Bienvenu Atawa

Natália T Correia

Nicolas Couvrat

Frédéric Affouard
email: frederic.affouard@univ-lille.fr

Gérard Coquerel

Eric Dargent

Allisson Saiter

Molecular Mobility of Amorphous N-Acetyl-α-Methylbenzylamine and Debye Relaxation evidenced by Dielectric Relaxation Spectroscopy and Molecular Dynamics Simulations

The present work focusses on the molecular mobility characterization of amorphous N-acetyl--methylbenzylamine (Nac-MBA) by Broadband Dielectric Relaxation Spectroscopy (DRS) coupled to Fast Scanning Calorimetry (FSC) and Molecular Dynamics (MD) Simulations covering over 12 decades in the frequency range. This study reveals another example of a secondary amide that shows a very intense Debye-like contribution (almost 90% of the global dielectric intensity) in addition to the structural -relaxation and secondary Johari-Goldstein -relaxation). The D-and -relaxations are separated by about one decade (in frequency) and their relaxation times follow a near parallel temperature evolution (Vogel-Fulcher-Tammann-Hesse). The micro-structure of Nac-MBA has been investigated from MD simulations. It is shown that intense Debye-like process emanates from the formation of linear intermolecular H-bonding aggregates (precursors of the crystalline structure) generating super-dipole moments.

Introduction

Glass forming materials are omnipresent in our daily life and play an essential role in modern technology. Their physical understanding is still controversial and remains an unsolved problem of condensed matter physics. [START_REF] Angell | Glass-Formers and Viscous Liquid Slowdown since David Turnbull: Enduring Puzzles and New Twists[END_REF][START_REF] Debenedetti | Metastable Liquids: Concepts and Principles[END_REF] One of the most intriguing phenomenon is the extraordinary increase of the structural relaxation time in the supercooled liquid state when approaching the glass transition temperature (Tg), the non-Debye character of the frequency or time domain response and non-Arrhenius behavior of the temperature dependence. [START_REF] Richert | Disorder Effects on Relaxational Processes[END_REF][START_REF] Ediger | Supercooled Liquids and Glasses[END_REF][START_REF] Angell | Relaxation in glassforming liquids and amorphous solids[END_REF][START_REF] Ngai | Dynamic and thermodynamic properties of glass-forming substances[END_REF][START_REF] Donth | The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials[END_REF][START_REF] Descamps | Disordered Pharmaceutical Materials[END_REF] The dynamical behavior of glass-forming liquids and thus of their relaxational processes cover a wide range of time scales from local vibrations to the highly cooperative motions as the temperature approaches Tg. [START_REF] Stickel | Dynamics of glass-forming liquids. II. Detailed comparison of dielectric relaxation, dc-conductivity, and viscosity data[END_REF] When it comes to molecular mobility investigation in glass forming liquids or disordered systems exhibiting a permanent dipole moment, Dielectric Relaxation Spectroscopy (DRS) remains one of the most powerful and useful tool to achieve this goal. [START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF][START_REF] Lunkenheimer | Glassy dynamics[END_REF][START_REF] Brás | Molecular Motions in Amorphous Ibuprofen As Studied by Broadband Dielectric Spectroscopy[END_REF][START_REF] Schammé | Molecular Relaxations in Supercooled Liquid and Glassy States of Amorphous Quinidine: Dielectric Spectroscopy and Density Functional Theory Approaches[END_REF][START_REF] Grzybowska | Disordered Pharmaceutical Materials[END_REF][START_REF] Rijal | Dynamic Heterogeneity and Cooperative Length Scale at Dynamic Glass Transition in Glass Forming Liquids[END_REF][START_REF] Floudas | Molecular Dynamics of Glass-Forming Systems: Effects of Pressure[END_REF][START_REF] Kremer | The Scaling of Relaxation Processes[END_REF] Generally, for most glass forming liquids, the main process localized at the lowest frequency is recognized as the structural relaxation process (α-relaxation) associated to the dynamic of glass transition and molecular cooperative motions in an environment of few nanometers. 7,18- 20 The frequency and temperature dependency of dielectric features of the α-relaxation present a non-Debye and non-Arrhenius character respectively. However, some glass formers display a slower dielectric relaxation than α-process with a Debye-like behavior and a non-Arrhenius temperature dependency of the relaxation times.

The so-called D-process has been observed in many hydrogen-bonded liquids such as water, [START_REF] Floriano | On the relaxation between Debye and nonexponential relaxation in supercooled monohydric alcohols and water: A solution study[END_REF][START_REF] Kaatze | Dielectric and structural relaxation in water and some monohydric alcohols[END_REF] monohydroxy alcohols, [START_REF] Stickel | Dynamics of glass-forming liquids. II. Detailed comparison of dielectric relaxation, dc-conductivity, and viscosity data[END_REF][START_REF] Dannhauser | Dielectric Study of Intermolecular Association in Isomeric Octyl Alcohols[END_REF][24][25][26][27][28] secondary amides N-mono-substituted amides, [29][30][31][32] and pharmaceuticals. [START_REF] Brás | Molecular Motions in Amorphous Ibuprofen As Studied by Broadband Dielectric Spectroscopy[END_REF][33][34][35] Furthermore, for monohydroxy alcohols and secondary amides, the dielectric strength usually exceeds by several orders of magnitude the expected value predicted from the molecular dipole moment basis. [START_REF] Dannhauser | Dielectric Study of Intermolecular Association in Isomeric Octyl Alcohols[END_REF]29 Surprisingly, the D-process does not contribute in the calorimetric signature 36,37 although a very intense dielectric response. Likewise, viewed from earlier shear mechanical spectroscopy experiments, no significant contribution in the mechanical shear signal was associated to D-process. Since then, Gainaru et al 38 evidenced by high resolution shear mechanical spectroscopy the mechanical signature of D-process in two monohydroxy alcohols close to the glass transition region. Moreover, the rheological behavior is similar to what is observed for short chain polymers. 38 Nevertheless, the origin of the D-process has been a matter of discussions for many decades and still remains an unsolved puzzle in condensed matter physics. 24,[39][40][41] In H-bonded liquids, the D-process often originates from structural correlation between several dipoles. 42 Numerous models have been proposed to explain the microscopic origin of the D-process in these systems: wait and switch models of dipole reorientation 25,43,44 (breaking of H-bonds in a chain and formation of a new chain leading to end-to-end fluctuations of the dipole moment), dipole inversion by cooperative rotation of the OH-groups 45 and self-restructuring transient-chain model, where molecules break off from or add to the ends of the chain promoting slow rotations of the effective dipoles of the chain. 46 So far, none of these models can give a universal account of the origin of the Debye behavior. Thus, a more fundamental and predictive model is still expected. 42 Nevertheless, the postulated models figure out the important role played by H-bonds in the expression of D-process.

In order to shed some light on the microscopic origin of the peculiar D-process, we have performed a thorough combined investigation of physical properties of an H-Bond (HB) glass forming liquid, N-acetyl-α-methylbenzylamine, C10H13NO (Figure 1), by means of Dielectric Relaxation Spectroscopy (DRS), Temperature Modulated Differential Scanning Calorimetry (TMDSC), Fast Scanning Calorimetry (FSC), and Molecular Dynamics (MD) Simulations. Nacetyl-α-methylbenzylamine (Nac-MBA hereafter), is a chiral mono-substituted amide derivative from α-methylbenzylamine (a widely used chiral resolving agent for optical resolution procedures 47 ).

In this study the investigated Nac-MBA system (Figure 1) is actually a mixture of 25% of the pure R(+) enantiomer with 75% of the counter S(-) enantiomer, since the optically pure enantiomers of Nac-MBA present an enhanced crystallization tendency (forthcoming paper). 2. Materials and methods.

R (+) enantiomer S (-) enantiomer

Synthesis of R(+) and S(-) Nac-MBA

Optically pure R(+) or S(-) Nac-MBA was obtained by acetylation of R(+) or S(-)

α-methylbenzylamine purchased from Aldrich (99%) using acetyl chloride. 48,49 The method (see ref 50 for more details) consists to dilute the amine in dichloromethane and mixed the diluted solution with an aqueous solution of sodium carbonate initially prepared. Slight excess of diluted acetyl chloride in dichloromethane is slowly added to the previous solution. After stirring, the reaction mixture is decanted and the unreacted products are removed by successive washings with acid and basic aqueous solutions. The washed solution is dried with MgSO4 and evaporated (yield almost 90%). The white solid obtained is doubly recrystallized from toluene or purified by column chromatography. 50 The mixture studied in the present work (25% R(+) plus 75% S(-)),corresponds to an enantiomeric excess, ee (i.e. the absolute difference between the mole fractions of each enantiomer) of 50%.

Temperature Modulated Differential Scanning Calorimetry (TMDSC)

TMDSC and conventional DSC analyses were carried out with a Q100 DSC from TA instruments equipped with a refrigerated cooling system allowing a cooling down to 203 K. All measurements were performed under nitrogen gas flow of 50 mL/min. Baseline (empty cell) was calibrated at a chosen heating rate from 203 to 423 K. Temperature and enthalpy calibrations were achieved using indium and benzophenone as reference materials. An empty aluminum pan, identical to that used for the sample, was used as reference. Specific heat capacity was calibrated using two sapphire discs as a reference materials.

TMDSC experiments were carried out in the "heat only" modulation mode with 1) an oscillation amplitude of ±0.318 K and oscillation period of 60 s, during heating at 2 K/min and 2) an oscillation amplitude of ± 1 K and oscillation period of 60 s during heating at 0.5 K/min.

The oscillation period refers to the time needed to complete one temperature modulation cycle.

The modulation scanning was performed on an initially amorphous sample obtained during a previous cooling at 70 K/min carried out on the conventional mode.

Sample Preparation:

In order to prepare the DSC samples with an enantiomeric excess of 50% in the S(-) enantiomer, 75 mg of S(-) Nac-MBA and 25 mg of R(+) Nac-MBA were manually grounded in a mortar and 5 mg of this physical mixture was encapsulate in a Tzero aluminum pan. Each experiment was repeated at least twice using fresh samples.

Fast Scanning Calorimetry (FSC)

FSC measurements were conducted with a power compensation twin-chip namely Flash DSC 1 (FSC) from Mettler-Toledo. Cooling was ensured down to 203 K by an intracooler model Huber TC100. Approximately 35 ng of sample was loaded onto the central position of a sensor chip using an optical microscope and a long fiber for sample positioning. The sample was submitted to a nitrogen purge gas flow at a rate of 20 mL/min. Successive conditioning and corrections of the chip sensor were performed according to the manufacturer and prior to measurements. 51 As highlighted in previous work on FSC, 52 it remains ambiguous to ensure an effective composition of ee=50% even though the sampling (dozens of ng) were taken from a 10 mg mixture of both enantiomers in such way that the average ee=50%. Here, we considered the ee as intermediate between enantiomerically pure and racemic composition (i.e. scalemic). Sample mass was estimated by comparing the amplitude of the heat capacity step at the glass transition with values obtained from classical DSC. A pre-melting step was launch to ensure homogeneous distribution of the sample on the chip area. With regards to determine the cooling rate dependency of the fictive temperature Tf, a cyclic heat-cool thermal program was applied. This protocol consisted in quenching the liquid at different cooling rates from 1500 K/s down to 5 K/s followed by a heating scan at 1500 K/s. To ensure reproducibility, all measurements were performed at least twice.

Dielectric Relaxation Spectroscopy (DRS)

a -Low frequency range 0. (Eq. 2)

The complex frequency-dependent permittivity 𝜀 * (𝑓) = 𝜀 ′ (𝑓) -𝑖𝜀 ′′ (𝑓) (where 𝜀 ′ =storage permittivity and 𝜀 ′′ = loss permittivity) was measured over the range 10 -1 Hz -2MHz using an Alpha-A analyzer from Novocontrol Technologies GmbH. Sample preparation and amorphization steps were performed ex situ. Almost 150 mg of the crystalline powder was deposited on the IE then heated above the liquidus temperature at 383 K and annealed 5 minutes to guarantee complete melting. The melt was then promptly quenched in liquid nitrogen in order to ensure total amorphization (according to previous work by some of us on the amorphization conditions of Nac-MBA). 50 Dielectric spectra were collected isothermally from 153 K up to 273 K with a suitable consecutive step: measurements were performed from 241 K to 273 K with an increment of 1 K; for T< 241 K an increment of 2 or 5 K was considered.

b -High frequency range 1 MHz -1 GHz

Measurements were performed in the range 1 MHz -1 GHz by mean of a coaxial line reflectometer based on the impedance analyzer HP 4191 from Novocontrol Technologies GmbH. Preceding measurements, two successive calibration steps where achieved: Line and sample cell calibrations. Samples were prepared in a parallel plate geometry between two gold-plated electrodes of 10 mm diameter, separated by silica spacers of 50 μm thickness. The prepared sample cell was connected to the end of the coaxial line, then melted in situ and annealed for 5 minutes at 383 K. Isothermal frequency scans were carried out upon cooling from 383 K down to 273 K, by steps of 10 K.

A Quatro Cryosystem (Novocontrol Technologies GmbH) was used to control the temperature with a stability of ± 0.2 K for both low and high frequency domains.

c -Data Analysis

In order to access the time scale of each of the four processes expressed in Nac-MBA (ee=50%) system, their temperature dependencies, the shape parameters, and the dielectric strength, the isothermal dielectric spectra (𝜀 ′ and 𝜀 ′ ′) were fitted with a sum of Havriliak-Negami model functions 56 and a conductivity contribution for 𝜀 ′ ′, writing as followed:

𝜺 * (𝝎) = 𝜺 ′ (𝒇) -𝒊𝜺 ′′ (𝒇) = 𝜺 ∞ + ∑ [

Molecular Dynamics Simulations (MD)

Molecular dynamics (MD) simulations have been performed using the DL_POLY package 57 and the GAFF force field (General Amber Force Field). 58 This force field was chosen due to its capability of reproducing successfully large number of experimental data of low molecular weight molecules. 59 As recommended for the GAFF force field, 58 atomic partial charges were calculated from ab-initio computations using the Gaussian 03 programme 60 at the HF/6-31G* level with the restrained electrostatic potential (RESP) fitting approach. For the electrostatic interactions, a pairwise damped shifted method developed by Wolf 61 was used. For both van der Waals and electrostatic interactions, the same cutoff radius of 10 Å was employed.

Periodic boundary conditions were applied in all directions. The time step to integrate Newton's equation of motion was chosen to 0.001 ps. The modelled system possesses a total number of N = 64 molecules composed of 48 S(-) Nac-MBA and 16 R(+) Nac-MBA molecules in order to obtain an enantiomeric excess of 50%. Simulations have been conducted either in the NPT or NVT statistical ensemble where P is the pressure, T the temperature and V the volume.

Pressure and temperature were controlled with a Nose-Hoover barostat and thermostat respectively. All NPT simulations were realized at atmospheric pressure. The Nosé Hoover thermostat and barostat relaxation times have been set at 0.2 and 2.0 ps respectively. The equilibrated volume of the simulation box during the NPT simulation was considered to compute the averaged density of system and used to perform the subsequent production simulation in the NVT ensemble. MD simulations were performed at five temperatures 500 K, 450 K, 400 K, 360 K and 340 K. The equilibration/production times range from 10/200 ns to 20/800 ns from the highest to the lowest investigated temperature. The Nac-MBA densities determined in the present study are shown in figure 2. The monotonic behavior of the density as function of the temperature confirms that the system is equilibrated over the simulated times. The calculated density of the liquid extrapolated to a low temperature near below the glass transition temperature (𝜌 (235 𝐾) = 1.1 𝑔/𝑐𝑚 3 ) is consistent regarding the crystalline counterpart: 𝜌 = 1.13 𝑔/𝑐𝑚 3 . 47 Indeed, typically, the density of the glassy state is above 90% of the density of the crystal 62-64 (i.e. 𝜌 = 1.017 𝑔/𝑐𝑚 3 ).

Results and discussions

Thermal behavior characterization of Nac-MBA by TMDSC and FSC

Figure 3a) shows conventional DSC thermograms of a physical mixture of Nac-MBA (ee=50%) for the first (curve 1) and the second (curve 2) heating scans at 2 K/min. When heating the crystalline powder mixture, three successive endotherms are evidenced and associated to the eutectoid (Teutectoid= 338 K), eutectic (Te= 348 K) and liquidus (TL= 353 K) phenomena, respectively. At the eutectoid temperature, the physical mixture of both enantiomers (conglomerate) converges towards a co-crystal of both enantiomers (racemic compound).

These results are in good agreement with the phase diagram between both enantiomers of Nac-MBA proposed in the work of Druot et al. 47 During liquid quenching from 383 K down to 203 K at circa 70 K/min (not shown) no sign of crystallization is observed. The subsequent heating scan (curve 2) shows the characteristic heat flow step for the calorimetric glass transition at circa 240 K along with the typical enthalpy recovery peak (zoom in insert). It is a proof that an amorphous glass has been obtained during the earlier stage of melt quenching.

Therefore, under this condition, Nac-MBA can be considered as a good glass forming liquid.

Upon further heating well above Tg, cold-crystallization of the supercooled liquid is evidenced at TCr on = 273 K (ΔHCr = 70 ± 5 J/g). This exothermal peak is followed by a double endothermal event related to the eutectic and liquidus events of the corresponding racemic compound. No thermal event revealed the existence of the conglomerate equilibrium. Consequently, from the amorphous glassy state Nac-MBA (ee=50%) evolves towards metastable equilibrium.

TMDSC allows to measure the heat capacity and kinetic components (usually referred as reversing and nonreversing components) of the total heat flow signal obtained in conventional DSC. 65 Fast scanning calorimetry is used to characterize the glass transition in terms of the variation of the fictive temperature over a wide range of cooling rates. The fictive temperature concept was introduced by Tool and Eicitlin 66 and defines the temperature at which any properties related to the structural relaxation are at the equilibrium state. supercooled liquid its magnitude increases with decreasing cooling rate as classically observed in literature for physical aging phenomena (for increasing aging time). [67][68][69] When 𝑞 𝑐 = 𝑞 ℎ = 1500 𝐾/𝑠, 𝑇 𝑔 is shifted toward higher temperatures for more than 10 K compared to standard DSC results obtained for 𝑞 ℎ = 2 𝐾/ min = 0.033 𝐾/𝑠. This shift is the expression of the kinetical character of the glass transition phenomenon observed for such kind of thermal protocol. 70 The fictive temperature (𝑇 𝑓 ) calculation was performed according to the equal area rule method 71 (details on 𝑇 𝑓 calculation procedure are described in ref 52 ). The change in the fictive temperature at variable cooling rates can be used to describe the temperature dependence of the structural -relaxation. 67,72,73 The empirical approach proposed by Frenkel-Kobeko-Reiner (FKR) 72 illustrates the link between the cooling rate 𝑞 𝑐 (K/s) and the structural relaxation time 𝜏 (s) express as follow:

𝒒 𝒄 𝝉 = 𝑪 (Eq. 5)
where 𝐶 (Kelvin) is a shift factor usually assume to be a constant 67,74 . Consequently it should be considered that:

𝒍𝒐𝒈 (𝒒 𝒄 ) = -𝒍𝒐𝒈(𝝉) + 𝒍𝒐𝒈 (𝑪) (Eq. 6)
This proposal is generally used to combine DSC and/or FSC data with features of the dynamic glass transition as probed from heat capacity or dielectric relaxation spectroscopy measurements. 67,[74][75][76][77] Superposition of the temperature dependence of the cooling rate and the dielectric -relaxation time will be reported in the relaxation map (Figure 9a)); section 3.2.2).

Molecular Mobility of Nac-MBA (ee = 50%) probed by DRS

As highlighted by calorimetric results, amorphous glass Nac-MBA (ee=50%) can be obtained by melt quenching. For dielectric characterization of the glassy and supercooled liquid state near and above Tg, the amorphous sample was prepared ex-situ to guarantee a sufficient high cooling rate in order to avoid crystallization. Measurements of the complex permittivity *(f) were then performed isothermally, during heating, in the temperature range [153 K, 273 K] with a frequency scan from 0.1 Hz to 2 MHz for each selected temperature. Isothermal, high frequency (1 MHz -1 GHz) measurements of *(f) were carried out when cooling the liquid from 383 K down to 283 K with regards to access the molecular mobility at higher temperatures in the supercooled and liquid state (see experimental section 2.4 for details).

Identification and analysis of dipolar relaxations

The temperature dependence of the real permittivity 𝜀 ′ at some representative frequencies (taken from the isothermal dielectric data acquired upon heating) is shown in Figure 5a). Near the calorimetric glass transition a frequency dependent step increase is observed. For temperatures higher than 260 K a steep decrease is visible in the 𝜀 ′ (T). This drop in the dielectric permittivity is frequency independent and represents the manifestation of coldcrystallization. Melt crystallization was also observed at circa 273 K during the isothermal high frequency (RF) measurements (not shown). Therefore, in the following only the dielectric spectra obtained for the fully amorphous samples will be taken for analysis and discussion (i.e., data taken at 260 K < T < 283 K are excluded). As shown in the inset of figure 5a), a very intense dielectric loss 𝜀 ′′ response is observed in the supercooled liquid, associated to the frequency dependent step 𝜀 ′ increase when the sample is heated from the glassy state. A close up of the 𝜀 ′′ (T) trace at the lowest measured frequency (0.1 Hz) is depicted in Figure 5b).

A quick analysis of this spectrum using two Gaussian functions allows clear evidence for two relaxations: a very low intensity peak with maximum at Tmax = 240.9 K and a predominant one at Tmax = 246.3 K. For further information on these processes, Figure 5c) shows the first temperature derivative of the heat capacity Cp´(T) curve. The real part of the complex heat capacity (Cp´) of a system is related to the reversing heat capacity measured from TMDSC. [START_REF] Rijal | Dynamic Heterogeneity and Cooperative Length Scale at Dynamic Glass Transition in Glass Forming Liquids[END_REF]78,79 The Cp´ curve shown in the inset of figure 5c) was obtained by TMDSC during heating at 0.5 K/min with an oscillation period of 60 s (f = 1.7x10 -2 Hz). The use of such scanning conditions allow a thermal treatment (and time scale) that reasonably compares to the dielectric relaxation experiments (equilibrium isothermal measurements were performed with temperature steps that correspond to an average heating rate of around 0.3 K/min). The temperature derivative of Cp´(T) curve (proportional to the imaginary component of the heat capacity, Cp´´) is perfectly described by a single Gaussian function with a full width at half maximum (FWHM) of ~3.9 K and centered at Tmax= 240.4 K. This temperature corresponds to the middle temperature of the sigmoidal shaped C´p(T), i.e., the kinetic glass transition temperature (Tg mid = 240.40.5 K). The low temperature shoulder in the 𝜀 ′′ (T) trace (Tmax = 240.9 K; FWHM = 3 K) is clearly associated with the measurable change of heat capacity and therefore it is recognized as the signature of the structural (primary) -relaxation. In spite of its high dielectric magnitude, the high temperature relaxation (Tmax = 246.3 K, FWHM = 5 K)

shows no calorimetric activity. Similar features are observed for hydrogen bonded liquids where a dominant Debye-like process (D-process) slower than the smaller -relaxation is observed. 31,32,38,41 For the Nac-MBA system here studied the high amplitude relaxation is also identified as a Dprocess taking into account that its dielectric loss frequency dependency can be well described by a single Debye function as it is illustrated later on (Figure 7b). Figure 6a) depicts the dielectric loss spectra as a function of the frequency for Nac-MBA at given temperatures below the glass transition temperature. The asterisks are visual guides delimiting three clearly distinct dispersion regions labelled -, 𝛽and ""-, in order of increasing frequency. As expected, at temperatures close to the dynamic glass transition (Tg mid = 240 K; f ~2x10 -2 Hz) the high frequency tail of the -relaxation becomes visible in the available frequency range. At higher frequencies than the -relaxation, a secondary 𝛽-process appears as a shoulder that clearly shift towards higher frequencies as temperature increases.

Deep in the glassy state, the 𝛽-relaxation seems to leave the frequency range and only a very broad dispersion (""-relaxation) remains. Figure 6b) shows the isochronal plot of log(𝜀 ′′ ) at chosen frequencies that allows a further evidence for the appearance of this broad ""relaxation at lower temperatures than the -relaxation (see also the close up in insert). For frequency f=5x10 [START_REF] Richert | Disorder Effects on Relaxational Processes[END_REF] Hz a crossover between the straight lines drawn to qualitatively describe the temperature dependence of ε ′′ (T) of the 𝛽and -process occurs at circa 240 K, close to the glass transition temperature, Tg mid . This feature is an indication of a Johari-Goldstein 𝛽relaxation. 80 frequency side f<10 3 Hz of the ε ′′ (𝑓) spectrum. The faster relaxation previously identified as the -relaxation is somewhat obscured under the high-frequency flank of the large amplitude D-process. In order to improve the frequency resolution of the overlapping dielectric loss peaks, the derivative method developed by Wübbenhorst and van Turnhout 81 is applied:

𝜺 𝒅𝒆𝒓 ′′ = - 𝝅 𝟐 𝝏𝜺 ′ (𝒇)
𝝏 𝐥𝐧(𝟐𝝅𝒇) (Eq. 7)

This method has the advantage to considerably improve the resolution of overlapping peaks and also suppress Ohmic conduction because in that case 𝜀 ′ is independent of frequency. 81

Relaxation map

The temperature dependence of the relaxation time of all the four detected processes (Debye, ,  and ) is summed up in relaxation map (-log vs 1/T) shown in Figure 9. For sake of simplicity and clarity, the Arrhenius plot will be analyzed from the glassy state towards the supercooled liquid. The evaluated parameters from the applied model functions (Arrhenius and VFTH) used to describe the temperature dependence of the relaxation times of the various processes are summarized in table 1. As generally observed for localized, non-cooperative motions observed in the glassy state, the relaxation time of both 𝛾and 𝛽processes follows a linear temperature dependence of log τ vs 1000/T well described by the Arrhenius relation:

𝜏 (𝑇) = 𝜏 ∞ 𝑒𝑥𝑝 ( 𝐸 𝑎 𝑅𝑇 ) (Eq. 9)
where 𝜏 ∞ is the pre exponential factor, 𝐸 𝑎 the activation barrier energy and R the ideal gas constant. Given the value of the activation barrier energy of the 𝛾-process (𝐸 𝑎 𝛾 ~39 𝑘𝐽/𝑚𝑜𝑙) it is reasonable to suggest that this secondary relaxation may originate from intramolecular, small angular fluctuations of polar side groups of Nac-MBA molecules that could persist deep in the glassy state. Nevertheless, it should also mentioned that small amount of impurities can also be at the origin of such manifestation. 82 The higher activation energy found for the 𝛽-process (𝐸 𝑎 𝛽 ~72 𝑘𝐽/𝑚𝑜𝑙) can be compatible with a complete rotation of the whole molecule as suggested for the 𝛽 Johari-Goldstein (𝛽 𝐽𝐺 ) process when analyzed in the framework of the Coupling Model (CM). 83 The temperature evolution of 𝜏 𝛽 strongly deviates from the Arrhenius behavior at the vicinity of Tg. This peculiar change of 𝜏 𝛽 (T) has been highlighted for many glass formers, [START_REF] Brás | Molecular Motions in Amorphous Ibuprofen As Studied by Broadband Dielectric Spectroscopy[END_REF]78,84,85 and it is recognized as a manifestation of the 𝛽 𝐽𝐺 -process. According to the CM, the 𝛽 𝐽𝐺 -process is considered as located near an independent or primitive motion (that implies the rotation of the whole molecule) precursor of the structural α-relaxation. The relaxation time of the 𝛽 𝐽𝐺 process is approximatively the same as that of the characteristic time of the primitive motion (𝜏 0 ) and correlated to 𝜏 𝛼 through the following equation:

𝜏 𝐽𝐺 (𝑇) ≈ 𝜏 0 (𝑇) = 𝑡 𝑐 𝑛 [𝜏 𝛼 (𝑇)] 1-𝑛 (Eq. 10)
where n is the coupling parameter (𝑛 = 1 -𝛽 𝐾𝑊𝑊 ) and 𝑡 𝑐 is the time characterizing the crossover from independent to cooperative fluctuations proven to be close to 𝑡 𝑐 = 2 × 10 -12 𝑠 for molecular glass formers. 86 From the experimental data it is very difficult to estimate a 𝛽 𝐾𝑊𝑊 value due to the poor frequency resolution of the α-relaxation in the dielectric loss spectrum. Though, a value 𝛽 𝐾𝑊𝑊 =0.66 can be estimated from the obtained HN shape parameters of the structural -relaxation ( 𝛼 𝐻𝑁 = 0.59 and 𝛽 𝐻𝑁 = 1) using the empirical relation proposed by Alegria and coworkers: 87

𝛽 𝐾𝑊𝑊 ≈ (𝛼 𝐻𝑁 𝛽 𝐻𝑁 ) 1 1.23 (Eq. 11) Taking into account the 𝛽 𝐾𝑊𝑊 value in the CM relation and 𝜏 𝛼 = 100 s at 𝑇 𝑔 , a value of 𝜏 𝐽𝐺 (𝑇 𝑔 ) = 2.22 × 10 -3 s is obtained. This value is in good agreement with the relaxation time derived from the analysis of the dielectric loss data as shown in figure 9a). We must note that the most obvious manifestation of this secondary relaxation (a weak shoulder in the dielectric loss peak) takes place in a very narrow temperature range (almost 15 K). These difficulties increase the uncertainties for a direct confirmation of the 𝛽 𝐽𝐺 nature of this slower secondary process. Therefore, empirical approaches such as the proposal of Ngai and Capacciolli 88 can be applied:

𝐸 𝑎 𝑅𝑇 𝑔 = 2.303 (2 -13.7𝑛 -𝑙𝑜𝑔𝜏 ∞ )
(Eq.12)

In the present case, the ratio 1/T). For the temperature range common to calorimetric and dielectric technique's ([240 K;

253 K] -the TMDSC raw point is also included), a strong similarity of the slope of the two set of data is clearly evidenced which without a doubt confirms the evaluation of 𝜏 𝛼 using the fitting procedure described previously. The nonlinearity of the temperature dependence (or super-Arrhenius behavior) of the -relaxation time is often described by the empirical Vogel-Fulcher-Tamman-Hesse (VFTH) equation given by: 89-91

𝜏(𝑇) = 𝜏 ∞ 𝑒𝑥𝑝 ( 𝐵 𝑇-𝑇 0 ) (Eq. 13)
where 𝜏 ∞ is the value of the relaxation in the high temperature limit, B is an empirical parameter representative of the material that accounts for the deviation of linearity and 𝑇 0 is the so-called Vogel temperature or ideal glass transition temperature ( 0 𝐾 < 𝑇 0 < 𝑇 𝑔 ). In the present case of Nac-MBA, the experimental data alone (Figure 9a)) or when combined with MD simulations (Figure 9b)) are fairly well described by a single VFTH function (see table 1)

with physically acceptable values of the pre-exponential factor corresponding to phonon lifetime ( 𝜏 ∞ ~ 10 -14 -10 -13 s). Note that the VFTH fit allows one to estimate, for = 9. Usually, the temperature evolution of the -relaxation time of all glass formers do not express the same deviation from an Arrhenius behavior near Tg. The quantification of this deviation is at the origin of the distinction between fragile and strong systems. This deviation near Tg can be quantified by the so called fragility (or steepness) index 𝑚 defined as follow: 92

𝑚 = [ 𝑑𝑙𝑜𝑔𝜏 𝛼 𝑑(𝑇 𝑇 𝑔 ⁄ ) ]
𝑇=𝑇 𝑔 (Eq. 14)

A glass former is considered as strong when 𝑚 ≤ 30 and fragile when 𝑚 ≥ 100. For the particular case where 30 < 𝑚 < 100 the liquid can be considered as an intermediate glass former. Combining Eq. 13 and Eq. 14, 𝑚 can be expressed as:

𝑚 = 𝐵𝑇 𝑔 𝑙𝑛10(𝑇 𝑔 -𝑇 0 ) 2 (Eq. 15)
Considering the values of B and 𝑇 0 reported in table 1 (for DRS and DRS+MD data set) an average value of 𝑚 = 111 ± 7 is obtained, allowing one to classify Nac-MBA as a fragile glass former. The secondary amide Nac-MBA here studied appears a much more fragile system than N-ethylacetamide and its mixtures with N-methylformamide for which a fragility index m = 65 has been reported. 31 As depicted in Figure 9b), where DRS and MD data obtained in the supercooled liquid and liquid state are combined, the temperature dependence of the D-process relaxation times, 𝜏 𝐷 also be described by a VFTH equation. The obtained values of the B parameter and of the Vogel temperature 𝑇 0 much resemble the ones of the VFTH used to describe the temperature dependence of 𝜏 𝛼 (table 1). These similarities between both relaxations could possibly mean that D-process may be controlled or governed by 𝛼-relaxation. [START_REF] Brás | Molecular Motions in Amorphous Ibuprofen As Studied by Broadband Dielectric Spectroscopy[END_REF] The non-Arrhenius temperature dependence of 𝜏 𝛼 is highly correlated with a non-Debye relaxation dynamics.

However the Debye-like process is not liable to any dispersion of the relaxation time and consequently cannot originates from the same microscopic phenomena as the α-process. 31,93 The ratio of the two times scales 𝜏 𝐷 /𝜏 𝛼 = 8 at T(𝜏 𝐷 = 100 s) = 241 K (~ 2 K above Tg DRS ); near the melting temperature Tm (~30 at T = 340 K from MD) and for 𝑇 → ∞, 𝜏 𝐷 /𝜏 𝛼 =25. The obtained ratio is in agreement with those observed in N-ethylacetamide and its mixture with N-methylformamide, 31,32 and some monohydroxy alcohols. 36,41,94 The microscopic origin of D-process and its implications in the dielectric response of the investigated Nac-MBA system (ee = 50 %) are probed by MD simulations and presented in the following section.

MD simulations investigations

Static dipolar properties

The A Kirkwood factor G K greater than 1 is often found in monohydroxy alcohols 25,43,96,97 and secondary amides 31,32 for which values may range from 3 up to 4. It is interpreted as the capability of these liquids to form HB associating structures as linear chains showing HB aggregates with parallel dipoles orientation. This type of HB associations has been often detected from the present MD investigations from snapshots of instantaneous configuration of the system, and will be discussed below. An example of an n = 6 Nac-MBA molecules HB aggregate is displayed in figure 11. It should be also noted that the G K values increase with decreasing temperature suggesting an increasingly parallel correlation of dipoles which can be seen as a precursor of the crystalline structure that may ultimately formed as observed many times in HB molecular liquids. [START_REF] Brás | Molecular Motions in Amorphous Ibuprofen As Studied by Broadband Dielectric Spectroscopy[END_REF]98 The crystalline structure of Nac-MBA suggests that the strongest intermolecular link is a HB between 𝑁 -𝐻 ⋯ 𝑂 atoms with 𝑑(𝑁 -𝑂) = 2. In other words, the latter hypothesis is consistent with the main feature of the crystalline structure (being the steric accessibility of the NH group leading to the presence of infinite linear H-bond chains of molecules between 𝑁 -𝐻 ⋯ 𝑂 atoms). 49 Though in the liquid or

Dynamical dipolar properties

Molecular mobility can be probed from the time dependence of the total electric dipole moment 𝑀 ⃗⃗⃗ (𝑡) at time t. 99 The total electric dipole moment is defined as 𝑀 ⃗⃗⃗ (𝑡) = ∑ 𝜇 ⃗ 𝑖 𝑁 𝑖=1 (𝑡) where 𝜇 ⃗ 𝑖 (𝑡) is the individual dipole moment of the molecule i at time 𝑡 and N = 64 the total number of molecules. In a first approximation, the complex frequency dependent permittivity 𝜀 * (𝑓) = 𝜀 ′ (𝑓) -𝑖 • 𝜀 ′′ (𝑓) measured experimentally by DRS is actually proportional to the Fourier transform of the time-dependent correlation function Φ 𝑀 (𝑡) = 〈𝑀 ⃗⃗⃗ (𝑡). 𝑀 ⃗⃗⃗ (0)〉 of the total dipole moment. [START_REF] Lunkenheimer | Glassy dynamics[END_REF] This function allows probing the collective dynamics of Nac-MBA dipoles and has been calculated at different temperatures (see figure 12). The long time behavior of the Φ 𝑀 (𝑡) functions is dominated by a single exponential decay which can be well ] in order to reproduce the behavior of the Φ 𝑀 (𝑡) function over the complete investigated time range.

β 𝑓𝑎𝑠𝑡 (𝑀𝐷) ≈ 0.82 is found to provide a reasonable adjustment in the fitting procedures.

The characteristic times τ 𝐷 (𝑀𝐷) and τ 𝑓𝑎𝑠𝑡 (𝑀𝐷) obtained at different temperatures have been

Hydrogen bonding structures

HB statistics allow determining the population of different HB associating structures and probing the structure of the investigated liquid with the aim to clarify experimental and numerical dipolar properties. The investigated molecule Nac-MBA possesses both a -NH and C=O group from which they may form intermolecular HBs (see figure 11). In the present study, two Nac-MBA molecules are considered to be H-bonded if i) the nitrogen-oxygen distance is less than 3.4 Å and ii) the (O⋯H-N) angle is larger than 150 deg. This criterion is classically used in MD simulations and allows including more deformed and weaker HBs in statistics. 100

The fraction 𝑃 𝑛 of HB aggregates composed of n Nac-MBA molecules detected on average during the MD simulation is displayed in figure 13 at the highest T = 500 K and the lowest T = 340 K investigated temperatures. A monotonic trend of the HB statistics is found from 500 K down to 340 K and data obtained at 450, 400 and 360 K are not shown for clarity. Figure 13 clearly shows that a large variety of HB aggregates are theoretically possible ranging from isolated molecule (n = 1) to very large aggregates made of n = 20 molecules. Additional even larger associated structures could be also transiently observed but their fraction remains really very small at all investigated temperatures. As expected from results obtained on dipolar correlations in the present study, the dominant intermolecular HB association originates from the formation of linear chain. As an example, a snapshot of a n = 6 HB aggregate is displayed in figure 11. Upon decreasing temperature, the fraction of isolated molecules (n = 1) and dimers (n = 2) progressively decreases. The population of trimers (n = 3) remains stable while the population of n > 3 aggregates increases. At the lowest investigated temperatures T = 340 K, associations as dimer (n = 2) and trimer (n = 3) are seen as the most probable structures. However, it may be speculated that the trend found in the investigated temperature range will continue somehow at lower temperatures at which larger structures n > 3 might become the most probable. It is confirmed by simply plotting the number of molecule n corresponding to the maximum of 𝑃 𝑛 as a function of temperature (data not shown). This evolution shows a monotonic increase of n upon decreasing temperature.

Additional MD simulations are clearly required that unfortunately extends beyond the scope of this study due to the difficulty to equilibrate liquid states at temperatures below T = 340 K from MD because of the slow dynamics of the dominant D process (see below).

In addition to the population 𝑃 𝑛 of a given HB aggregate of size n, the average total dipolar moment < 𝜇 𝑛 > for this aggregate may also be computed (see figure 13). The evolution of < 𝜇 𝑛 > as a function of n clearly shows that < 𝜇 𝑛 > increases with n, which is well in line with the linear nature of the HB aggregates (see snapshot on figure 11) and the Kirkwood correlation factor G K > 1 (see figure 10). Cyclic aggregates would cause < 𝜇 𝑛 > to decrease or even to go to zero. Interestingly, it may be also mentioned that the evolution of < 𝜇 𝑛 > is quite temperature-independent. All < 𝜇 𝑛 > data at T = 340 and 500 K overlap onto the same curve as well as T = 360, 400 and 450 K data (data not shown). It suggests that the same types of association may potentially form at all temperature whose the exact conformation -linear chain above all -does not change very much with temperature. From both 𝑃 𝑛 and < 𝜇 𝑛 >, it is also possible to calculate the product 𝑃 𝑛 •< 𝜇 𝑛 > which provides an estimation of the "dipolar weight" of an aggregate made of n molecules. At T = 340 K, although the population of aggregates made of n = 4, 5 or 6 molecules is weaker than n = 2 or 3, they could contribute to a larger extend to the dipolar response since they possess a total higher dipole moment.

Conclusions

Molecular mobility investigations in amorphous Nac-MBA system were achieved by combining experimental and simulation tools such as DRS, FSC and MD simulations covering over 12 decades in the frequency window. As many glass formers, Nac-MBA revealed a complex relaxational landscape. But interestingly a low frequency high intensity Debye-like process in the supercooled and liquid is observed by DRS and MD simulations, similarly to monohydroxy alcohols and secondary amides.

A secondary 𝛽-relaxation is detected in the glassy state and presents some characteristics (including a good agreement of its characteristic time at Tg with that predicted from the CM model) that suggest that is a genuine Johari-Goldstein relaxation type.

Due to the high amplitude of the dominant D-relaxation, the visualization and localization of the α-relaxation in the frequency dependence of dielectric loss data seems fastidious.

Somehow, the derivative method applied on experimental raw 𝜀´ data permitted improving the frequency resolution of α-process. The temperature dependence (fictive temperature) of the cooling rate performed by FSC mimics with an excellent agreement 𝜏 𝛼 (T) in the common probed temperature range. 𝜏 𝛼 (T) follows a single VFTH temperature evolution and the evaluated dielectric T g DRS agrees well with the kinetic Tg mid from TMDSC. The deviation from the Arrhenius behavior at 𝑇 𝑔 measured by the fragility index (𝑚 = 111 ± 7) revealed that Nac-MBA can be classified as a fragile glass forming liquid.

The relaxation time of the highly intense monodispersive D-process that appears at lower frequencies in comparison with the structural relaxation is well describe by a VFTH law and the fit parameters show some similarities with that of α-relaxation. This may be an indication that the dynamics of D-process is influenced by the α-relaxation. The Debye-like process contributes almost for 90% in the global dielectric polarization and does not present calorimetric signature.

The Debye like process is clearly observed from MD simulations and its characteristic relaxation times 𝜏 𝐷 are in very good agreement with high frequency DRS measurements confirming the validity of the calculations.

MD simulations also allowed to evidence in the liquid state the existence of HB aggregates of different size n involving dimers, trimers, tetramers (e.g., n= 2, n=3, n= 4) and so on which can be mainly described as HB linear chains. As temperature decreases, longer H-bonded chains become more probable. It is suggested that this trend could persist even in the supercooled liquid state and that HB linear chains of finite size could be thus seen as precursors of the future crystalline structure to grow composed of infinite HB chains.

HB linear chains possess parallel dipole orientation consistently with the Kirkwood correlation factor 𝐺 𝐾 > 1 like the behavior reported for monohydroxy alcohols and secondary amides.

Upon decreasing temperature, the Kirkwood correlation factor 𝐺 𝐾 increases well in line with the formation of longer H-bonded linear chains. Effective ''super-dipoles'' are actually generated by these chains and contribute to the dielectric response observed experimentally.

Although longer HB chains are less probable than shorter chains, they could contribute to a larger extend to the dipolar response since they possess a higher ''super-dipole'' moment.

Interestingly, the value of the ''super-dipole'' of a given HB linear chain of size n does not vary much with temperature implying that the "linearity" of these chains remains similar at all temperatures. This result nicely confirms that the hydrogen bond induces very strong directional intermolecular interactions which are absent in other non-associated liquids.

Globally, the results presented here for the secondary amide NAC-MBA, reinforce and confirmed the idea proposed in the literature for amides and monohydroxy alcohols that the high intense single exponential dielectric response originates from its rich microstructure due to hydrogen bonds. In the present study, only one excess enantiomeric ratio has been investigated ee=50%, it should be interesting to investigate other ratios to evaluate the impact of chirality on the Debye process which possibility could lead to additional insights on its still unclear origin.

Figure 1 :

 1 Figure 1: Developed formula of Nac-MBA enantiomers.

Figure 2 :

 2 Figure 2: Temperature evolution of the density of Nac-MBA (ee=50%) determined from MD

  Figure 3b) depicts the reversing specific heat capacity obtained for amorphous Nac-MBA upon heating at 2 K/min: the thermal glass transition region is very narrow with onset temperature 𝑇 𝑔 𝑜𝑛𝑠𝑒𝑡 = 240.7 K and Tg determined from the mid-step position 𝑇 𝑔 𝑚𝑖𝑑 = 242.4 K. The heat capacity change at Tg is ΔCP = 0.45 ± 0.01 (J/(g.K). It is worth pointing out that while the crystallization ability is strongly impacted by the enantiomeric concentration (forthcoming communication), the glass transition temperature and ΔCP are independent of the enantiomeric excess. 50

Figure 3 :

 3 Figure 3: a) Thermal behavior (normalized total heat flow versus temperature) of Nac-MBA (ee=50%) during heating at 2K/min (standard DSC); b) Reversing specific heat capacity (Cp) as function of temperature (at 2 K/min) in the glass transition domain (TMDSC).

Figure 4

 4 depicts the temperature evolution of the normalized heat flow during heating at rate 𝑞 ℎ = 1500 𝐾/𝑠, after several cooling scans with a rate (𝑞 𝑐 ) ranging from 5K/s up to 1500 K/s. The enthalpy recovery peak observed at the glass transition region shifts towards higher temperatures and

Figure 4 :

 4 Figure 4: FSC heating curves of Nac-MBA obtained at 𝒒 𝒉 = 𝟏𝟓𝟎𝟎 𝑲/𝒔 after cooling from the supercooled liquid state down to the glassy state at cooling rate 𝒒 𝒄 ranging from 5 K/s to 1500 K/s.

Figure 5 :

 5 Figure 5: a) Temperature dependence of the real part of complex permittivity (𝜺 ′ ) and dielectric loss 𝜺 ′′ (insert) of Nac-MBA (ee = 50%) for the indicated frequencies. b) 𝜺 ′′ (T) for f = 0.1 Hz (circles) in the glass transition temperature range [232 K-252 K]. Solid line -global fit using two Gaussian functions (dashed lines). c) First temperature derivative (d(C'p)/dT) of the Cp´ curve shown in inset (~reversing heat capacity obtained by TMDSC upon heating at 0.5 K/min with temperature oscillation period 60 s = 1.7x10 -2 Hz). Solid line: Gaussian curve fit to d(Cp´)/dT data.

Figure 6 :

 6 Figure 6: a) Frequency dependence of dielectric loss, 𝜺 ′′ of glassy Nac-MBA (ee = 50%) at the indicated temperatures. Lines connect data points. b) Temperature dependence of log(𝜺 ′′ ) at indicated frequencies (data taken from isothermal data). The insert shows a close up for T< Tg mid =240 K.

Figure 7a )

 7a Figure 7a) and 7b) show illustrative dielectric spectra (ε ′ and ε ′′ respectively) as a function of the available frequency range in the glassy state near below Tg and in the supercooled liquid state. The slower D-relaxation visible as a step in ε ′ or as a very intense dielectric response ε ′′ (almost 90% of the dielectric strength) arises at temperatures above 246 K. At T > 253 K dc conductivity (not shown for clarity reasons) rises at the lower frequency side of the D-process. In Fig 7b) it is shown that for T =259 K a Debye curve (solid red line) matches only the low

Figure 7c )Figure 7 :

 7c7 Figure 7c) shows that the 𝛼-relaxation appears much more clearly from the derivative expression at intermediate frequency range between the slower D-process and the faster 𝛽process. The conduction free dielectric loss 𝜀 𝑑𝑒𝑟 ′′ data was analyzed with a sum of derivative HN-functions in order to obtain the relaxation times corresponding to the frequency of maximal dielectric loss (fmax) for the D-and 𝛼relaxations. The calculated parameters were then used as starting values for a fitting procedure resulting in a reliable description of the raw data using a sum of Debye (𝛼 𝐻𝑁 = 1, 𝛽 𝐻𝑁 = 1) and Cole-Cole (𝛽 𝐻𝑁 = 1) model functions (equation 3). For high temperatures, a dc conductivity contribution was also added to describe the raw dielectric loss data at the lower frequency side of the D-process. Illustrative example of the fitting procedure used for a semi-quantitative analysis of the current data is given in figure 8 for T = 259 K. The D-process for T > 246 K is well described by a Debye model function (𝛼 𝐻𝑁 = 1, 𝛽 𝐻𝑁 = 1) corresponding to a single exponential behavior in the time domain. Concerning the structural 𝛼-relaxation (T > 241 K) the parameter that quantifies the symmetric broadening remains constant in the supercooled liquid domain(𝛼 𝐻𝑁 = 0.59 ± 0.01). Normally, the fitting procedure permits to derive information about the shape parameters, the dielectric strength and the relaxation time of the different processes. In the present case such fitting procedure is somehow challenging and daring regarding poorly resolved or overlapping relaxations phenomena, and low magnitude processes. The uncertainties of the width parameters (and dielectric strengths) and their temperature dependencies obtained from the fits, especially for the 𝛽 and 𝛾 relaxations, are rather large and must be taken indicative (see Supplementary materials). The temperature dependence of the relaxations times are now presented and discussed.

Figure 8 :

 8 Figure 8: Fitting procedure of the dielectric loss peak in the supercooled liquid at 259 K.The dash-dot lines represent the individual fitting functions (Cole-Cole for the -relaxation and Debye for the D-process). Solid lines are the total fit functions (see text for details).

Figure 9 :

 9 Figure 9: a) Relaxation map (-log vs 1000/T) for the -, -and -relaxations of Nac-MBA probed by DRS (vertical left-hand axis). The red filled circle corresponds to the kinetic glass

-

  Figure9a) includes data obtained from TMDSC (red full circle) and FSC (gray full circles) techniques allowing comparison with the dielectric -relaxation time 𝜏 𝛼 (red open circles). For the FSC data, a vertical shift factor (after the FKR relation (eq. 5), 𝑙𝑜𝑔 𝐶 = 0.67 ± 0.12 or C= (4.7 ± 1.3) K, is necessary in order to superimpose the temperature dependence of the cooling rate (-log(𝑞 𝑐 ) vs. 1/Tf) with the temperature behavior of the -relaxation time (-log(𝜏 𝛼 ) vs.

  agrees well with Tg mid = 240.4  0.5 K obtained from the specific heat capacity curve (TMDSC) at the equivalent frequency (~0.02 Hz and heating rate 0.5 K/min). The dielectric glass transition temperature Tg DRS (100 s) = 238.8 ± 0.4 K is obtained by extrapolating the VFTH fit to the relaxation time 𝜏 = 100 𝑠.

  individual dipole moment of a given molecule at time 𝑡 is calculated from the classical expression 𝜇 ⃗(𝑡) (𝑡) where 𝑞 𝛼 and 𝑟 ⃗ 𝛼 (𝑡) are respectively the fixed charge localized on atom 𝛼 and its position at time 𝑡, and N a =25 is the number of atoms in the considered Nac-MBA molecules C10H13NO. Fixed charges are directly taken from the ab-initio calculation (see section 2.5 MD simulations details). In the present investigation, the average dipole moment < 𝜇 > of Nac-MBA molecules computed from MD is found to be about 3.93 +/-0.01 Debye in the liquid state, and does not significantly vary with temperature. The symbol < > indicates an average on both time and molecules. It could be noted that this value is close to the value 3.74 Debye directly obtained from the ab-initio calculations of a single molecule in the gas phase (see section 2.5). Dipolar correlations are well described by the so-called Kirkwood correlation factor G K given by the following equation 95 𝐆 𝐊 = 𝟏 + (𝐍 -𝟏)〈𝝁 ⃗ ⃗⃗ 𝒊 . 𝝁 ⃗ ⃗⃗ 𝒋 〉/〈𝝁 𝟐 〉 (Eq. 16) where 𝜇 ⃗ 𝑖 is the dipole moment of the molecule 𝑖, 〈𝜇 2 〉 the average dipole moment, and 〈𝜇 ⃗ 𝑖 . 𝜇 ⃗ 𝑗 〉 indicates an average over distinct pairs of molecules (𝑖 ≠ 𝑗). The Kirkwood correlation factor accounts for the orientational correlation of neighboring dipoles. Parallel and antiparallel dipoles orientation lead to G K > 1 and G K < 1, respectively. If orientations of individual dipole are completely random, one finds G K = 1. The Kirkwood correlation factor G K was calculated at different temperatures from Eq. 16 and it is shown in figure 10. The Kirkwood correlation factor monotonically increases from about 1.6 at T = 500 K to about 2.6 at T = 340 K.

Figure 10 :

 10 Figure 10: Temperature evolution of the Kirkwood correlation factor GK of Nac-MBA (ee=50%) determined from MD simulations.

  87 Å, 𝑑(𝐻 ⋯ 𝑂) = 2.10 Å; ((𝑁 -𝐻 ⋯ 𝑂) = 171.8 deg. The structure reveals infinite syndiotactic linear chains of H-bonded molecules alternately parallel to 〈110〉 and 〈1 -10〉 in successive (004) slices. More details about the crystalline structure can be found in the work of Druot et al. 47

  this infinite intermolecular associations cannot be envisaged (lack of long range order), it is very likely that as temperature fluctuation increases the number of molecules implicated in these associations diminishes. The high dielectric intensity values may emanate from the linear nature of the H-bonding chains and linked to the number of molecules involved in these intermolecular associations.

Figure 11 :

 11 Figure 11: Snapshot extracted from an MD simulation run at T = 400 K showing an HB aggregate composed of n = 6 Nac-MBA molecules. This HB aggregate possesses a linear shape and the net dipole < 𝝁 𝟔 > of this aggregate is also schematically represented. Intermolecular HB are displayed as bold line.

  is the characteristic time associated with this long time decay. This behavior is well in line with the Debye like process observed by DRS. The single exponential law allowed us to reproduce the evolution of Φ 𝑀 (𝑡) almost over the whole investigated time domain except at relatively short times (10 < t < 100 ps) where an additional fast relaxational process is present (see figure12). This fast process possesses a very low intensity and a reliable characterization is difficult from the present MD data. It has thus just been taken into account considering an additional law ~exp[-(𝑡 τ

  represented in Figure 9b) for comparison with the DRS experimental data. It clearly shows that the characteristic times determined from MD simulations are in very fair agreement respectively with the relaxation times of the D-and -processes obtained from DRS experiments. Both processes are separated by about one to 1.5 decades in time and closely follow each other's in the investigated temperature domain very much like the behavior observed in monohydroxy alcohols and secondary amides.

Figure 12 :

 12 Figure 12: Time-dependent of the total dipole correlation function 𝚽 𝑴 (𝒕) = 〈𝑴 ⃗⃗⃗⃗ (𝒕). 𝑴 ⃗⃗⃗⃗ (𝟎)〉 for Nac-MBA represented at different temperatures T = 340, 360, 400, 450, and 500 K. The solid lines indicate a fit

Figure 13 :

 13 Figure 13: Evolution of the population 𝑷 𝒏 (bottom), the average net dipole < 𝝁 𝒏 > (middle) and the product 𝑷 𝒏 • 𝝁 𝒏 (top) for HB aggregates made of n Nac-MBA molecules at T = 340 (circles) and 500 K (squares).

  𝑁 gives the exponent of the frequency dependence of 𝜀 ′′ ; k sums over the different relaxation processes, 𝜏 𝐻𝑁 is a characteristic relaxation time regards to the frequency of maximal loss 𝑓 𝑚𝑎𝑥 ; ∆𝜀 is the dielectric relaxation strength of the process under investigation; 𝜀 ∞ the high frequency limit of the real permittivity ε′(f). 𝛼 𝐻𝑁 and 𝛽 𝐻𝑁 are fractional shape parameters which detail the symmetric and asymmetric broadening of the complex dielectric response in such a way that 0 < 𝛼 𝐻𝑁 , 𝛼 𝐻𝑁 .𝛽 𝐻𝑁 ≤ 1.[START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF] From the derived values of 𝜏 𝐻𝑁 , 𝛼 𝐻𝑁 , 𝛽 𝐻𝑁 , a model free relaxation time 𝜏 of each relaxation process is

	∆𝜺 𝒌 (𝟏+𝒊(𝟐𝝅𝒇𝝉 𝑯𝑵 𝒌 ) where 𝜎 𝑑𝑐 is the specific dc-conductivity; 𝜀 0 the permittivity of the vacuum (𝜀 0 = 𝜶 𝑯𝑵 𝒌 ) 𝜷 𝑯𝑵 𝒌 ] 𝒏 𝒌 -𝒊 ( 𝝈 𝒅𝒄 𝟐𝝅𝒇𝜺 𝟎 ) 𝑵 (Eq. 3) -𝟏 𝟏 8.854 × 10 -12 𝐹/𝑚); 𝑓 the frequency; given by: 𝝉 = 𝟏 𝟐𝝅𝒇 𝒎𝒂𝒙 = 𝝉 𝑯𝑵 [𝐬𝐢𝐧 ( 𝜶 𝑯𝑵 𝝅 𝟐+𝟐𝜷 𝑯𝑵 )] 𝜶 𝑯𝑵 [𝒔𝒊𝒏 ( 𝜶 𝑯𝑵 𝜷 𝑯𝑵 𝝅 𝟐+𝟐𝜷 𝑯𝑵 )] 𝜶 𝑯𝑵 (Eq. 4)

Table 1 : Fit parameters of the Arrhenius and VFTH functions used to describe the temperature evolution of the relaxation time of

 1

	𝑻 < 𝑻 𝒈	𝑬 𝒂 (𝒌𝑱. 𝒎𝒐𝒍 -𝟏 )	𝝉 ∞ (s)
	𝜸-process	39.4 ± 1.5	(6.6 ± 4.5) × 10 -14
	𝜷-process	71.8 ± 8.0	(5.37 ± 5.36) × 10 -17
	𝑻 > 𝑻 𝒈	B (K)	𝜏 ∞ (s)	𝑇 0 (K)
	𝜶-process VFTH (DRS)	1310 ± 90	(1.3± 0.5) ×10 -14	203 ± 2
	VFTH (DRS + MD)	1123± 66	(8 ± 2) ×10 -14	207 ± 2
	D-process VFTH	1024 ± 38	(2.0 ± 0.2) ×10 -12	209 ± 1

𝜸and 𝜷-secondary relaxations and 𝜶and D-processes respectively.

-Molecular mobility below Tg: 𝜸and 𝜷relaxations

Acknowledgments

The authors are grateful to FEDER and Region Normandie for the financial support of this work through FEDER MACHI project. This project has received funding from the Interreg 2 Seas program 2014-2020 co-funded by the European Regional Development Fund under subsidy contract 2S01-059_IMODE. The authors acknowledge the use of the facilities of the CRI (Villeneuve d'Ascq, France) where calculations were carried out.

Conflicts of interest

There are no conflicts to declare.