Modified power law

The release mechanism is governed by a modified power law which takes into account an initial release phase that differs from the main release phase 5... 

It should be noted that the effects of fillers may be incorporated into the cure and shear-rate effects. The main forms of combined-effects model consist of WLF, power-law or Carreau shear effects, Arrhenius or WLF thermal effects and molecular, conversion or empirical cure effects. Nguyen (1993) and Peters et al. (1993) used a modified Cox-Merz relationship to propose a modified power-law model for highly filled epoxy-resin systems. Nguyen (1993) also questions the validity of the separability of thermal and cure effects in the derivation of combined models. 

Kuroki et al (1999) examined the chemorheology of an epoxy resin with 90 wt.% silica. Gelation was predicted by independence of tan, power-law behaviour in dynamic moduli and GPC. A modified power-law-Arrhenius chemorheology model fitted the data well. 

The modified power law models (eqs 7 and 8) have previously been less useful for rig site use as the constants K and n have to be obtained by trial-and-error methods. The use of rig site computers has overcome these difficulties. 

Analysis of the critical behavior of thermodynamic properties is complicated by the fact that simple asymptotic-power law behavior is restricted to temperatures very close to the critical point. Investigations of metals in this range are limited by the severe experimental difficulties. It is necessary, therefore, to consider not only the asymptotic critical behavior, but also the corrections needed to analyze the experimental data over a larger range. Wegner (1972) worked out an asymptotic series expansion for critical behavior. The translation of Wegner s expansion into the terminology of fluids shows that measured properties should obey modified power laws of the form... 

Columns I to K are prepared by inputting known data on the polymers for TG, E, and L2, whereas Columns N to Q are used to store model parametric values for the General Rheological model given by Eq. (6.4) (see Rg. 7.1b). Thus, columns N and O contain n and K values for the modified power-law model, whereas columns P and Q contain P and t) X MFI. Because the values of K and iv, x MFI are typically lai], it is better to use scientific notation through... 

In this work, the viscosity of the shear-thinning fluids is modeled by a modified power-law according to... 

In some polymer/rock systems, e= . For example, Cannella et al. 63 and Teeuw and Hesselink report good agreement between tie and n for xanthan biopolymers in porous media used in their studies. Cannella et al. report permeability reduction from biopolymer retention was less than a factor of 0.9. They were able to correlate the apparent viscosity with a modified power-law expression of the form given by... 

As a material-dependent expression relating mechanical properties, molecular structure and engineering design, one may consider the tensile relaxation modulus, E elCt), which characterizes the deformation response of a polymer through a broad time or temperature scale. The relaxation modulus for a given polymeric material may be produced by well-established testing procedures [6] and then be approximated by a five-parameter modified power law [7]. One form of this relationship is... 

Table I demonstrates the application of the five-parameter relationship in the Interaction Matrix format. After inserting the five modified power-law parameters along the top of the vertical columns, one may then proceed to list molecular and microstructural characteristics along the left side of the horizontal rows. The list of characteristics is fairly representative of measurable features common to polymer science and technology. Individual characteristics may not be mutually independent, and additions or refinements are likely to be used for specific cases. The intersecting columns and rows may then...

Molecular Characteris tics Symbol Modified Power Law Parameters ... 

The modified power law may then be expressed in terms of Ng, the chain stiffness index, for purposes of examining the dependence of the design variables upon this particular molecular parameter. 

Note the modified power law approximation for the relaxation modulus (1) can be deduced from the relaxation spectrum... 

It is concluded that the modified power laws for spin-lattice relaxation dispersion in the mesophase reflect a modified behavior of chain modes rather than collective fluctuations of ensembles of molecules in ordered domains. This conclusion is corroborated by the identical frequency dispersion of the spin-lattice relaxation times Ti and Tip in the laboratory and in the rotating frames, respectively. If the order in the PDFS mesophase would be of a nematic nature and the fluctuations causing dispersion region II consequently would be of the ODF type, the frequency dispersion of Tip should vanish while that of Ti is retained [22]. 

The polymer melt is modeled as a pmely viseous fluid, where the shear-rate- and temperature-dependent viscosity of the polymer melt is described via a modified power-law model ... 

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