Recoverable shear compliance

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

Note 5 Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance. The definitions of such terms are outside the scope of this document. 

If a system is to possess elasticity, it must possess a physical mechanism for storing energy. In flow, droplet distortion causes an increase in surface free energy (aAA) which is released on cessation of flow, manifesting as, for example, recoverable shear compliance. The magnitude of the (dimensionless) droplet distortion is proportional to the Weber number, We, which is the ratio of the deforming stress (n y) to the restoring Laplace stress (4(j/dd). Thus, the drop Weber number based on drop radius is defined as ... 

FIGURE 5.17 Logarithmic presentation of the recoverable shear compliance, Jr(t), of Epon 100 IF as a function of the logarithm of time I at nine temperatures as indicated. Dramatic loss of long-time viscoelastic mechanisms is evident when temperature is decreased toward Tg. 

With all of the viscoelastic functions it is important to note the limiting values or forms which are qualitatively independent of the molecular structure. For a viscoelastic liquid, lini, /(f) = Jg, lim, /(f) = tlr], and lim,, Jr t) = J t) -thr] = Jg + Jd = /j.The last Umiting value Js is called the steady-state recoverable shear compliance. It is the maximum recoverable strain per unit stress, which reflects the maximum configurational orientation achievable at the present stress. 

In the behavior of polymeric liquids two quantities are important. These are steady-state recoverable shear compliance, (as shown above) and steady-state viscosity at zero shear rate, r o- These quantities are related ... 

At low shear rates, polymeric liquid properties are characterized by two constitutive parameters zero shear rate viscosity t]o and recoverable shear compliance Jq, which indicates fluid elasticity. At higher shear strain rates, rheological behavior is measured with a viscometer. Extensional strain viscosity, associated with extensional flow, occurs with film extrusion. 

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 dynamic moduli G (co) and G"(co) are linked to G(t) through Eq. (3.7). From those expressions, the zero-shear viscosity and recoverable shear compliance can be obtained from the low-frequency limiting behavior through Eqs. (3.10) and (3.11) ... 

For a Newtonian low molar mass liquid, knowledge of the viscosity is fully sufficient for the calculation of flow patterns. Is this also true for polymeric liquids The answer is no under all possible circumstances. Simple situations are encountered for example in dynamical tests within the limit of low frequencies or for slow steady state shears and even in these cases, one has to include one more material parameter in the description. This is the recoverable shear compliance , usually denoted and it specifies the amount of recoil observed in a creep recovery experiment subsequent to the unloading. Jg relates to the elastic and anelastic parts in the deformation and has to be accounted for in all calculations. Experiments show that, at first, for M < Me, Jg increases linearly with the molecular weight and then reaches a constant value which essentially agrees with the plateau value of the shear compliance. 

At higher strain rates even more complications arise. There the viscosity is no longer constant and shows a decrease with increasing rate, commonly called shear-thinning . We will discuss this effect and related phenomena in chapter 7, when dealing with non-linear behavior. In this section, the focus is on the limiting properties at low shear rates, as expressed by the zero shear rate viscosity , ryo, and the recoverable shear compliance at zero shear rate,... 

The two material parameters which characterize polymeric fluids at low strain rates, the viscosity, j, and the recoverable shear compliance, Je, can be directly determined, rj follows from the measurement of the torque under steady state conditions, Jq shows up in the reverse angular displacement subsequent to an unloading, caused by the retraction of the melt. From the discussion of the properties of rubbers we know already that simple shear is associated with the building-up of normal stresses. More specifically, one finds a non-vanishing... 

The agreement implies a relationship between the zero shear rate value of the recoverable shear compliance, following from 7 as... 

As we can see, both are independent of the strain rate 7. Hence, as a first conclusion. Lodge s equation of state cannot describe the shear thinning phenomenon. Equation (7.147) is in fact identical with Eq. (5.107) derived in the framework of linear response theory. The new result contributed by Lodge s formula is the expression Eq. (7.148) for the primary normal stress difference. It is interesting to note that the right-hand side of this equation has already appeared in Eq. (5.108) of the linear theory, formulating the relationship between G t) and the recoverable shear compliance. If we take the latter equation, we realize that the three basic parameters of the Lodge s rubber-like liquid, rjo, and 1,0 are indeed related, by... 

Figure 6.10 Creep compliance (/) as a function time (t). Calculation of zero-shear-rate viscosity and steady-state recoverable shear compliance.

Flow properties are very strongly dependent on molecular architecture, i.e. molar mass and chain branching. Figures 6.13 and 6.14 illustrate the effect of molar mass on the zero-shear-rate viscosity (//q) and on the steady-state recoverable shear compliance. 

Figure 6.14 The logarithm of the steady-state recoverable shear compliance plotted versus the logarithm of molar mass. Schematic curve.

The Rouse theory is clearly not applicable to polymer melts of a molar mass greater than (M ) for which chain entanglement plays an important role. This is obvious from a comparison of eqs (6.40)-(6.42) and experimental data (Figs 6.13 and 6.14) and from the basic assumptions made. However, for unentangled melts, i.e. melts of a molar mass less than (M ), both the zero-shear-rate viscosity and recoverable shear compliance have the same molar mass dependence as was found experimentally (Figs 6.13 and 6.14). The Rouse model does not predict any shear-rate dependence of the shear viscosity, in contradiction to experimental data. 

---