Modified Cox-Merz rule

Yang and Rao (1998a) used a modified Cox-Merz rule (Equation 8.46) to determine the parameters relating the dynamic and steady shear data, and a TR model for parent viscosity was derived. Equation 8.47. 

Figure 8-9 Modified Cox-Merz Rule for a Gelatinized Starch Dispersion (Yang and Rao, 1998b).

The relationship between steady-shear viscosity and dynamic-shear viscosity is also a common fundamental rheological relationship to be examined. The Cox-Merz empirical rule (Cox, 1958) showed for most materials that the steady-shear-viscosity-shear-rate relationship was numerically identical to the dynamic-viscosity-frequency profile, or r] y ) = r] m). Subsequently, modified Cox-Merz rules have been developed for more complex systems (Gleissle and Hochstein, 2003, Doraiswamy et al., 1991). For example Doriswamy et al. (1991) have shown that a modified Cox-Merz relationship holds for filled polymer systems for which r](y ) = t] (yco), where y is the strain amplitude in dynamic shear. 

Figure 4.7. Comparison of the Cox-Merz rule and the modified Cox-Merz rule for a highly filled epoxy-novolac moulding sample used for computer-chip encapsulation.

Figure 4.7 shows the steady- and dynamic-viscosity profiles as functions of shear rate for a filled reactive epoxy-resin moulding compound. Here, interestingly, the Cox-Merz rule provides a better correlation than does the modified Cox-Merz rule. 

The complex viscosity can be related to the steady-shear viscosity rf) via the empirical Cox-Merz rule, which notes the equivalence of steady-shear and dynamic-shear viscosities at given shearing rates ri y) = rj (co). The Cox-Merz rule has been confirmed to apply at low rates by Sundstrom and Burkett (1981) for a diallyl phthalate resin and by Pahl and Hesekamp (1993) for a filled epoxy resin. Malkin and Kulichikin (1991) state that for highly filled polymer systems the validity of the Cox-Merz rule is doubtful due to the strain dependence at very low strains and that the material may partially fracture. However, Doraiswamy et al. (1991) discussed a modified Cox-Merz rule for suspensions and yield-stress fluids that equates the steady viscosity with the complex viscosity at a modified shear rate dependent on the strain, ri(y) = rj yrap3), where y i is the maximum strain. This equation has been utilised by Nguyen (1993) and Peters et al. (1993) for the chemorheology of highly filled epoxy-resin systems. 

The measurement of yield stress at low shear rates may be necessary for highly filled resins. Doraiswamy et al. (1991) developed the modified Cox-Merz rule and a viscosity model for concentrated suspensions and other materials that exhibit yield stresses. Barnes and Camali (1990) measured yield stress in a Carboxymethylcellulose (CMC) solution and a clay suspension via the use of a vane rheometer, which is treated as a cylindrical bob to monitor steady-shear stress as a function of shear rate. The effects of yield stresses on the rheology of filled polymer systems have been discussed in detail by Metzner (1985) and Malkin and Kulichikin (1991). The appearance of yield stresses in filled thermosets has not been studied extensively. A summary of yield-stress measurements is included in Table 4.6. 

The authors also note the application of the modified Cox-Merz rule relating dynamic and steady viscosities, r] y ) = r] j (D). 

Han et al (1997) examined the chemorheology of a highly filled epoxy-resin moulding compound that is characterized by a modifed slit rheometer. Results show that a modified Cox-Merz rule relating dynamic and steady viscosities is established, >7(7 ) = (Tm )-Also the material was shown to exhibit a yield stress at low shear rates and power-law behaviour at higher shear rates. The temperature dependence of the viscosity is well predicted by a WLF model, and the cure effects are described by the Macosko relation. 

Spoelstra et al (1996) examined the chemorheology of a highly filled epoxyresin moulding compound and showed that the modified Cox-Merz rule (as defined in... 

However, while the Cox-Merz rule is valid for general unfilled and low concentration filled systems, it was discovered that the rule fails to hold for concentrated suspensions. It is important at this point to recognize that MIM or PIM feedstock is essentially a concentrated suspension at elevated temperature. A modified Cox-Merz rule was developed for cmicentrated suspensions by Doraiswamy et a [20]. The modified rule stated that the complex viscosity versus shear rate amplitude plot is equal to the corresponding viscosity versus shear rate plot. That is... 

Recent literature has shown that the modified Cox-Merz rule could fit e rimental data very well [20-22], Isayev and Fan [22] of University of Akron investigated both steady and oscillatory shear flow behavior of silicon-polypropylene ceramic compoimd. They reported that steady shear measurement with either parallel plate or capillary rheometers posed a... 

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