Cross model

To give Newtonian regions at both low and high shear rates. Cross (i%S) proposed 

Typically % so when Uho) = y is very small, goes to %. At intermediate y the Cross model has a power law region 

At very high shear rates the right-hand side of eq. 2.4.15 becomes very small, and t) goes to the high shear rate Newtonian limit,.  

The parameter t]0 is the limiting viscosity at low co. The reciprocal (t-1) of relaxation time % marks the midpoint co for the transition from a power-law exponent of 0 at low co to - 1 at high co. Interpretation of these low strain amplitude parameters in nonlinear fabrication shear flows is enabled by the Cox-Merz rule [43]. 

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The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

Polypropylene molecules repeatedly fold upon themselves to form lamellae, the sizes of which ate a function of the crystallisa tion conditions. Higher degrees of order are obtained upon formation of crystalline aggregates, or spheruHtes. The presence of a central crystallisation nucleus from which the lamellae radiate is clearly evident in these stmctures. Observations using cross-polarized light illustrates the characteristic Maltese cross model (Fig. 2b). The optical and mechanical properties ate a function of the size and number of spheruHtes and can be modified by nucleating agents. Crystallinity can also be inferred from thermal analysis (28) and density measurements (29). 

Fig. 4. Graphic representations (viscosity vs shear rate) of Cross model with different values for d.

Zeno paradox. On the other hand, recovering these interferences from a single path leads to excessive correlation, as evidenced by the highly oscillatory results obtained with TSH for Tully s third, extended coupling with reflection, model. This is remedied effortlessly in FMS, and one may speculate that FMS will tend to the opposite behavior Interferences that are truly present will tend to be damped if insufficient basis functions are available. This is probably preferable to the behavior seen in TSH, where there is a tendency to accentuate phase interferences and it is often unclear whether the interference effects are treated correctly. This last point can be seen in the results of the second, dual avoided crossing, model, where the TSH results exhibit oscillation, but with the wrong structure at low energies. The correct behavior can be reproduced by the FMS calculations with only ten basis functions [38]. 

Under conditions of steady fully developed flow, molten polymers are shear thinning over many orders of magnitude of the shear rate. Like many other materials, they exhibit Newtonian behaviour at very low shear rates however, they also have Newtonian behaviour at very high shear rates as shown in Figure 1.20. The term pseudoplastic is used to describe this type of behaviour. Unfortunately, the same term is frequently used for shear thinning behaviour, that is the falling viscosity part of the full curve for a pseudoplastic material. The whole flow curve can be represented by the Cross model [Cross (1965)] ... 

Anderssen, E., Dyrstad, K., Westad, F., Martens, H. Chemom. Intell. Lab. Syst. 84, 2006, 69-74. Reducing over-optimism in variable selection by cross-model validation. 

The disadvantage of the power law model is that it cannot predict the viscosity in the zero-shear viscosity plateau. When the zero-shear viscosity plateau is included, a nonlinear model must be specified with additional fitting parameters. A convenient model that includes the zero-shear viscosity and utilizes an additional parameter is the Cross model [30] ... 

The forbidden retro-[ls -I- 2s]-cycloaddition can now be treated using a simple curve-crossing model analagous to the Marcus-Hush theory of electron-transfer [11]. The ground state at the quadricyclane-like geometry is the... 

L. Gidskehang, E. Anderssen and B.K. Alsberg, Cross model validation and optimisation of bihnear regression models, Chemom. Intell. Lab. Syst., 93, 1-10 (2008). 

Within a local complex potential curve crossing model, the cross section for the simple DEA reaction e + AB AB A + B, where AB is a diatomic molecule, may be expressed as [18]... 

This avoided crossing model for describing a reaction barrier is striking in its generality. It has been applied in a detailed theoretical analysis to reaction... 

Carrean-Yasnda Model (full curve) Cross Model (full curve) For portions of the Cross equation that predict portions of the complete flow curve h-ih, 1 ho-1 . [1 + (k1y)2P 2 110-11 =(KiT Tl-11 1 oo... 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

The models used are typically either the Cross model or the Carreau-Yasuda model (UNIT Hl.l), if a complete curve is generated. A complete curve has both plateaus present (zero and infinite shear see Figure HI.1.4). 

If a logarithmic ramp is performed, then the data should not be fit with linear models (unit m.i). These data should be plotted as viscosity versus shear rate on logarithmic axes and the Carreau-Yasuda or Cross models (or subsets) should be used instead. It is unlikely that the zero-shear plateau will be seen in these types of tests. For a complete flow curve, the equilibrium tests described in Basic Protocol 2 should be used. 

Despite our natural satisfaction with this explanation, we noted that it nevertheless does not provide a clear physical mechanism for the effect. For example, it was not evident why the frequency exaltation is state selective, i.e., it is not observed in any other state, and the frequency of all other modes are reduced upon excitation. It was deemed necessary, therefore, to articulate the avoided crossing model, derive a clear physical origin of the frequency exaltation of the Kekule mode, its state, and mode specificity, and establish the connection of the phenomenon to the jr-distortivity in the ground state. This was achieved in 1996.209... 

Phulkar S, Rao BSM, Schuchmann H-P, von Sonntag C (1990) Radiolysis of tertiary butyl hydroperoxide in agueous solution. Reductive cleavage by the solvated electron, the hydrogen atom, and, in particular, the superoxide radical anion. Z Naturforsch 45b 1425-1432 Pross A, Yamataka H, Nagase S (1991) Reactivity in radical abstraction reactions - application of the curve crossing model. J Phys Org Chem 4 135-140 Rao PS, Flayon E (1975) Reaction of hydroxyl radicals with oligopeptides in aqueous solutions. A pulse radiolysis study. J Phys Chem 79 109-115... 

The polymer melt viscosity was simulated using a cross-model given by... 

We acknowledge Dr. Dongyun Ren, who, with the help of Mr. B. J. Jeong and Drs. J. Guo and Linjie Zhu, evaluated, via mathematical regression, the Power Law model, Carreau model, and Cross model parameters. 

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