms of the tube model, is related to the idealised Doi-Edwards relaxation time for component i in a matrix of fixed obstacles, tde. by, Xi = (1/2)tde. Hence, in the double reptation model, the effect of constraint release is to half the relaxation time (if single exponential decay is assumed), from that predicted for a polymer in a fixed matrix. In the heterogeneous blends considered here, the tj are the tube survival times for chains of species i in an idealised environment, in which the chemical heterogeneity matches that of the blend, but all chains share the same relaxation time. That is, double reptation accounts for mutual effects in topological stress relaxation, but not for direct effects of local composition on the monomeric friction factors. The parameters of the double reptation model should be treated as phenomenological, to be determined from independent linear rheology experiments in the one phase region (see for example reference [61]). 

Garcia-Franco and Mead [142] proposed the use of the parameters of Eq. 5.65 to describe the behavior of polyethylenes prepared by means of anionic polymerization, gas-phase metallocene catalysis, and Ziegler-Natta catalysis. They reported that Eqs. 5.67 gave a good fit of their data and suggested that it is valid for all linear, flexible polymers with monomodal molecular weight distributions except in the terminal zone. They found that except in the terminal zone it provides a representation of the data that is similar to that given by the double-reptation model. They... 

Figure 6.15 Comparison of the predictions of the double reptation model (lines) to experimental data (symbols) for (a) the storage modulus G, and (b) the loss modulus G", for bidisperse polystyrenes (MW = 160,000 and 670,000) at 160 °C [35]. The volume fractions of the high molecular weight component ((j ) from right to left are 0.0,0.05,0.1,0.2, and 0.5, and 1.0, respectively.The parameter values are G 5 = 2 10 Pa and /f = 4.6 10 s/(mol) " The latter value, obtained by a best fit to the data for monodisperse samples, is almost identical to the value (K = 4.55 -10" s/(mol) ) obtained using Eq. 7.3 with =0.00375. Adapted from Pattamaprom and Larson [19].

Figure 6.17 (a) Comparison of the predictions of the dual constraint model (solid lines) and the double reptation model (broken lines) to experimental data (symbols) for the storage modulus, G, and the loss modulus, G", for monodisperse linear polystyrene (M = 363,000) at 150 °C. The parameter values are G 5 = 2 -10 Pa.and = 0.05 s,the latter value being obtained as a best fit. From this value of Tg,after multiplying it by the correction factor ofO.375 in footnote (g) of Table 7.1, the value K = 2.275 10" s/(mol) for the double reptation model is obtained from Eq. 7.3 (from Pattamaprom and Larson [19]).(b)The same as (a), except the sample is a polydisperse polystyrene M = 357,000 = 2.3) constructed from 11... 

Nobile and Cocchini [33] used the double reptation model to calculate the relaxation modulus, the zero-shear viscosity and the steady-state compliance for a given MWD. They compared three forms of the relaxation function for monodisperse systems the step function, the single integral, and the BSW. In the BSW model, they set the parameter j8 equal to 0.5, which gives /s° G equal to 1.8. The molecular weight data were fitted to a Gex function to facilitate the calculations (see Section 2.2.4 for a description of distribution functions). For the step function form of the relaxation function is given by Eq. 8.37. 

As a consequence, the proper evaluation of segment diffusion at long diffusion times turned out to be more complicated than often anticipated. However, even with flip-flop spin diffusion theoretically taken into account, it is not possible to fit the formulas predicted by the tube/reptation model to the experimental data with respect to both the time and molecular weight dependences in a consistent way and without assuming unrealistic parameters [12]. 

Figure 44 shows typical echo attenuation curves recorded with PEO in a solid PHEMA matrix [186]. The good coincidence of the theoretical curves calculated on the basis of Eqs. 76-79 for all accessible values of the experimental parameters k and t for different molecular weights [11] corroborates the validity of the tube/reptation model for linear polymers confined to arti-... 

A value of c equal to 0.3, previously used to describe FT selectivity data on Ru catalysts (4), was also chosen here to describe the behavior of cobalt catalysts. This equation for hydrocarbon diffusion in melts reflects the strong influence of molecular size in reptation and entanglement models of transport in such systems (IJ6). Our model also requires the input of intrinsic values for jSn (given by the asymptotic j8r), jSo, j8r, and j8s, measured independently. After such parameters are specified, the model yields a non-Flory carbon number distribution of increasingly paraffinic hydrocarbons that agrees well with our experimental observations (Fig. 16). 

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