## Abstract

The rheological characterization of glass-forming liquids is challenging due to their extreme temperature dependence and high stiffness at low temperatures. This study focuses on the special precautions that need to be taken to accommodate high sample stiffness and torsional instrument compliance in shear rheological experiments. The measurement errors due to the instrument compliance can be avoided by employing small-diameter parallel plate (SDPP) rheometry in combination of numerical instrument compliance corrections. Measurements of that type demonstrate that accurate and reliable rheological data can be obtained by SDPP rheometry despite unusually small diameter-to-gap (*d*/*h*) ratios. Specimen preparation for SDPP requires special attention, but then experiments show excellent repeatability. Advantages and some current applications of SDPP rheometry are briefly reviewed. SDPP rheometry is seen as a simple and versatile way to measure rheological properties of glass-forming liquids especially near their glass transition temperature.

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It is also possible to estimate the value of

*J*_{ i }by performing theoretical calculations. However, experience has shown that these calculations are rather complicated and do not always provide a reliable estimate of*J*_{ i }. Therefore, it is often recommended to determine*J*_{ i }experimentallyAs of writing, it is often recommended that the default compliance values for the motor shaft and the temperature control unit (provided by the rheometer manufacturer) should not be modified by the rheometer user. In case there are any inaccuracies in these default compliance values, those are taken into account in the measured compliance value for the measurement fixture.

The rheological studies of McKenna et al. and Pogodina et al. were performed mainly with 8-mm-diameter parallel plate geometries that are not ideal for the measurement of rheological properties at very high stiffness levels (parallel plates with an even smaller diameter are typically preferred). However, they carefully corrected their measurement data for instrument compliance and therefore obtained reliable rheological data even when |

*G**| > 1 GPaTo be accurate, this is strictly true only for single-head rheometers, in which the effect of the instrument compliance is instantaneous and fully recoverable (Läuger 2010). In the case of dual-head rheometers, on the other hand, the control routines for the torque transducer will lead to an additional viscous compliance contribution. It has been experimentally shown that the inelastic compliance contribution may result in a notable frequency dependence of instrument compliance (Farrar et al. 2015). However, the more complicated instrument compliance analysis of dual-head rheometers is omitted here due to the fact that the vast majority of modern rotational rheometers are single-head rheometers (however, an interested reader is referred to Franck (2006)).

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## Acknowledgements

Financial support from the Vilho, Yrjö, and Kalle Väisälä Foundation is gratefully acknowledged. The author also wishes to thank Anton Paar GmbH and Malvern Instruments Ltd. for technical support and equipment loans, Prof. Gregory McKenna and Dr. Stephen Hutcheson for providing plate diameter-dependent rheological data of glycerol, and Prof. H. Henning Winter for helping in the preparation of the manuscript.

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## Appendices

### Appendix A

### Derivation of the instrument compliance correction equations

The angular displacement measured at the optical encoder (*θ*
_{
m
}) is composed of the sample angular displacement (*θ*
_{
s
}) and the instrument angular displacement (*θ*
_{
i
}).

The torsional stiffness (*K*) is defined as the ratio of the angular displacement (*θ*) and the torque (*M*):

By combining Eqs. (5) and (6), the relationship between the measured torsional stiffness (*K*
_{
m
}) and the torsional stiffness of the sample (*K*
_{
s
}) and of the instrument (*K*
_{
i
}) can be written as follows:

where the torsional stiffness of the instrument is the inverse of the instrument compliance:

By rearranging Eq. (7), the torsional sample stiffness can be obtained as:

The torsional stiffness (*K*) can be converted into the shear modulus (*G*) by introducing the geometry conversion factor (*k*
_{
g
}):

For a parallel plate geometry with the plate radius *R* and the gap between the plates *h*, the geometry conversion factor is calculated as follows:

Due to the viscoelastic nature of samples, the measured and sample stiffness and moduli are written in the complex form hereafter. On the other hand, the instrument stiffness and modulus are scalars since the rheometer setup can be assumed to be purely elastic.^{Footnote 4} The substitution of Eqs. (8) and (10) into Eq. (9) yields:

which can also be written in terms of the storage (*G′*) and loss (*G″*) moduli:

Finally, the storage and loss moduli of the sample can be solved from Eq. (13):

and the loss tangent of the sample becomes:

### Appendix B

### Analysis of the effective measurement range of SDPP rheometry

The effective measurement range of SDPP rheometry is limited at low modulus values by the minimum torque of the rheometer. According to Ewoldt et al. (2015), the minimum measurable viscoelastic moduli *G*
_{min} in oscillatory experiments is defined by the following expression:

where *F*
_{
σ
} is a geometry factor that correlates the applied torque *T* to the shear stress *σ*, *T*
_{min} is the minimum torque of the rheometer in oscillation, and *γ*
_{0} is the strain amplitude. *G*
_{min} refers to either *G′* or *G″*, whichever is smaller. For the parallel plate geometry:

where *R* is the plate radius. Figure 7 shows dynamic oscillatory data for a petroleum oil, measured with 4-mm-diameter parallel plate geometry. This data set was measured with a stress-controlled Anton Paar MCR 301 rheometer, having a minimum torque limit *T*
_{min} = 0.01 μNm in oscillation. The strain amplitude was varied in the range of 0.01 to 0.05%, being 0.05% at the highest measurement temperatures (corresponding to the lowest moduli values). Therefore, using Eqs. (14) and (15), the minimum measurable moduli can be calculated to be *G*
_{min} = 1.59 kPa in this case. The dynamic moduli master curves of Fig. 7a demonstrate the good quality of the 4-mm-diameter parallel plate data down to this *G*
_{min} value, the terminal slopes of *G′* and *G″* being close to 2 and 1, respectively. However, it should be kept in mind that the value of *G*
_{min} varies on a case-by-case basis, depending on *T*
_{min} and *γ*
_{0}. The low-torque sensitivity limit can be lowered, when possible, by using a more sensitive rheometer (lower *T*
_{min} limit) and/or by increasing *γ*
_{0}. Eventually, larger parallel plate geometries need to be used if glass-forming liquids are to be characterized in a low-viscosity liquid state (below *G*
_{min} limit).

The upper stiffness limit of the effective measurement range of SDPP rheometry is practically determined by the recommendation that the ratio between the angular displacement due to instrument compliance and the angular displacement due to sample deformation should be less than 10. If we assume typical values of *J*
_{
i
} = 0.01925 rad/Nm, *R* = 2 mm, and *h* = 1.5 mm in Eq. (12), the critical |*G**| value at which the aforementioned ratio becomes equal to 10 is 27.9 GPa. This value is much higher than the glassy modulus of any known glass-forming liquid. Therefore, in most cases, there is no practical upper stiffness limit for using SDPP rheometry.

Figure 7b shows this same data set plotted in the van Gurp-Palmen plot of phase angle versus log|*G**|. The purpose of this plot is to demonstrate that even very liquid-like samples can be characterized by SDPP rheometry, i.e., phase angle values close to 90° can be reliably measured. Therefore, it can be concluded that this technique is applicable for the rheological characterization of glass-forming liquids even well above the glass transition temperature, which is not possible by using torsion bar or DMA techniques.

Another factor limiting the effective measurement range of oscillatory rheological experiments is instrument inertia (Läuger and Stettin 2016). In the case of SDPP rheometry, the effect of instrument inertia is magnified due to the small plate diameter and large gap. Although all modern rheometer models correct measurement data for instrument inertia, inertial errors may still exist due to the inaccuracies in the instrument inertia determination. The effect of imperfect instrument inertia corrections on the effective measurement range can be assessed from the following equation (Ewoldt et al. 2015):

where *G*
_{iner,inst} is artificial inertial moduli, *ε* is the error in the instrument inertia value used for the corrections, *F*
_{
γ
} is a geometry factor that correlates the applied angular deformation *θ* to the shear strain *γ*, *I* is the combined moment of inertia of the rotating parts of the rheometer (motor, air bearing, optical encoder, motor axis with geometry coupling, and measuring geometry), and *ω* is the angular frequency. In the case of the parallel plate geometry:

Since *G*
_{iner,inst} ∼ *ω*
^{2}, the instrument inertia effects are most pronounced at high frequencies. For example, if there is a 1% error in the instrument inertia corrections (*ε* = 0.01) and we assume typical values of *I* = 13.4 μNm.s^{2}, *R* = 2 mm, and *h* = 1.5 mm, the artificial inertial moduli obtain a value of *G*
_{iner,inst} = 79.8 kPa at *ω* = 100 rad/s. For any practical purposes, this modulus contribution can be considered negligible as compared to the true material moduli (typically, at this frequency, *G′* and *G″* values are in excess to 1 MPa for materials that are to be characterized by SDPP rheometry). However, it should be noted that instrument inertia values can vary significantly depending on the rheometer model, and therefore the inertia limits for the effective measurement range should be evaluated on a case-by-case basis.

Lastly, the effective measurement range of oscillatory experiments may be affected by fluid (sample) inertia (Läuger and Stettin 2016). In order to be sure that fluid inertia does not influence rheological measurement data, the wavelength *λ*
_{
s
} of a propagating shear wave should be much larger than the geometry gap *h*. The wavelength of a linear viscoelastic shear wave between a moving boundary and a fixed reflecting boundary can be calculated as follows (Ewoldt et al. 2015):

where *ρ* is the density of the sample. Linear viscoelastic wave propagation has been analyzed in detail by Schrag (1977) who suggested the criterion *λ*
_{
s
}/*h* ≥ 40 for high-precision measurements in which errors must be less than 1% in |*G**| and l° in phase angle. By substituting the Schrag criterion into Eq. (18) and by rearranging it, we obtain:

As the critical value for |*G**
_{iner,fluid}| scales with *ω*
^{2}, the fluid inertia effects are most pronounced at high frequencies. If we assume typical values of *δ* = 89°, *ρ* = 1000 kg/m^{3}, and *h* = 1.5 mm, the criterion for having fluid inertia-free measurement data becomes |*G**
_{iner,fluid}| > 464 Pa at *ω* = 100 rad/s. Again, this modulus limit is significantly lower than the |*G**| values typically measured by SDPP rheometry at *ω* = 100 rad/s. Therefore, the fluid inertia effects can be considered negligible in SDPP experiments.

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Laukkanen, OV. Small-diameter parallel plate rheometry: a simple technique for measuring rheological properties of glass-forming liquids in shear.
*Rheol Acta* **56**, 661–671 (2017). https://doi.org/10.1007/s00397-017-1020-5

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DOI: https://doi.org/10.1007/s00397-017-1020-5