The Zero-Shear Viscosity

In Chapter 4 it was explained that the linear elastic behavior of molten polymers has a strong and detailed dependency on molecular structure. In this chapter, we will review what is known about how molecular structure affects linear viscoelastic properties such as the zero-shear viscosity, the steady-state compliance, and the storage and loss moduli. For linear polymers, linear properties are a rich source of information about molecular structure, rivaling more elaborate techniques such as GPC and NMR. Experiments in the linear regime can also provide information about long-chain branching but are insufficient by themselves and must be supplemented by nonlinear properties, particularly those describing the response to an extensional flow. The experimental techniques and material functions of nonlinear viscoelasticity are described in Chapter 10. 

The direct measurement of rjg is often practically impossible, especially for polydisperse samples. This is because standard melt rheometers are often unable to provide reliable data at sufficiently low shear rates to reach the region of Newtonian behavior. While this issue is discussed in Section 10.8, it is important to note here that the use of empirical equations for the viscosity function rj y) to extrapolate data is an unreliable procedure. Sometimes it is found that within a given family of polymers (same structure and shape of the MWD) a 

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The dynamic viscosity is related to the loss component of the shear modulus through the result 77= G"/co As co 0, the dynamic viscosity approaches the zero shear viscosity of an ordinary Uquid, 77 -... 

Polymer solutions are often characterized by their high viscosities compared to solutions of nonpolymeric solutes at similar mass concentrations. This is due to the mechanical entanglements formed between polymer chains. In fact, where entanglements dominate flow, the (zero-shear) viscosity of polymer melts and solutions varies with the 3.4 power of weight-average molecular weight. 

Here r/g is the "zero-shear" viscosity limit, is an activation... 

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to Mw whereas above this critical value r 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

Taking into account the relevance of the range of semi-dilute solutions (in which intermolecular interactions and entanglements are of increasing importance) for industrial applications, a more detailed picture of the interrelationships between the solution structure and the rheological properties of these solutions was needed. The nature of entanglements at concentrations above the critical value c leads to the viscoelastic properties observable in shear flow experiments. The viscous part of the flow behaviour of a polymer in solution is usually represented by the zero-shear viscosity, rj0, which depends on the con-... 

On the basis of a relationship between T Sp and the dimensionless product c [rj], simple three-term equations can be developed to correlate the zero-shear viscosity with the concentration and molar mass. 

The viscosity level in the range of the Newtonian viscosity r 0 of the flow curve can be determined on the basis of molecular models. For this, just a single point measurement in the zero-shear viscosity range is necessary, when applying the Mark-Houwink relationship. This zero-shear viscosity, q0, depends on the concentration and molar mass of the dissolved polymer for a given solvent, pressure, temperature, molar mass distribution Mw/Mn, i.e. 

The zero-shear viscosities measured in toluene solution are listed in Table 1 together with the values of r 0(theor) calculated from Eq. (14). The percentage deviation of the theoretical from the measured viscosities is given in column 8. 

The influence of the molar mass and concentration above the zero-shear viscosity has been described. In the following sections the influence of these parameters in the region of Newtonian and non-Newtonian regimes will be discussed. 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, q0. Flow curves (plots of log q vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

Steady shear flow properties are sensitive indicators of the approaching gel point for the liquid near LST, p < pc. The zero shear viscosity rj0 and equilibrium modulus Ge grow with power laws [16]... 

Fig. 10. Experimental values of the gel stiffness S plotted against the relaxation exponent n for crosslinked polycaprolactone at different stoichiometric ratios [59]. The dashed line connects the equilibrium modulus of the fully crosslinked material (on left axis) and the zero shear viscosity of the precursor (on right axis)...

We can also calculate other viscoelastic properties in the limit of low shear rate (linear viscoelastic limit) near the LST. The above simple spectrum can be integrated to obtain the zero shear viscosity 0, the first normal stress coefficient if/1 at vanishing shear rate, and the equilibrium compliance J... 

The zero-shear viscosity and the dynamic viscosity (at low frequencies) diverge at high concentration, while they are constant at low concentration [99,100,102-105],... 

Galgali and his colleagues [46] have also shown that the typical rheological response in nanocomposites arises from frictional interactions between the silicate layers and not from the immobilization of confined polymer chains between the silicate layers. They have also shown a dramatic decrease in the creep compliance for the PP-based nanocomposite with 9 wt% MMT. They showed a dramatic three orders of magnitude drop in the zero shear viscosity beyond the apparent yield stress, suggesting that the solid-like behavior in the quiescent state is a result of the percolated structure of the layered silicate. 

Figure 14.7 Dependence of the zero-shear viscosity, uo, on molecular weight, M, for different dendrimer systems. (1) Dendrimers of different chemical composition but in the same state (i.e. PAMAM, PPI and PBzE dendrimers in bulk D, C and E, respectively). (2) Compositionally identical dendrimers (i.e. PAMAMs) in solutions and in the bulk state (A, B and D, respectively). (3) Compositionally identical dendrimers and linear polymers of comparable molecular weights (i.e. PAMAMs in the bulk state D and F, respectively)...

Also recalling that sinh x - x as x -> 0 gives us the zero shear viscosity as... 

Few dispersions in everyday use are monodisperse and this will mean modification to Equation (3.54). A general expectation resulting from polydispersity is that denser packing may be achieved.22 The simplest case occurs for bi- or multimodal systems with very large size differences between each mode, several orders of magnitude for example. To calculate the zero shear viscosity of systems containing 1, 1 3, and... 

Now the functions for doing simple power law-dependent simulations are developed. The zero-shear viscosity, //o. is 1.268 x 10 Pa-s as shown by Fig. 3.22 and the viscosity data in Table 3.6. This holds for all shear rates in the plateau range. For the power law fit, the last six entries in Table 3.6 are used to develop a regression fit, and then the line is extrapolated back to lower shear rates. The regression fit is as follows ... 

The disadvantage of the power law model is that it cannot predict the viscosity in the zero-shear viscosity plateau. When the zero-shear viscosity plateau is included, a nonlinear model must be specified with additional fitting parameters. A convenient model that includes the zero-shear viscosity and utilizes an additional parameter is the Cross model [30] ... 

Figure 11 Logarithmic plot of the zero shear viscosities of PS-(S03Li)2 against the concentration in cumene, samples as given in Table II.

Note 5 Extrapolation of rj or /app for non-Newtonian liquids to zero y gives the zero-shear viscosity, which is given the symbol rja. 

The melt flow index is a useful indication of the molar mass, since it is a reciprocal measure of the melt viscosity p. p depends very strongly on 77 ( ) (doubling of results in a 10.6 times higher 77 ). This relation is valid for the zero-shear viscosity the melt index is measured at a shear stress where the non-Newtonian behaviour, and thus the width of the molar mass distribution, is already playing a part (see MT 5.3.2). The melt index is a functional measure for the molar mass, because for a producer of end products the processability is often of primary importance. 

The whole curve is thus shifted upward by a factor 10 (one unit on the log scale). Broadening of the molar mass distribution shifts the curve to the left the zero-shear viscosity does not change. 

For the unfilled polystyrene melt at low elongational rates a constant value of Tjg is achieved given by three times the zero shear viscosity according to Trou-... 

The zero-shear viscosity r 0 has been measured for isotropic solutions of various liquid-crystalline polymers over wide ranges of polymer concentration and molecular weight [70,128,132-139]. This quantity is convenient for studying the stiff-chain dynamics in concentrated solution, because its measurement is relatively easy and it is less sensitive to the molecular weight distribution (see below). Here we deal with four stiff-chain polymers well characterized molecu-larly schizophyllan (a triple-helical polysaccharide), xanthan (double-helical ionic polysaccharide), PBLG, and poly (p-phenylene terephthalamide) (PPTA Kevlar). The wormlike chain parameters of these polymers are listed in Tables... 

Another often used representation of the viscoelastic flow behavior utilizes normal stress coefficients P/ = Ni/y. Figure 10 depicts flow curves of a family of PAA/water solutions differing in concentrations and therefore in their viscosities. Normalized by the zero-shear viscosity fiQ and by a constant shear rate /q shear stress value of to= 1 N/m they produce master curves for viscosity and the normal stress coefficient. The preparation... 

This article builds upon an earlier review on the same subject by Porter and Johnson in 1966 (14), and on the recent treatise on viscoelasticity in polymers by Ferry (15). We have generally tried to maintain the same nomenclature as the latter. Recent reviews on the relation between the zero-shear viscosity and molecular structure (16), crosslinked networks (17), and flow birefringence (18) in this same journal cover portions of the subject. We have tried to minimise redundancy with these works while at the same time making the review reasonably self-contained. 

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