The Cox-Merz Rule

Oscillatory shear measurements provide more precise data than capillary rheometry and require smaller samples. In addition, capillary viscometers are hmited to use at moderate to high shear rates and do not provide accurate results at shear rates below 10 s for most materials. It would thus be useful to be able to estimate the viscosity curve using data from oscillatory shear measurements. Based on their data for two polystyrenes, Cox and Merz [133] reported that the curve of apparent viscosity versus shear rate, as determined using a capillary viscometer, lay very close to the curve of complex viscosity versus frequency. While the apparent viscosity measured in a capillary viscometer is usually defined as the apparent wall shear stress divided by the apparent wall shear rate, it is likely that the entrance pressure drop was nearly negligible for the polystyrenes studied by Cox and Merz. Therefore, what they observed can be expressed as follows  

This implies that to obtain the curve of viscosity versus shear rate from that of complex viscosity versus frequency, the latter curve should be shifted to the right on the frequency axis as indicated by the Rabinowich correction, which converts the apparent wall shear rate to the true one. If we use the simpler Schtlmmer approximation of this correction [9, p. 304,128] we find that what Cox and Merz observed can be expressed as follows the viscosity at a shear rate of 0.79co is approximately equal to the complex viscosity at the frequency O). 

However, in practice, the Cox-Merz rule is nearly always interpreted to mean  

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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 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 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... 

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 ... 

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. 

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. 

Steady Shear Viscosity and Dynamic Viscosity Data Neat HDPE rheology data fairly well correspond to each other when obtained by both capillary and rotational rheometers. This actually means that HDPE melt obeys the Cox-Merz rule [26]. The... 

Figure 17.7 shows that shear rate and frequency are almost identical for neat polymers that is, the Cox-Merz rule (see above) is valid for these systems. The power-law index n, calculated from the slope of the two viscosity curves in Figure 17.7, would be the same (excluding data at very low frequencies, at so-called Newtonian plateau). But for filled plastics this rule is not applicable, and one will get different values of the power-law index from the two viscosity curves. In other words, one will obtain much higher viscosity data from a parallel plate rheometer. 

Typically, WPC based on polypropylene and polyethylene show deviation from the Cox-Merz rule. This is due to the different nature of flow. Capillary flow is a pressure-driven flow, including entrance and exit effects, wall slip, friction in the barrel, and orientation effects. Parallel-plate flow is pure drag shear flow, in which particle-particle and matrix-particle interactions result in higher viscosities for filled polymers. In other words, a straightforward question is a 100-fold increase in shear rate and 100-fold increase in frequency result in the same effects the answer would be yes for neat polymers, and no for wood-filled composites. 

Small amplitude dynamic viscoelastic properties of apple butter, mustard, table margarine, and mayonnaise were compared to their respective properties in steady shear flow in the range of shear rates and frequencies of 0.1 to 100 sec" (Bistany and Kokini, 1983). Comparisons of dynamic and steady viscosities showed that dynamic viscosities (tj ) are much greater than steady viscosities (17). Consequently, the Cox-Merz rule is not obeyed (Bistany and Kokini, 1983). This phenomenon can be explained by a signifi-... 

The Cox-Merz rule states that the shear-rate dependence of the steady-state viscosity equals the frequency dependence of the complex viscosity,... 

In ordinary polymer solutions in which polymers interact by nonspecific van der Waals type potentials, it is known that the phenomenological relation = r]st y) (the Cox-Merz rule) often holds [35]. Disagreement between the complex viscosity and the stationary viscosity at finite frequencies is one of the common features of the hydrogen-bonded networks. 

Fig. 9.7 Deviation from the Cox-Merz rule. The frequency-dependent linear viscosity r](co) (solid lines) is compared to the nonlinear stationary viscosity rjsiiy) (broken lines). The quadratic /3(r) is assumed. The parameter /3q is varied from curve to curve. (Reprinted with permission from Ref. [17].)...

The rheology of blends of linear and branched PLA architectures has also been comprehensively investigated [42, 44]. For linear architectures, the Cox-Merz rule relating complex viscosity to shear viscosity is valid for a large range of shear rates and frequencies. The branched architecture deviates from the Cox-Merz equality and blends show intermediate behavior. Both the zero shear viscosity and the elasticity (as measured by the recoverable shear compliance) increase with increasing branched content. For the linear chain, the compliance is independent of temperature, but this behavior is apparently lost for the branched and blended materials. These authors use the Carreau-Ya-suda model. Equation 10.29, to describe the viscosity shear rate dependence of both linear and branched PLAs and their blends ... 

The importance of t] is that many polymers obey the Cox-Merz rule,05) which states that T] (a)) = T](7) when co (rad/sec) = y (sec). Thus, dynamic shear measurements may be useful for estimating steady shear viscosity [t] (7)] for crosslinked polymer systems since it has the following correlations06) ... 

Dynamic viscosity data can be used to approximate the steady shear viscosity by taking advantage of an empirical relationship known as the Cox-Merz rule (Cox and Merz 1958), which relates the magnitude of the complex viscosity at frequency co to the steady shear viscosity at a shear rate y equal to co ... 

Dynamic viscosity measurements generally are easier to perform over a wide range of frequencies than are steady shear measurements over a wide range of shear rates, while steady shear viscosity is of more value in the analysis of polymer processing. Thus, the Cox-Merz rule is frequently assumed and t data are used in place of steady shear viscosity data. 

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