Extensional geometries

Fig. 22. Schematic diagram of the opposite jets device with some of the associated streamlines (the x marks the location of the stagnation point). It has been determined that a ratio of d/(2 r0) w 1 — 1.4 constitutes the optimum geometry for extensional viscosity measurements [104]...

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

In summary, we have commented briefly on the microscopic applications of NMR velocity imaging in complex polymer flows in complex geometries, where these applications have been termed Rheo-NMR [23]. As some of these complex geometries can be easily established in small scales, NMR velocimetry and visc-ometry at microscopic resolution can provide an effective means to image the entire Eulerian velocity field experimentally and to measure extensional properties in elastic liquids non-invasively. 

An excellent reference describing appropriate ways of measuring the piezoelectric coefficients of bulk materials is the IEEE Standard for Piezoelectricity [1], In brief, the method entails choosing a sample with a geometry such that the desired resonance mode can be excited, and there is little overlap between modes. Then, the sample is electrically excited with an alternating field, and the impedance (or admittance, etc.) is measured as a function of frequency. Extrema in the electrical responses are observed near the resonance and antiresonance frequencies. As an example, consider the length extensional mode of a vibrator. Here the elastic compliance under constant field can be measured from... 

Fig. 3.13 Schematic of three prototype industrial flow (PIF) geometries showing shaded extensional flow regions for each geometry. [Reprinted by permission from J. F. Agassant et al., Intern. Polym. Proc., 17, 3 (2002).]...

Fig. 9. Cartoon which reviews variations in the geometry of fault damage zones. The simple elliptical geometry (a) is likely to be modified in reality by strain flelds caused by (b), deviations away from a planar plane (c) at the fault splay points (d) from extensional tip strains and (e) variations in the strengths and architecture of lithologies present.

Figure 3.81. Extensional-flow rheometer geometries. Adapted from Figure 7.1.2 (Macosko, 1994). Copyright (1994). Reprinted with permission of John Wiley and Sons, Inc.

The mechanical analog of equation (3-2) is the dashpot element (Figure 3-1). This is merely a leaky piston in a cylinder filled with a liquid of viscosity 77 for shear or t)e for extension. (Remember, the details of geometry, etc. are not important we are looking only for qualitative behavior by analogy. Thus we will use the models of Figure 3-1 for both extensional and shear properties.) Integration of equation (3-2b) for constant shear stress cr0, yields... 

There are several direct methods of measurement of yield stress. The constant stress rheometer is most frequently used to determine value in shear. Dzuy and Boger [1983, 1985] used a rotational vane viscometer. Yield stresses in compression can be calculated from the unrelaxed stress values in parallel plate geometry. Its value in elongation has been directly measured as the critical stress value below which no sample deformation was observed during 30 minutes of straining in an extensional rheometer. 

In order to maximize blend properties, three factors must be controlled LCP domain orientation, LCP domain morphology (geometry), and interfacial adhesion. The first two factors depend on the rheology of the matrix and reinforcement phase and on the deformation fields to which they are exposed. As with neat LCPs, extensional flow fields have a greater influence on orientation than do shear fields. The LCP should ideally have an equal or lower viscosity than the matrix to ensure deformation. 

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