Viscosity, apparent stress dependence

With most homopolymers and copolymers the apparent viscosity is less dependent on temperature and shear stress (up to 10 dyn/cm ) than that of the polyolefins, thus simplifying die design. On the other hand the melt has a low elasticity and strength and this requires that extruded sections be... 

Viscosity and Plasticity—Viscosity and plasticity are closely related. Viscosity may be defined as the force required to move a unit-area of plane surface with unit-speed relative to another parallel plane surface, from which it is separated by a layer of the liquid of unit-thickness. Other definitions have been applied to viscosity, an equivalent one being the ratio of shearing stress to rate of shear. When a mud or slurry is moved in a pipe in more or less plastic condition the viscosity is not the same for all rates of shear, as in the case of ordinary fluids. A material may be called plastic if the apparent viscosity varies with the rate of shear. The physical behavior of muds and slurries is markedly affected by viscosity. However, consistency of muds and slurries is not necessarily the same as viscosity but is dependent upon a number of factors, many of which are not yet clearly understood. The viscosity of a plastic material cannot be measured in the manner used for liquids. The usual instrument consists of a cup in which the plastic material is placed and rotated at constant speed, causing the deflection of a torsional pendulum whose bob is immersed in the liquid. The Stormer viscosimeter, for example, consists of a fixed outer cylinder and an inner cylinder which is revolved by means of a weight or weights. 

Shear viscosity (rj) is the most important intrinsic property determining extrudability. Since the apparent viscosity is highly dependent on temperature and shear stress (hence on pressure gradient), these variables, together with the extruder geometry, determine the output of the extruder. 

The test described above and the results obtained at a single stress value do not reveal anything about the stress dependence of flow rate. The apparent viscosity calculated from them is 3.6 x 10 dPa s. 

In a Newtonian liquid the viscosity does not depend on the stress or the strain rate applied, but many liquids are non-Newtonian. Many liquid dispersions are strain rate thinning, i.e., the apparent viscosity decreases with increasing strain rate. Some of these dispersions are also thixotropic the apparent viscosity decreases during flow at constant strain rate and slowly increases again after flow has stopped. 

An example is shown in Figure 6.6, lower curve. At very low shear rate, the solution shows Newtonian behavior (no dependence of r] on shear rate), and this is also the case at very high shear rate, but in the intermediate range a marked strain rate thinning is observed. The viscosity is thus an apparent one (/ ,), depending on shear rate (or shear stress). It is common practice to give the (extrapolated) intrinsic viscosity at zero shear rate, hence the symbol [ri]0 in Eq. (6.6). The dependence of t] on shear rate may have two causes. 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

A non-Newtonian fluid is one whose flow curve (shear stress versus shear rate) is non-linear or does not pass through the origin, i.e. where the apparent viscosity, shear stress divided by shear rate, is not constant at a given temperature and pressure but is dependent on flow conditions such as flow geometry, shear rate, etc. and sometimes even on the kinematic history of the fluid element under eonsideration. Such materials may be conveniently grouped into three general elasses ... 

Stress dependence of the apparent viscosity /(, at temperatures between 170 C and 270°C for a moulding grade of potyfmethyl methaoylate). Measurements by capillary flow (after CkrgswelO. 

Above all other parameters, it is the relative molecular mass of a polymer which determines the apparent shear viscosity (7.N.3). Figure 7.14 shows the stress dependence at 170°C of four BPE polymers with relative molecular masses ranging from extremely high (MFI 0.2) to extremely low (MFI200). Note that the shape of the curves (the shear thinning characteristics) is little changed by variations in relative molecular mass. 

In a study on the rheological behavior of a concentrated gelatin solution, using a rotation viscometer, it is found that in a certain concentration range the apparent viscosity q pp depends on the shear stress a as follows ... 

Fig. 7.14. Stress dependence of the apparent viscosity /i, at 170 C for four branched polyethylene resins ranging from a very high relative molecular mass, MFI — 0.20. to a very low relative molecular mass, MFI = 200. Measurements were made by capillary flow. Note the thousandfold change in /i, produced by changes in relative molecular mass (after Cogswell).

Fluid flow past a surface or boundary leads to surface forces acting on it. These surface forces depend on the rate at which fluid is strained by the velocity field. A stress tensor with nine components is used to describe the surface forces on a fluid element. The tangential component of the surface forces with respect to the boundary is known as shear stress. The nature or origin of shear stress depends on the nature of flow, i. e., laminar or turbulent. The stress components for a laminar flow are functions of the viscosity of the fluid and are known as viscous stresses. The turbulent flow has additional contributions known as Reynolds stresses due to velocity fluctuation, i. e., the stresses of a laminar flow are increased by additional stresses known as apparent or Reynolds stresses. Hence, the total shear stresses for a turbulent flow are the sum of viscous stresses and apparent stresses. In a turbulent flow, the apparent stresses may outweigh the viscous conponents. 

At higher shear rates, three types of deviations are observable when compared to ideal Newtonian flow (see Fig. 2.1).The first kind of deviation relates to the existence of a flow threshold (yield point). In the case of a Bingham fluid, flow occurs only when the yield stress is exceeded. The second type of deviation is shear thickening, observed where the viscosity increases with shear rate. This is the case for a dilating fluid, behaviour which is seldom apparent in polymers. Last, where viscosity decreases with increase in shear rate, fluxing is observed and such fluids are usually referred to as pseudoplastic fluids. This last phenomenon is a general characteristic of thermoplastic polymers. Flow effects may also be time dependent. Where viscosity does not depend only on the shear rate, but also on the duration of the applied stress, fluids are thixotropic. Polymers in a molten state thus behave as pseudoplastic fluids having thixotropic characteristics. 

Instead, the viscosity dependsonshear stress and/or time. Non-Newtonian fluids areclas-sifled as being time-independent, time-dependent, and viscoelastic. They exhibit characteristics where the apparent viscosity either increases as the rate of shear increases, such as polymer melts, paper pulp, wall paper paste, printing inks, tomato pur, mustard, rubber solutions, protein concentrations, and are known as pseudoplastic, or decreases as the rate of shear increases. Examples of the latter are comparatively rare but include TtOj suspensions, comflour/sugar suspensions, cement aggregates, starch solutions, and certain honeys. 

Rheology. Both PB and PMP melts exhibit strong non-Newtonian behavior thek apparent melt viscosity decreases with an increase in shear stress (27,28). Melt viscosities of both resins depend on temperature (24,27). The activation energy for PB viscous flow is 46 kj /mol (11 kcal/mol) (39), and for PMP, 77 kJ/mol (18.4 kcal/mol) (28). Equipment used for PP processing is usually suitable for PB and PMP processing as well however, adjustments in the processing conditions must be made to account for the differences in melt temperatures and rheology. 

Apparent viscosity Ratio of shear stress to shear rate. It depends on the rate of shear. 

Figure 4 Plots showing dependence of apparent viscosity (17) on apparent shear stress (t) [33].

Many fluids, including some that are encountered very widely both industrially and domestically, exhibit non-Newtonian behaviour and their apparent viscosities may depend on the rate at which they are sheared and on their previous shear history. At any position and time in the fluid, the apparent viscosity pa which is defined as the ratio of the shear stress to the shear rate at that point is given by ... 

It will be noted that the dimensions of k are ML-IT" 2, that is they are dependent on the value of n. Values of k for fluids with different n values cannot therefore be compared. Numerically, k is the value of the apparent viscosity (or shear stress) at unit shear rate and this numerical value will depend on the units used for example the value of A at a shear rate of 1 s-1 will be different from that at a shear rate of 1 h 1. 

For a Newtonian fluid, the shear stress is proportional to the shear rate, the constant of proportionality being the coefficient of viscosity. The viscosity is a property of the material and, at a given temperature and pressure, is constant. Non-Newtonian fluids exhibit departures from this type of behaviour. The relationship between the shear stress and the shear rate can be determined using a viscometer as described in Chapter 3. There are three main categories of departure from Newtonian behaviour behaviour that is independent of time but the fluid exhibits an apparent viscosity that varies as the shear rate is changed behaviour in which the apparent viscosity changes with time even if the shear rate is kept constant and a type of behaviour that is intermediate between purely liquid-like and purely solid-like. These are known as time-independent, time-dependent, and viscoelastic behaviour respectively. Many materials display a combination of these types of behaviour. 

The second category, time-dependent behaviour, is common but difficult to deal with. The best known type is the thixotropic fluid, the characteristic of which is that when sheared at a constant rate (or at a constant shear stress) the apparent viscosity decreases with the duration of shearing. Figure 1.21 shows the type of flow curve that is found. The apparent viscosity continues to fall during shearing so that if measurements are made for a series of increasing shear rates and then the series is reversed, a hysteresis loop is observed. On repeating the measurements, similar behaviour is seen but at lower values of shear stress because the apparent viscosity continues to fall. 

There is an expression that does not truly fit either class of behaviour, for power law fluids which can be expressed in terms of stress, rate or apparent viscosity with relative ease. They can describe shear thickening or thinning depending upon the sign of the power law index n ... 

The durability of the particle network structure imder the action of a stress may also be time-dependent. In addition, even at stresses below the apparent yield stress, flow may also take place, although the viscosity is several orders of magnitude higher than the viscosity of the disperse medium. This so-called creeping flow is depicted in Fig. 11 where r (. is the creep viscosity. In practice this phenomenon is insignificant in the treatment of filled polymer melts, but may be relevant, for example, in consideration of cold flow of filled elastomers. 

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