Steady shear viscosity

The exponents a and a+ depend not only on the relaxation exponent n, but also on the dynamic exponents s and z for the steady shear viscosity of the sol and the equilibrium modulus of the gel. 

The transient viscosity f] = T2i(t)/y0 diverges gradually without ever reaching steady shear flow conditions. This clarifies the type of singularity which the viscosity exhibits at the LST The steady shear viscosity is undefined at LST, since the infinitely long relaxation time of the critical gel would require an infinitely long start-up time. 

The divergence of the longest relaxation time does not perturb the measurement. In comparison, steady state properties (the steady shear viscosity, for instance) would probe an integral over all relaxation modes and, hence, fail near the gel point. 

Measurement of the diverging steady shear viscosity is an appealing experiment because of its simplicity. Even the torque on a processing machine might serve as an estimate of the diverging viscosity. It has, however, severe disadvantages that need to be considered [149] ... 

Figure 14.9 Dependence of steady shear viscosity, rj, on shear rate, y, for the first seven generations of PAMAM dendrimers at 70°C in the bulk state...

Rheologists sometimes use an empirical functionality called the Cox-Mertz rule, which states that the complex viscosity value at a given frequency is equal to the steady shear viscosity at the same shear rate [14] ... 

Fig. 8.1. Dynamic viscosity t] (a>) and steady state viscosity f](y) for undiluted narrow distribution polystyrenes. The data are plotted in reduced form to facilitate comparison. The dimensionless shear rate or frequency is t]0Mwy/gRT >r r/ M co/gRT. [See Eq.(8.3)]. The dynamic viscosities are for Mw = 215000 (O) and Mw = 581000 ( ) at 160° C (312). The steady shear viscosity is for Mw = 411000 (A) at 176° C (313). The shapes in the onset region are similar for the three curves, but the apparent limiting slope for the dynamic...

Fig. 2.5. Steady-state and dynamic oscillatory flow measurements on a 2 wt. per cent solution of polystyrene S 111 in Aroclor 1248 according to Philippoff (57). ( ) steady shear viscosity (a) dynamic viscosity tj, ( ) cot 1% from flow birefringence, (A) cot <5 from dynamic measurements, all at 25° C. (o) cot 8 from dynamic measurements at 5° C. Steady-state flow properties as functions of shear rate q, dynamic properties as functions of angular frequency m. Shift factor aT which is equal to unity for 25° C, is explained in the text, cot 2 % and cot 8 are expressed in terms of shear (see eqs. 2.11 and 2.22)...

Steady shear viscosities can be measured with two different instruments. The System IV can measure polymer viscosities from about 0.001 to 10 sec 1 while the Gottfert Capillary Rheometer is capable of obtaining viscosities from 0.1 to 100,0001/s. In steady shear, the strains are very large as opposed to the dynamic measurements that impose small strains. In the capillary rheometer, the polymer is forced through a capillary die at a continuously faster rate. The resulting stress and viscosity are measured by a transducer mounted adjacent to the die. A schematic of the system is illustrated in Figure 5. 

A typical evolution of equilibrium mechanical properties during reaction is shown in Fig. 6.1. The initial reactive system has a steady shear viscosity that grows with reaction time as the mass-average molar mass, Mw, increases and it reaches to infinity at the gel point. Elastic properties, characterized by nonzero values of the equilibrium modulus, appear beyond the gel point. These quantities describe only either the liquid (pregel) or the solid (postgel) state of the material. Determination of the gel point requires extrapolation of viscosity to infinity or of the equilibrium modulus to zero. 

Steady shear viscosity measurements are very simple and are often used in practice. Very often a viscosity of 103 Pas is arbitrarily identified with the gel point. But the determined gelation time, tgei, depends on the shear rate, and extrapolation to zero shear rate meets the following difficulties ... 

Figure 6.1 Schematic evolution of steady-state mechanical properties of a thermoset as a function of reaction time or conversion. Representative properties are the steady shear viscosity for the liquid state and the equilibrium modulus for the solid state.

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

There exists a substantial history of interest in flow and deformation properties of monolayers. Perhaps, the first is the theoretical formulation of hydrodynamic coupling between the monolayer and subphase by Harkins and Kirkwook in 1938 [129], in determination of steady shear viscosity of mono-layers, which has since been augmented by Hansen [130] and Goodrich [131]. A variation of the method based on the Maxwell model was proposed by Mannheimher and Schechter [132] in an oscillatory mode in a canal. Experimentally, the method was implemented by joint efforts in our laboratories for determinations of steady shear viscosity of monolayers through the canal... 

Gortemaker (et al.), 1976). In Fig. 15.12, the dynamic moduli are plotted vs. reduced angular frequency. From these results the complex viscosity rf and its components // and rf were calculated. They were plotted vs angular frequency in Fig. 15.13, where also experimental values of the steady shear viscosity are shown. The agreement between rj q) and rf co) is clearly visible. This relationship between steady shear and sinusoidal experiments... 

Fig. 19. The steady-shear rheological behavior for a series of delaminated nanocomposites ol polydimethylsiloxane with dimethyl ditallow montmorillonite at 25 °C. The steady-shear viscosity shows an increase with respect to that of the pure-polymer at low shear rates but still obeys Newtonian type behavior, even at the highest silicate loadings examined. From Ref. [5].

The steady shear viscosities in Fig. 40 demonstrate the primary features of the phenomena ... 

Fig. 40. Steady shear viscosities of aqueous dispersions of polystyrene latices in nonadsorbing dextran solutions (Patel and Russel, 1989b) (a) a/r, = 6.9, 0 = 0.20. A, single phase, 4nR J/3pb = 0.15 B, two-phase, 4jtR /3pb = 0.30 C, two-phase, 4jtRj/3pb = 0.45 D, two-phase, 4jtRj/3pb = 0.65. (b) a/R, = 1.9, 0 = 0.10. F, single phase, 4jtR /pb = 0.65 G, fluid-fluid, 4jtR /3pb = 0.75 H, fluid-solid, 4nR /3p = 0.95 I, fluid-solid, 4jiR3/3p = 1.25.

This section draws heavily from two good books Colloidal Dispersions by Russel, Seville, and Schowalter [31] and Colloidal Hydrodynamics by Van de Ven [32] and a review paper by Jeffiey and Acrivos [33]. Concentrated suspensions exhibit rheological behavior which are time dependent. Time dependent rheological behavior is called thixotropy. This is because a particular shear rate creates a dynamic structure that is different than the structure of a suspension at rest. If a particular shear rate is imposed for a long period of time, a steady state stress can be measured, as shown in Figure 12.10 [34]. The time constant for structure reorganization is several times the shear rate, y, in flow reversal experiments [34] and depends on the volume fraction of solids. The viscosities discussed in Sections 12.42.2 to 12.42.9 are always the steady shear viscosity and not the transient ones. 

FIGURE 12.12 Steady shear viscosity as a function of Peclet number for polystyrene lattices of radii of 54, 70, 90, 37, and 55 nm at 50% Iqr volume in different solvents (—, H2O O, benzyl alcohol and, meta-cresol), where tjq = 24.7tjn, >, = 13.97, . Data from Kreiger [42]. 

Rheological Behavior. Figure 4 shows the room-temperature steady-shear viscosity as a function of shear rate for PDM-PMAS polymers and their precursors. Polymers with Ciq, C12, and C14 side chains exhibit Newtonian behavior over the range of shear rates monitored. 

Figure 4, Steady-shear viscosity versus shear rate for PDM-PMAS. Key O, precursor A, Nc = 10 O, = 12 , Nc = 14 A, Nc = 16 and 9, Nc = 18. Nc is the number of side-chain carbon atoms.

Choi, G. R. and Krieger, 1. M. 1986. Rheological studies on sterically stabilized model dispersions of uniform colloidal spheres II. Steady-shear viscosity. J. Colloid Interface Sci. 113 101-113. 

The superimposition of the shear rate dependence of steady shear viscosity, that is, t]a(o)), and of the frequency dependence of the complex viscosity, that is, i ( >), at equal values of frequency and shear rate was first reported by Cox and Merz (1958) for polystyrene samples, and is known as the Cox-Merz rule. 

The Cox-Merz rule (Cox and Merz, 1958) is useful in predicting steady shear viscosity from complex viscosity and vice versa ... 

A common feature observed was the departure between the steady shear viscosity ( 7a) and the real component of the dynamic viscosity (/ ) at large values of shear rate and frequency, with the expected more rapidly decrease of t] with frequency than rja does with shear rate (Bird et al., 1977a), which can be attributed to the very different molecular motions involved in the dynamic and steady shear at high m and y (Ferry, 1980). Because of the relatively high value of strain amplitude used in our tests (36%) and the two-phase nature of our HM dispersions, the observations with respect tor) and t)a are in agreement with those of Matsumoto et al. (1975). 

Comparison of the steady shear viscosity with the complex and... 

Experimental detection of the gel point is not always easy since the equilibrium shear modulus is technically zero at the gel point and any applied stress will eventually relax, but only at infinite time. From the classical theory, the attributes of the gel point are an infinite steady-shear viscosity and a zero equilibrium modulus at zero frequency limit (Figure 6-3) (Flory, 1953). These criteria have been widely employed to detect the gel point of chemical gels. However, because continuous shearing affects gel formation, accurate information from viscosity measurement is not possible in the close vicinity of the gel point. Further, information regarding the transition itself could only be obtained by extrapolation, thereby introducing uncertainties in the determination of the gelation moment. 

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