Shear flow curve

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

It appears that we should use the non-Newtonian shear flow curve to calculate the strain rates. However, the pressure flow is much smaller than the drag flow, and its shear rates are superimposed on the higher shear rate Vcos Olh in the drag flow. Therefore, we can assume that the melt is approximately Newtonian for the pressure flow, with a viscosity equal to the apparent viscosity rja in the drag flow, and put... 

For a length L = 0.2 m, a density p = 750kgm and g = 9.8 ms , the stress a = 1.5 kPa. To avoid significant stretching, the tensile strain rate must be less than 0.2 s , which means that the tensile viscosity must exceed 7500Nsm. When these two conditions are imposed on a typical shear flow curve in Fig. 5.18, it is clear that the melt must be highly non-Newtonian. A similar process cannot be used for Newtonian silicate glasses... 

The analysis of mould filling requires rheological and thermal data for the plastic, and the mould dimensions. Polymer manufacturers usually provide shear flow curves at a range of temperatures these can be approximated by a power law relationship over a limited range of shear strain rates. In the days before computer analysis, flow lengths of short shots were determined in spiral test cavities, as a function of the injection pressure. However, the geometry of this constant cross section mould differs so much from most other moulds that the flow lengths in the two types of mould do not correlate well. 

Figure 7.10 Shear flow curves of the original polymers (PA and EPM) and of their extruded...

Table 18.5 showed the power law index and the consistency of flow of STR5L/EPDM and STR5L/BEPDM blends. Shear flow curves of the pure rubbers and their blends illustrated the pseudoplastic property as shear thinning behaviour with a power law index napparent shear viscosity of the two sets of blends decreased as the shear rate increased as shown in Figures 18.22 and 18.23 for STR5L/EPDM blend and STR5L/BEPDM blend, respectively. 

Linear polymers that exhibit sharkskin typically also exhibit a discontinuity in the shear flow curve known as slip-stick. The phenomenon is illustrated in Figure 12.12 for an LLDPE in a piston-driven capillary rheometer. At a critical wall stress there... 

If the particular product is not very non-Newtonian, a single-point measurement might be sufficient. Even then however we must be sure that the flow type is the same. For instance if our experience off-line is built up from steady-state, simple-shear flow curves, then a too-fast measurement might mean the measurement is not made under steady-state conditions. Equally if the on-hne flow has an appreciable extensional component, then problems can arise for some hquids, especially pol3nner solutions. Also if a vibrational mode is used, then it is probable that some non-hnear oscillatory function is being measured. Ah these facts could mean that we end up with no simple one-to-one correlation between on-hne and off-hne. Hence the safest way is to duphcate on-hne what is done off-hne. This is possible nowadays for most situations. 

Fig. 2.31 Polypropylene extrusion at 315°C steady state shear flow curves [46].

Figure 7.2 (a) Steady shear flow curves, reduced to 100 °C, of unfilled elastomers. 

Pseudoplastic fluids have no yield stress threshold and in these fluids the ratio of shear stress to the rate of shear generally falls continuously and rapidly with increase in the shear rate. Very low and very high shear regions are the exceptions, where the flow curve is almost horizontal (Figure 1.1). 

Fig. 2. Flow curves (shear stress vs shear rate) for different types of flow behavior.

Viscosity is equal to the slope of the flow curve, Tf = dr/dj. The quantity r/y is the viscosity Tj for a Newtonian Hquid and the apparent viscosity Tj for a non-Newtonian Hquid. The kinematic viscosity is the viscosity coefficient divided by the density, ly = tj/p. The fluidity is the reciprocal of the viscosity, (j) = 1/rj. The common units for viscosity, dyne seconds per square centimeter ((dyn-s)/cm ) or grams per centimeter second ((g/(cm-s)), called poise, which is usually expressed as centipoise (cP), have been replaced by the SI units of pascal seconds, ie, Pa-s and mPa-s, where 1 mPa-s = 1 cP. In the same manner the shear stress units of dynes per square centimeter, dyn/cmhave been replaced by Pascals, where 10 dyn/cm = 1 Pa, and newtons per square meter, where 1 N/m = 1 Pa. Shear rate is AH/AX, or length /time/length, so that values are given as per second (s ) in both systems. The SI units for kinematic viscosity are square centimeters per second, cm /s, ie, Stokes (St), and square millimeters per second, mm /s, ie, centistokes (cSt). Information is available for the official Society of Rheology nomenclature and units for a wide range of rheological parameters (11). 

Thixotropy and Other Time Effects. In addition to the nonideal behavior described, many fluids exhibit time-dependent effects. Some fluids increase in viscosity (rheopexy) or decrease in viscosity (thixotropy) with time when sheared at a constant shear rate. These effects can occur in fluids with or without yield values. Rheopexy is a rare phenomenon, but thixotropic fluids are common. Examples of thixotropic materials are starch pastes, gelatin, mayoimaise, drilling muds, and latex paints. The thixotropic effect is shown in Figure 5, where the curves are for a specimen exposed first to increasing and then to decreasing shear rates. Because of the decrease in viscosity with time as weU as shear rate, the up-and-down flow curves do not superimpose. Instead, they form a hysteresis loop, often called a thixotropic loop. Because flow curves for thixotropic or rheopectic Hquids depend on the shear history of the sample, different curves for the same material can be obtained, depending on the experimental procedure. 

At the same time it is not surprising that polymer melts are non-Newtonian and do not obey such simple rules. Fortunately, if we make certain assumptions, it is possible to analyse flow in certain viscometer geometries to provide measurements of both shear stress (t) and shear rate (7) so that curves relating the two (flow curves) may be drawn. 

In fact with a Newtonian liquid y = 4Q/ nR. This latter expression, viz. 4Q/ uR, is obviously much easier to calculate than the true wall shear rate and, since they are uniquely related and the simple expression is just as useful, in design practice it is very common when plotting flow curves to plot against... 

A plot of apparent viscosity against shear rate produces a unique flow curve for the melt is shown in Fig. 5.3. Occasionally this information may be based on the true shear rate. As shown in Section 5.4(a) this is given by... 

Note that rotational viscometers give true shear rates and if this is to be used with Newtonian based flow curves then, from above, a correction factor of (4n/3 + 1) needs to be applied to the true shear rate. 

From the flow curves for polypropylene at 210 C, at this shear rate, T) = 130 Ns/m, so the shear stress, t, is given by... 

Since the power law equations are going to be used rather than the flow curves, it is necessary to use the true shear rates. 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

For scaly fillers the increase of relative viscosity with filler concentration is not as pronounced as in case of fibrous fillers [177,178]. Owing to filler orientation, the flow curves for systems with different concentrations of a fibrous and a scaly filler may merge together at high shear rates [181]. In composites with a dispersed filler the decrease of the effective viscosity of the melt with increasing strain rate is much weaker. 

In a number of works (e.g. [339-341]) the authors sought to superimpose graphically the flow curves of filled melts and polymer solutions with different filler concentrations however, it was only possible to do so at high shear stresses (rates). More often than not it was impossible to obtain a generalized viscosity characteristic at low shear rates, the obvious reason being the structurization of the system. 

During dynamic measurements frequency dependences of the components of a complex modulus G or dynamic viscosity T (r = G"/es) are determined. Due to the existence of a well-known analogy between the functions r(y) or G"(co) as well as between G and normal stresses at shear flow a, seemingly, we may expect that dynamic measurements in principle will give the same information as measurements of the flow curve [1],... 

A similar thing takes place when we consider flow curves obtained at different temperatures. As seen from Fig. 7, if we take a region of low shear rates, then due to the absence of the temperature dependence Y, the apparent activation energy vanishes. At sufficiently high shear rates, when a polymer dispersion medium flows, the activation energy becomes equal to the activation energy of the viscous flow of a polymer melt and the presence of the filler in this ratio is of little importance. 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

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