Correlation between entropy fluctuations and the dielectric relaxation of glass-forming systems: The central role of dipolar–dipolar cross correlations Open Access

The glass transition is one of the most intriguing phenomena in solid state and soft matter physics. Starting from an equilibrium state, upon decreasing temperature, the molecular mobility of glass-forming materials progressively decreases until, at some point, equilibrium cannot be maintained on the time scale of the cooling rate, resulting in vitrification. This arrest of molecular dynamics accompanied by the transition from an equilibrium to non-equilibrium state is known as the glass transition phenomenon.^1,2 This still puzzling problem has been approached using many different experimental methods.^3–5 Although various techniques basically agree in a common picture, a detailed comparison of particular materials or samples is not always trivial or consistent. Most of the time difficulties come from the fact that different experimental techniques record diverse observables and correlation functions that relate or couple to the motion of the molecules (the underlying physical phenomenon, which is common for all the techniques) in a different way, complicating the comparison of different sets of data.

Dielectric spectroscopy (DS) has been widely used to characterize the glass transition and the associated dynamics.⁶ Molecular motions are reflected in the relaxation of the macroscopic polarization under an applied electric field, which directly results from dipole dynamics at the microscopic level. By means of dielectric spectroscopy, the relaxation associated with the glass transition phenomenon, i.e., the α-relaxation, can be easily monitored across a broad temperature and frequency range, which explains the technique’s popularity in the field. While there is ample evidence showing that dipolar relaxation is generally a reliable probe for structural relaxation, there could be certain specific circumstances in which dipolar relaxation might not be fully representative of the overall dynamics of the system. This could occur, for example, as a result of heterogeneity between polar and apolar segments or if dipolar dynamics becomes decoupled from the structural dynamics.

In particular, the spectral shape of the dielectric α-relaxation in relatively polar systems has recently attracted considerable attention.^7–20 When measured using nuclear magnetic resonance (NMR) or light scattering (LS) techniques, many systems exhibit similar time distributions or spectral shapes for their structural relaxation.^21–26 In dielectric spectroscopy (DS), however, the shape of the primary structural relaxation or α-relaxation is comparable to that measured by NMR or LS only for systems of low dielectric strength.^23,25,26 Furthermore, it is well-established experimentally that the dielectric spectral shape and the dielectric strength of a system are correlated such that narrower responses correspond to higher dielectric strengths.⁷ This property was explained by Ngai et al.⁷ as arising from dipole–dipole intermolecular interactions, making the overall potential more harmonic. Moreover, according to theoretical works by Déjardin et al.,¹³ positive dipole–dipole correlations lead to an additional narrow process in the susceptibility response of polar samples. These ideas are experimentally supported, for example, by the observation that the narrow dielectric signal of the polar liquid tributyl phosphate (TBP) broadens upon dilution with an apolar solvent.⁹ Computation of the dipole correlation function in model systems first¹⁴ and that of the dielectric response in glycerol later¹⁵ further support this view. Both studies show that the cross correlation term exhibits a narrower shape compared to that of a self-correlation term, although the timescales for both contributions in these works are indistinguishable or very close (∼0.3 and 0.1 decades apart in Refs. 14 and 15, respectively). Exploiting the fact that photon-correlation spectroscopy is largely insensitive to orientational cross correlations, comparisons of LS and DS results by Blochowicz et al. have revealed the significant impact of cross correlations on the dielectric response of several polar glass-forming systems.^8,9,19,27 In these works, it is proposed that dipole–dipole cross correlations contribute with a narrow response to the overall dielectric signal, while a relatively less intense and faster dielectric self-part would exhibit the rather universal spectral shape observed by LS. The timescale separation between the self- and cross-components, experimentally deduced from this comparison approach, however, is considerably larger than the separation observed in computational studies to date.

In this context, the interpretation of the dielectric response of high dielectric strength systems and the role of the dipole–dipole interactions, in particular, remain a topic of ongoing debate. One perspective embraces the idea that there is a fundamental generic spectral shape for the structural relaxation, focusing primarily on the self-contribution to the signal, relegating the presence of a narrow cross term to a particularity of the dielectric spectroscopy probe. In this sense, the identification of the dielectric self-component as the “structural α-relaxation” (corresponding to the α-relaxation observed by LS) and the designation of the slower cross component as a distinct Debye process remain contentious.^{8–10,16,19,27} This view generates controversy in that it is difficult to reconcile the idea that the structural relaxation—a collective phenomenon—can be best and solely described in terms of self-correlation functions, overlooking the cross components in the case of dielectric spectroscopy.¹⁰ In practice, from an experimental point of view, the type of correlation functions accessible by a given technique is limited by the physics behind the measuring principle. However, from a conceptual point of view, or any given dynamic event, the self- and cross-components represent just two different types of correlation functions, two different mathematical tools to follow dynamics that highlight the best distinct aspects of the phenomenon under study. Therefore, unless multiple dynamic events are considered, both self- and cross-components offer different perspectives on the same process and, in the author’s opinion, both should be regarded as part of the “α-relaxation.” Incorporating dipolar interactions and the resulting cross-contributions to the dielectric response as part of the physics behind the structural relaxation, however, necessarily implies that there is no fundamental, generic spectral shape for the relaxation of the structure or, at least, not one that applies universally to all probes and correlation functions. The work of Gainaru et al. supports this later view by showing that glassy physical aging²⁸ is better described by the overall dielectric response.^10,16

Considering all, the emergence of a narrow shape dipolar cross contribution in the dielectric response of polar systems has raised some questions on the feasibility of DS for characterizing the primary structural relaxation in glass-forming systems, a controversy that requires a solid and definitive answer. Do the overall dielectric α-relaxation in general and the cross correlation term dominating the DS signal of high dielectric strength systems in particular faithfully probe the primary structural relaxation? This work reclaims the significance of the DS technique in studying primary structural relaxation. Evidence is presented that the overall dielectric α-relaxation in general and the slower cross contribution dominating the dielectric response in particular univocally render the primary structural relaxation identified as entropy fluctuations in equilibrium, demonstrating the critical role of dipolar cross correlations in structural relaxation.

The study of the glass transition through the recording of entropy fluctuations—providing information on the motion of all degrees of freedom, regardless of their polarity—is considered one of the most direct measurements at the very basis of the definition of the glass transition.²⁹ Its results have often been used as an argument to identify α-relaxation in the cases where the presence of more than one close process made the identification non-trivial.^30,31 There is substantial experimental evidence supporting a one-to-one correspondence between the glass transition observed in calorimetry and the dynamic arrest observed upon cooling as measured by various techniques and especially by dielectric spectroscopy. In fact, there are examples in the literature where this relationship allows for the prediction of the dielectric (or calorimetric) response based on experimental calorimetric (or dielectric) data from complex systems.^32–34 Consequently, the comparison of calorimetric and dielectric data results very interesting in the context of understanding and interpreting dielectric data and can help to settle the debate over which contributions to the signal measured by different techniques play a relevant role in the glass transition process.

One of the most common methods for contrasting data that characterize certain molecular motion is by comparing the characteristic time of the dynamic process as a function of temperature. In practice, when comparing the characteristic times obtained from different approaches, the temperature dependence of the dynamics observed by various techniques tends to agree very well. However, the absolute times derived from analyzing different observables often show slight variations between techniques.^35–37 This discrepancy is a natural consequence of the fact that different techniques measure distinct observables and correlation functions, not to mention the variety of options that could be available to parameterize a given property. In general, a temperature-independent shift can reconcile the characteristic times from different datasets, and for certain pairs of magnitudes, such a shift can even be sample-independent.

In this work, some of the questions outlined in the Introduction are addressed by combining results from dielectric spectroscopy and calorimetry techniques. The approach was first to develop a phenomenological protocol to quantitatively compare the characteristic times for structural relaxation as obtained from both techniques. Later, this comparison is used to identify and discuss which contributions to complex dielectric response are univocally related to entropy fluctuations in equilibrium and, thus, to structural relaxation. Accordingly, the content of this paper is organized as two main parts. Section III is dedicated to the systematic and quantitative experimental verification of the premise that the calorimetric and dielectric α responses share a common origin and there is one-to-one correspondence in the context of structural relaxation dynamics. To this end, an experimental protocol is developed, tested, and validated for comparing the characteristic times of entropy fluctuations and dielectric α-relaxation with the goal of establishing a material-independent phenomenological quantitative relationship. The validation of this relationship is approached at multiple levels, considering factors such as dynamic simplicity, macromolecular structure, and the presence of hydrogen bonds. Section III A tests low dielectric strength systems exhibiting a simple shear response (single Maxwell-like relaxation) where the assignment of the dielectric α process to the primary structural relaxation is almost trivial. Section III B (polymers) and Sec. III C (monoalcohols) cover systems showing more than one dynamic process (and timescales) but where the identification of the dielectric α relaxation among these processes is beyond reasonable doubt. The final part of Sec. III D provides a critical evaluation of the limits of this relationship within experimental accuracy. In Sec. IV, on the other hand, the focus shifts to the dielectric response of polar systems. The same relationship, as discussed earlier, is now validated for polar glass-forming systems, taking into account factors such as structural and dynamic simplicity, as well as the presence of hydrogen bonds. Finally, the implications of the obtained results for interpreting the dielectric response of polar systems are explored and discussed, with particular attention to the role of dipolar cross correlations in the primary structural relaxation process.

Most of the samples used in this work were commercial products purchased from various suppliers. 1,1-bis(4-methoxyphenyl)cyclohexane (BMMPC), polystyrene (PS, Mw = 500 gr/mol), and polypropylene glycol (PPG, Mw = 2000 gr/mol) were provided by Polymer Source. 3-fluoroaniline (3FAN) and glycerol (GC) were from Thermo Scientific Chemicals. Pentaethylene glycol (5EG) was from Fluka. Tetramethyltetraphenyltrisiloxane (DC704) was from abcr Germany. Diglicylether of bisphenol A (DGEBPA), phenyl salicylate (Salol), propylene carbonate (PC), 2-ethyl-1-hexanol (2E1H), 2-butyl-1-octanol (2B1O), 3,7-dimethyl-1-octanol (37dM1O), 1-phenyl-1-propanol (1P1P), 3-phenyl-1-propanol (3P1P), propylene glycol (PG), dipropylene glycol (2PG), triethylene glycol (3EG), 1,2,4-butanetriol (124ButOH), 1,2,6-hexanetriol (126HexOH), trisbutoxyethyl phosphate (3BoxEP), and o-terphenyl (OTP) were obtained from Sigma-Aldrich. Bisphenol-C-dimethylether (BCDE) was provided by the Max Planck Institute for Polymer Research, Mainz, Germany. Whenever possible, the very same sample was tested across different setups. When this was not feasible, paired samples were used and the experiments were conducted following identical protocols within 24–48 h period to minimize possible variability arising from moisture uptake, minor reactions with air, batch impurities, or other sources of experimental uncertainty. Liquid samples at ambient temperature were stored with activated molecular sieves to minimize traces of ionic impurities and reduce moisture absorption.

Differential Scanning Calorimetry (DSC) measurements were performed using TAInstruments DSC Q2000 equipped with a liquid nitrogen cooling system. The baseline and temperature calibration were performed through a series of runs, involving measurements of empty pans, sapphire standards, and the melting of indium. The sample chamber was purged with dry helium at a flow rate of 25 ml/min. Specimens were prepared sealing ∼10 mg of sample in dedicated aluminum pans, and the thermal history of the samples was fully erased between different cooling scans. Basically, two main approaches are commonly used to report the temperature dependence of the characteristic times in undercooled glasses: (i) methods that rely on the variation of the glass transition temperature with a cooling rate in a vitrification experiment and (ii) methods that measure entropy fluctuations under equilibrium conditions. It is important to note that these two approaches are not equivalent, as the first involves testing the system under out-of-equilibrium conditions, while the second examines the system in equilibrium.^39,40 In the present work, since dielectric experiments were performed under isothermal equilibrium conditions, the second approach was followed. The dynamics of entropy fluctuations in equilibrium were characterized using Temperature Modulated Differential Scanning Calorimetry (TMDS) under a quasistatic condition. The experiments involved very slow cooling scans with superimposed faster temperature modulation. Average cooling rates ranged from 0.1 to 0.3 K/min, while oscillation periods, P, of 10, 25, 40, 60, and 100 s were used. For each period, the oscillation amplitude and the base cooling rate were selected so that the modulated local rate was at least one order of magnitude faster (in logarithmic scale) than the base cooling rate. This allows us to relate the frequency response (temperature modulation) dependence of the heat capacity to fluctuations in a quasi-equilibrium state in a first approximation. In addition, the amplitude of the temperature oscillations ranged from 0.15 to 1.25 K, well below the typical thermal fluctuations found by TMDSC for many glass-forming systems,⁴¹ ensuring that the systems remained within the linear regime. TMDS enables the separation of the overall response into reversing and non-reversing contributions. The temperature dependence of the time scales of spontaneous entropy fluctuations was determined from the temperature point of the maximum change rate of the reversing specific heat (i.e., maximum of the derivative of reversing C_p—arrows in Fig. 1) measured at different oscillation periods according to 2π/P = ω = 1/τ_dsc.

Isothermal broadband dielectric spectroscopy (DS) experiments were performed using a commercial Novocontrol Alpha A setup, with temperature control provided by a nitrogen jet stream, ensuring a temperature stability of ±0.1 K. Samples were sandwiched between two parallel gold-plated electrodes using Teflon thin strips as a spacer when necessary. Dielectric spectroscopy allows for the characterization of molecular dynamics across a wide range of temperatures and frequencies. Among the different magnitudes accessible through dielectric measurements, in practice, the characteristic timescale of dipolar motion is typically identified with the frequency position of the changes observed in the complex permittivity ϵ*(ω) = ϵ′(ω) − iϵ″(ω) [such as steps in ϵ′(ω) or peaks in ϵ″(ω), which relate by Kramers–Kroning relations]. It is important to note that, depending on the shape of the relaxation, there is no unique way to parameterize the characteristic times of the variation of ϵ*(ω) with frequency (e.g., most frequent time, Havriliak-Negami time, average time and others). To remain as model-independent as possible and minimize data dispersion arising from fitting biases, all characteristic times for the dielectric relaxation (or its individual components) presented in this work were determined from the frequency at the maximum value of the derivative of the real part of the permittivity, −dϵ′/d(log ω). This later magnitude shows narrower relaxation peaks than those observed in the imaginary part of the permittivity, making the identification of the frequency at maximum both easier and more accurate.³⁸ The relation established in this article is derived by analyzing permittivity data, i.e., retardation times τ_ϵ. Note that the relation between τ_ϵ and the characteristic time of the dielectric modulus τ_M [with M*(ω) = 1/ϵ*(ω)] is strongly dependent on the dielectric strength of the material under test.⁶ As a result, the same correlation cannot hold for both τ_ϵ and τ_M simultaneously, except in systems with low dielectric strength.

The comparison between the timescales observed for dielectric relaxation (or its individual components) and those obtained for entropy fluctuations in equilibrium could offer new insights into the questions discussed in the Introduction. Before that, however, it is necessary to establish a solid foundation for a quantitative comparison. To this end, in the following, the characteristic times obtained from dielectric spectroscopy and calorimetry around the glass transition temperature are systematically compared for several types of glass-forming liquids, including samples with varying intermolecular interactions and dynamic complexity. The first set of examples focuses on simpler systems, where the identification of the dielectric α-relaxation is beyond reasonable doubt, allowing us to confidently establish the sought correlation. Subsequent examples examine the selectivity of the procedure by comparing the calorimetric and dielectric responses of systems with complex dynamics. Finally, experimental accuracy is tested by studying a couple of complex systems for which, until recently, the interpretation of their dielectric relaxation remained uncertain.

The systems included in this section exhibit low dielectric strength and “simple” shear response. On the one hand, low dielectric strength implies that no significant dipolar correlations or cross contributions to the dielectric response are expected. Actually, for these systems, the agreement between the shapes of the α-relaxation as seen by different techniques is reasonably good.^23,26 On the other hand, the systems included here all show a “simple” rheological response, “simple” understood here in the sense that their shear relaxation is monomodal and very close to a Maxwell model (one characteristic time, secondary relaxations aside). As a result, the terminal relaxation regime (where G′ ∼ ω² and G″ ∼ ω) sets in at a frequency close to the intersection frequency of the real and imaginary shear moduli, ω_x (see Fig. 1S of the supplementary material). It has been established that, for such “simple” systems, there exists a quantitative correlation between ω_x and the frequency of the maxima of the dielectric relaxation.¹² This is consistent with the idea that the same underlying physics, i.e., the same molecular motions or the primary structural relaxation in this particular case, governs both dielectric and shear relaxation, and, a priori, given the simple dynamics, the calorimetric response too.

Figure 2 shows the characteristic times obtained by dielectric spectroscopy (black symbols, left y axis) and those obtained from temperature modulated calorimetry (open diamonds—left y axis—and solid diamonds, right y axis) for (a) OTP, (b) DC704, (c) BMMPC, and (d) BC-DME. Since the magnitudes measured and the parameters chosen to characterize timescales in each technique differ, we do not expect the data to match on an absolute scale (black symbols and open diamonds in the left y axis). Dielectric spectroscopy primarily probes rotational degrees of freedom based on their dipole moment through the measurement of polarization, whereas in calorimetry, all degrees of freedom are evenly weighed by measuring heat transfer. However, if both responses arise from the same phenomenon, we expect to observe the same activation energy when determined by different techniques,^35–37 and depending on the experimental magnitudes compared, a material independent factor can even exist between two different datasets.^11,12 As shown in Fig. 2, both conditions are satisfied: (i) the two sets of data show the same activation energy and (ii) the timescale shift bringing together DSC and DS data—black symbols and solid diamonds—remains constant for different samples (note that left and right y-axes scales are the same for all panels).

This section shows results on systems that show a “complex” rheological response, understood as the presence of more than one process or characteristic time in their shear relaxation (secondary relaxations aside). Due to their macromolecular nature, polymers present “complex” dynamics, i.e., in addition to the segmental dynamics associated with the glass transition, they exhibit other dynamic processes related to the relaxation of their chain macromolecular structure. These dynamics can be followed by either shear rheology or dielectric experiments (the later for type A polymers only), and there exists indeed a correlation between various signals observed by both techniques, substantiating that these signals share the same origin or phenomena. The dielectric α-relaxation of polymers correlates with the crossing of shear moduli (as in simple systems), while the slower so-called dielectric normal mode correlates to the onset of the terminal behavior in the shear relaxation. We have tested a couple of glass-forming polymers, representative of type A and B polymers, PPG and PS, respectively. In Fig. 3, the black symbols (left axis) represent dielectric data—filled symbols for the α-relaxation and open symbols for the normal mode—and diamonds (open, right axis and solid, left axis) represent characteristic times obtained from calorimetry. As expected, calorimetric relaxation times nicely correlate with the faster dielectric α-relaxation times and not with the slower normal mode, which is separated by less than two decades in this particular PPG sample. Thus, for polymers as well, the dielectric α and calorimetric times match on an absolute scale under the same shift (0.8 decades) applied in Sec. III A.

Monoalcohols represent another interesting class of complex glass-forming systems. In general, monoalcohols tend to form chain-like aggregates by hydrogen bonding. Related to this phenomenon, they exhibit additional dynamic processes to primary structural relaxation that can be quite overlapped.⁴² One well-known feature of the dielectric relaxation in these systems is the so-called “Debye” process, which is often very prominent relative to the faster dielectric α-relaxation contribution. Contrary to the initial belief, the phenomenon underlying the dielectric Debye relaxation in monoalcohols can also be followed by shear rheology,^43,44 and one-to-one correlations are identified between the various dynamic modes or components observed by shear and dielectric relaxation techniques.^45,46 In particular, a correlation between the dielectric α-relaxation and the crossing of shear moduli can be clearly established so that the dielectric α-relaxation of monoalcohols merges together with that of other “simple” glass-forming systems when the frequency axis is normalized to the crossing frequency ω_x of their shear moduli.^11,12

Figure 4(a) shows the dielectric response for three different monoalcohols, namely, 2E1H, 2B1O, and 37dM1O. As can be seen, various processes in monoalcohols can overlap significantly. As a consequence, the derivative of the real part of the permittivity (black squares) provides a clearer resolution of different components compared to the more conventional representation of the imaginary part of the permittivity (empty circles). Figures 4(b)–4(d) contain characteristic times for the dielectric Debye relaxation—represented by open black symbols—and for the α relaxation—represented by filled black symbols—of these three monoalcohols. The timescale separation between the Debye and the α-relaxation characteristic times in these samples around the glass transition temperature is roughly 2.5, 2, and 1.5 decades for 2E1H, 2B1O, and 37dM1O, respectively. For these complex systems too, Figs. 4(b)–4(d) show that the characteristic times for the calorimetric relaxation clearly correlate with the faster dielectric α-relaxation times using the same shift as in the previous examples. These results are consistent with the correlation observed between the crossing frequency of shear moduli and the dielectric α-relaxation, as discussed earlier, and with many other works devoted to unravel the nature and origin of the dielectric α component in monoalcohols,^30,31,42 which definitely relate this faster component with the primary structural relaxation.

The results in Secs. III A–III C have demonstrated that when following the proposed procedure, there is a quantitive relationship between the characteristic times from entropy fluctuations and the α-relaxation measured by dielectric techniques whether in simple or complex systems. This suggests that, in principle, one could use this approach to discriminate between the origins of different dielectric components or predict timescales for primary structural relaxation by comparing dielectric and calorimetric data. The question now is as follows: what is the “resolution” of the proposed methodology and how sensitive can it be within practical experimental limits? The answer to this question is not unique, as, while the underlying cause–effect relationship is universal, the dispersion of results depends on repeatability (operator and instrumental), instrument quality and calibration, and/or experimental procedures and data analysis methodologies. As a consequence, prior to using the proposed comparison methodology to draw solid conclusions about the origin of a particular dielectric component, it is essential to validate the specific measuring and analyzing protocols applied, assessing the reproducibility and sensitivity of the instruments in use to reliably discriminate between different processes and situations. The procedures for data acquisition and analysis throughout this work, starting from paired samples to the choice of characteristic time parameterization, were carefully designed from the beginning to minimize both systematic and non-systematic instrumental and operational errors. In Secs. III A–III C, we have validated a procedure for quantitatively comparing dielectric and calorimetric timescales measuring a diverse group of glass-forming systems with well-identified α-relaxation components. With the procedure established, in the following, we will test its robustness and sensitivity by comparing the response of a couple of challenging examples.

In the monoalcohol samples discussed in Sec. III C, different components of the dielectric relaxation overlapped, with their relative separation varying between ∼2.5 and 1.5 decades near the glass transition temperature. Among monoalcohols, phenyl monoalcohols represent an extreme case in the sense that various processes get so overlapped that, for some time, it was believed that this family of monoalcohols did not exhibit any hydrogen bond-related slow relaxation, presumably due to steric hindrance from phenyl rings. As shown in (a) of Fig. 5, the imaginary part of the dielectric permittivity (empty circles) of 3P1P and 1P1P shows just a single peak. Even the derivative of the real part of the permittivity (black squares) does not show a differentiated maximum, but just some subtle change in slope on the high-frequency side of the curve [see dashed lines in Fig. 5(d)]. Recently, however, several studies have supported the presence of a slow Debye contribution in the dielectric signal of phenyl alcohols: first by comparison of dielectric and light scattering results;⁴⁷ after by field-induced enhancing of the Debye-like contribution relative to that of the primary structural relaxation;⁴⁸ on extensive works characterizing hydrogen bonds in these systems by different techniques;^49,50 and finally by bringing to light the non-simple shear response of phenyl alcohols and its relation with the dielectric response.⁵¹ In this context, these systems are a good test for the sensitivity of the proposed method and its potential to unravel complex dynamics and aid in the interpretation of intricate dielectric data.

Panels (c) and (d) of Fig. 5 show the comparison of the characteristic times for the main dielectric peak (empty squares) and for entropy fluctuations in 3P1P and 1P1P samples. Unlike the cases presented in previous sections, the characteristic times obtained from the maximum of the overall dielectric relaxation and those obtained from calorimetry do not align well. Instead, the timescales for entropy fluctuations remain clearly below those of the main dielectric maxima, in agreement with recent studies reporting a faster dielectric α component at the high-frequency wing of the overall signal. In this scenario, one can attempt to estimate from dielectric data some characteristic times for the postulated faster α relaxation. For the 1P1P sample, where the change in slope is subtle, a tentative faster dielectric component was determined by fitting the low-frequency part and the maximum of the overall relaxation to a Debye function and taking the maximum frequency of the remaining signal [depicted by solid color lines in panel (b) of Fig. 5]. The resulting tentative dielectric α relaxation times are represented by solid squares in panels (c) and (d) of Fig. 5. As can be seen, in both cases, entropy fluctuations nicely match the dielectric α times estimated in this way. Furthermore, the calorimetric relaxation and dielectric α times also agree well with the α times [green triangles in Figs. 5(c) and 5(d)] estimated from the location of the crossing frequency of shear moduli ω_x based on the shear-dielectric correlation established for “simple” systems in previous works.^11,12 Figure 6 compares the dielectric and shear relaxation of 3P1P and 1P1P (details on shear relaxation experiments are provided in the supplementary material). The non-simple shear response of 3P1P and 1P1P is subtle, but clear when a careful analysis is performed paying attention to the onset of the terminal behavior.¹¹ For simple liquids, pure viscous flow is shortly established after the modulus crossing frequency ω_x, whereas for 3P1P and 1P1P, the separation is larger (see Fig. 2 of the supplementary material). These results on the non-simple shear response of phenyl monoalcohols are in agreement with previously published data by Mikkelsen et al.⁵¹ On the other hand, the non-associative (simple) shear behavior of 3P1P reported in another recent study⁵² could be due to the use of time-temperature superposition and/or the use of large scale graphs to represent and analyze data. Similar to other monoalcohols, the results for 1P1P and 3P1P show that the main dielectric peak locates around the crossover frequency to the terminal shear behavior, while entropy fluctuations and the overlapped faster dielectric component correspond well with the crossing of shear moduli.⁴⁵ With all, the above discussion on phenyl monoalcohol case studies demonstrates that by a quantitative comparison of dielectric and entropy fluctuation characteristic times, it is possible to identify complex dynamic behavior and to discriminate structural relaxation among overlapped processes exhibiting characteristic timescales separated less than a decade.

The quantitative relationship established in Sec. III, in principle, allows for the prediction of the characteristic time of the dielectric α relaxation from those determined for the entropy fluctuations. In practice, this provides a means to discriminate the primary structural relaxation component from other contributions in the dielectric response and can be a valuable tool for addressing the question of the role of dipolar intermolecular correlations in the primary structural relaxation.

Figure 7 shows a comparison of the timescales obtained by DS and DSC for (a) DGEBPA, (b) Salol, (c) 3FAN, and (d) PC. These liquids are low molecular weight glass-forming systems with a “simple” rheological response and moderate to high dielectric strength (about 5, 7, 45, and 70 for DGEBPA, Salol, 3FAN, and PC, respectively). Having simple viscoelastic behavior, the maximum of the dielectric relaxation and the crossing of shear moduli are quantitatively correlated.^11,12 Given the dynamic simplicity of these systems, the assignment of the relaxation observed by different techniques to the primary structural relaxation is, in principle, straightforward. As can be seen in Fig. 7, for these samples too, there is a clear correlation between the two sets of characteristic timescales. Moreover, the same shift as before brings together dielectric and calorimetric times within experimental uncertainty. Having moderate to high dielectric strength, the dielectric response of these materials is relatively narrow, which, according to the general belief, reflects the increasing contribution of dipole–dipole cross correlations to the overall dielectric signal. Therefore, for these samples, the overall dielectric signal is, in principle, dominated by a slow and narrow spectral shape cross contribution, with this effect becoming more pronounced as the dielectric strength increases. Nevertheless, and irrespective of the dielectric strength of the glass-forming systems, the same quantitative correlation is observed between the main dielectric peak maximum and entropy fluctuations. This result is systematically reproduced in other systems where strong dipolar interactions are expected, such as in TBP and 3BoxEP phosphate liquids (see Fig. 8) or in hydrogen-bonded systems, such as dialcohols and trialcohols (see Figs. 9 and 10, respectively).

Calorimetry is considered one of the most direct methods for probing structural relaxation, and it has traditionally been used as a reference for identifying the structural α-relaxation.²⁹ Fluctuations of entropy in equilibrium are, in principle, sensible to all the configurational degrees of freedom relevant for the primary structural relaxation. In this sense, a systematic comparison of calorimetric and dielectric data is particularly valuable in the context of understanding and interpreting the dielectric response of glass-forming systems. Throughout this paper, it has been demonstrated that there is a clear, quantitative, and material-independent relationship between the α-relaxation characteristic times determined from the permittivity response and the entropy fluctuation times derived from the reversing heat capacity. This relationship extends beyond the temperature dependence of the characteristic times so that the shift bringing together the times measured by these two approaches is the same for all the examples shown and under the experimental procedures followed here. Note that the particular value of the shift (0.8 decades) is not relevant itself because this is a figure that depends on the particular parameters used and analysis procedures followed to determine characteristic times, as well as on the particular pair of instruments and their relative thermocouple calibration. For example, in calorimetry, it is possible to parameterize the glass transition at the onset temperature, inflection point, middle point, and so on. In the case of dielectric spectroscopy as well, characteristic times can be obtained, among others, by analyzing retardation time—permittivity—or time—modulus—and for each of these approaches, one can choose to parameterize the timescales using the mean time, most frequent time, Havriliak–Negami time, and so on. At this point, it is important to emphasize that the relation was established by analyzing permittivity data, i.e., retardation times τ_ϵ. The relation between τ_ϵ and the characteristic time of the dielectric modulus τ_M [with M*(ω) = 1/ϵ*(ω)] strongly depends on the dielectric strength of the material under study. As a result, the quantitative correlation described here does not hold for τ_M, except for low dielectric strength systems.

These results, on the one hand, experimentally verify the premise that the dielectric α relaxation has a one-to-one correspondence relation with entropy fluctuations in equilibrium near the glass transition and reclaim the suitability of dielectric spectroscopy to echo the structural relaxation of glass-forming systems. On the other hand, from a practical point of view, establishing a robust experimental protocol quantitatively relating the characteristic times of these two measurements provides a powerful phenomenological tool to (i) discriminate the origin of diverse dielectric components and (ii) quantitatively predict timescales for primary structural relaxation. Thus, careful measuring protocols (designed to minimize systematic and operational errors) and well-tested analysis methods (that check the reproducibility and sensitivity of the instruments and procedures) can provide valuable means for a phenomenological comparison, guiding the interpretation of diverse data with high sensitivity and accuracy.

In Subsection III A, calorimetric and dielectric timescales were compared for cases where the assignment of the dielectric relaxation to the primary structural α process is beyond reasonable doubt. For those glass-forming systems, the dielectric strength is low, and they exhibit unimodal shear response. As a result, no significant dipolar correlations, special interactions, or structurally driven effects are expected. In the absence of any additional factors that could complicate the interpretation, the assignment of the dynamics observed by both techniques to the primary α relaxation is relatively straightforward. Moreover, for these systems, the agreement between the shapes of the α relaxation observed by different techniques is reasonably good.^23,26 Samples in Subsections III B and III C were selected to validate the comparison approach in more challenging scenarios, where the presence of additional interactions and or structural characteristics leads to a complex dynamic response. When more than one dynamic process is present, the calorimetric relaxation follows the component of the dielectric response previously identified with the primary α-relaxation in the literature. This point was tested for type A and B polymers, as well as for several monoalcohols with varying degrees of overlap between their Debye and α processes. The final part of Sec. III, Subsection III D, was dedicated to critically assessing the limits of the aforementioned quantitative relationship within experimental accuracy. The mismatch between the timescales from the overall dielectric maximum and the entropy fluctuations in equilibrium in 3P1P and 1P1P accurately signals the complex nature of the dielectric response in these systems, demonstrating the ability of the proposed procedure to discriminate structural relaxation among overlapping processes with characteristic timescales separated by less than a decade. It is important to note that, even though the underlying cause–effect relationship is universal, the dispersion of results depends on repeatability (operator and instrumental), instrument quality and calibration, and/or experimental procedures and data analysis methodologies. As a consequence, prior to using the proposed comparison methodology to draw solid conclusions on the origin of a certain dielectric component, one should (i) validate the applied measuring and analyzing protocols and (ii) determine the reproducibility and sensitivity of a particular given pair of instruments in discriminating between different scenarios.

As discussed in the Introduction, there is an ongoing debate regarding the interpretation of dielectric relaxation in relatively polar systems, particularly in relation to the dynamic response obtained by other techniques that are less sensitive to cross correlation contributions. There is good consensus in that dipolar cross correlations are at the origin of the relatively narrow dielectric response of high dielectric strength glass-forming systems. This contribution, however, challenges the idea of a universal spectral shape for the primary structural relaxation. This issue has raised some questions on the feasibility of DS for characterizing the primary structural relaxation of glass-forming systems, a controversy that requires a solid answer. In this context, Sec. IV focused on comparing the dielectric and calorimetric responses of moderate to high dielectric strength glass-forming systems. Having large dielectric strength, the systems analyzed in this section exhibit a relatively narrow dielectric response, which according to the general belief reflects the increasing contribution of dipole–dipole cross correlations to the overall dielectric signal. Nevertheless, and regardless of the dielectric strength of the system, the same quantitative correlation between dielectric and calorimetric responses remains valid.

First, in Fig. 7, calorimetric and dielectric timescales were compared for systems showing moderate to high dielectric strength and “simple” shear response. For these systems, having simple viscoelastic behavior, the maximum of the dielectric relaxation and the crossing of shear moduli are quantitatively correlated. This along with the relation between entropy fluctuations and dielectric signal makes the assignment of the dielectric response to the primary α-relaxation highly robust. The result that the relatively narrow dielectric relaxation of polar systems can be univocally related with the primary structural relaxation challenges the notion of a universal spectral shape for this dynamic process.

In a scenario where the dielectric response of polar systems is interpreted as the sum of (i) a self-component identified with the α-relaxation and (ii) a slower cross component identified as a dipolar Debye process, one could argue that, for simple systems, entropy fluctuations correlate with the dominating Debye process due to indistinguishable timescales for both self- and cross-contributions. That reasoning, however, does not apply to other systems such as TBP in Fig. 8(b). TBP was one of the first systems where dielectric and photon-correlation response data were directly compared, showing that the former was narrower and slower than the later. Since photon-correlation spectroscopy is largely insensitive to orientational cross correlations, the dielectric signal of TBP was then decomposed into (i) a fast self-component identified with the α-relaxation and the light scattering response and (ii) a slower cross component identified as a dipolar Debye process. This view raised some controversy in that, by referring solely to the self-part as α-relaxation, the role of dipolar intermolecular interactions was somewhat relegated to a peculiarity of the dielectric response. On comparing calorimetric and dielectric data for TBP, entropy fluctuations correlate with the slower dielectric component attributed to dipole–dipole cross correlations and not with the faster self-component. This crucial observation indicates that cross contributions to the dielectric response are integral to the physics of structural relaxation and that they should also be regarded as α-relaxation. Yet TBP is not an isolated case. For example, the dielectric response of propylene glycol has also been reported to be narrower and ∼20 times slower than the response measured by photon-correlation spectroscopy, which presumably captures the self-contributions to the structural relaxation.²⁷ For propylene glycol as well, calorimetric relaxation times correlate well with the dielectric maximum [see Fig. 9(a)]. The same can be said about the dielectric response of glycerol, reported to be about three times slower than that measured by photon-correlation spectroscopy⁸ [see Figs. 10(c) and 10(d)]. These findings confirm the conclusions of previous studies by Moch et al.^10,16 in TBP and PG and further extend them by generalizing the results to many other systems.

In conclusion, the premise that the dielectric α-relaxation has a one-to-one correspondence with entropy fluctuations in equilibrium near the glass transition has been verified, as demonstrated through a robust empirical quantitative relationship. This correspondence extends beyond the temperature dependence of the characteristic times, and quantitatively, the shift bringing together the times measured by these two approaches is the same for all the examples shown and under the experimental procedures followed. The present systematic comparison provides strong evidence that the dielectric α relaxation of glass-forming systems truly reflects the primary structural relaxation of the systems, at least when structural relaxation is understood as entropy fluctuations in equilibrium. More importantly, this correlation holds true even in cases where dipolar cross correlations contribute significantly or even dominate the dielectric signal. This highlights the crucial role of dipolar intermolecular interactions on the primary structural relaxation and underscores the necessity of appropriately referring (also) to the cross component as the α relaxation. Finally, the present findings do not support the existence of a generic spectral shape for the structural relaxation valid for all types of susceptibility functions. The shape universality, however, might still be valid for susceptibility functions relying on self-correlations.

The supplementary material encompasses details on the acquisition and analysis of shear data to determine whether a given liquid is classified as having “simple” or “complex” viscoelastic behavior in the context of the paper.

I acknowledge financial support under Grant No. PID2021-123438NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and Grant No. IT1566-22 by the Basque Government. I would also like to acknowledge Daniele Cangialosi (CSIC) and Angel Alegría (UPV/EHU) for helpful discussions, useful comments, and support.

The author has no conflicts to disclose.

S. Arrese-Igor: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.