Viscosity linear region

Below a critical concentration, c, in a thermodynamically good solvent, r 0 can be standardised against the overlap parameter c [r)]. However, for c>c, and in the case of a 0-solvent for parameter c-[r ]>0.7, r 0 is a function of the Bueche parameter, cMw The critical concentration c is found to be Mw and solvent independent, as predicted by Graessley. In the case of semi-dilute polymer solutions the relaxation time and slope in the linear region of the flow are found to be strongly influenced by the nature of polymer-solvent interactions. Taking this into account, it is possible to predict the shear viscosity and the critical shear rate at which shear-induced degradation occurs as a function of Mw c and the solvent power. 

A good diagnostic for creep and stress relaxation tests is to plot them on the same scales as a function of either compliance (J) or modulus (G), respectively. If the curves superimpose, then all the data collected is in the linear region. As the sample is overtaxed, the curves will no longer superimpose and some flow is said to have occurred. These data can still be useful as a part of equilibrium flow. The viscosity data from the steady-state part of the response are calculated and used to build the complete flow curve (see equilibrium flow test in unit hi.2). 

Some regularities, similar to Refs. 1718 21>, of viscosity variation in time under conditions of x = const were observed by the authors of Refs. 23,24,32-341 in extension of polyethylene and butadiene rubber (BR). Note that in Ref.34) the linear region of strain reaching the stationary flow was attained in extension of butadiene rubber. With further step-wise increase of x the effective viscosity grew and the time during which the stationary flow was attained increased significantly. References 23,24) will be discussed in more detail below. 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

In practice it appears that in a figure where log rj is plotted vs. log q and log q vs. log co, both lines coincide not only in the linear region but also, to some extent, in the region where both viscosities decrease. The agreement is even better if log rj instead of log rf is plotted vs. log co. This is demonstrated for polystyrene in Figs. 15.12 and 15.13... 

It is obser ed that upon reaching the isothermal temperature, the viscosity profiles exhibit two relatively linear regions as demonstrated by the 115 and 135 C isothermal profiles. The viscosity corresponding to t=0 at each isothermal temperature is calculated using a linear least squares analysis on the linear viscosity region directly after the point of minimum viscosity. An Arrhenius plot of the values versus 1/T is constructed to ob-... 

Region C-D is a linear region of Newtonian compliance in which the units as a result of rupture of the bonds flow past one another. The compliance Jn and the viscosity are related by ... 

The viscosity obtained from the above equation in the linear region of a creep experiment can be used to extend the low-shear rate region of apparent viscosity versus shear rate data obtained in a flow experiment by about two decades (Giboreau et al., 1994 Rayment et al., 1998). The low shear rate region of about 10 -10 is often used for the characterization and differentiation of structures in polysaccharide systems through the use of stress controlled creep and non destructive oscillatory tests. The values of strain (y) from the creep experiment can be converted to shear rate from the expression y t) = y t)/t. 

In other words, independently of the viscoelastic history in the linear region, the tensile compliance function can readily be obtained from both the shear and bulk compliance functions. For viscoelastic solids and liquids above the glass transition temperature, the following relationships hold when t oo J t) t/T[ [Eq. (5.16)], D t) = y Jt [Eq. (5.21)], and D t)J t)/ >. These relations lead to r 3t that is, the elongational viscosity is three times the shear viscosity. It is noteworthy that the relatively high value of tensile viscosity facilitates film processing. 

Dynamic mechanical strain-controlled measurements for both concentrated fabric softeners are shown in Figure 4.26. There are significant differences between the two products as regards the magnitude of the complex viscosity and complex modulus components and their strain dependence. Product B exhibits a higher viscosity and markedly longer linear region. The zero shear viscosity of product B is approximately 95 mPa s whereas that of product A is approximately half of this value at 50 mPa s. 

Continuing in the pseudoplastic region it is often found that an upper threshold can be reached beyond which no further reduction in viscosity occurs. The curve then enters a second linear region of proportionality the slope of which is the second Newtonian viscosity. 

In the linear region, the viscosity is independent of the rate-of-strain and has a constant value, and is often referred to as the zero-shear-rate viscosity or zero-shear viscosity and denoted by rjo. 

Viscosity is one of the most frequently applied method to study the polymer surfactant interaction. The hydrodynamic data are expressed in various ways viscosities relative to the solvent (water) or to the surfactant solution furthermore, specific viscosity measured as a function of polymer concentration at constant surfactant concentration and as a function of surfactant concentration at constant polymer content as well as measurements at a constant polymer/surfactant concentration ratio can be found in the literature. In some cases, efforts were made in order to determine the intrinsic viscosity of the polymer-surfactant complex by extrapolation from the linear region of the Tigp/cp vs. Cp function in spite of the fact that at low polymer concentration it shows anomaly. ... 

Terms of higher order in Eq. (4.1) cannot be neglected for rising polymer concentrations due to the increasing intermolecular interactions (see Concentration on molar mass in Chap. 5). The linear region determined by the first two terms in Eq. (4.1) is only observed at low concentrations. In addition to this, measurement errors at low concentrations are more likely due to the small rise of the viscosity with the concentration. 

Fig. 4.6. Reduced viscosity plotted as a function of the concentration c according to the Huggins equation and as a function of the specific viscosity according to the Schulz-Blaschke equation. Two poly(acrylamide) samples with different molar masses are shown, data from [89]. In the shown examples, the longer linear region of the Schulz-Blaschke plot allows for better extrapolation to c- 0 for the determination of the intrinsic viscosity...

At the same time, the slope of the curves in the linear region increases with the solvent quality and the onset of the non-linear behavior is shifted to lower concentrations. The slope of the reduced viscosity is equivalent to the product K x [ of the Huggins constant and the intrinsic viscosity squared according to the Huggins Eq. (4.9). The slope is also formal equivalent to a second virial coefficient like A2 in the equation of the reduced osmotic pressure H/c ... 

The viscosity in the linear and non-linear regions for cellulose solutions at different concentrations. 

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