Modified Cross model

Shen (1990) reviews the flow modelling of injection moulding of thermosets. The kinetic model was the Kamal model (as shown above in Equation (6.31)) and the chemoviscosity model is given by a modified WLF model for low shear mtes (as shown by Equation (6.35)) and a modified Cross model for high shear (as shown by Equation (6.36)), namely... 

Wan and Isayev (1996) examined a hybrid approach of control-volume finite-element and finite-difference modelling of injection moulding of rubber compounds. The effect of vulcanization on viscosity and yield stress during cavity filling is reported. On comparing two versions of the modified Cross viscosity models - with and without the effect of cure - the use of a viscosity model that accounts for the cure was found to improve the accuracy of the cavity-pressure-prediction models. When the modified Cross model was further extended to include the yield stress and was implemented in the simulation program a significant improvement in the prediction of cavity pressure was obtained in the case of low injection speed. 

The (much) modified Cross model [11] has seen increasing application in recent... 

In the melt flow curve (viscosity versus shear rate), a Newtonian plateau at very low shear rate is usually considered as the ZSV. For polymers of very low molecular weight, this ZSV can be obtained directly from the shear rheology experiment (Dealy and Wissburn 1990). However, this is really difficult to obtain for high molecular weight polymers as well for LCPs. From the experimental data presented in Fig. 4.5, it was impossible to determine the ZSV, since there was no distinct plateau detected at a lower shear rate up to 0.01 s . Therefore, a modified Cross model (4.6) was used to determine the zero shear viscosity of the LCPs. Data from low and high shear rates were combined to fit into the Cross model predictions (Fig. 4.5). The model prediction showed good agreement with experimental data. 

Fig. 4.5 Steady shear viscosity of LCP-A at (filled diamond) 350 °C and (filled square) 360 °C using modified Cross model, —) and steady shear experiment (Rahman 2013)...

Figure 15,8 shows fits of the modified Cross model to the data in Fig 15.6a. (Note the magnitudes of those viscosities ) The constants C and n are common... 

Modified-Cross model with Arrhenius temperature dependence is employed to describe the viscosity of polymer melt ... 

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). 

The novel L-lysine modified cross-linked chitosan resin containing imine linkers [15] has been studied by the 13C CP MAS spectroscopy.31 However, as some ambiguous results were obtained and the resin was insoluble, instead of the resin a soluble model compound, derivative of chitosan oligosaccharide, has been obtained and investigated. 

Fig. 8.18. Modified Thornton-model for RF sputtered ZnO Al films according to Kluth et al. [96] Structural properties and etching behavior as function of deposition pressure and substrate temperature Ts. The SEM micrographs show cross-sections and the surface in the as-deposited state (left) and after etching in hydrochloric acid (right). The matrix of substrate temperature and deposition pressure contains structure zones according to the Thornton model [120,121]. Reprinted with permission from [96]...

M.A. Uddin, A.K.E Haque, K.R. Karim, A.K. Basak, F.B. Malik, Modified Kolbenstvedt model for the electron impact K-shell ionization cross sections of atoms and ions, Eur. Phys. J. D 37 (2006) 361. 

Both the Carreau and the Cross models can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given in Equation (2.16) was employed in the study of the rheological behavior of glass-filled polymers (Poslinski et al., 1988) ... 

Apparent viscosity-shear rate data of food polymer dispersions have been reviewed by Launay et al. (1986), Lopes da Silva and Rao (1992) and others. The general log versus log y curve, discussed in Chapter 2, has been used to characterize food polymer dispersions. For example, Lopes da Silva et al. (1992) found that both the modified Carreau and the Cross models, wherein the infinite shear viscosity was considered to be negligible, described the apparent viscosity-shear rate data of locustbean (LB)... 

Creep equations can be appropriately modified as models of intermediate and final stages of sintering. For simplicity, the matter transport during creep of a dense solid is considered first. For a pure single crystal solid with cubic structure, which is a rod with a cross section of length L. Normal stresses pa nre applied to the rod on the sides, as shown in Fig. 5.23a. It is assumed that self-diffusion within the crystal... 

This behavior is reminiscent of the e((o) for the localization-modified Drude model of Section 15.2.2. The zero crossing of s o)) at Wpj is therefore due to localized electrons. 

The modified Cross equation gives the temperature dependence of viscosity at finite shear rates. With other equations, the temperature dependence of all the constants would have to be known. Another approach has used an equation that is independent of any model for the flow curve. Since the viscosity is a function of temperature and shear stress or shear rate ... 

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