Carlos Fernandez-De-Alba

Andrew M Jimenez

Mozhdeh Abbasi

Sanat K Kumar

Kay Saalwächter
email: kay.saalwaechter@physik.uni-halle.de

Guilhem P Baeza
email: guilhem.baeza@insa-lyon.fr

On the Immobilized Polymer Fraction in Attractive Nanocomposites: Tg Gradient vs. Interfacial Layer

We study the slowing down of polymer dynamics in canonical nanocomposites made of strongly interacting mixtures of polyvinyl-pyridine (P2VP) and silica nanoparticles (radius =7 2 nm) by means of low-field NMR relaxometry. We demonstrate that this technique enables the accurate quantification of the fraction of immobilized (irreversibly adsorbed) polymer on to the filler surface, in a manner that complements data from broadband dielectric spectroscopy (BDS) and differential scanning calorimetry (DSC). We rationalize the slowing down by using two previously developed approaches to model the immobilized layer. The first one is based on a glass transition temperature, , gradient, while the second one assumes a single interfacial component with a distribution of relaxation times. While both models convincingly fit the NMR data providing us with robust and complementary conclusions, the former approach is found to fit more accurately previously published DSC experiments.

Quantitatively, NMR results indicate that the segmental relaxation of the immobilized layer is ca.

1 decade slower than in the bulk polymer while earlier BDS analyses were equivocal on this point, indicating either 1 or 2 decades. These discrepancies highlight the difficulties and potential model dependencies in quantifying the dynamics of the minority polymer fraction in the immobilized layer, necessitating a systematic multi-technique approach to properly characterize dynamical heterogeneities in polymer-based materials.

Introduction

Polymer based nanocomposites (PNCs) have attracted considerable interest due to their ability to significantly enhance the physical properties of polymers, e.g., mechanical, optical and electrical properties. 12,34 A well-known example concerns the application of PNCs in tires where the addition of nanoparticles improves the wear resistance while diminishing the vehicle's fuel consumption. [START_REF] Zhang | Materials development for lowering rolling resistance of tires[END_REF] Such properties enhancements have been assumed to be governed by an immobilized layer comprised of polymers that are essentially irreversibly bound to the nanoparticles (NPs) surface; with increasing NP loading, these layers percolate through the sample to form a network that significantly alters PNC macroscopic properties. [START_REF] Mujtaba | Detection of surface-immobilized components and their role in viscoelastic reinforcement of rubber-silica nanocomposites[END_REF] Among all nanofilled polymers, "attractive nanocomposites" made of a strongly interacting polymer-filler combination have been particularly investigated for their ability to form strong networks resulting in complex dynamics. A particularly notable case is of poly-2-vinyl pyridine (P2VP) filled with nano-silica where the two components interact favorably through multiple hydrogen bonds ensuring individual particle dispersion. [START_REF] Harton | Immobilized polymer layers on spherical nanoparticles[END_REF] The resulting network, based on silica NPs bridged by P2VP strands, was shown to noticeably increase the shear modulus of the material by more than two orders of magnitude when filled with ca. 30 vol.% (50 wt.%) of 14 nm diameter nanoparticles. 89 In contrast, the reinforcement in non-attractive nanocomposites (e.g. tires) filled with a similar fraction of nano-objects is limited to a factor of 20-30, i.e., 5-10 times less than the attractive case. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Mujtaba | Mechanical properties and cross-link density of styrene-butadiene model composites containing fillers with bimodal particle size distribution[END_REF] This difference in behavior is assigned to the immobilized (or interfacial) layer surrounding the NPs in the attractive case. This ensures stress transmission through "glassy bridges", [START_REF] Chen | Mechanical reinforcement of polymer nanocomposites from percolation of a nanoparticle network[END_REF] i.e., through adjacent polymer domains of slow segmental (or " ") dynamics. This "interfacial" layer must be distinguished from the "bound" layer which refers to a larger polymer sub-ensemble made of chains segments connected to the particles but located far enough from them not to have their average segmental motion perturbed. [START_REF] Popov | Strong Reduction in Amplitude of the Interfacial Segmental Dynamics in Polymer Nanocomposites[END_REF] The bound layer however has a major role on longer time dynamics such as those probed with rheology. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] Other authors found similar trends in other nanocomposites. [START_REF] Yang | Decoupling the polymer dynamics and the nanoparticle network dynamics of polymer nanocomposites through dielectric spectroscopy and rheology[END_REF] The presence of the immobilized layer was further evidenced through differential scanning calorimetry (DSC) 914 and broadband dielectric spectroscopy (BDS). [START_REF] Harton | Immobilized polymer layers on spherical nanoparticles[END_REF][START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF][START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF][START_REF] Gong | Segmental dynamics of polymer melts with spherical nanoparticles[END_REF][START_REF] Genix | Enhancing the mechanical properties of glassy nanocomposites by tuning polymer molecular weight[END_REF] Both of those methods revealed a clear broadening of the "calorimetric" glass transition and segmental relaxations with increasing the filler content, respectively. A seemingly well-separated slower relaxation mode ( ) was also observed by Cheng et al. with BDS. [START_REF] Cheng | Unexpected molecular weight effect in polymer nanocomposites[END_REF] In similar systems, Holt et al. rationalized the broadening of the segmental relaxation through a bicomponent Havriliak-Negami function representing the contributions of the neat matrix ( ) and a more sluggish interfacial layer ( ). [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] This approach is consistent with a fraction of polymer with reducedmobility shifted in timescale by about two decades, but with a similar activation energy as the neat P2VP. In contrast, in the context of a different study, the same group analyzed new BDS data on virtually the same samples and concluded that the slowing down of the interfacial layer was one decade only. [START_REF] Genix | Enhancing the mechanical properties of glassy nanocomposites by tuning polymer molecular weight[END_REF] This quantitative contradiction, i.e., the one or two decades dynamical slow down, illustrates that BDS, although having a large dynamic range, may suffer from significant fitting ambiguities.

In the same vein, while the interfacial polymer fraction (or corresponding interfacial layer thickness) was also determined from BDS measurements by using the amplitude of the process, e.g., [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF][START_REF] Cheng | Unexpected molecular weight effect in polymer nanocomposites[END_REF][START_REF] Cheng | Correlation between the temperature evolution of the interfacial region and the growing dynamic cooperativity length scale[END_REF] ), we acknowledge that its estimation is severely challenged by the presence of the interfacial polarization mechanism known as "Maxwell-Wagner-Sillars" (MWS). Because this phenomenon depends not only on the NPs structure and their surface chemistry, but also the nature and (temperature dependent) diffusion coefficient of the charge carriers within the PNC, it is hard to quantify. This makes the joint fitting of the real and imaginary part of the electric permittivity (respectively denoted and ) equivocal at best. Beyond quantitative estimations of the dynamical slow down and fraction of immobilized layer, the present work addresses the more fundamental question of whether the immobilized layer should be treated from a continuous or a bi-component approach. This question can be illustrated by the following example. By using BDS on similar nanosilica/P2VP composites, Holt et al. [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] and later Baeza et al. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] measured a similar broadening of the segmental relaxation with increasing the filler fraction. However, while in the former case the authors used a bi-component approach ( and ) to fit the imaginary part of the electric permittivity, a continuous broadening of the relaxation was considered in the latter case in addition to an MWS process.

This resulted in two different pictures being respectively (i) a distinct interfacial layer characterized by a segmental relaxation 10-100 times longer than the bulk's one, and (ii) a continuous slowing down of the polymer chains quantified by the drop of the "symmetric exponent" in the Havriliak-Negami function (denoted in ref. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] ), providing a quasi-identical average segmental relaxation time regardless of the filler content. While, both scenarios seem reasonable and well-argued, the best option is difficult to identify because of the presence of the MWS contribution, requiring therefore an alternative experimental technique.

Similar studies were conducted on poly-vinyl acetate (PVAc) [START_REF] Cheng | Unexpected molecular weight effect in polymer nanocomposites[END_REF][START_REF] Cheng | Correlation between the temperature evolution of the interfacial region and the growing dynamic cooperativity length scale[END_REF] and poly-ethylene-acrylate (PEA) [START_REF] Papon | Low-field NMR investigations of nanocomposites: polymer dynamics and network effects[END_REF][START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] based PNCs. These works verify the presence of an immobilized polymer layer at the filler surface, one that is characterized by strongly retarded dynamics. In ref. [START_REF] Papon | Low-field NMR investigations of nanocomposites: polymer dynamics and network effects[END_REF] , Papon et al. used low-field nuclear magnetic resonance (LF-NMR) to extract both the fraction and the dynamics of such immobilized chains (then termed "glassy layer", a term we henceforth avoid). In this work, the material was subdivided into three components, a mobile bulk material, an intermediate and a strongly immobilized polymer fraction, respectively; the latter two constitute the reducedmobility fraction. A significant advantage of NMR is that its signal is strictly proportional to the number of protons, thus allowing us to "count" monomers, and avoid some of the ambiguities of BDS (see above). Further, [START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] combining NMR and DSC data allowed the authors to prove that both sets of results were consistent with a continuous gradient in , such as imagined originally by Keddie and Jones, [START_REF] Keddie | Size-dependent depression of the glass transition temperature in polymer films[END_REF] for the PEA silica composites:

(

where is the distance from the particle surface, is the characteristic length of the gradient that must not be understood as the interfacial layer thickness, is the glass transition temperature in the bulk and describes the gradient shape. Note that this model should be considered simplified because it implies that other parameters in the Vogel-Fulcher equation describing the temperature dependence of the dynamics are unchanged (i.e., the -dependence of the segmental relaxation times is taken to be -independent). In addition, the cited work was subject to the more significant limitation that there is a relatively thick 1-2 nm base layer of the silanization agent on the NPs, which is treated as the inner shell before the onset of the gradient in the PEA. Here, we focus on the P2VP silica model nanocomposites, which do not contain any additional components (apart from a small amount of antioxidant). Relying on the "proton counting" feature, we use NMR in conjunction with DSC as unbiased approaches for quantifying the material with reduced-mobility and test whether the gradient or single interfacial layer models can best describe the data.

Materials and Methods

Nanocomposite preparation

PNC samples (loaded with 10, 30 and 50 wt.% in silica corresponding respectively to volume fractions 6, 18 and 31 vol.%) were prepared following the protocol used for the samples studied by BDS in ref. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] . Thermogravimetric analysis data is provided in Supporting Information (SI) section 1. Hydrophilic silica nanoparticles (radius =7 2 nm) were donated by Nissan Chemical Corporation, USA (MEK-ST). Methyl ethyl ketone (MEK, or 2-butanone, ACS reagent >99.0%) and pyridine (anhydrous, 99.8%) were purchased from Sigma-Aldrich, GER. The 100 kDa P2VP (purchased through Scientific Polymer Products, USA) was dissolved overnight in MEK at a concentration of 50 mg mL -1 together with a small amount of Irganox antioxidant (BASF, GER) ( 0.5 wt.% with respect to the polymer). The NPs, stabilized by the manufacturer in MEK, were mixed with pyridine at a ratio of 4:1 and sonicated for 10 minutes before being added into the polymer solution. The mixture was then vortexed for one hour and cast at room temperature in a Teflon dish for three days. Annealing was performed at 150 °C in a vacuum oven for five days prior to any testing. Transmission Electron Microscopy (TEM) was performed on a JEOL 1400 Flash microscope at 120 keV on samples ultramicrotomed at room temperature.

Micrographs in Figure 1 confirm uniform NP dispersion at all filler contents considered. Small angle X-ray scattering (SAXS) on low loading samples confirm uniform NP dispersion (not shown here, see ref. [START_REF] Harton | Immobilized polymer layers on spherical nanoparticles[END_REF] ). An additional vacuum drying step was used at 80 °C for 12 hours before the NMR experiments were conducted. Low-Field NMR NMR measurements were performed on a Bruker Minispec mq20, at a 20 MHz proton resonance frequency. The temperature was lowered from 473 K to 403 K with a BVT 3000 heater working with nitrogen gas (the glass transition temperature of P2VP is =373 K). Prior to each measurement, the temperature was stabilized for 10 minutes. The experiments were then repeated from low to high temperature to insure data reproducibility. Neat P2VP was further measured down to 313 K to extract the second moment of the dipolar-broadened NMR spectrum (see Equation 3 in section 3 and SI section 2).

In this work, we choose to use a methodology previously followed by others combining free induction decay (FID) and magic sandwich echo free induction decay (MSE-FID) pulse sequence. [START_REF] Papon | Low-field NMR investigations of nanocomposites: polymer dynamics and network effects[END_REF] While FID was acquired with a single π/2 excitation, the accurate fitting of the rigid component fraction (immobilized polymer) is challenged by the dead time (15 μs ) inherent to the technique. To solve this issue, we refocused the strong dipolar interactions (fast decay contributions) by using the MSE-FID sequence. [START_REF] Maus | A robust proton NMR method to investigate hard/soft ratios, crystallinity, and component mobility in polymers[END_REF] This signal suffers some intensity loss, but also allows for an accurate fitting of the most rigid components. For all the materials, both FID and MSE-FID signals at short times were adjusted to be between 60 and 100 % intensity and limited to 200 μs to avoid any field inhomogeneity which might affect the signal at longer acquisition times. Importantly, the gain was identical for both FID and MSE-FID experiments.

Results and Discussion

3.1 Preliminary considerations on the interfacial layer from DSC results.

Figure 2a shows our previous data on a PNC loaded with 31 vol.% in silica, where modulated DSC was used to quantitatively determine the temperature-dependent heat capacity of the PNCs. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] The plotted data is obtained by subtraction of the silica contribution (with known and near-linear temperature dependence) around the region of the P2VP glass transition. The plot also shows the step for neat P2VP, which is the "base function" from which the fits to the trace of the PNC was constructed. For each assumed contribution (i.e., spherical shell) with shifted , the neat response was shifted by the required , not just along the x-axis but along a line with the slope defined by the glassy-state heat capacity, i.e., the slope of the initial part of the P2VP trace. For fixed =1, we obtained values between 0.01 and 0.02 nm for increasing silica fraction, where the result for larger fractions must be considered more reliable due to the larger deviations from neat P2VP (see, e.g., Figure 5a). We observed a significant improvement of fitting quality by allowing to vary freely. A simultaneous fit to all data sets (5-31 vol.% loadings), shown in Figure 2b, provides the best-fits results in the same figure. The gradient corresponding to the so-obtained and values, respectively 0.069 nm and 1.45, is the one shown in the inset of Figure 2a (calculated from Equation 1).

In Figure 2b we also show a shifted version (for clarity) of the data for the 31 vol.% sample that we fit assuming the simple superposition of the neat response and a single -shifted component (solid green line). This fit, which is based on a relatively simpler approach, is visually inferior, particularly regarding the overestimation of at the glass transition endpoint and its slope at higher temperature. We thus conclude that the sample is unlikely to contains two actual fractions with well-separated as suggested on the basis of BDS data in several publicationssee e.g. ref. [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF][START_REF] Cheng | Unexpected molecular weight effect in polymer nanocomposites[END_REF][START_REF] Cheng | Correlation between the temperature evolution of the interfacial region and the growing dynamic cooperativity length scale[END_REF] by Sokolov et al. Given these facts we believe it is not appropriate to describe the BDS data, taken at higher temperature, where the segmental relaxations are faster and presumably more homogeneous (diffusion-averaged), with two distinct components. We note that the width of the BDS-detected second alpha relaxation ( '') in the nanocomposites is not significantly wider than the one for the main relaxation (see e.g., ref. [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] ).

Analysis of the neat P2VP from low field NMR

In Figure 3a, we present the evolution of the free induction decay signal "FID" ( ) measured on the neat P2VP as a function of the NMR time for temperatures varying from 403 to 473 K. As expected, a slower signal decay is observed at high temperatures, i.e., when the chains have extra-mobility (short segmental relaxation time); steeper decays indicate more sluggish dynamics corresponding to slower segmental relaxations. In addition to the FID signal, Figure 3b shows the magic-sandwich-echo free induction decay signal "MSE-FID" ( ) which allows us to measure the material's response at times shorter than the spectrometer dead time (ca. 15 μs). Due to -relaxation effects during the MSE-FID measurement, is somewhat reduced, and goes through a minimum when the segmental correlation time is of the order of the inverse static limit dipolar coupling (see inset). This method thus probes a somewhat smaller dynamic sub-ensemble, which can be quantified from the measured relative intensities. This fact is considered in our fits. 3) considering a log-normal distribution of (see text). The inset shows the expected non-monotonic variation of the MSE-FID signal at short time with the temperature (see refs. [START_REF] Papon | Low-field NMR investigations of nanocomposites: polymer dynamics and network effects[END_REF][START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] ).

The calorimetric glass transition temperature ( of P2VP is 373 K. However, glass transition effects on the NMR observables only set in several tens of degrees higher (typically from +30 °C) , i.e., when the segmental dynamics reaches correlation times ( -relaxation times) of 0.1- 0.01 ms, corresponding to the range where it is of the order of the inverse of the average dipolar coupling constant in the static limit (i.e., ≈ 2 30 kHz). [START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] Unlike in BDS, the relationship between the measured NMR response and the timescale of segmental relaxation is highly non-linear, and all available theories involve approximations, rendering the extraction of semi-quantitative. However, the fact that we are able to represent the neat-polymer data quantitatively at any given temperature means, that we will be nevertheless be able to evaluate quantitatively the amount of reduced-mobility material and the degree of slowdown in our nanocomposites (section 3.3).

As mentioned, the decay of the NMR signal (the "apparent" T2 relaxation time) reflects the strength of through-scape dipolar couplings, which become averaged out due to segmental motion when it is fast enough to interfere with the distance-and orientation-dependent dipolarcoupling effect. We thus detect motions through any change in dipolar couplings in the dense proton system, arising from either rotations of internuclear vectors (both intra-and intersegmental) or changes in distance (by actual diffusing or by rotation of a neighboring segment).

Given the featureless data, it is impossible to disentangle these contributions. Assuming rather simplistically that these effects are governed by a single correlation time , the and were simultaneously fitted with the Andersen-Weiss (AW) function, originally developed for use in electron paramagnetic resonance, [START_REF] Anderson | Exchange narrowing in paramagnetic resonance[END_REF] and successfully used recently to model NMR data [START_REF] Papon | Low-field NMR investigations of nanocomposites: polymer dynamics and network effects[END_REF][START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF][START_REF] Szymoniak | Spatial inhomogeneity, Interfaces and Complex Vitrification Kinetics in a Network Forming Nanocomposite[END_REF] parameters are defined below:

. ( 2
)
This model includes the option for locally anisotropic (e.g. uniaxial) dynamics as quantified by the order parameter . Indications of anisotropic dynamics in the interfacial layer were recently discussed by Sokolov and coworkers [START_REF] Popov | Strong Reduction in Amplitude of the Interfacial Segmental Dynamics in Polymer Nanocomposites[END_REF] based on works by Liu et al. [START_REF] Liu | Study of the kinetics of the pancake-to-brush transition of poly (N-isopropylacrylamide) chains[END_REF] and Ou-Yang et al. [START_REF] Ou-Yang | A pancake-to-brush transition in polymer adsorption[END_REF] evidencing that the most probable conformation of an adsorbed chain is a pan-cake like structure. This option, which is further discussed in Section 3.3.1., was used to perform the fits of our composites' NMR data by varying from 0 to 0.3 as illustrated in Figure 8, resulting in no major impact on the value. For example, applying Equation 2 within the IFL model (see section 3.3.1) to fit data of the nanocomposite loaded with 50 wt.% at 453 K resulted in 6.8 µs and 4.9 µs for =0 and =0.3 respectively (with fixed =0.5, see Equation 4).

Besides, when =0, i.e, in the case of isotropic dynamics (considered true for the neat matrix), Equation 2 simplifies into:

(

In Equations 2-3, the instrument related prefactor and the apparent correlation time were the (unshared) fitting parameters. is the second moment of the dipolar-broadened spectrum, which is a measure of the average static low-temperature limit dipole-dipole coupling in the system, . Even without motion, this quantity decreases upon heating due to thermal expansion (SI section 2 for details on its determination). Upon heating, the segmental relaxation time decreases, and when it reaches a value of about , i.e., about 10 µs, the mentioned averaging effects set in and the decay of the observed relaxation function starts to lengthen (the effective transverse relaxation time starts increasing). Note that a of 10 µs is usually found above + 30 K (while it is 100 s at ). In this range, can be fitted stably and values for higher temperatures were estimated by extrapolation (see SI section 2). In addition, was systematically assumed to follow a log-normal distribution ,

where the logarithmic standard deviation was a fit parameter. We found that the resulting were very similar for the and signals -for the neat P2VP it varies systematically from 0.50 to 0.88, from 403 to 473 K. Accordingly, values were found to vary from 27.2 μs to 0.3 μs for , and from 28.6 μs to 0.133 μs for , showing the consistency of our dual data set. This minimal model provides a near-perfect fit to the data, which serves the important purpose of parametrization/interpolation of bulk P2VP data beyond a possible interpretation of the fit parameters. While the single-assumption is unrealistic in the light of the complex relaxation map of P2VP (see below), it is not possible to use a more complicated model for fitting without challenging a robust fit. The question of the molecular-level origin of the so-observed relaxation actually opens opportunities for future experimental investigations. In fact, we envision dual BDS vs. LF-NMR analyses performed on series of model polymers in which the relative amplitude and/or the relaxation time of segmental and local processes could be varied progressively. This philosophy was notably used in the past to investigate series of methacrylatebased polymers where the length of the side group was correlated in a systematic way with the segmental relaxation and nano-structuration by means of BDS, DSC and mechanical spectroscopy, [START_REF] Beiner | Relaxation in poly (alkyl methacrylate) s: crossover region and nanophase separation[END_REF] as well as neutron spin echo. [START_REF] Arbe | Dynamics in poly (nalkyl methacrylates): A neutron scattering, calorimetric, and dielectric study[END_REF] Also, selective deuteration of the side groups is certainly of great interest in this context, enabling to discriminate contributions coming from segmental or local motions on the apparent " relaxation" by using LF-NMR only.

We can now compare the values of with the P2VP segmental relaxation time ( obtained from BDS, as well as with the rheological shift-factor at the appropriate reference temperature,

. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] This comparison is reported in Figure 4a, where reasonable agreement is observed between the three techniques, yet noting a shift towards too-large at higher temperatures and an apparently lower apparent activation energy for the NMR data at low temperature, resulting in a seemingly Arrhenius-type trend. To understand these deviations, we stress again the simplicity of the AW model, notably through (i) the assumption of an exponential loss of orientation correlation, (ii) the log-normal distribution of segmental relaxation time and (iii) the slight temperature dependence of . These issues will impart systematic errors on the fitted . Beyond this inherent uncertainty, the two discrepancies can be qualitatively understood in view of the complexity of the relaxation processes in P2VP. Figure 4b shows results from a more complete analysis of dielectric data. [START_REF] Papadopoulos | Origin of glass transition of poly (2-vinylpyridine). A temperature-and pressure-dependent dielectric spectroscopy study[END_REF] It is a processed version of Figure 6 of this reference, scaled to match the representation in Figure 4a. In addition to the  relaxation, there is a slower process related to the glass transition and attributed to a final isotropization of monomer sequences following a partly anisotropic process, in analogy to earlier observations on poly(nalkylmethacrylates) claiming a "conformational memory. [START_REF] Wind | Structural relaxation of polymers at the glass transition: Conformational memory in poly (n-alkylmethacrylates)[END_REF] In addition, there is a β process associated with the pyridine ring (possibly  flips). Now, Equations (2,3) imply significant changes of the NMR signal when is in the range of 1 to 100 μs. The additional processes feature broad correlation time distributions, [START_REF] Papadopoulos | Origin of glass transition of poly (2-vinylpyridine). A temperature-and pressure-dependent dielectric spectroscopy study[END_REF] such that they must be expected to contribute in this dynamic range, thus biasing the fitted and explaining qualitatively the deviations from .

Importantly yet, the main contributor to is the segmental relaxation, such that in the hightemperature range, where its temperature dependence is reflected correctly, we can expect to obtain a good estimate of the extent of slowdown in the nanocomposite samples. Finally, we remind that the most important purpose of using Equations (2-4) is the parametrization/interpolation of the measured data, used below primarily for a reliable quantification of the amount of interfacial material. performed by using the zero-time limit from the IFL model fit as shown in Figure 6.

Impact of the silica loading on P2VP dynamics

Based on the preliminary results obtained on neat P2VP as a reference, our aim is to quantify the impact of the silica loading on polymer dynamics. Two quantities are particularly targeted here:

(i) the fraction of P2VP adsorbed onto the silica surface (denoted by below), and (ii) the change of the segmental relaxation time of the immobilized layer, or in other words, its dynamical shift with respect to the bulk P2VP. Normalized data obtained from the neat P2VP and the nanocomposites loaded with 6, 18 and 31 vol.% silica are shown in Figure 5, where the immobilized layer contribution (faster decay, i.e., shorter ) clearly appears at high temperatures, i.e., far from the bulk P2VP glass transition (Figure 5a). Conversely, when measuring at lower temperature, the NMR signals from the PNCs tend to collapse progressively onto the neat matrix response indicating that the whole polymer phase is becoming glassy from an NMR point of view (Figure 5b-c). For this reason, we restricted the fitting procedures below to the 433-473 K range, in which a reasonable distinction could be made between the bulk and the interfacial layer dynamics.

While the signals from the PNCs loaded with 6 and 18 vol.% in silica are rather close over the whole temperature range, the 31 vol.% sample signal appears much stiffer from an NMR point of view, indicating a significantly higher . This variation parallels the initially limited, then stronger broadening of the -relaxation observed in BDS when increasing the filler content. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF][START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] Also, it is reminiscent of a similar WLF trend of the rheological shift factor in PNCs loaded with 10 vol.% before a clear transition towards an Arrhenius-like behavior at higher filler content. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] (For the sake of completeness, the whole data set containing all the temperatures and filler fraction dependences of both and signal is provided in SI section 3).

NMR data fitting with the single Interfacial Layer (IFL) model

The interfacial model consists of adding the bulk P2VP contribution (determined in the previous section) to an AW term (described in Equation 2-3) for the immobilized layer.

(

) 5
is the fraction of immobilized layer; and being, respectively, the AW fit performed on the matrix data (black lines in Figure 3) and the additional contribution from the immobilized layer. Note that Equation 5 assumes that FID data are normalized to 1 at short time while MSE-FID ones are systematically lower by a factor that depends on the temperature (see the inset of Figure 3b). This method applied jointly on and enables a straightforward and joint evaluation of and thus limiting the number of fit parameters. The result of this approach (considering =0) is presented in Figure 6 together with the gradient model (see next section) on a nanocomposite filled with 31 vol.% in silica at 453 K, highlighting a large contribution of the immobilized layer with respect to the bulk. b)

F / R (nm -1 ) a) 2.0 2.1 2.2 2.3 2. 4 2.5 -7 -6 -5 -4 31 vol.
E a =106 kJ/mol E a =74 kJ/mol E a =103 kJ/mol E a =96 kJ/mol
The main results of the IFL model are summarized in Figure 7 where the immobilized polymer fraction and its corresponding segmental dynamics are shown for different temperatures and filler fractions. In Figure 7a, is found to span values from ca. 0.1 (6 vol.% in silica) to 0.32 (31 vol.% in silica), in qualitative agreement with rheology results. [START_REF] Chen | Mechanical reinforcement of polymer nanocomposites from percolation of a nanoparticle network[END_REF][START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] In fact, while up to 6 vol.% in silica, the dynamical moduli as well as the corresponding master-curve shift factors were found unchanged, a severe broadening of the whole relaxation spectrum (including therelaxation) was observed at 31 vol.%. More interestingly, NMR estimates of are in fair agreement with BDS data from our previous work ( = 0.42 and 0.50 respectively). [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF] We believe that the discrepancy originates from the intricate fitting procedure of BDS data, caused by the emergence of the MWS process in the same frequency window as the -relaxation. Note that Sokolov et al. reported =0.4 for P2VP filled with 26 vol.% in silica [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] (with slightly bigger particles =12.5 nm), falling well in the (broad) range of immobilized layer fraction, =0.32-0.5

(see also a summary of various systems from the same authors in Table 2 in ref. [START_REF] Cheng | Interfacial properties of polymer nanocomposites: Role of chain rigidity and dynamic heterogeneity length scale[END_REF] ). On top of this, appears to be independent of temperature, apparently contrary to Sokolov et al. who found that followed a (weakly) decreasing linear trend with increasing the temperature in PVAc (at similar distance from than us) [START_REF] Cheng | Correlation between the temperature evolution of the interfacial region and the growing dynamic cooperativity length scale[END_REF] and in P2VP and PVAc (closer to their . [START_REF] Cheng | Interfacial properties of polymer nanocomposites: Role of chain rigidity and dynamic heterogeneity length scale[END_REF] At this point, we cannot exclude that systematic deviations arise from the fact that our fitting approach only provides a reliable lower limit of . Further efforts relying on the gradient model are proposed in the next section.

Figure 7b also unambiguously reveals the slowing down of the immobilized phase relative to the bulk P2VP. The corresponding broadening of the P2VP glass transition with increasing silica content has been widely investigated through DSC and BDS, where it is treated either as the appearance of a separate relaxation assigned to the adsorbed polymer (" " [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] or " " [START_REF] Cheng | Unexpected molecular weight effect in polymer nanocomposites[END_REF] ), or a continuous gradient of mobility. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF][START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] Here, LF-NMR shows a strong up-shift of when the addressed theoretically by Oyerokun and Schweizer who predicted that an anisotropic conformation increases the correlation between segmental motions (intrachain) resulting in an increase of at the interface. 38 In this context, it is logical to extend our IFL model to the case with to check the validity of our approach, as already investigated in the past in nacre-mimetic composites [START_REF] Eckert | Structure, Mechanical Properties, and Dynamics of Polyethylenoxide/Nanoclay Nacre-Mimetic Nanocomposites[END_REF] and semi-crystalline polymers. [START_REF] Ur | Intracrystalline jump motion in poly (ethylene oxide) lamellae of variable thickness: A comparison of NMR methods[END_REF] The result of this approach is represented in Figure 8 where the quality ( of the IFL fit is systematically plotted as function of different fixed and for the 31 vol.% nanocomposite at 463 K and 443 K (the rest of the data set is presented in SI section 4). The 463 K map (Figure 8a) indicates that best fits are obtained for close to 0 for reasonable relaxation time distributions ( =0.6-0.8). However, this is clearly not the case for the 443 K map; rather the IFL model better describes the data for =0.15-0.2. While this duality also emerges from the fits performed at different temperatures and filler fractions, we find more frequently that a value significantly above 0 enhances the fitting quality. Although it is impossible at this stage to conclude quantitatively on a true value of , these fits support the notion that the anisotropy of segmental motions is fully consistent with our LF-NMR experimental data, justifying the use of the simple IFL model in future investigations in this field.

NMR data fitting with the gradient model

The gradient model relies on the continuous variation of the glass transition temperature as a function of the distance ( ) between the polymer and the filler surface. This model, successfully applied in the past by Papon et al., [START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] is employed here to fit both DSC (see Figure 2) and NMR (see Figure 6) data:

where follows Equation 1 and is the mean polymer volume per nanoparticle. The integration is performed over spherical shells, i.e., , up to the value corresponding to . To apply this fitting method to NMR data, we choose to follow the AW expression (Equation 3) as this function successfully fits the whole neat P2VP data set (see Figure 3). Our conclusions will thus be independent of the failure of the AW approach to provide a fully quantitative , as we ultimately model the composite data as a superposition of temperature-shifted neat-P2VP responses (see ref. [START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] ).

Practically, the continuous integration in Equation 6 is implemented as a discrete sum of the neat P2VP signals in spherical shells at distance from the surface and thickness . Neat signals were measured every 10 K between 413 K and 473 K. The only fitting parameters are (appearing in the expression of ) and possibly when it is not forced to be 1. Note that this "slicing" method was also used by Papon in the Equation 6of ref. [START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] to incorporate a maximum shifted-cutoff. The glass transition temperature profile resulting from the fit performed on the 31 vol.% nanocomposite NMR data at T=453 K is provided in Figure 9 as a decreasing function (Equation 1) with nm ( . This value is found to vary slightly between 0. Corresponding values belong to the intervals 0.029-0.047 and 0.01-0.026 for Φ=10 vol.% and 18 vol.% respectively. Variations are erratic with the temperature.

As presented in Figure 2, the gradient model can also be used to rationalize successfully DSC data. In this case we performed the simultaneous fit on the whole data set ( measurements). Practically, the situation is different from the previous (NMR) case since the data are obtained from a continuous variation of temperature (it is every 10 K in NMR). This results is defining somehow arbitrarily the thickness of the slices ( ) considered for the fitting, each slice being characterized by a DSC profile identical to the neat P2VP one, but shifted "diagonally" by 10 K along the line defining the in the glassy regime (see Figure 2a inset).

Several fits were performed assuming =0.1, 0.2 and 0.5 nm, but they provided a similar if not identical result: with 1.45. (This is as expected.) The corresponding profile is the significant effect of reducing the interparticle distance on the polymer segmental motion by using small NPs ( =2 nm). [START_REF] Cheng | Big effect of small nanoparticles: a shift in paradigm for polymer nanocomposites[END_REF] At last, we want to highlight that although Equation 6 is expected to fail for very high filler fractions resulting in an overlap of the slowdown polymer shells surrounding the NPs, this hypothesis is not relevant in the present paper. In fact, the average wall-to-wall interparticle distance, approximated by , is 12.0, 7.3 and 3.8 nm respectively for 10, 18 and 31 vol.%. These distances must be compared with the shape of the gradient presented in Figure 9, clearly showing that no significant slowdown happens beyond 2 nm, confirming therefore that Equation 6 is consistent to describe our data.

Conclusions

LF-NMR relaxometry experiments have been performed on P2VP/silica attractive nanocomposites with varying the filler fraction (6-31 vol.%) and temperature (423-473 K) with the aim of investigating the immobilized polymer chains, both in terms of fraction and dynamical slow down. Two approaches have been critically examined to fit the whole data set: (i)

the single interfacial layer model (IFL) and (ii) the gradient model. In the former case, the immobilized layer is seen as a single shell within which the segmental relaxation time follows a log-normal distribution while in the latter case the polymer is taken to continuously increase as one proceeds closer to the filler surface, resulting in polymer "slices" of increasing relaxation time. In qualitative agreement with previous BDS investigations, we show in the present work that both approaches can be implemented to fit successfully NMR data. Quantitatively, our NMR analyses support the finding of Genix et al. [START_REF] Genix | Enhancing the mechanical properties of glassy nanocomposites by tuning polymer molecular weight[END_REF] who evaluate recently the dynamical slowdown of the interfacial layer to be ca. 1 decade on average by means of BDS and Brillouin light scattering.

However, they are in contrast with the earlier work of Holt et al. [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] who found a 2-decades slowdown with BDS in similar systems -pointing the difficulties to fit unequivocally BDS data.

The discrimination between the IFL and the gradient approaches becomes possible only when applying these models to fit previously measured DSC data. Obviously, the IFL model cannot fit the heat capacity of the polymer phase in PNCs as convincingly as the gradient model (with consistent outputs with respect to NMR data). However, it is worth noting that the implementation of the IFL model is also compatible with the existence of anisotropic segmental motions at the interface with the filler making the two approaches complementary to describe the whole material's physics. This phenomenon, as well as related changes in NMR relaxation, can also explain reduced fitting quality and fitting bias resulting in slight discrepancies in the gradient model (while NMR gives 1, DSC suggests 1.45).

Apart from rheology, BDS, DSC and NMR, other techniques such as X-ray photoelectron spectroscopy (XPS) and Sum Frequency Generation (SFG) were recently used to complement the information about the immobilized polymer. [START_REF] Voylov | Unraveling the molecular weight dependence of interfacial interactions in poly (2-vinylpyridine)/silica nanocomposites[END_REF] Further methods enabling the characterization of local and collective dynamics of nanofilled polymers are discussed in a recent review by Bailey and Winey. [START_REF] Bailey | Dynamics of Polymer Segments, Polymer Chains, and Nanoparticles in Polymer Nanocomposite Melts: A Review[END_REF] The unification of experimental results coming from these various probes (including numerical simulations) appears as a central challenge for the coming years.

Figure 1 :

 1 Figure 1: Transmission electron micrographs of PNCs loaded with a) 6, b) 18 and c) 31 vol.% in silica. Black bars are 500 nm.

Figure 2 :

 2 Figure 2: a) Modulated DSC data adapted from ref. 9 (heat capacity of the neat P2VP and polymer phase in nanocomposite filled with 31 vol.% in silica). Thin solid lines are shifted version of the neat P2VP measurement and illustrate the gradient in around the nanoparticle.

Figure 3 :

 3 Figure 3: a) Raw signal measured from the neat P2VP between 403 and 473 K. b)

Figure 4 :

 4 Figure 4: Temperature dependence of the neat P2VP segmental relaxation time. Panel a) shows date from our previous publication probed by dielectric spectroscopy ( and rheology ( , 9

Figure 5 :

 5 Figure 5: Normalized signals for the net P2VP and the nanocomposites loaded with 6, 18 and 31 vol.% in silica measured at a) 463 K, b) 443 K and c) 423 K. Intensity normalization is

Figure 6 :

 6 Figure 6: and raw signals measured at =453 K. Light blue and red solid lines stand

Figure 7 :

 7 Figure 7: a) Fraction of immobilized layer extracted from FID experiments performed between 433 and 473 K on composites loaded with 6, 18 and 31 vol.% in silica as a function of the reduced filler fraction representing the total interfacial surface. BDS based data from Baeza et al. (2016) and Holt et al. (2014) are shown for comparison. The dashed line is a guide to the eye connecting the 463 K data. b) Segmental relaxation time as function of the reciprocal temperature for the neat P2VP and the immobilized phase in nanocomposites of various filler content. values are extracted from FID data (MSE-FID ones are almost identical). The solid gray line is the same WLF law as in Figure 2. Other solid lines are linear fit to the data to extract the activation energies.

Figure 8 :

 8 Figure 8: Fitting quality ( as a function of and representing respectively the anisotropy of the segmental motion and the breadth of the segmental relaxation time distribution in the IFL model. Fits are performed on data measured on the nanocomposite loaded with 31 vol.% in silica at a) 463 K and b) 443 K.

  017 and 0.025 nm when changing the temperature from 443 to 473 K, supporting the consistency of the gradient model. Corresponding fits are displayed in Figure 9b-c. In the same interval of (trustworthy) temperatures, the 18 vol.% nanocomposite data fitting results in varying between 0.01 nm and 0.025 ( . The analogous values for the 10 vol.% composite, for which the NMR signal is very close to the neat P2VP one, are in the 0.04-0.045 nm interval. Although values for 10 and 18 vol.% are thus in fair agreement with the 31 vol.% sample, we believe that the results extracted from the latter sample are more accurate because of a larger fraction of immobilized layer. Overall, emerges from the gradient model, with no significant improvement of the fit when is varied in the 0.8-1.5 range.

Figure 9 :

 9 Figure 9: profile as a function of the distance from the filler surface ( ) resulting from the

Acknowledgements

All the authors warmly thank Walter Chassé (Univ. Munster) for its help with the LF-NMR routine and Paul Sotta (INSA-Lyon) for technical support on the minispec. The authors also thank Ahmed Belhadj and Pierre Alcouffe (INSA-Lyon and Univ. Lyon 1, respectively) for the TGA and TEM measurements. TEM was performed at the "Centre Technologique des Microstructures" at University of Lyon 1. G.P.B. thanks Aurélien Doitrand (INSA-Lyon) for his help with the IFL model fitting implementation.

Funding sources

polymer is immobilized, with this shift being almost independent of the filler content.

Quantitatively, this shift appears to be close to a decade at 473 K. This value is lower than the two decades in time reported by Holt et al. [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] (see also Bailey and Winey 32 ) whereas it matches well the more recent investigations by Genix et al., [START_REF] Genix | Enhancing the mechanical properties of glassy nanocomposites by tuning polymer molecular weight[END_REF] both measured from BDS on similar systems. We believe that this discrepancy originates from ignoring (or limiting) the amplitude of the MWS process occurring in PNC at lower frequency than the segmental motion. In fact, this would necessitate a leftward-shift and enhance (the immobilized segments contribution) to compensate for the lower permittivity in this frequency range. [START_REF] Holt | Dynamics at the polymer/nanoparticle interface in poly (2vinylpyridine)/silica nanocomposites[END_REF] Nevertheless, it is worth noting that both LF-NMR and BDS [START_REF] Cheng | Correlation between the temperature evolution of the interfacial region and the growing dynamic cooperativity length scale[END_REF] indicate an unchanged temperature dependence of the segmental relaxation of bulk and interfacial P2VP segments. The corresponding activation energies range between ca. 80 and 100 kJ mol -1 for LF-NMR, while it is ca. twice as much in the BDS (see Figure 2). Note that no evidence for a dead layer, which would be represented by a signal contribution resembling that of bulk P2VP measured below 400 K, was found in the present work. [START_REF] Napolitano | Dielectric signature of a dead layer in ultrathin films of a nonpolar polymer[END_REF][START_REF] Napolitano | Unusual deviations from bulk behavior in ultrathin films of poly (tert-butylstyrene): Can dead layers induce a reduction of T g?[END_REF] Further, since such nanocomposites are not expected to be thermodynamically equilibrated, [START_REF] Oh | Initial solvent-driven nonequilibrium effect on structure, properties, and dynamics of polymer nanocomposites[END_REF][START_REF] Genix | Understanding the static interfacial polymer layer by exploring the dispersion states of nanocomposites[END_REF] differences in the studied samples may well arise due to potential differences in preparation procedures.

This discussion to this point on the IFL model is based on the hypothesis that the segmental dynamics of the interfacial layer is isotropic ( in Equation 2), as in the bulk polymer.

However, as reported by Sokolov et al. in refs. [START_REF] Popov | Strong Reduction in Amplitude of the Interfacial Segmental Dynamics in Polymer Nanocomposites[END_REF][START_REF] Holt | Controlling interfacial dynamics: covalent bonding versus physical adsorption in polymer nanocomposites[END_REF] , it has been demonstrated that the polymer dynamic suppression at the interface is not only impacted by the strength of the interaction with the filler but also by the average local chain orientation in the vicinity of the NPs. Concretely, because the chains are expected to adopt a "pancake-like" conformation, near the NPs, their resulting segmental fluctuations become anisotropic. (Note that a similar logic can be adopted in the case of covalently grafted particles where the polymer segments close to the anchoring point would exhibit a preferred orientation normal to the NP surface). This phenomenon has been reported in Figure 9 for comparison with its NMR counterpart and previous NMR data by Papon et al. on PEA where larger silica particles were used. [START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF] The main message provided by Figure 9 is that the gradient model can be used successfully on both NMR and DSC data providing similar profiles; our analysis procedure is thus quite robust. We believe that the slight discrepancies in and are due to the simplicity of the model, notably 9). We assign this difference to a particle size effect: Papon et al. dealt with particles of radius 21.5 nm while it is only 7 nm in our case. In fact, reducing the NP radius reduces the thickness of the "slowed down" polymer layer. In ref. [START_REF] Harton | Immobilized polymer layers on spherical nanoparticles[END_REF] , Harton et al. reported a slowed down polymer layer thickness of 1 nm in P2VP/silica PNCs containing particles or 15 nm diameter (i.e., identical to the present work);

this is in excellent agreement with the profile we report in Figure 9. On the other hand, they showed that this thickness could reach ca. 5 nm (closer to the Papon et al. profile in Figure 9) in the case of a flat interface.

Apart from specific surface considerations, other authors showed the impact of the mean surface-to-surface distance between two adjacent NPs. Chen et al. [START_REF] Chen | Mechanical reinforcement of polymer nanocomposites from percolation of a nanoparticle network[END_REF] notably showed the formation of "glassy bridges" for particles of 7 nm radius from 10 vol.%, while this threshold was shifted to ca. 45 vol.% with particles of 25 nm radius. [START_REF] Chen | Mechanical reinforcement of polymer nanocomposites from percolation of a nanoparticle network[END_REF] In agreement with these results, Baeza et al. found a (progressive) dynamical transition from WLF to Arrhenius behavior when the filler content was increased in identical materials. [START_REF] Baeza | Network dynamics in nanofilled polymers[END_REF]