Shear rate amplitude

Using the cone and plate rheometer the angle Q is forced in a sinusoidal manner, leading to linear strain being introduced in the polymer. The shear strain, y, is a sinusoidal function of time t with a shear rate amplitude of % as follows ... 

However, while the Cox-Merz rule is valid for general unfilled and low concentration filled systems, it was discovered that the rule fails to hold for concentrated suspensions. It is important at this point to recognize that MIM or PIM feedstock is essentially a concentrated suspension at elevated temperature. A modified Cox-Merz rule was developed for cmicentrated suspensions by Doraiswamy et a [20]. The modified rule stated that the complex viscosity versus shear rate amplitude plot is equal to the corresponding viscosity versus shear rate plot. That is... 

X Dimensionless freezeline height Eq. 9.155 Yo Shear rate amplitude Eq. 3.18... 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

Fig. 22. Radius of drops produced by capillary breakup (solid lines) and binary breakup (dotted lines) in a hyperbolic extensional flow for different viscosity ratios (p) and scaled shear rate (p,cylo) (Janssen and Meijer, 1993). The initial amplitude of the surface disturbances is ao = 10 9 m. Note that significantly smaller drops are produced by capillary breakup for high viscosity ratios.

Tumbling regime At very low shear rates, the birefringence axis (or the director) of the nematic solution tumbles continuously up to a reduced shear rate T < 9.5. While the time for complete rotation stays approximately equal to that calculated from Eq. (85), the scalar order parameter S,dy) oscillates around its equilibrium value S. Maximum positive departures of S(dy) from S occur at 0 n/4 and — 3n/4, and maximum negative departures at 0 x — k/4 and — 5it/4, while the amplitude of oscillation increases with increasing T. 

Fig. 2.51 Effect of reciprocating shear (strain amplitude, k = 200%) on the ODT of an /pep = 0.55 PEP-PEE diblock (Koppi etal. 1993). Here y denotes the shear rate.The equilibrium order-disorder transition (, A) and disordered state stability limit (A.O) are shown. The upper curve is a fit to the scaling relation Tom y2- The lower curve represents the. scaling rs(A) A-i,3Todt> where A = y/y, with y an adjustable, parameter. Points given by and O were obtained at fixed temperature by varying y, while those represented by A and A were determined by varying the temperature at fixed y.

Fig. 4.28 Dynamic shear moduli as a function of temperature for a PS-PI diblock (Afw = 60kgmoL1, 17wt% PS) (points) in dibutyl phthalate (

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

Fig. 15 Undulation amplitude due to shear. The amplitude of the undulations A is given as a function of the strain rate y. At a shear rate y 0.01 undulations set in. The amplitude of these undulations grows continuously with increasing shear rate. The dashed line shows a fit to data points starting at y > 0.02 assuming a square root dependence of the amplitude above the undulation onset. Fig. 5.6 of [54]...

The conclusion is that Lodge s rheological constitutive equation results in relationships between steady shear and oscillatory experiments. The limits y0 0 (i.e. small deformation amplitudes in oscillatory flow) and q >0 (i.e. small shear rates) do not come from Lodge s equation but they are in agreement with practice. These interrelations between sinusoidal shear deformations and steady shear flow are called the relationships of Coleman and Markovitz. 

The data in Fig. 12 actually collapse onto a master curve when the wall stress o is rescaled by temperature T and the nominal shear rate y is normalized by a WLF factor aT [29]. Thus Eq. (6) for the critical stress oc is supported by the data in Fig. 12, where V does not change with T. Another feature of the transition is that the amplitude of the flow discontinuity does not vary with T. In other words, the extrapolation length bc, which is evaluated according to Eq. (4a) at the transition, is a constant with respect to T. Thus for a given surface, bc is more than just a material property such as the melt viscosity r. It essentially depends only... 

Another important point is the question whether static offsets have an influence on strain amplitude sweeps. Shearing data show that this seems not to be the case as detailed studied in [26] where shear rates do not exceed 100 %.However, different tests with low dynamic amplitudes and for different carbon black filled rubbers show pronounced effects of tensile or compressive pre-strain [ 14,28,29]. Unfortunately, no analysis of the presence of harmonics has been performed. The tests indicate that the storage (low dynamic amplitude) modulus E of all filled vulcanizates decreases with increasing static deformation up to a certain value of stretch ratio A, say A, above which E increases rapidly with further increase of A. The amount of filler in the sample has a marked effect on the rate of initial decrease and on the steady increase in E at higher strain. The initial decrease in E with progressive increase in static strain can be attributed to the disruption of the filler network, whereas the steady increase in E at higher extensions (A 1.2. .. 2.0 depending on temperature, frequency, dynamic strain amplitude) has been explained from the limited extensibility of the elastomer chain [30]. 

Figure 5. The stress curves for BK-60 associated with a sine wave strain of amplitude 1.5 x 10 radians and frequency of 0.06 Hz before (A) and-after (B) shearing with steady shear flow (shear rate 60 sec ) for 1 minute.

As a result it is of interest to examine some of the general characteristics of this component of the total response. If the real part of the dynamic shear viscosity is calculated for this component, tit, it is found to be a strong function of the strain amplitude and frequency. Data were obtained for BK-60 at 6 different amplitudes and 9 different frequencies. In an effort to systematize this Information, the dependence of on the shear rate was examined by plotting log (n] ) against the time average shear rate during the oscillatory cycle (Figure 8)... 

When departures from the Cox-Merz rule are attributed to structure decay in the case of steady shear, the complex viscosity is usually larger than the steady viscosity (Mills and Kokini, 1984). Notwithstanding this feature, the relation between magnitudes of T a and T can be dependent on the strain amplitude used (Lopes da Silva et al., 1993). Doraiswamy et al. (1991) presented theoretical treatment for data on suspensions of synthetic polymers. They suggested that by using effective shear rates, the Cox-Merz rule can be applied to products exhibiting yield stresses. The shift factors discussed above can be used to calculate effective shear rates. 

In Equation 3.116, is rigorously defined as [(an - 022)I(S 2 > 1 is the sum of a constant term and two oscillating terms, accounted by ijr[ and y i is the strain rate amplitude. Equations 6 to 8 suggest that oscillatory shear stress data are related to oscillatory primary normal stress difference data (Ferry, 1980). Youn and Rao (2003) calculated values of (co) for starch dispersions is applicable to oscillatory shear fields. 

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). 

Because of the existence of yield stress as well as time-dependent rheological behavior of mayonnaise, it would seem reasonable to expect that traditional relationships between steady shear properties on one hand and small amplitude dynamic properties on the other that were found for polymeric liquids will not hold for mayonnaise. Bistany and Kokini (1983) showed that the Cox-Merz rule and other relationships at low shear rates and frequencies did not hold for mayonnaise and... 

Figure 3.15 The frequency-dependent in-phase and out-of-phase components of the dynamic viscosity, rj and rj in small-amplitude oscillatory shear, along with the shear-rate dependence of the first normal stress coefficient hi (y) for a 0.05 wt% solution of polystyrene of molecular weight 2.25 X 10 in a solvent of oligomeric styrene. The lines through the data show the predictions of the Zimm theory for r and 2r)"f(o and the Zimm theory for hi(y) modified to account for finite extensibility, as discussed in Section 3.6.2.2.I. The dashed lines are the contributions of the individual Zimm relaxation modes to 2rj"((o) /

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

E4,B E4, 60% Temperature Ramp Shear Rate lOr Shear Amplitude 100%... 

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