Rheol Acta (2011) 50:537–546 DOI 10.1007/s00397-011-0560-3
ORIGINAL CONTRIBUTION
Instrument compliance effects revisited: linear viscoelastic measurements Chen-Yang Liu · Minglong Yao · Ronald G. Garritano · Aloyse J. Franck · Christian Bailly
Received: 21 March 2011 / Revised: 22 April 2011 / Accepted: 13 May 2011 / Published online: 4 June 2011 © Springer-Verlag 2011
Abstract Torsional compliance of the torque transducer can be an important issue in linear viscoelastic (LVE) measurements when the sample stiffness is high relative to the instrument stiffness. We evaluated compliance effects of the ARES 2K-FRT on LVE measurements by systematically comparing the results of the frequency sweep mode obtained with 25 and 8 mm plates, respectively. In addition to the transducer, the test fixtures do contribute significantly to the system compliance. Without correction, the upper limit for the complex modulus |G∗ | is approximately close to 4 × 105 Pa at 10% uncertainty, when using 25 mm plates. This limit is lower than the plateau modulus G0N of
most polymers. Therefore, instrument compliance can lead to significant errors for G0N and wrong scaling for G in the plateau and Rouse regions. The respective roles of transducer and tool compliances are discussed. The FRT transducer compliance is corrected in real time in the instrument firmware. Tool compliance is a common problem for all rheometers when measuring stiff samples. Keywords Compliance · Dynamic moduli · Viscoelasticity · Rheometer
Introduction
This paper is dedicated to Professor Helmut Münstedt, Friedrich-Alexander Universität Erlangen-Nürnberg on the occasion of his 70th birthday. C.-Y. Liu (B) Beijing National Laboratory for Molecular Sciences, CAS Key Laboratory of Engineering Plastics, Joint Laboratory of Polymer Science and Materials, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China e-mail:
[email protected] C.-Y. Liu · C. Bailly Unité de Chimie et de Physique des Hauts Polymères, Université catholique de Louvain, 1348 Louvain-La-Neuve, Belgium M. Yao · R. G. Garritano TA Instruments, 109 Lukens Drive, New Castle, DE 19720, USA A. J. Franck TA Instruments, Helfmann Park 10, 65760 Eschborn, Germany
Linear viscoelasticity (LVE) measurements are widely used for the study of polymer dynamics (Ferry 1980). It determines the relaxation of polymer stress, and is closely related to the molecular structure such as molecular weight, molecular weight distribution, and molecular architecture (star, H-, comb, ring. . . ) (Graessley 1974, 1982; Watanabe 1999; McLeish 2002). Thus, viscoelasticity plays an essential role in linking polymer structure and physical processes. High-precision experiments will provide the foundation for physical theories or models of polymer dynamics. Torsional compliance of the torque transducer can be an important issue in LVE measurements when the sample stiffness is high relative to the instrument stiffness. When the commended strain is imposed on a sample, the true strain on the sample may be lower than the measured strain due to the tools or transducer compliances. The errors of strain measurements can lead to significant errors in the LVE properties. For example, Fetters et al. (1991) showed that there were different
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results obtained with plates of different diameter for the same sample (see Fig. 1 reproduced from Fetters et al. 1991 (Fig. 5)). Geometry dependence of LVE tests is one evident characteristic of instrument compliance problems. Since the sample stiffness is a function of the sample geometry and is proportional to 4-th power of plate diameter, instrument compliance effects are much larger for 25 mm plate than for 8 mm plates. Since the absolute values and slopes (as a function of frequency) of the elastic modulus G and the loss modulus G in the plateau and Rouse regions have important physical meaning for polymer dynamics in the melt, an accurate measurement is a prerequisite to study polymer dynamics. Gottlieb and Macosko (1982) previously discussed the effect of instrument compliance (shear and compression) on dynamic rheological measurements and possible corrections for mechanical transducers. They extensively analyzed problems in dynamic measurements with the eccentric rotating disks rheometer that was the most common at that time. However, for the forced oscillation rheometer, they used an approximation for the compliance correction which underestimates the compliance effects. In fact, the error correlates with the ratio of the sample stiffness and the instrument stiffness, and is independent of the command stain or the measured torque. At the same time, Sternstein (1983) discussed the compliance error and correction (tension or shear) of LVE measurements for solids. For the force rebalance torque (FRT) transducers, Mackay and Halley (1991) found that the angu-
Fig. 1 Instrument compliance observed for the LVE test performed with 25-mm parallel plates [from Fetters et al. (1991); reprinted from Macromolecules with permission of the American Chemical Society]
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lar compliance of the FRT could severely affect the dynamic moduli, because no feedback control servo system is instantaneous, and a combination of high frequency and torque (actually high stiffness of sample) can lead to FRT compliance errors. To our knowledge, no detailed description of the instrument compliance effects for both test fixtures and FRT exists. In the present paper, we revisited the instrument compliance effects of the ARES 2K-FRT rheometer on LVE measurements by systematically comparing the results of the frequency sweep mode obtained with 25 and 8 mm plates, respectively. In the next section, we describe the samples and experimental conditions. Then, we present experimental phenomena of compliance effects—geometry dependence of frequency sweep measurements in “Geometry dependence of LVE measurements”. “Instrument compliance” extensively analyzes the instrument compliance problems for both, test fixtures and FRT, and discusses the limitations for different geometries. Compliance correction procedures are applied to dynamic moduli data. “Importance of instrument compliance problems for measurements on polymers” provides some examples of the importance of instrument compliance problems for measurements on polymer by analyzing some results in the literature.
Experiment Two commercial polymers were used to perform linear viscoelasticity measurements. Polyisobutylene (PIB, Oppanol B 50: Mv = 400 kg/mol, Mv/Mn = 3.3) was kindly supplied by BASF AG (Dr. H.M. Laun). Polycarbonate (PC, A-2700: Mw = 35 kg/mol, Mw/Mn = 2.1) was supplied by Idemitsu Petrochemical Co. (Liu et al. 2004). The measurements were made using a TA ARES rheometer which is equipped with force rebalance torque transducer (2K-FRTN1) that can detect torques within the range 0.002–200 m Nm. An air oven with liquid N2 cooling covers a broad temperature range of −150◦ C to 600◦ C. The motor is a low shear motor with a frequency range from 10−5 to 500 rad/s and an angular displacement range deformation of 0.005 to 500 mrad. Frequency sweep were performed under nitrogen atmosphere between 0.1 to 100 rad/s at −20◦ C for PIB with 25- or 8-mm diameter parallel plate geometry, respectively. A gap is 1 mm was selected and the strain amplitude was below 5%. Frequency sweeps were performed between 0.1 to 100 rad/s at 155◦ C for PC with 8-mm diameter plates, but with different gap (0.6 vs. 2.6 mm).
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Data of PIB at −20 C with 25- and 8-mm diameter parallel plate geometry are presented in Fig. 2. Since the sample stiffness is proportional to 4-th power of the plate diameter, the influence of instrument compliance can be neglected in the case of 8 mm plates, and the measured strain is almost equal to the command strain and independent of frequency. Therefore, the data obtained with 8 mm plates may be referred to as “true” sample data. For the data with 25 mm plates, the measured strain deviates from the command strain at high modulus in Fig. 2a, as a result of FRT compliance (will be discussed in “Influence of FRT compliance on LVE”). At the same time, the absolute values of the complex modulus G∗ are smaller than those obtained
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Fig. 3 Geometry dependence of LVE test for PC A-2700 at 155◦ C by using 8 mm diameter: 2.6 mm vs 0.6 mm gap. a Complex modulus |G∗ | and measured strain γ . b Elastic modulus G and loss modulus G
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ω (rad/s) Fig. 2 Geometry dependence of the LVE test for BASF PIB50 at -20◦ C: 25 mm diameter (f illed symbols) vs. 8 mm diameter (open symbols and line). a Complex modulus |G∗ | and measured strain γ . b Elastic modulus G and loss modulus G . There is a crossover point between G and G for the data with 8 mm plates but not for the data with 25 mm plates
from tests performed with 8 mm plates. Figure 2b shows the data of the elastic modulus G and the loss modulus G . G is slightly underestimated and G is severely underestimated as well as tan δ = G /G in the experiments performed with 25 mm plates, identical to the findings in Fig. 1 (Fetters et al. 1991). Therefore, there is a crossover point between G and G for the data with 8 mm plates but not for the data with 25 mm plates. The crossover point is connected to the basic relaxation time τ e in tube models (McLeish 2002; Liu et al. 2007). A similar geometry dependence of LVE measurements can be found in Fig. 3. In this case PC (polycarbonate) samples were measured at 155◦ C with 8-mm diameter plates, but with different gap settings (0.6 vs. 2.6 mm). The measured strain is smaller than the command strain at high modulus for both gap settings, which implies that both measurements need to be corrected for compliance. In summary, all results in
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T
Figs. 1–3 indicate an effect of instrument compliance on the measured data. Z
Instrument compliance β
To detect the torque in a sheared sample, traditional rheometers are usually equipped with torque transducers using a compliant member such as a torsion spring. Even very stiff torsion bars must twist slightly to record the torque (Gottlieb and Macosko 1982). The angular movement is defined as angular compliance. For stiff samples, this spring compliance can lead to significant errors in the strain imposed on a sample, hence in the LVE properties. The most popular rheometer ARES (rheometrics/TA) is equipped with the force rebalance torque transducer. The instrument compliance for this configuration is more complex. First, the torque transmitted through the rotating shaft lead to a slight angular twist of the “tools” (including the instrument frame and the test fixtures) and the error due to the tool compliance is similar to the error of the spring compliance for mechanical transducers. Although the transducer deflection can be eliminated by a feedback control servo in the FRT transducer, the servo system is not instantaneous and leads to FRT transducer compliance (Mackay and Halley 1991). Since the time response of the servo system is a function of frequency and stiffness of the sample in an oscillation measurement, a complex compliance (rather than a real number) is required to describe and correct the FRT compliance. Because the test fixtures are metal and only very slight angular displacements are imposed, a reasonable assumption is that the test fixtures behave purely elastic during LVE measurements. Hence, tool compliance Kθ is a real number (contrarily, to a complex number for FRT compliance) and the angular displacement of the test fixture is in phase with the torque. Tool compliance is a common problem for all rheometers (including DMA) and therefore the simple case—the influence of tools compliance on LVE, is discussed first. Influence of tool compliance on LVE: limits and corrections A phasor diagram of angular displacement and torque is given in Fig. 4 (Sternstein 1983). We assume (quasi-)infinite stiffness for the FRT transducer here— thus the transducer deflection can be neglected. The total angular displacement is X, the tool angular displacement due to the tool compliance Kθ is Z, and the
δ
X
Y=X–Z Fig. 4 The vector relationships between the torque T, the measured angular displacement X, the sample (tool corrected) angular displacement Y, the tools angular displacement Z, the apparent (measured) phase angle β, and the true phase angle δ, are shown
true angular displacement for the sample is given by Y, where Y=X−Z
(1)
and X, Y, and Z are complex numbers. Since the tool compliance Kθ is a real number, the tool angular displacement Z is in phase with the torque T and not the total angular displacement X as shown in Fig. 4. Z is given by Z = Kθ T = Kθ S∗ X = Kθ BG∗ X (2) S∗ is the “apparent” dynamic stiffness, B is the sample geometry factor, and G∗ is the apparent (measured) dynamic modulus (= G + iG ). For the case of parallel plates, B = π R4 /32h, R and h are the plate diameter and gap. In addition, BG∗c Y = T (3) and the true (corrected) dynamic modulus G∗c (= Gc + iGc ) is obtained by solving Eqs. 1–3 (Sternstein 1983), G 1 − Kθ BG − G Kθ BG (4) Gc = D Gc =
G D
tan δ =
tan β 1 − Kθ BG 1 + tan2 β
(5)
(6)
where β is the apparent (measured) phase angle, and δ is the true phase angle as shown in Fig. 4, and where 2 2 (7) D = 1 − Kθ BG + Kθ BG
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and |Y| √ = D |X|
(8)
The true sample displacement amplitude is always smaller than the apparent (measured) one, therefore the absolute value of G∗ is always smaller than the absolute value of Gc∗ . The compliance correction factor can be defined as the ratio of the apparent sample stiffness BG∗ (or the components of BG and BG ) and the tool stiffness 1/Kθ . If accurately known, the tool compliance can be readily corrected for. Usually, the compliance correction is small, but can become significant when the product of the sample stiffness and test fixture compliance is large. Note that the correction factors are independent of torque, which is in contrast with the view of Mackay and Halley (1991). The correction for the loss modulus in Eq. 5 is quite simple. The loss modulus may be severely underestimated, if the compliance correction is not performed. On the other side the correction for the elastic modulus is relatively small, since the numerator in Eq. 4 is always smaller than the apparent elastic modulus G , and the denominator D is smaller than 1. These results are consistent with experimental observations in Fig. 2b. Finally, the correction for loss factor tanδ is nonlinear with respect to the apparent loss factor tanβ. Since the sample stiffness is the product of the sample geometry factor and the dynamic modulus, changing the sample geometry may eliminate the compliance effect (see Figs. 1–3). To estimate the upper test limits of the complex modulus G∗ we refer to the most common geometry used for LVE measurements— 25-mm diameter plates and 1 mm gap. In the plateau region for polymer melts, the samples are highly elastic (low loss), so G >> G and the absolute value of G∗ ∼ G . If the tool compliance Kθ is 7.0 mrad/Nm (a procedure to estimate the tools compliance Kθ is suggested below), G = 0.4 MPa and according to Eq. 7, D ∼ 0.8. Therefore, the measured loss modulus G is 20% lower than the true value in Eq. 5. This is true for the data at frequencies above 1 rad/s in Fig. 2. Strictly speaking, the upper limit for LVE measurements is below 0.4 MPa, when using 25 mm diameter plates and 1 mm gap. It is surprisingly low, and is smaller than the plateau modulus G0N of most polymers (PI 0.4 MPa, PBD 1.15 MPa, PE 2.0 MPa, and PC 2.3 MPa). Therefore, neglecting tool compliance can lead to significant errors in G0N (Liu et al. 2006a) and wrong scaling of G in the plateau and Rouse regions. Some examples in literature are shown in “Importance of instrument compliance problems for measurements
on polymers”. When using 8-mm diameter plates and 1 mm gap, because the sample stiffness is proportional to 4-th power of plate diameter, the upper limit of the modulus is about 100 times higher than for 25-mm diameter plates, i.e., 40 MPa. It is worth noting that D and tanδ are the function of G and G instead of |G∗ | according to Eqs. 6 and 7. This means that corrections are slightly different even if two samples have the same |G∗ | but different G and G . Therefore, the |G∗ | limits are only approximate values, and the correction factor (1/D) for G is always larger than the correction factor (1/D1/2 ) for |G∗ |. Tool compliance correction is reliable only if Kθ is known accurately. Since the sample stiffness when using 8-mm diameter plates is about 1% of the value when using 25-mm diameter plates, the influence of tool compliance can be neglected in the range below 40 MPa. The data obtained with 8 mm plates therefore can be used as a calibration reference. With Eqs. 4 and 5, the tool compliance Kθ can be calculated from the true elastic modulus Gc and loss modulus Gc and the measured G and G obtained with 25 mm plates. Comparing the corrected data with the reference data (8 mm), a compliance value of 7 mrad/Nm is obtained (Fig. 5a). The corrected data match very well with the data obtained with the 8 mm plates. There is a slight difference only when the G is higher than 2 MPa, as the correction factor is large in this range. Measured G and G using 8 mm plates are also compared with the corrected values (8 mm) using Eqs. 4 and 5 in Fig. 5b. As expected, there is no difference between the two data sets because the modulus is much smaller than 40 MPa. Lastly, correction factors for |G∗ | (as |G∗ |/|Gc∗ |), G (as G /Gc ), and the phase angle δ (as δ – β) vs |Gc∗ | for data with 25 mm plates are given in Fig. 5c. It is clear that without correction, the upper limit for the complex modulus |G∗ | is close to 4 × 105 Pa at 10% uncertainty for this sample, when using 25 mm plates, and the correction factor for G is much larger than that for |G∗ | in the high modulus region. Old data obtained without the tool compliance correction can be easily corrected with this procedure; the one condition is that the tool compliance Kθ is known accurately. It is worth noting that in Fig. 3 the data depend on geometry, even when using 8 mm plates with different thickness (2.6 vs. 0.6 mm), because in the glass region the 40 MPa limit for the modulus is exceeded. The measured G and G with 2.6 mm are compared with corrected values using Eqs. 4 and 5 and Kθ = 9.0 mrad/Nm in Fig. 6a. There are evident differences between the corrected data and the measured data. The compliance correction is required even with 8 mm plates in the glass
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Fig. 5 Tool compliance correction for PIB-50 data (Fig. 2b) using Eqs. 4 and 5 and a tools compliance of Kθ = 7.0 mrad/Nm. a Corrected G and G with 25 mm plates vs. measured G and G with 8 mm plates as a calibration reference. b Corrected G and G with 8 mm plates vs. measured G and G with 8 mm plates. c Corrections for |G∗ |, G , and the phase angle δ as a function of |G∗c | for the data with 25 mm plates
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Fig. 6 Tool compliance correction for PC-A2500 data with 8 mm plates (Fig. 3b) using Eqs. 4 and 5 and a tools compliance of Kθ = 9.0 mrad/Nm. a Corrected G and G with 2.6 mm gap vs. measured G and G with 2.6 mm gap. b Corrected G and G with 0.6 mm gap vs. corrected G and G with 2.6 mm gap. c Corrections for both of |G∗ |, G , and the phase angle δ as a function of |G∗c | for the data with 2.6 mm gap
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T
Y = X − Z
Z’
(9)
with
Y’ = X – Z’ Fig. 7 The vector relationship among the torque T, the measured angular displacement X, the sample (FRT corrected) angular displacement Y , the FRT angular displacement Z , the FRT displacement phase angle β , are shown. Note the FRT angular displacement Z is not in phase with the torque T
region, because the sample apparent stiffness is close to the tool stiffness. Corrections also are performed for the measured G and G with 0.6 mm using Kθ = 9.0 mrad/Nm, as shown in Fig. 6b. The corrected G and G for the two data sets with different gaps are in good agreement between each other, which implies that the selection of Kθ = 9.0 mrad/Nm is proper. The tool compliance Kθ for 8 mm plates is slightly larger than 7.0 mrad/Nm for 25 mm plates, since the end part of 8 mm plates is smaller. Correction factors for |G∗ | (as |G∗ |/|Gc∗ |), G (as G /Gc ), and the phase angle δ (as δ – β) vs |Gc∗ | for the data with 2.6 mm gap are also given in Fig. 6c. Without correction, the upper limit for the complex modulus |G∗ | is more than 0.1 GPa at 10% uncertainty for this PC sample, when using 8 mm plates, but G has a larger error here. A non-linear behavior is observed for the correction of the phase angle δ in Fig. 6c, which can be explained by Eq. 6. The important physics implication of the correction in the glass region will be discussed in “Importance of instrument compliance problems for measurements on polymers”. Influence of FRT compliance on LVE A key advantage of FRT transducers is their wide torque range, of approximately six decades, with good low torque resolution, 10−7 Nm or less. However, FRT transducer compliance cannot be omitted at high frequency and high sample stiffness (Mackay and Halley 1991), because the servo system does not respond instantaneously. The FRT compliance is complex since the time response of the servo is a function of frequency and stiffness. A phasor diagram of angular displacements and torque is given in Fig. 7. In case the compliance of the
Z = K FRT T
(10)
Since the FRT compliance K FRT is a complex number, the FRT angular displacement Z is not in phase with the torque T, and the FRT displacement phase angle β is not same as the apparent (measured) phase angle β. In the ARES rheometer, a real-time hardware compliance correction is applied. This means that the transient sample deformation is calculated by subtracting the transducer displacement from the motor angular
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ω (rad/s) Fig. 8 a Normalized strains are compared with each other for the data of PIB-50 at −20◦ C with 25 mm plates (Fig. 2a). Command strain (as 100%), measured strain (with FRT compliance correction), corrected strain (with tool correction). b Ratio of strains
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displacement in the time domain. The tool compliance is taken into account in order to obtain the true sample deformation according to “Influence of tool compliance on LVE: limits and corrections”. The complex transfer functions for the sample and the FRT transducer are obtained by correlating the FRT displacement with the motor displacement and the FRT power consumption with the FRT displacement. Knowing the FRT phase and magnitude, the correct sample modulus and phase can be calculated using simple vector arithmetic. In Fig. 8a, the command strain (as 100%), the measured strain, and the corrected strain are normalized and compared to each other for the data of PIB-50 at −20◦ C obtained with 25 mm plates (Fig. 2). The measured strain obtained from the ARES hardware is
already corrected for FRT compliance. The corrected strain is a result of correction of the measured strain by using Eq. 8. The measured strain is smaller than the command strain due to the FRT compliance; the corrected strain is smaller than the measured strain due to the tool compliance. Furthermore, the magnitude of the FRT compliance becomes larger than the tool compliance, with increasing frequency, which can be observed in Fig. 8b, where the ratio of the measured strain to the command strain is smaller than the ratio of the corrected strain to the measured strain. The data from tests performed with 25 mm plates in Fig. 5a are corrected for both FRT and tools compliances. Since values of FRT and tool compliances are of the same magnitude (Fig. 8b) and both FRT and
Fig. 9 Results in literature. a The master curve of poly(vinyl methyl ether) measured by using rheometrics solids analyzer (RSA-II) in the shear sandwich geometry (15.95 × 12.65 × 0.5 mm). b The master curve of polyethylene measured by using a rheometrics dynamic analyzer II (RDA II) in 25-mm parallel plate geometry with a gap of 1 mm. c The master curve of SEB measured by using rheometrics ARES rheometer in 25-mm
parallel plate geometry with a gap of about 1.5 mm. d The master curves of linear and star polybutadienes measured by using rheometrics ARES rheometer in 25-mm parallel plate geometry with a gap of about 1.3 mm. Discussion in the text [from Kannan and Lodge 1997; Wood-Adams et al. 2000; Yoshida and Friedrich 2005; Zamponi et al. 2010; reprinted from Macromolecules with permission of the American Chemical Society]
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tools compliances have hardly any influence on the LVE measurement when using 8 mm plates as shown in Fig. 2a, where the measured strain is almost equal to the command strain and independent of frequency, the good agreement between the corrected data of 25 mm plates and the reference data of 8 mm plates in Fig. 5a implies the validation of FRT transducer correction.
Importance of instrument compliance problems for measurements on polymers Without tool compliance correction, for the 25-mm diameter plates and 1 mm gap the upper limit for LVE measurements is below 0.4 MPa. Since the plateau modulus G0N of most polymers (e.g., PI 0.4 MPa, PBD 1.15 MPa, PE 2.0 MPa, and PC 2.3 MPa) is higher, omitting tool compliance correction will lead to significant errors in the plateau region and underestimate the values of G0N which is an important polymer property (Ferry 1980; Liu et al. 2006a). Furthermore, since the correction of the loss factor tanδ in Eq. 6 is nonlinear with respect to the apparent loss factor tanβ, the slopes of G data in the plateau and Rouse regions can be severely altered as shown in Figs. 1–3, and hence there is not a crossover point between G and G in the transition region for the data with 25 mm plates. Similar phenomena (reproduced in Fig. 9) are found in literature in which different rheometers were used (Kannan and Lodge 1997 (Fig. 1); Wood-Adams et al. 2000 (Fig. 18); Yoshida and Friedrich 2005 (Fig. 3a); Zamponi et al. 2010 (Fig. 6)). The slope of the G data has a significant physical meaning (Doi and Edwards 1986; Milner and McLeish 1998; Liu et al. 2006b). In addition, the crossover point between G and G is the experimental definition for the basic relaxation time τe in the frame of tube models (McLeish 2002; Liu et al. 2007). Therefore, it is a must to have correct measured data in the plateau and transition region since they provide the foundation of the physical theories or models of terminal dynamic or “Rouse like” dynamics. A similar case is related to the structural αrelaxation of glass-formers. In the glass region, G relaxation peaks are normally fitted to the Kohlrausch– Williams–Watts (KWW) stretched exponential function (Williams and Watts 1970): α t G (t) = exp − (11) τ using G (ω) = ω
0
∞
G (t) cos (ωt) dt
(12)
The fractional exponent α in the KWW expression is related to the breadth of the structural α-relaxation, and represents the non-exponential nature of the structural relaxation of the glass-former. However, the geometry dependence of oscillation tests using 8 mm plates is shown in Figs. 3 and 6a exhibiting a potential error of the G peak in the glass region. Therefore, the instrument compliance problem must be excluded before any explanation of α for the dynamic properties of glassforming substances can be presented.
Conclusions Neglecting instrument compliance can lead to significant errors in LVE measurements (especially in the transition region and glass region). Strictly speaking and without tool compliance correction, the upper limit for the complex modulus |G∗ | is close to 0.4 MPa at 10% uncertainty, when using the most common geometry—25 mm plates and 1 mm gap. Since the sample stiffness is proportional to 4-th power of plate diameter, the upper limit for 8-mm diameter plates and 1 mm gap is about 40 MPa. Tools compliance is a common problem for all rheometers when measuring the stiff samples. Proper tools compliance corrections are incorporated in the TA Orchestrator or TRIOS software. The magnitude of the FRT compliance is similar to that of the tool compliance, which can be observed in Fig. 8b. The transducer compliance is done at the instrument level (firmware) in real time. TA Orchestrator or TRIOS operation software receives the final (inertia and compliance corrected) torque and displacement amplitude and phase from the instrument processor. Acknowledgements This work has been supported by National Natural Science Foundation of China (grant no. 20874109).
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