Comparative analysis of dielectric, shear mechanical and light scattering response functions in polar supercooled liquids

The studies of molecular dynamics in the vicinity of liquid–glass transition are an essential part of condensed matter physics. Various experimental techniques are usually applied to understand different aspects of molecular motions, i.e., nuclear magnetic resonance (NMR), photon correlation spectroscopy (PCS), mechanical shear relaxation (MR), and dielectric spectroscopy (DS). Universal behavior of molecular dynamics, reflected in the invariant distribution of relaxation times for different polar and weekly polar glass-formers, has been recently found when probed by NMR, PCS, and MR techniques. On the other hand, the narrow dielectric permittivity function ε*(f) of polar materials has been rationalized by postulating that it is a superposition of a Debye-like peak and a broader structural relaxation found in NMR, PCS, and MR. Herein, we show that dielectric permittivity representation ε*(f) reveals details of molecular motions being undetectable in the other experimental methods. Herein we propose a way to resolve this problem. First, we point out an unresolved Johari–Goldstein (JG) β-relaxation is present nearby the α-relaxation in these polar glass-formers. The dielectric relaxation strength of the JG β-relaxation is sufficiently weak compared to the α-relaxation so that the narrow dielectric frequency dispersion faithfully represents the dynamic heterogeneity and cooperativity of the α-relaxation. However, when the other techniques are used to probe the same polar glass-former, there is reduction of relaxation strength of α-relaxation relative to that of the JG β relaxation as well as their separation. Consequently the α relaxation appears broader in frequency dispersion when observed by PCS, NMR and MR instead of DS. The explanation is supported by showing that the quasi-universal broadened α relaxation in PCS, NMR and MR is captured by the electric modulus M*(f) = 1/ε*(f) representation of the dielectric measurements of polar and weakly polar glass-formers, and also M*(f) compares favorably with the mechanical shear modulus data G*(f).

A general property found by DS is the co-invariance of logτ α (P, T) − logτ JG (P, T) and β K (P, T) to variations of P and T while keeping τ α (P, T) constant. The term JG β-relaxation was chosen for such secondary relaxation to distinguish it from other and usually intramolecular secondary relaxations. The JG β-relaxation is predicted to be present in all glass-formers since the omnipresent primitive relaxation of the CM is a part of the distribution of processes in the JG β-relaxation, and the primitive relaxation time τ 0 is approximately equal to the most probable JG β-relaxation time τ JG , i.e., τ JG ≈ τ 0 24 . This approximate relation was one of the criteria commonly used to check if a resolved secondary relaxation is the JG β-relaxation or not. For those polar and highly polar glass-formers with larger β DS K , the JG β-relaxation is not resolved because (logτ α − logτ JG ) according to Eq. (2) is small, and hence it is not well separated from the dominant α-relaxation. Nevertheless, the dielectric loss data cannot be accounted entirely by the Fourier transform of a Kohlrausch function. There is an excess loss on the high-frequency flank of the Kohlrausch fit, and in addition, an excess wing shows up at higher frequencies in some cases such as propylene carbonate, glycerol 25,26 , quinaldine 27 , and picoline 28 . The excess wing should be distinguished from the nearly constant dielectric loss ε″(f) ∝ f −λ with λ small and positive, which is due to loss while molecules are mutually caged by the anharmonic intermolecular potential. There are several facts supporting that the excess loss and the excess wing come from the unresolved JG β-relaxation, although this is still not universally accepted. (1) Long-term aging experiments performed on propylene carbonate, propylene glycol and glycerol 29,30 show the excess wing was transformed to a broad shoulder making the JG β-relaxation partially resolved. (2) The relation in the frequency of the excess loss/excess wing to the α-loss peak remains unchanged with variations of P and T while the α-loss peak frequency is kept constant in propylene carbonate, aroclor (polychlorinated biphenyls), salol, and other polar and highly polar glass-formers, in accord with the property of JG β-relaxation given by Eq. (3). (3) The separation in frequency between the excess wing and the α-loss peak agrees with that calculated by the right-hand-side of Eq. (2) using the dielectric β DS K for β K therein. (4) Highly polar glass-formers with larger β DS K such as quinaldine 31 , picoline 32 , and cyanobenzene 33 have no resolved secondary relaxation at all, and methyltetrahydrofuran (MTHF) 34,35 and diethyl phthalate (DEP) 36 have a non-JG secondary relaxation. A JG β-relaxation belonging to all of these glass-formers was resolved by mixing with a higher T g non-polar component. These experiments indicate that the JG β-relaxation is present in these highly polar glass-formers but located too close to the dominant α relaxation and not resolved. The root cause is the more harmonic and hence weaker intermolecular interaction resulting from the dipole-dipole interaction contribution to the attractive part of intermolecular potential 15 . The resultant more harmonic and weaker intermolecular potential is consistent with the larger values of β DS K for the α-relaxation observed by dielectric relaxation and molecular dynamics simulations 23 .
From the narrative given above, we postulate the presence of an unresolved JG β-relaxation in polar and highly polar glass-formers having dielectric strength small compared to the α-relaxation. It shows up as the excess loss/ excess wing on the high-frequency flank of the narrow dielectric α-loss peak. Notwithstanding, it does not alter the frequency dispersion of the α-relaxation, and thus the dielectric Kohlrausch exponent β K truly reflects the dynamic heterogeneity and cooperativity of the α-relaxation. With this done, we are ready to suggest the cause of the dramatic broadening when probed by shear modulus (SM), PCS and NMR. A priori, there is no reason to expect the responses of the JG β-relaxation relative to the α-relaxation observed in susceptibility by DS is exactly preserved when probed by any of the other methods simply because the correlation functions are different. Moreover, dielectric susceptibility ε* is compliance and shear mechanical G* is the modulus, and the difference is another reason for expecting a change. As we shall show by actual dielectric data, the much larger strength Δε(T g ) of the α-relaxation relative to Δε JG (T g ) of the unresolved JG β-relaxation in polar glass-formers shown in permittivity becomes much reduced when represented in electric modulus, resulting in broadening of the former by the latter. Therefore a heuristic explanation of the broader α-relaxation observed by the other methods than DS in polar glass-formers is a decrease of the relaxation strength of the α-relaxation relative to that of the JG β-relaxation. The reduction of the relaxation strength of the α-relaxation when probed by the other methods with little or no change of the JG β-relaxation is plausible. This is because the cooperative many-body α-relaxation is more sensitive to change of correlation function and/or change from compliance to modulus than the JG β-relaxation. This heuristic explanation needs to be tested by experimental data. The results are presented in the following sections.

Experimental verifications
We have proposed a heuristic explanation of why the narrow dielectric α-loss peak with large β DS K at temperatures near T g of polar glass-formers becomes a broader loss peak with smaller β K when probed by the other methods. In supporting this explanation, we have made new dielectric and shear modulus measurements of several glassformers and also have collected and reanalyzed previously obtained data. All the polar glass-formers showing the difference in α-relaxation dispersion of dielectric and PCS considered by Körber et al. are covered here. Additionally, we added more cases not included in their paper. The results are reported below, and the explanation is reiterated wherever deemed necessary.
Highly polar glass-formers. We have mentioned that the ratio of the relaxation strengths of the α and JG β relaxations of polar glass-formers with large Δε = (ε 0 − ε ∞ ) can be reduced when converted to electric modulus representation, resulting in a broader modulus loss peak. The most direct test is to compare ε*(f) with the electric modulus M*(f) = 1/ε*(f). Actually, ε*(f) and its time-domain correspondent ε(t) should be referred to as dielectric retardation. The true dielectric relaxation is the modulus M*(f) and M(t). M(t) can be directly determined by   Fig. 1A) 39 . The data of G″(f) at the same temperature show a slightly narrower peak, and the Kohlrausch function used to fit has β G K = 0.58. The time dependence of the VH light scattering intensity autocorrelation functions from Kahle et al. 40 was fitted to the Kohlrausch function. The exponents β PCS K (T) , shown in Fig. 1C, decrease with temperature and assume the value of β PCS K = 0.51 at 318 K. The correlation function of PCS is the second order Legendre polynomial and the susceptibility χ ,, PCS (f ) is a compliance and not modulus. Nevertheless, it is much broader than ε″(f) and its β PCS K = 0.55 is significantly smaller than β DS K = 0.76. More comparison of χ ,, PCS (f ) data from PCS with ε″(f) and M″(f) of polar glass-formers will be given later. In Fig. 1D we compare the ε″(f), M″(f), and G″(f) data of KDE at 318 K. Compared with ε*(f), it is well known that M*(f) is shifted to higher frequencies by a factor of about ε s /ε ∞ . To observe the decrease of the relaxation strength of the α-relaxation with little or no change of the JG β-relaxation in M″(f) and G″(f), we shift the M″(f) data vertically as well the scaled G″(f) data to superpose their high frequency data with that of ε″(f). The ε″(f) together with the vertically shifted M″(f) and G″(f) are presented in Fig. 1D. It shows, when probed as electric modulus or shear modulus, the maximum of the α-loss peak in ε″(f) is reduced by about one decade in M″(f) and G″(f), while the excess loss/excess wing representing the unresolved JG β-relaxation is unchanged. Hence when KDE is presented by electric modulus M″(f) or shear modulus G″(f) formalisms, the α-loss peak is distorted by the presence of by the JG β-relaxation. The shift of the α-loss peaks of M″(f) and G″(f) to higher frequencies from that of ε″(f) by the factor ε s /ε ∞ is slightly larger than one decade. The shift reduces the separation of the α-relaxation from the JG β-relaxation, and it also enhances the merge of the latter with the former. Consequently, the α-loss peaks of M″(f) and G″(f) become broader than that found in ε″(f), and explains why the Kohlrausch exponents β M K = 0.58 and β G K = 0.58 are smaller than β DS K = 0.76 (see Fig. 1A).

Phenolphthalein-dimethyl ether (PDE). The difference between ε″(f) and the electric modulus M″(f) and shear
modulus G″(f) in the frequency dispersion and strength of the α-loss peak of KDE is general for all highly polar glass-formers, and we have more data to show. Figure 2A shows  40 . As shown in the inset the Kohlrausch exponent β PCS K is temperature dependent and the value of 0.51 at 301 K is close to β M K of M″(f). This suggests that the cause of the broader dispersion of the α-relaxation seen by PCS than by dielectric spectroscopy is the same as M″(f).
Polychlorinated biphenyl (Aroclor1242). Polychlorinated biphenyls also known as Aroclor is another highly polar glass-former having a narrow dielectric loss peak. The dielectric loss spectra of Aroclor 1242 41   Plazek et al. 42 made shear recovery compliance J r (t) measurements of Aroclor 1248 having a slightly higher molecular weight than Aroclor 1242 and T g two degrees higher. From the measurements, the complex dynamic compliance was computed. The imaginary part, J" − 1/ωη , shown in Fig. 3C is fitted by the Fourier transform of the Kohlrausch function with β J K = 0.54. The agreement of β J K = 0.54 with β G K and β M K = 0.53 indicates that mechanical spectroscopies (employing either the modulus or compliance modes) broaden the α-relaxation in the same way.
PCS was performed by Rizos et al. 43 on Aroclor 1242 at temperatures near 249.1 K and higher but not at lower temperatures. The value of β PCS K reported is 0.64 and temperature independent, which is practically the same as 0.63 for β M K and we made this clear in Fig. 3. The agreement between β PCS K and β M K is like that found in Fig. 2 for PDE. It indicates that the broadening of the α-relaxation seen in dielectric permittivity when probed by PCS is due to a decrease of the relaxation strength of the α-relaxation relative to the unresolved JG β-relaxation close by, enabling the latter to broaden the frequency dispersion of the former.
Tributyl phosphate (TBP). The family tributyl phosphate (TBP), triethyl phosphate (TEP), and triphenyl phosphate (TPP) are highly polar glass-formers. TBP has Δε = 20 and narrow dielectric α-loss peak as shown in the upper panel of Fig. 4 and the Kohlrausch fit requires a large β DS K = 0.84 and similar values for the other members (shown in Fig. S1 for TEP and Fig. S2 for TPP) 21,22,44,45 . This property is like KDE, PDE, aroclor, and glycerol, as well as the other highly polar glass-formers 16,17 conforming to the correlation of β DS K with Δε found 15 . However unlike KDE, PDE, and glycerol, TBP and the other members have a prominent dielectric secondary γ-relaxation but it is not the JG β-relaxation, which is unresolved as suggested by the location of the primitive frequency f 0 at 146 K indicated by the arrow in the figure. This difference of TBP and other examples such as diethyl phthalate 36 , dibutyl phthalate 46 , and higher members. TBP and the others do not fall into the class of the so called "type A glass formers", defined as liquids with dielectric spectra that do not display a discernible secondary relaxation peak (β-relaxation) at temperatures above T g 47 . Nevertheless, these polar glass-formers have larger β DS K and correlate with Δε as well.
The comparison of the frequency dispersion of the α-loss peak from ε″(f) and M″(f) shows the reduction of the relaxation strength of the α-relaxation relative to the excess loss representing the unresolved JG β-relaxation as well as the resolved non-JG γ-relaxation. This is the cause for the broader α-loss peak in M″(f). In the lower panel of Fig. 4 we compare M″(f) with the susceptibility χ″(f) from PCS (or DLS) obtained by Pabst et al., which has β PCS K = 0.49 21,22 . There is excellent agreement in frequency dependence between M″(f) and χ″(f).  The fact that f 0 is much higher than the α-loss peak frequency suggests the broadening in going from ε″(f) to M″(f) or G″(f) and χ″(f) from PCS is not large. This is consistent with the small difference between β PCS K = 0.51 and β DS K = 0.53.

New ε″(f), M″(f), and G″(f) experimental data of highly polar glass-formers.
To bolster the experimental support of the explanation, we made new measurements of G″(f) over the range, 10 -2 < f < 20 Hz, of several highly polar glass-formers for which ε″(f) were also measured and represented together in Fig. 6. These include propylene carbonate (PC) and its three derivatives: S-methoxy PC, 4-vinyl-1,3-dioxolan-2-one (VPC) and 4-ethyl-1,3-dioxolan-2-one (EPC). The dielectric experimets of PC-derivatives in a frequency range from 10 -3 Hz to 10 7 Hz were carried out by means of dielectric spectrometer (alpha Novo-Control GMBH with novocool system). The stainless steel capacitor (diameter = 15 mm; distance of 0.098 mm provided by quartz) was used for measurements. ARES G2 Rheometer was used to determine the mechanical properties of PC-derivatives. The shear modulus measurements were performed by means of aluminum parallel plates of diameter = 4 mm. The width of the dielectric α-relaxation in S-methoxy PC is narrowest and its value of 0.85 for β DS K at T g is the largest recorded for highly polar glass-formers consistent with its large Δε = 230. Dielectric loss peaks of the The ε″(f) changes by more than two decades from the α-loss peak to the excess wing. By contrast, the corresponding change in M″(f) and G″(f) is about one decade. Again the difference means that the α-relaxation seen by dielectric relaxation and characterized by the larger β DS K is real because it is not modified by the much weaker JG β-relaxation despite the latter is close by. On the other hand, the broader modulus peaks characterized by smaller β M K and β G K is unreal because of the reduction in the disparity between the relaxation strengths of the two processes. www.nature.com/scientificreports/ formers in this paper, giving the value of β DS K = 0.60. On the other hand, the fit in the earlier work was not shown and a smaller value of 0.51 was reported and used by Körber et al. 19 . The calculated M″(f) at 246 K is presented in Fig. 7A. The M″(f) data are shifted vertically to have the excess loss in M″(f) coalescing with that in ε″(f) like done before for KDE in Fig. 1D. The difference in the height of the α-loss peak between the shifted M″(f) and ε″(f) is a factor of 1.5 compared to 10 in the case of KDE. The width of the α-relaxation in M″(f) is still larger than in ε″(f) as reflected by β M K = 0.53 compared to β DS K = 0.60 53 . The PCS data we consider are not from Meier et al. but from a later work published by the same group in Mainz by Patkowski et al. 55 , again because the more accurate data and analysis performed. The value of β PCS K for PCS decreases on lowering temperature and by extrapolation of the trend its value at T g = 247 K is estimated to be 0.52-0.53 55 . Hence there is good agreement between β M K = 0.53 and β PCS K = 0.52-0.53 of BMPC, in accord with the prediction.

1,1′-bis(p-methoxyphenyl) cyclohexane (BMMPC).
The glass-former BMMPC, also referred to in the literature as bis-kresol-C-dimethylether (BKDE) is closely related in chemical structure to BMPC and its T g is 263 K. It has Δε = 0.82 and the value of 0.55 for the dielectric Kohlrausch exponent β DS K at 264.1 K 56 . The smaller value of β DS K =0.55 implies the JG β-relaxation is widely separated from the α-relaxation. The dielectric loss ε″(f) at 271 K is shown in Fig. 7B together with the calculated M″(f). The vertically shifted M″(f) remarkably has the same frequency dependence as ε″(f) except for the slight horizontal shift due to change from susceptibility to modulus. This feature validates the prediction of no broadening in going from ε″(f) to M″(f) because the JG β-relaxation is well separated from the α-relaxation and has no effect in broadening the α-relaxation in M″(f). PCS data of BMMPC were published by Patkowski et al. 55  www.nature.com/scientificreports/ dispersion of weakly polar glass-formers with smaller dielectric β DS K is not broadened when probed by shear modulus or PCS.
Polybutadiene. The polymer polybutadiene (PB) with a molecular weight of 5000 g/mol has Δε = 0.15 was studied by dielectric and shear modulus 48 . The frequency dispersion of the α-loss peaks in ε″(f) and G″(f) are shown in Fig. 8B together with M″(f) we calculated from ε*(f). The frequency dispersions of ε″(f) and M″(f) are identical, and the exponents β DS K and β M K are equal to 0.35. Such a small value of β DS K leads to JG β-relaxation widely separated from the α-relaxation, and is ideal for testing the prediction. It follows that the JG β-relaxation has no effect in changing the frequency dispersion of the α-relaxation in going from ε″(f) to M″(f). Moreover, the frequency dispersion in G″(f) is only slightly narrower with the exponent β G K = 0.40, slightly larger than 0.35 of β DS K and β M K . Like DC704, the data of polybutadiene provide strong support for the prediction for weakly polar glass-formers.
Poly(methylphenylsiloxane) (PMPS). PMPS is another weakly polar polymer. Dielectric permittivity measurements on a sample with a molecular weight of 23,360 g/mol were made by Paluch et al. 60 . PCS measurements on a slightly higher molecular weight of 28,500 g/mol were made by Boese et al. 61

Discussion and conclusion
A serious challenge to a verity of the dynamics of polar glass-formers obtained by dielectric permittivity spectroscopy (DS) was issued by the recent publications by Körber et al. 19 , Gabriel et al. 62 , and Pabst et al. 22 . The challenge is that the narrow frequency dispersion of the intense dielectric α-loss peak in ε″(f) becomes much broader when the same material is probed by other spectroscopies including PCS (depolarized light scattering) and NMR 19,36 and also shear mechanical modulus G″(f) 21 . The exponent β DS K of the Kohlrausch fit to the dielectric loss peak in ε″(f) is significantly larger than the exponents, β PCS K , β NMR K , and β G K . We confirmed this discrepancy in several polar glass-formers from the literature as well as by making our own dielectric ε*(f) and G*(f) measurements on additional compounds (see Fig. 9). On the other hand, the relaxation times τ DS α of DS, though different from τ PCS α and τ G α due to the difference in correlation functions, all have similar temperature dependence. This discrepancy, however, does not occur in weekly polar glass-formers where the width of ε*(f) function is approximately the same as those of the others techniques.
As it stands, the general finding casts serious doubt on the verity of the narrow frequency dispersion and the large value of β DS K of polar glass-formers taken by dielectric spectroscopy. Potentially. the worth of dielectric spectroscopy in the study of the dynamics of glass-formers, and validity of the results from DS accumulated over the past hundred years, as well as the recently found correlation of β DS K with Δε found by Paluch et al., are questioned 15 and repeatedly verified by others 18,19 . The seriousness of the situation requires an in-depth consideration of the dynamics of polar glass-formers, not only the structural α-relaxation but also the presence of the accompanying and universal JG β-relaxation. Empirically the excess loss/excess wing in ε″(f) data of polar glass-formers indicates the JG β-relaxation is present although unresolved, and it is located nearly the α-relaxation on its high-frequency flank. This property of the JG β-relaxation as seen by DS is consistent www.nature.com/scientificreports/ with the correlation of log(τ α /τ JG ) with (1 − β K ) given by Eq. (3) from the Coupling Model (CM). The dielectric strength of the unresolved JG β-relaxation is small compared to the α-relaxation. Thus it has no effect on the full-width at half-maximum of the frequency dispersion of the α-relaxation or the value of the exponent β DS K of the Kohlrausch fit. Thus the larger value of β DS K from dielectric permittivity truly characterizes the dynamics of the α-relaxation of polar glass-formers.
It is essential to consider not just the change of the α-relaxation alone but also the JG β-relaxation altogether when going from ε″(f) of dielectric permittivity of polar glass-formers to G″(f) of shear modulus or χ″(f) of PCS and NMR. The changes in the relaxation strengths of the two processes are not necessarily uniform. Thus the narrow frequency dispersion of the α-relaxation seen in ε″(f) can change substantially when probed by the other spectroscopies. We substantiate this possibility by changing the representation of dielectric measurements from ε″(f) to the electric modulus M″(f). For the polar glass-formers, we found the narrow frequency dispersion of the α-loss peak in ε″(f) becomes a broad peak in M″(f) (see Fig. 9A,B). The cause is traced to the more significant reduction of the dielectric strength of the α-relaxation relative to that of the JG β-relaxation. Since the two relaxations in polar glass-formers are not widely separated already in ε″(f), the disparity in the changes of their relaxation strengths in conjunction with the additional decrease in the separation of their relaxation time gives rise to the broadening of the α-loss peak in M″(f). This explanation of broadening of the α-relaxation in M″(f) applies verbatim to G″(f) since both are modulus and is supported by M″(f) and G″(f) from experiments having either nearly the same frequency dispersion in a number of glass-formers shown in Fig. 9. So is the good agreement of χ″(f) from PCS with M″(f), or the Kohlrausch exponents β DS K and β PCS K being about the same, in some glass-formers. By explaining the broadening of the dielectric α-loss peak of polar glass-formers when probed by other techniques, we have two crucial conclusions. The narrow width of the dielectric loss peak in ε″(f) and the associated larger β DS K truly reflect the heterogeneous and cooperative molecular dynamics of the α-relaxation in polar glass-formers because it is unaffected by the much weaker JG β-relaxation despite it is nearby. Thus there is nothing wrong with dielectric spectroscopy in applying it to study the dynamics of polar glass-formers. By contrast, the broadened 'α'-relaxation observed by G″(f) or by PCS and NMR has the α-relaxation admixed with the JG β-relaxation, and its smaller exponents, β G K , β PCS K , and β NMR K do not characterize the genuine α-relaxation of polar glass-formers. In other words, for polar glass-formers having narrow dielectric α-loss peak and larger β DS K , the broader frequency dispersion of the α-relaxation deduced from shear modulus, PCS, and NMR are not factual. Needless to say, the correlation of β DS K with Δε found by dielectric spectroscopy and the theoretical rationalization 15 remain valid.
Our explanation of the effect found in polar glass-formers with narrow α-loss peak implies that the dielectric α-relaxation is not broadened in glass-formers having the JG β-relaxation widely separated from the α-relaxation, whether they are polar or not. According to Eq. (3), the separation, log(τ α /τ JG ), is proportional to (1 − β DS K ). Hence a corollary of the explanation is the absence of a significant change of the frequency dispersion in glassformers with larger (1 − β DS K ) or smaller β DS K . Molecular glass-formers having smaller β DS K and larger log(τ α /τ JG ) are usually non-polar like OTP, TNB, and toluene, studied before by DS 47,59,63,64 , PCS 65,66 , shear compliance J(t) 67 . The dielectric β DS K of these three glass-formers have values of about 0.51 close to those of β PCS K and β J K , and thus verifying the prediction directly. We have more non-polar glass-formers with smaller β DS K in showing first the broad frequency dispersion of ε″(f) is either unchanged or hardly changed when replaced by M″(f). Furthermore, β DS K and β M K are nearly the same as β PCS K or β G K , whichever is available. The amount of data confirm the predicted behavior of weakly polar glass-formers with smaller β DS K to be different from the polar glass-formers with larger β DS K , and strengthens the explanation for the polar glass-formers. The ubiquitous presence of the JG β-relaxation and the inseparable relations of its relaxation times to that of the α-relaxation (Eq. (3)) are supported by many corroborative evidences such as given in Refs. 34,[68][69][70][71][72][73] are critical in restoring the verity of the dynamics obtained by using dielectric permittivity spectroscopy of polar glass-formers. On the other hand, the presence of the JG β-relaxation and its relations to the α-relaxation was not recognized in the papers of Körber et al. 19,74 , Gabriel et al. 21,62 , and Pabst et al. 22 , and the consequence is that they were not able to reach the same conclusion.