Dual constraint model

Figure 6.14 Loss modulus data for bidisperse polybutadienes (MW = 70,900 and 335,000) at 30 °C on a log-linear scale (from Rubinstein and Colby [26]).The volume fractionsofthe high molecular weight component (X ) from right to left are 0.0,0.638,0.768,0.882,and 1.0, respectively.The lines are predictions of the "dual constraint model" Adapted from Pattamaprom etal. [10].

Figure 6.16 Comparison of the predictions of the dual constraint model (lines) to experimental data (symbols) in Fig. 6.15. The parameter values are = 2 10 Pa and % = 0.01 s. The latter value gave the best fit to the data for the monodisperse samples. Adapted from Patta-maprom 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... 

We also note that similar predictions can be obtained for other polydisperse linear melts, including polyethylene see, for example. Figs. 7.13 and 9.5a. And other, related models appear to give predictions roughly equivalent to those of the dual constraint model. Of particular note is the work of Marin and coworkers [20,21 ], whose model is described in more detail in Chapter 8, and the double reptation model with a more complex kernel relaxation function F(t) [29]. 

Figure 7.11 shows predictions of C and G" from the dual constraint model compared with experimental data for nearly monodisperse hydrogenated polybutadiene [28,25] of molecular weights 4.39,6.02,7.15,11.9,17.4,20.2, and 35.9 10. In this case, the value of the parameter % = 7 10 s was obtained by fitting the model to the data. For hydrogenated 1,4-polybuta-diene, the value of the plateau modulus, = 2.31 10 Pa [28], is a little lower than that for... 

As shown in Fig. 7.13a, the tube model (in this case the dual constraint model) is also reasonably successful in predicting the linear viscoelastic response of polydisperse polyethylene melts, as long as they do not contain long side branches. The rheological properties are extremely... 

Pattamaprom, C., Siriwat, A., Larson, R. G. Determination of the molecular weight distributions of polymers from their rheological properties using the dual constraint model. (2005) in preparation... 

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]. 

Figure 9.12 Predictions of the dual constraint model (lines) compared to experimental data (symbols) for (a) the storage modulus, and (b) the loss modulus, for bimodal star polyisoprenes M = 2.8 10 /1.44 10 ) at the reference temperature 25 °C (from Blotti4re etal., [22]).The volume fraction of the high molecular weight component (x,) from right to left are 0.0,0.2, 0.5, and 0.8, respectively. The model parameters were obtained from fits to data for linear polyisoprenes at the same temperature = 1.25-10 Pa and = 2 10" s. From Patta-mapromefa/. [48].

Wasserman and Graessley [30] showed that if two high-molecular-weight components with molecular weights of around 3.8 and 4.5 million are added at a total concentration of only 1% to the mixture of 11 components described above, then the terminal region of the G curve is measurably altered compare G for the 11-component mixture Ml with the 13-component mixture M2 in Fig. 6.18. Very importantly, the G curves predicted for both Ml and M2 by the dual constraint theory match the experimental data very well. Similar sensitivity to high molecular weight components is shown by the double reptation model [27]. 

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. 

If the nature of spacetime involves the interference of dual wave fronts of two dimensions, then there are two wave fronts, each of two dimensions, that constructively and destructively interfere, but that are determined by the same symmetry space. Gravitation can be described by the set of diffeomorphisms of a two-dimensional surface and SU(2) x SU(2) x SU(3) plus gravity involving a space of nine dimensions. The additional dimensions to spacetime are purely virtual in nature. A field dual to QCD would require a large space of 12 dimensions, and an additional constraint is required in order for this theory to satisfy current models of supergravity. 

In a number of papers [16-23,25], the discrete variant of the PCAO-model is considered the chain is modeled by a random walk on the lattice with spacing a and the topological constraints are placed on the dual lattice with period c. 

At the tip contact, the motion and constraint vector spaces may be defined using the general joint model discussed in Section 2.3. For convenience, we will assume that the two dual bases used to partitiai the spatial acceleration and force vectors at the tip. 

The assumption of the Energetic Kinetic Theory (EKT) is that under nonisothermal conditions the total free energy remains that of the equilibrium at the corresponding temperature (i.e., RT In ( 2/ 7) in this example), and that there is a transfer of populations between the/and b types of bonds to render this constraint feasible. The kinetic duality between the b and/units within a closed but split statistical ensemble is the reason for the nomenclature "Split Dual Kinetics," but since the free energy remains as the driving force to determine the tme kinetic expression, we favor the expression "Energetic Kinetic Theory" to describe our new model. It will become apparent shordy that the latter expression has a more general sense. The set of equations for the new statistics... 

The optimization problem from Eq. [20] represents the minimization of a quadratic function under linear constraints (quadratic programming), a problem studied extensively in optimization theory. Details on quadratic programming can be found in almost any textbook on numerical optimization, and efficient implementations exist in many software libraries. However, Eq. [20] does not represent the actual optimization problem that is solved to determine the OSH. Based on the use of a Lagrange function, Eq. [20] is transformed into its dual formulation. All SVM models (linear and nonlinear, classification and regression) are solved for the dual formulation, which has important advantages over the primal formulation (Eq. [20]). The dual problem can be easily generalized to linearly nonseparable learning data and to nonlinear support vector machines. 

To demonstrate the validity of the ReaxFF potential for modeling catalytic V/O/C/H interactions, the authors simulated the oxidative dehydrogenation of methanol over the 205(00 ) surface. They conducted a 250 ps NVT-MD simulation of a three-layer oxide slab surrounded by 30 gas-phase methanol molecules in a 20 x 20 x 20 periodic box. A dual temperature constraint was... 

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