In this section we describe procedures used to check empirically the limits of a rotational rheometer. First, we address the viscosity determination in steady shear experiments. If a low viscosity fluid is tested in a strain-controlled instrument, the minimum reliably measurable torque created by the sample is decisive. For a stress-controlled machine, the smallest torque that can be reliably imposed on the sample is important. This torque is not identical to the minimum torque of the instrument specification, because friction contributions from the bearing do not necessarily remain constant over time. There is also an upper limit on applicable shear rates, either before the maximum torque is reached or before side effects like sample loss at the rim, melt fracture, and too high normal forces (modifying the gap geometry) come into play. In conclusion, a map exists for each rheometer, tool geometry, and type of material (!), indicating the regime of reliable viscosity measurements (Fig. 9). This map can be explored using reference liquids with known viscosity or viscosity function $$\eta \mathrm{(}\dot{\gamma}\mathrm{)}.$$ Typically, a shear rate (or shear stress) sweep is performed on each of the reference liquids. The resulting viscosity function is compared to the specification of the reference material.

Fig. 9 Schematic of effects that limit the reliable measurement range of a rotational rheometer. There may be additional effects not listed in the scheme.

Figures 10 and 11 present examples for the Haake Viscotester 500 (a strain-controlled viscometer mainly for quality control applications). The manual specifies for the angular velocity range: (0.5–800) rpm [(0.52–84) rad·s^{–1}] and for the torque range: (0.0001–0.03) N m. Two different CP systems were used, one with 28 mm diameter and the cone angle *α* = 1° (PK1, 1°) and another one with 50 mm diameter and the same cone angle (PK5, 1°). Based on the velocity and torque ranges, the expected limits for shear rate and shear stress were calculated ().

Fig. 10 Viscosity versus shear rate using the Haake Viscotester 500 for the viscosity reference materials listed in , see the small numbers in italics in the figure. Dotted lines represent the specified viscosity of the standards. Full and broken lines indicate the nominal measurement regimes for CP tools PK1, 1° and PK5, 1°, respectively (see ). Full symbols stem from PK1, 1°, and the unfilled symbols represent PK5, 1°.

Fig. 11 Viscosity versus shear rate using tool PK1, 1° for three of the reference materials in more detail (two runs for each sample). For each sample, the arrows indicate the shear rate regime in which the measured viscosity deviates <10 % from the specified value of the standard (so-called “good data” shown in Fig. 10).

Table 2 Nominal shear rate and shear stress limits of the Haake Viscotester 500 for two different plate diameters and cone angle 1° based on the angular velocity and torque specifications of the rheometer.

summarizes the list of Newtonian liquids used to check these theoretical limits. Not all of them are certified viscosity standards. For instance, the Baysilone^{®} oils are not. However, the producer provides a viscosity specification.

Table 3 List of Newtonian liquids used to check rheometer limits.

In a plot of viscosity versus shear rate (see Fig. 10) these theoretical limits from define two boxes with vertical left and right borders (here identical for both geometries because of the constant cone angle) and upper and lower borders of slope –1 in the double logarithmic representation (here different for the two geometries). The tools using the smaller diameter (PK1, 1°) are represented by the full lines, tools PK5, 1° by broken lines. Only the regimes of “good data” (see Fig. 11) for the various liquids and the two tool geometries are shown in the figure. The specified viscosities of the viscosity reference materials are represented by dotted lines. At least for the PK1, 1° tool, the upper stress and shear rate limits fit the instrument specification of the manufacturer. Obviously, the rheometer provides reasonable data down to substantially lower shear rates than those indicated in the instrument specification!

Figure 11 shows in more detail the viscosity versus shear rate data for three of the reference materials, measured with geometry PK1, 1° over a broad shear rate range. For each fluid, a shear rate range can be defined, where the measured viscosity agrees with the specified value of the standard within ±10 % of its specified viscosity (so-called “good data”).

Compared to measurements of the steady-state shear viscosity function $${\eta}_{s}\mathrm{(}\dot{\gamma}\mathrm{)},$$ where the control parameter is the shear rate and the viscosity is the target quantity, the situation becomes more complex in small amplitude oscillatory shear experiments. In principle, there are now two control parameters: the angular frequency *ω* and the shear amplitude *γ*_{0}. The target quantities are the absolute value of the complex modulus ∣*G*^{*}(*ω*)∣, – which, divided by the angular frequency, yields the absolute value of the complex viscosity ∣*η*^{*}(*ω*)∣ – and the phase angle *δ*. Alternatively, the storage and loss moduli *G*′(*ω*) and *G*″(*ω*) can be evaluated. The limits of the ∣*G*^{*}(*ω*)∣ measurement depend on the torque range and bearing friction of the instrument. Furthermore, it is required that the shear amplitude is chosen such that the measurements are performed in the linear viscoelastic range.

Each rheometer allowing oscillatory measurements can cover only a certain frequency range of operation, *ω*_{min} to *ω*_{max}. The upper machine limit *ω*_{max} is governed by the inertia of the moving parts, the maximum torque imposed by the motor, and also the sampling rate and response time of the drive control. The lower machine limit *ω*_{min} depends on the angular resolution and the minimum controllable angular velocity. In reality, there are additional effects that may significantly reduce the frequency range in which reliable data are obtained. As in steady shear experiments, one limitation in the low-frequency range can be that the oscillatory torque is too small. Although oscillatory measurements provide the advantage of detecting the oscillating torque even if there is a high level of noise, there remains a practical limit of resolution. Furthermore, the necessary measurement time may also limit the range of accessible low frequencies. Assuming that the electronics require half a cycle to evaluate amplitude and phase angle of a sine wave, the necessary measuring time for *ω* = 0.01 rad·s^{–1} is calculated to be *t*=314 s. Furthermore, a stationary state in the oscillation experiments has to be achieved. For *ω* = 0.001 rad·s^{–1}, the measurement time increases to 53 min! This effect is on one hand relevant for samples that are not stable over the required time at measurement temperature and environment, and on the other hand one tends to avoid long measurement times in routine testing. Typically, a sweep from high to low frequencies will be used, and the total measurement time is determined by the number of data points per decade and the lowest frequency selected.

The measurement times in the high-frequency regimes are conveniently short, and the torque amplitudes are high. By reducing the shear amplitude, too high torques may be avoided. Principally, this should allow data acquisition with high accuracy. Two effects, however, need to be taken into account. Firstly, if low-viscosity liquids are submitted to high-frequency oscillatory shear, inertia may cause the shear strain in the sample to deviate from the linear vertical profile (the sample “swashing back and forth”). This leads to an apparent increase in the storage modulus *G*′ [35]. In a PP configuration, this effect may be reduced by reducing the gap height. Secondly, the torque transducer has a certain angular compliance. The part of the tool attached to the transducer does not remain stationary, but exhibits an angular deflection proportional to the momentary torque. As a consequence, the true shear in the sample is smaller than the nominal one, causing an apparently smaller modulus ∣*G*^{*}∣. Furthermore, there occurs an error in the measured phase angle. A subsequent correction of the measured data from oscillatory shear is possible, provided the instrument compliance is known [36].

In summary, the map for reliable ∣*G*^{*}∣ or ∣*η*^{*}∣ measurements is of larger dimensions than for steady shear viscosity, and a complete exploration becomes rather cumbersome. Some methods are available for checking data quality and accessible ranges for reasonable test times:

Choice of the optimum tool dimensions: The accessible measurement range is strongly influenced by the tool geometry. To reach higher or lower stress levels in the sample, the tool diameter should be chosen accordingly. When using very small tool dimensions and a tiny sample volume, however, the influence of geometry misalignment and errors in loading becomes more severe. A plate diameter of about 5 mm appears to be the lower limit for most commercial rheometers and setups, but depending on the alignment, still represents an acceptable compromise for reliable measurements. The diameter can be increased to measure low-viscosity samples. By comparing results from various geometries and measurement modes, one may also estimate the magnitude of errors caused by misalignment, sample loading, etc.

Improving the signal-to-noise ratio: This can be achieved either by gradually increasing the imposed strain or stress amplitude or by subsequent statistical data analysis. Initially, the imposed amplitudes are chosen to be in a medium range. Then they can be optimized for the specific measurement task. By repeating an experiment several times and averaging the data, it is possible to evaluate and improve noisy data. For example, in oscillatory experiments the number of cycles and data points per cycle can be increased and the applied shear amplitude adjusted on a frequency-dependent basis (“auto-strain technique”). When the latter method is applied, care has to be taken to avoid increasing the amplitude beyond the limit of linear viscoelastic behaviour.

Time-temperature superposition: This method is often used to determine a material function for a reference temperature *T*_{0} based on measurements at various temperatures *T* [8], but over a limited frequency range. The master curve for *T*_{0} typically provides extended low-frequency data in less time and with higher accuracy than a direct measurement at the reference temperature itself. Furthermore, an inconsistent overlap of the master curve may indicate inaccurate instrument calibration. In principle, only insufficient thermal stability will limit this procedure for fluids in which the microscopic structure does not change with temperature. Some care needs to be taken when testing at various temperatures to avoid errors caused by poor sample temperature equilibration or thermal expansion. The latter effect may be compensated in PP geometry simply by adjusting the gap height via a measurement of the normal force. Some rheometers can do that automatically.

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