Cox—Merz rule

By comparing the viscosity of polystyrene samples obtained by oscillatory measurements and in steady-state conditions, Cox and Merz (50) found that 

Diluents and plasticizers in polymeric systems increase the steady-state compliance and decrease the zero shear rate viscosity. These two combined opposing effects give rise to a diminution in the value of Hence the critical value of the shear rate in dilute systems is shifted to higher values as the dilution increases (see Fig. 13.28). 

Comnarative plots showing the variation of the viscosity and the nn.nL viscosity, Ipl, as functions of the shear rate, k, and the 

The flow behavior of blends depends on the compatibility of their components. The viscosity of a compatible blend, for example polystyrene and polyoxyethylene, is the average of the viscosity of the components (52). Blends made up of high viscosity incompatible polymers may have lower viscosity than either component (53) (see Fig. 13.29). This effect may be the 

Fillers usually enhance the viscosity of the polymer melts. The viscosity of these systems depends not only on the characteristics of the melt but also to a great extent on the nature and volume of the filler. Several empirical relationships have been proposed between the viscosity of the melt and that of the melt filled with noninteractive particles for Newtonian flow. Among them, the empirical equation of Maron and Pierce (55) stands out  

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There is a relationship that is used to cross between time and shear rate dependence regimes and that is the Cox-Merz rule.5 The dynamic viscosity when plotted as a function of frequency, has a similar... 

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]. 

The extended Cox-Merz rule [49] can be successfully applied for HD and LLD. This nile states that the viscosity and elasticity coefficients for oscillatory and steady state shear flows are related, according to ... 

Relationship Among Rheological Parameters Cox Merz Rule... 

The superimposition of the shear rate dependence of steady shear viscosity, that is, t]a(o)), and of the frequency dependence of the complex viscosity, that is, i ( >), at equal values of frequency and shear rate was first reported by Cox and Merz (1958) for polystyrene samples, and is known as the Cox-Merz rule. 

The Cox-Merz rule is an empirical correlation that has been confirmed experimentally for several synthetic polymers (Ferry, 1980), and for solutions of several random-coil polysaccharides (Morris, 1981). However, many exceptions to the Cox-Merz rule have also been found for synthetic polymers (Matsumoto et al., 1975 Kulicke and Porter, 1980) as well as for biopolymer systems (Morris etal., 1981 Milts and Kokini, 1984 Yang and Rao, 1998) and foods (Bistany and Kokini, 1983 Rao and Cooley, 1992). Relationships between the parameters were determined for equal magnitudes of the shear rate, y, and the frequency, co (Bistany and Kokini, 1983). Based on... 

Deviation from Cox-Merz rule appears to be an indication of structural heterogeneity in a food. For example, significant deviations from Cox-Merz rule were found in dispersed systems, such as tomato concentrates (Rao and Cooley, 1992) and cross-linked waxy maize starch dispersions (da Silva et al., 1997 Tattiyakul and Rao, 2000). In contrast to the observation on several foods of dispersed nature, the Cox-Merz rule was found to be applicable to fluids with homogeneous structure, such as dispersions of guar gum (Mills and Kokini, 1984) and locust bean gum (Lopes da Silva et al., 1993). 

When departures from the Cox-Merz rule are attributed to structure decay in the case of steady shear, the complex viscosity is usually larger than the steady viscosity (Mills and Kokini, 1984). Notwithstanding this feature, the relation between magnitudes of T a and T can be dependent on the strain amplitude used (Lopes da Silva et al., 1993). Doraiswamy et al. (1991) presented theoretical treatment for data on suspensions of synthetic polymers. They suggested that by using effective shear rates, the Cox-Merz rule can be applied to products exhibiting yield stresses. The shift factors discussed above can be used to calculate effective shear rates. 

Doraiswamy, D., Mujumdar, A. N., Tsao, I., Beris, A. N., Danforth, S. C., and Metzner, A. B. 1991. The Cox-Merz rule extended a theological model for concentrated suspensions and other materials widi a yield stress. J. Rheol. 35 647-685. 

The Cox-Merz rule (Cox and Merz, 1958) is useful in predicting steady shear viscosity from complex viscosity and vice versa ... 

In addition, for dispersed polymer systems at small strain amplitudes, Matsumoto et al. (1975) observed a pronounced departure from the Cox-Merz rule, with r]a much smaller than i) at the same rate of shear and angular frequency. They also observed that rf decreased with increase in the strain amplitude such that for high strain amplitudes rj was smaller than a-... 

The Cox-Merz rule (Equation 4.51) was not applicable to gelatinized CWM starch dispersions in the range of concentrations studied in that the complex viscosity (ri ) was higher than the flow viscosity over the range of frequencies and shear rates studied (Figure 4-31). 

Figure 4-31 The Cox-Merz Rule was not to Applicable to Gelatinized Cross-Linked Waxy Maize Starch Dispersions in the Range of Concentrations Studied in that the Complex Viscosity (t] ) was Higher than the Flow Viscosity over the Range of Frequencies and Shear Rates Studied.

With automated rheometers it is relatively easy to obtain dynamic shear data than steady shear data. For this reason, the interrelationship between co and t) on one hand and y and t a on the other is of interest. The Cox-Merz rule, that is, equal magnitudes of i)a and if at equal values of y and co, respectively (Equation 5.17), was obeyed by several synthetic and biopolymer dispersions (Lopes da Silva and Rao, 1992). 

Because of the wide variability in tomato pastes due to different cultivars and processing methods, the constants in the structure-based correlations would not be universally applicable. However, the correlations can be used to obtain order of magnitude estimates of the rheological properties of tomato pastes without extensive and expensive experimentation. The various constants are valid within the ranges of the variables from which they were derived. More importantly, it is our hope that studies will be conducted to make use of the concepts presented here wi th respect to the weak gel nature of tomato pastes, shift factors for Cox-Merz rule, and structure based (d>s and = 1 — s) relationships for correlating the rheological parameters. 

Because of the existence of yield stress as well as time-dependent rheological behavior of mayonnaise, it would seem reasonable to expect that traditional relationships between steady shear properties on one hand and small amplitude dynamic properties on the other that were found for polymeric liquids will not hold for mayonnaise. Bistany and Kokini (1983) showed that the Cox-Merz rule and other relationships at low shear rates and frequencies did not hold for mayonnaise and... 

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).

A plot of viscosity versus shear rate for a model HUER polymer is shown in Fig. 5-20, and compared to the dynamic viscosity versus frequency. Note that the Cox-Merz rule (see Section 1.3.1.5) fails in that at the frequency (o> 1 sec ) where the dynamic viscosity-... 

The relationship between steady shear and complex viscosity is fairly well established. Cox and Merz " found that an empirical relationship exists between complex viscosity and steady shear viscosity when the shear rates are the same. The Cox-Merz rule is stated as follows ... 

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. 

Ao, Ai, A2, constants Cl, C2, constants To, reference temperature Assumes Cox-Merz rule is valid... 

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. 

Isothermal dynamic-shear sweep 7 — tliy > T) low to high / from Cox-Merz rule... 

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. 

An inherent assumption when using the above dynamic techniques is that the complex viscosity gives a good representation of the steady-shear viscosity during the curing reaction. This has been validated for many systems. However, care should be taken when relating the effects of cure on complex viscosity to the processing viscosity in other words the Cox Merz rule or a similar relationship must be validated. 

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. 

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