Trouton ratio

The ratio of extensional viscosity r e to shear viscosity r s is known as the Trouton ratio, which is three for Newtonian fluids in uniaxial extension and larger than three for non-Newtonian fluids. For a viscoelastic fluid such as a polymer in solution, the uniaxial extensional viscosity characterizes the resistance of the fluid... 

An elongational or extensional viscosity (%) develops as a result of a conformational transition when disperse systems are forced through constrictions, or compressed or stretched (Kulicke and Haas, 1984 Rinaudo, 1988 Barnes et al., 1989 Odell et al., 1989 Clark, 1992). The intuitive logic is that the random coils resist the initial distortion. % is believed to elicit the human sensation of stringiness (Clark, 1995). If shear viscosity is denoted iq, rheologists define a Trouton ratio as %/ti, wherein % > T) by a factor approximating 3 for uniaxial extension and 6 for biaxial extension. Alternatively stated, the Newtonian ly calculates to one-third to one-sixth % (Steffe, 1992). 

This is, according Eq. (15.64), three times the zero shear viscosity, which in agreement with practice. It is a result already obtained by Trouton in 1906. Accordingly, the Trouton ratio, defined as... 

The Trouton ratio, 7r, is defined as the ratio of the extensional viscosity to the shear viscosity ... 

A solution or melt composed of such stretchable molecules can be highly springy, especially in extensional flows (Tirtaatmadja and Sridhar 1993). The kinematics of an extensional flow are described in Section 1.4.1.2. From Eqs. (1-6) and (1-9), one can show that for a Newtonian fluid (for which a = 2rjD) the Trouton ratio Tr = f)u/hQ the uniaxial extensional viscosity to the zero-shear viscosity /jo is numerically equal to 3. For polymers, Tr can be much higher than this. Figure 3-2, for example, shows Tr for a... 

Figure 3.2 Trouton ratio, Tr, of uniaxial extensional viscosity to zero-shear viscosity jq after start-up of steady uniaxial extension at a rate of 1 sec i for a Boger fluid consisting of a 0.185 wt% solution of flexible polyisobutylene (Mu, = 2.11 x 10 ) in a solvent composed mostly of viscous polybutene with some added kerosene (solid line). The dashed line is a fit of a multimode FENE dumbbell model, where each mode is represented by a FENE dumbbell model, with a spring law given by Eq. (3-56), without preaveraging, as described in Section 3.6.2.2.I. The relaxation times were obtained by fitting the linear viscoelastic data, G (co) and G"(cu). The slowest mode, with ri = 5 sec, dominates the behavior at large strains the best fit is obtained by choosing for it an extensibility parameter of = 40,000. The value of S — = 3(0.82) n/C(x, predicted from the...

Comparisons of the predictions of the FENE dumbbell model with measurements of the extensional viscosity of dilute solutions have been fairly encouraging. Figure 3-2 compares the Trouton ratio predicted by a multimode FENE dumbbell model with experimental data for a Roger fluid Good agreement is obtained if one uses a value of the... 

Trouton ratio at finite and zero rate of deformation, respectively... 

Such a viscoelastic or memory liquid is another example of a non-Newtonian liquid. Nearly all viscoelastic liquids are also strain rate thinning, but not all strain rate thinning liquids show significant elasticity. Deformation can, of course, be in shear or elongation, etc. However, for viscoelastic liquids, the Trouton ratios [see Eq. (5.2)] are higher, often much... 

Symbols for (physical) quantities, be they variables or constants, are given by a single character (generally Latin or Greek letters) and are printed in italics, e.g., F (force), p (pressure), p (chemical potential), k (Boltzmann constant). Further differentiation is achieved by the use of subscripts and/or superscripts these are printed in italics if it concerns the symbol of a quantity, otherwise in roman type, e.g., cp (specific heat at constant pressure), hp (Planck s constant), Ffu (surface dilational modulus). For clarity, symbols are generally separated by a (thin) space, e.g., F=ma, not ma. Some generally accepted exceptions occur, such as pH, as well as symbols (or two letter abbreviations, rather) for the dimensionless ratios frequently used in process engineering, like Re for Reynolds number and Tr for Trouton ratio (in roman type). 

However, for anisometric particles the Trouton ratio, q / q is a strong function of p. For example, at ( ) = 0.01 extensional viscosity of rods with aspect ratio p = 1000 is 1000 times higher than that for suspension of spheres. 

Blends of LLDPE/PP = 50 50, with or without compatibilizing ethylene-propylene copolymer, EPR, were studied by Dumouhn et al. [1984]. In spite of the expected immiscibihty, the blends showed additivity of properties with good superposition of the stress growth functions in shear and elongation, as well as with the zero deformation rate Trouton ratio, Rj, = 1. In earlier work, blends of medium density PE (MDPE) with small quantities of ultra-high molecular weight polyethylene (UHMWPE) were studied in shear and extension. Again, SH and Rj, = 1 were observed. 

As we have seen, there exist a number of treatments of the increase in extensional viscosity of the solution corresponding to the coil-stretch transition. In particular, the Warner FENE dumbbell model (29) and the Kramers bead-rod model (30) predict increases in normalized extensional viscosity of the order of N, the number of statistical segment units in the flexible chain. The normalized extensional viscosity (T)e ) compares the increase in extensional viscosity due to the polymer to three times (the Trouton ratio) the corresponding increase in the simple shear viscosity and is given by... 

The earliest determinations of elongational viscosity were made for the simplest case of uniaxial extension, the stretching of a fibre or filament of liquid. Trouton [1906] and many later investigators found that, at low strain (or elongation) rates, the elongational viscosity he was three times the shear viscosity n [Barnes et al., 1989], The ratio Mb/M is referred to as the Trouton ratio, Tr and thus ... 

The value of 3 for Trouton ratio for an incompressible Newtonian fluid applies to values of shear and elongation rates. By analogy, one may define the Trouton ratio for a non-Newtonian fluid ... 

The definition of the Trouton ratio given by equation (1.26) is somewhat ambiguous, since it depends on both e and y, and some convention must therefore be adopted to relate the strain rates in extension and shear. To remove this ambiguity and at the sametime to provide a convenient estimate of behaviom in extension, Jones etal. [1987] proposed the following definition of the Trouton ratio ... 

It is important to note that the Trouton ratio, Tr, defined as the ratio of the extensional viscosity to the shear viscosity, involves the shear viseosity evaluated at the same magnitude of the second invariant of the rate of deformation tensor where s is the rate of extension, i.e. ... 

Viscoelasticity, Fig. 2 Extensional viscosity curve of polymeric solution and polymer melt, (a) Comparison between steady shear and elongation response (b) Trouton ratio as a function of total Hencky strain e = sot for polymeric solution... 

A commOTi behavior of polymeric solutions [9-11] and polymer melts [12] is that the Trouton ratio is greater than three at moderate and higher deformation rate. Their typical behaviors are illustrated in Fig. 2, which show that the Trouton ratio can be a function of both strain and strain rate. 

Since the Trouton ratio depends on both a and y in Eq. 10, there is some ambiguity in choosing their values. To provide a convenient estimate without this ambiguity, Jones et al. [8] proposed the following definition ... 

In qualitative agreement with the von Mises criterion, ctii y/ai2,5 was reported (Utracki 1984). The Trouton ratio... 

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