Cox-Merz relation

PLS nanoeomposites always exhibit significant deviations from the empirical Cox-Merz relation, while all neat polymers obey the empirical relation, which requires that, for 7 = w, the viscoelastic data obeys the relationship 77(7) = r] uj). Two possible reasons may be offered for the deviation from the Cox-Merz relation in the case of nanoeomposites. First, this rule is only applicable for homogeneous systems like homo-polymer melts, but nanoeomposites are heterogeneous systems. Second, the slrueture formation is different when nanoeomposites are subjected to dynamic oscillatory shear versus steady shear measurements. 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

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 ... 

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. 

For fluid dynamics and heat transfer investigations related to food processing, the necessaiy tja data may be obtained from models developed for i] data using relationships based on the Cox-Merz mle (Rao, 1992 Rao and Cooley, 1992). These results were described in detail elsewhere (Yang, 1997 Yang and Rao, 1998, 1998b, 1998c) and will be reviewed in brief here. 

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. 

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. 

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. 

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. 

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 ... 

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 ... 

Linear viscoelastic measurements can provide surprising insight into the steady shear behavior of many polymer melts through an empiricism known as the Cox-Merz rule. According to this relation, a plot of t] = [( ) + versus fre-... 

The complex viscosity can be related to the steady shear viscosity (fj) via the empirical Cox-Merz rule, which states that the shear rate-dependence of the steady-state viscosity, is equivalent to the frequency dependence of the complex viscosity. 

We note that this is like to the Cox-Merz relationship in that it relates a nonlinear viscometric function to linear behavior. 

This relation is known as the Cox-Merz rule (Cox and Merz, 1958). When i](y) is not available, the Cox-Merz rule serves as a usefiil way to obtain t](y), especially for linear polymers (i.e., those without branching). When dealing with filled polymers, polymer blends, fiber suspensions, or highly branched polymers, the Cox-Merz rule may not hold. 

An alternative to the Cox-Merz rule is Gleissele s mirror relation (Gleissele, 1980) ... 

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