# Influence of the Graft Length on Nanocomposite Structure and Interfacial Dynamics

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{8}) is studied. As a function of grafting density and particle content, polymer dynamics is followed by broadband dielectric spectroscopy and analyzed by an interfacial layer model, whereas the particle dispersion is investigated by small-angle X-ray scattering and analyzed by reverse Monte Carlo simulations. NP dispersions are found to be destabilized only at the highest grafting. The interfacial layer formalism allows the clear identification of the volume fraction of interfacial polymer, with its characteristic time. The strongest dynamical slow-down in the polymer is found for unmodified NPs, while grafting weakens this effect progressively. The combination of all three techniques enables a unique measurement of the true thickness of the interfacial layer, which is ca. 5 nm. Finally, the comparison between longer (C

_{18}) and shorter (C

_{8}) grafts provides unprecedented insight into the efficacy and tunability of surface modification. It is shown that C

_{8}-grafting allows for a more progressive tuning, which goes beyond a pure mass effect.

## 1. Introduction

_{18}) on interfacial polymer dynamics and particle dispersions, by using BDS and SAXS [12]. In the present article, the effect of a shorter C

_{8}-silane is studied using the same methods and analysis, varying both the grafting density and the particle volume fraction. By comparing to the previously studied C

_{18}-system, the impact of the alkyl-chain length is highlighted. In a second time, a recent methodological approach is further developed. Reverse Monte Carlo simulations provide sequences of particle configurations of scattering compatible with the observed SAXS signals. This has been combined with the BDS results, which provide the volume fraction of interfacial layers. The result is a more realistic estimation of the interfacial layer thickness because it avoids the idealized vision of perfectly dispersed particles underlying simple IPS equations [38], whereas our approach allows taking into account polydispersity and layer overlap caused by NPs in close vicinity. This combined method is now investigated further by analyzing the evolution of the interfacial layer thickness with its volume fraction for each sample, that is, for different types of experimentally observed dispersions.

## 2. Materials and Methods

_{0}= 8.4 nm, σ = 18%), leading to an average NP radius of R = 8.5 nm.

_{3}(CH

_{2})

_{7}Si(CH

_{3})

_{2}OCH

_{3}, termed C

_{8}) from Gelest. The grafting reaction was conducted at 323 K for 3 days in ethanol. To achieve different grafting densities ranging from zero (bare NPs) to ca. 3 nm

^{−2}, different amounts of silanes were added to the NP suspension: 50 mL of the suspension were mixed with 120, 750, 1200, and 1500 µL of C

_{8}silane. After the reactions were completed, the surface-modified NP suspensions were dialyzed against ethanol for 3 days. The grafting densities were determined by thermogravimetric analysis (TGA, TA instrument, Discovery, 5 K/min under air) using the weight loss between 473 and 873 K corresponding to the thermal decomposition of the grafted silanes [39]. The TGA curves are given in SI (Figure S1). The resulting grafting density of C

_{8}-molecules on the silica NPs, between 0.8 and 2.9/nm

^{2}, are given in Table 1.

^{−1}(polydispersity index = 1.07), was purchased from Scientific Polymer Products Inc. (Ontario, New York, United States), and used as received. The radius of gyration of the chain is 5.2 nm. The polymer dissolved in ethanol (66 mg/mL) and (bare or surface-modified) NP suspension also in ethanol (16 mg/mL) were mixed for at least 12 h, then filtered through a 200 nm Teflon filter. The final PNCs were formed by evaporating the solvent at room temperature followed by drying in a vacuum oven at 393 K for 2 days. All samples were hot-pressed at 423 K, and they were further annealed under vacuum at 393 K for 3 days before the SAXS and BDS measurements. The silica fractions in PNCs were obtained by TGA (20 K/min, under air) from the weight loss between 433 and 1073 K. The NP volume fractions, Φ

_{NP}, were determined by mass conservation using the density of neat P2VP (1.19 g·cm

^{−3}by pycnometry) [17] and silica (2.27 g·cm

^{−3}by SANS) [40]. They are reported in Table 1.

^{−2}to 10

^{7}Hz (ω = 2πf) using disk-shaped samples with a diameter of 20 mm and a typical thickness of 0.15 mm. The samples (without spacer) were sandwiched between two gold-plated electrodes forming a capacitor. They were first annealed for 1 h at 433 K in the BDS cryostat under nitrogen flow to ensure that both the real and imaginary parts of the permittivity became constant in the probe frequency range. Then, isothermal frequency measurements were performed at 423 K, and from 303 K down to 233 K with an interval of 10 K to specifically follow the β-relaxation of P2VP. A measurement at the lowest measurable temperature of 103 K was performed to normalize the permittivity values. After that, the samples were measured again at 293 and 433 K to check reproducibility. The normalization procedure of PNCs is described in detail in [17] considering two-phase heterogeneous materials [19] with the high-frequency limit of the real part ε

_{∞}= 3.05 and 3.9 for the polymer and silica, respectively. It allows for the removal of possible artifacts (mostly thickness variations) and leads to the dielectric spectra in absolute values.

_{HN}as the dielectric function of the neat polymer measured independently. The free parameters of the ILM are the volume fraction of interfacial layer, Φ

_{IL}

^{PNC}, and its dielectric function, ε

_{IL}*(ω), which is well-described by a symmetrical HN-process (δ = 1). In the following, the relaxation times are defined by τ

_{HN}related to the peak position in frequency f

_{max}, which is used to determine the relaxation time τ

_{max}= 1/(2πf

_{max}).

^{3D}from Xenocs, Grenoble, France). It delivered an ultralow divergent beam (0.5 mrad). The scattered intensities were measured by a 2D “Pilatus” pixel detector at a single sample-to-detector distance D = 1900 mm, leading to a q range from 4 × 10

^{−3}to 0.2 Å

^{−1}. The scattering cross-section per unit sample volume dΣ/dΩ (in cm

^{−1}), which we term scattered intensity I(q), was obtained by using standard procedures, including background subtraction and calibration [42].

_{box}= 2π/q

_{min}, where q

_{min}is the experimental minimum q-value, such that the total volume fraction Φ

_{NP}corresponds to the experimental value of the sample. The scattered intensity of the particles in the simulation box is calculated based on the individual size of each particle, and converted to absolute units using the contrast based on the scattering length densities of silica and the polymer: ρ

_{SiO2}= 19.49 × 10

^{10}cm

^{−2}, ρ

_{P2VP}= 10.93 × 10

^{10}cm

^{−2}. The calculation is split in the Debye formula [43] at high q, and a lattice calculation avoiding box contributions [35,44,45] at low q. Particles are then moved around randomly, while following a simulated annealing procedure leading to agreement of the theoretically predicted apparent structure factor S(q) with the experimental one. In all cases, the excluded volume of the particles is respected. Particle configurations that are compatible with the experimental intensity are saved regularly, and they can be analyzed a posteriori, for example, in terms of interparticle spacing. S(q) and any statistical measures are averaged over different particle configurations and represent the result of the simulation.

## 3. Results and Discussion

#### 3.1. Nanoparticles in Solvent and Polymer: Dilute Conditions

^{−2}is shown, and compared to a series with 10 times higher silica content. These pictures illustrate that the NPs are rather well dispersed under all conditions, that is, there are no large structural heterogeneities.

#### 3.2. Dynamical Properties of the NP-Polymer Interface Probed by BDS

_{18}instead of C

_{8}, see [12]): the nanoparticle surface induces a slow-down of the neighboring polymer chains, whereas modification of the same surface with small silane molecules counteracts this slow-down. We will now analyze the modification of the α-relaxation in terms of the ILM, which describes the total dielectric response in terms of a bulk (with unmodified dynamics with respect to the neat polymer) and an interfacial contribution, as well as their interferences. As a result, the loss and storage permittivity responses are quantitatively and simultaneously described, and the fits describe well the data in Figure 2 (as well as the other data shown in the SI). Details of the fits are displayed in Figure S4 in the SI, where the different contributions—MWS, α (including both IL and bulk contributions which are linked), β, and conductivity—to the dielectric loss are highlighted for two selected samples.

_{bulk}= 0), ε

_{IL}″(ω), is represented for PNCs with 30%v of silica, and different grafting densities as given in the legend. It corresponds to the sum of two HN functions for the α- and β-processes as deduced from the ILM fit.

_{IL}, given here with respect to the polymer part (Φ

_{IL}+ Φ

_{bulk}= 1), and its characteristic time. The first is plotted in Figure 3b. Two remarkable features can be seen: first, the volume fraction of the interphase is independent of the surface modification. Secondly, it increases with the silica volume fraction, that is, the available silica surface. Φ

_{IL}can thus be used to determine the thickness of the interfacial layer. A simple cubic model calculation, which ignores silica interactions and thus real particle arrangement gives a thickness of 3.7 ± 0.1 nm for all PNCs. A more accurate model taking into account the particle positions from scattering and simulations and the overlapping volume is discussed below. Whatever the exact value, it is striking to see that the thickness of the interfacial layer is independent of experimental conditions, like silica content and grafting. Therefore, it seems to reflect an intrinsic property of the polymer-surface couple.

_{g}, which is the closest temperature to T

_{g}where the overall segmental dynamics of P2VP is well-visible in the BDS frequency window. The increase of the ratio τ

_{IL}/τ

_{neat}thus represents the slow-down of the polymer dynamics in the vicinity of the silica (i.e., within ca. 4 nm of the surface, as discussed in the preceding paragraph). It is interesting to see that the PNCs with bare particles possess an interfacial layer with virtually the same relaxation time for all silica contents. As surface modification is introduced, the dynamics accelerates, that is, the ratio decreases, and seems to level off at around one, which corresponds to unperturbed segmental dynamics. The effect of silane grafting on the slow-down can thus be followed, and above approximately 1.5 nm

^{−2}, the pure P2VP dynamics is recovered. For comparison, we have superimposed the main result of a previous analysis with a similar but longer silane molecule, C

_{18}[12]. The effect on τ

_{IL}is considerably stronger, and with already 0.5 C

_{18}per nm

^{2}, the pure P2VP relaxation is reached. Although the statistics of the two curves in Figure 3c are insufficient for a precise determination of such a threshold value, it is clear from the decays that the relaxation time of the C

_{18}-samples has already joined the pure P2VP value at 0.5 nm

^{−2}, whereas the C

_{8}-PNCs will do the same somewhere between 1.5 and 2 nm

^{−2}. The ratio between the two thresholds is thus of the order of 3 or 4, which is considerably larger than the ratio of about two between the alkyl chain masses (18:8). As for a given surface grafting density, the total mass of alkyl chains surrounding a given NP is directly related to the molecular weight of the graft, it can be concluded that the effect on the dynamics is not simply related to the amount of CH

_{2}groups, but also to their spatial organization. In particular, one may speculate that the C

_{18}groups have a higher propensity to homogeneously cover the silica surface providing a more efficient screening effect from the polymer chains. The shorter C

_{8}might form locally dense regions due to a lower steric hindrance, leaving free silica zones, that is, covered by hydroxyl groups favoring polymer adsorption. Our findings are in qualitative agreement with recent results from atomistic molecular dynamics simulations of silica-filled polyisoprene, where planar silica substrates were covered with silane of different alkyl lengths (C

_{3}and C

_{8}, i.e., with 3 or 8 carbon atoms in the alkyl part) and different grafting densities [16]. It was found that the slow-down of the polymer dynamics due to polymer adsorption is weakened upon silane grafting with a stronger effect of the longest graft at high grafting density. This effect is concomitant with an increase of the diffusion coefficient of the adsorbed chains, which almost reaches the one of the bulk polymer chains.

#### 3.3. Structure of PNCs Studied by SAXS

^{−2}Å

^{−1}, the PNC intensities at these high concentrations begin to deviate. For bare NPs, or low grafting density up to 1.3 nm

^{−2}, the curves present a well-defined peak around 2.8 × 10

^{−2}Å

^{−1}. At the highest grafting of 2.9 nm

^{−2}, the curve shows a completely different spatial organization. As already visible at 2%v (see SI and discussion above), there is a deep correlation hole, and the intensity deviates from the form factor at higher q-vectors. At low q, a strong upturn is found. This low-q increase translates the attractive interactions between nanoparticles, inducing aggregation. They are triggered by the suppression of attractive polymer–silica interactions caused by surface modification, and thus of the steric protection against aggregation. Depletion interactions induced by the polymer chains (which are about a factor of two smaller than the particles) may also participate in generating interparticle attraction. All these features correspond to aggregation and large-scale spatial fluctuations induced by the high grafting density and they correspond to those reported for nanocomposite melts by Hall et al. [47] These authors experimentally varied the interfacial attraction via the polymer. They studied poly(ethylene oxide) and polytetrahydrofuran (PTHF)-systems, the latter being less attractive because of less hydrogen bonding with the silica. A decrease of the interfacial attraction in PTHF reduces local order and thus leads to a low-q increase, and a structure factor peak shifted towards higher q. PRISM integral equations describe these features, and provide a satisfactory description of polymer-mediated NP concentration fluctuations. The latter ultimately induces depletion aggregation and microphase separation. In our case, increasing coating coverage decreases the polymer–NP effective attraction [12] with a qualitatively similar behavior as predicted by PRISM in terms of peak shift and low-q upturn.

^{−2}can be seen at 15 and 20%v of silica, showing that grafting affects NP interactions in a progressive (and thus tunable) way. One can also follow the peak positions as a function of volume fraction for bare NPs and intermediate grafting (while it disappears at the highest grafting): at 15%v, the peaks correspond to center-to-center distances of 27–31 nm, which are larger than two particle radii, and indicates that NPs still have quite some space to re-organize. At 20%v, the upturn is more prominent at low q, but the peaks remain well-defined leading to distances of ca. 24 to 26 nm. At 30%v, finally, the peak position corresponds to a center-to-center particle distance of ca. 22.5 nm. This distance expresses the fact that particles interact repulsively due to their hard cores, and they do not have much space to reorganize at this concentration. There does not seem to be any systematic dependence with the amount of surface modification. Indeed, we found that the peak positions follow a Φ

^{−1/3}-law for the three series (see SI, Figures S6 and S7) and the slight variations observed upon grafting are compatible with the variation in volume fraction between samples (Table 1).

_{NP}= 17 nm. The same is observed for the lower volume fractions, 15 and 20%v. Our interpretation of this result is that the system has developed into a highly aggregated one, where the internal structure is probably optimized towards higher density by peculiar correlations between larger and smaller beads.

^{−2}seems to persist. In Figure 6b, there appears to be an artefact at 15%v, where an increase in the second-particle distance is seen at 1.3 nm

^{−2}. We have attempted to understand the origin of this increase and we have checked if it can be traced back to slight uncertainties in the absolute intensity. We have, therefore, repeated the same Monte Carlo analysis for three sample data sets at 1.3 nm

^{−2}, shifting two of them by ±2% which corresponds to the uncertainty in positioning the sample intensity with respect to the particle form factor (see SI, Figure S8). The resulting indicators in Figure 6, including the “two particle distances”, have been converted into an error bar. The increased value at 1.3 nm

^{−2}and 15%v of silica in Figure 6b is seen to persist within the error bar. We conclude that, while we do not have any physical explanation for such an effect, its origin does not lie in a wrong positioning of the scattering curve in absolute intensity. In the last plot, Figure 6c, a different analysis is proposed. Here a fixed distance, corresponding to the length of two C

_{8}molecules (2L = 2.5 nm) has been used as upper bound for the integral over the raw IPS. The result is the number of neighbors typically encountered up to this distance. This number obviously increases with the volume fraction, but also with the grafting density, and the latter effect is again stronger for the lowest silica fraction. Finally, one may note that Figure 6b,c are two sides of the same coin, expressed by different parameters.

^{−2}. Below this threshold, particle dispersions are only slightly affected by the grafting, whereas above it, a complete reorganization, with strong aggregation, is observed.

_{8}with the corresponding indicators for C

_{18}. For this purpose, we have superimposed the evolution of the three indicators with the grafting density of C

_{18}at 15%v of silica in Figure 6, that is, when the signature of aggregation with respect to hard spheres is most developed. The contact values are higher for C

_{18}than for C

_{8}in Figure 6a, indicating denser assemblies. In parallel, both the strong decrease of the distance between two NPs and the increase of the number of neighbors within a shell corresponding to the size of two silanes are clearly shifted towards lower grafting densities, in Figure 6b,c, respectively. As observed for the dynamical features in Section 3.2, the influence of the longer alkyl-chain length of C

_{18}is stronger than C

_{8}to favor NP aggregation by reducing the buffer effect of the polymer segments at the silica surface. As with the dynamics, by again comparing the threshold values in Figure 6, it appears that the C

_{18}effect is stronger than the mass effect of (18:8) expected from the ratio between the alkyl chain masses.

#### 3.4. Determination of the True Interfacial Thickness by Combining BDS, SAXS, and RMC

_{IL}

^{PNC}with a (hypothetical) interfacial layer thickness is plotted exemplarily for different types of dispersion, corresponding to PNC samples with 15%v silica content, with either bare particles or high silane grafting (2.9 nm

^{−2}). The silica content is also represented and is seen to meet the experimental volume fraction with high accuracy. Note that the silica and the layer volume fractions are determined by the same algorithm based on the positions of N particles in the simulation box, thereby providing a cross-check of the algorithm. Another verification lies in the fact that Φ

_{IL}

^{PNC}-curves saturate at values approaching 1 − Φ

_{NP}, which is why we have left the silica in the definition of Φ

_{IL}

^{PNC}(Φ

_{IL}

^{PNC}+ Φ

_{NP}+ Φ

_{bulk}= 1) as opposed to the pure polymer part discussed in Figure 3b.

_{IL}

^{PNC}-curves in Figure 7a follow different laws despite their close silica contents. For very small thicknesses, PNC with bare NPs display a steeper slope, meaning that the dispersion is better, and more interfacial volume is created with every Angstrom of thickness around the more individually dispersed particles. On the contrary, in the highly aggregated case, there is immediate overlap of interfacial layers, leading to a reduction of volume of the latter. For bare NPs, the maximum available polymer volume is thus reached with ca. 15 nm thickness, while 25 nm are needed to cover all the particle-free regions of the sample in the aggregated case. The evolution of the interfacial volume fraction with thickness shown in Figure 7a for different grafting densities thus characterizes the quality of the dispersion. In Figure 7a, the determination of the real thickness corresponding to the total interfacial layer volume fraction is exemplarily shown, and a thickness of 4.6 nm is found for the bare system, based on an interfacial layer volume fraction determined by BDS of Φ

_{IL}

^{PNC}= 0.31.

_{IL}

^{PNC}-curves are quite similar. It is instructive, however, to superimpose the prediction of the cubic model (including possible overlap between neighboring layers) used in the literature to the RMC-analysis of experimental data in Figure 7b. Clearly, the large interparticle distance between all spheres in the cubic model leads to a much stronger increase of the Φ

_{IL}

^{PNC}–function with thickness. It is concluded that it is by no means suitable for dense and possibly aggregated structures as studied here.

_{IL}

^{PNC}to the experimental value of the interfacial volume fraction determined by BDS can be generalized to all samples, and the corresponding thickness can be read off following the downward arrow. The interfacial layer thicknesses reported in Figure 8 for all nanocomposite samples, as a function of grafting density, represent a key result of the present study, together with Figure 3c where the time scale of the interfacial dynamics is plotted. It needs to be emphasized that the true thickness could only be obtained by a combination of static structural (SAXS) and dynamic methods (BDS), with the help of RMC simulations. As a result, the interfacial layer thickness is found to be remarkably constant with both the grafting density and the silica fraction in PNC. An average value of 4.9 ± 0.2 nm is found, where the error bar has been determined from the standard deviation and the number of points. Moreover, it is found to be compatible with the value obtained with C

_{18}-surface modification (5.0 ± 0.5 nm) with a lower error bar due to a lesser dispersion of the points [12]. In the presence or absence of surface modification of any type (C

_{8}or C

_{18}), the range of interactions between the polymer segments and the silica surface is thus constant, intrinsic to the polymer–surface couple, or possibly intrinsic to the transmission of cage constraints [13] from one polymer layer to the next. Depending on the grafting, however, the strength of the interaction varies, inducing a stronger (in the case of bare) or weaker (for grafted, longer molecules having a higher impact) slow-down of the segmental dynamics.

## 4. Conclusions

_{8}-grafts: the presence of surfaces of bare silica NPs has a strong impact on the segmental dynamics, resulting in several times longer relaxation times due to the possibility of hydrogen bonding between the pyridine ring of P2VP and the hydroxyl groups at the silica surface. Upon grafting, the interface appears to be screened from the polymer, and the interfacial slowdown of segmental relaxation vanishes. For C

_{8}-grafts, this happens above a typical grafting density of about 1.5 nm

^{−2}, whereas it happens much earlier (ca. 0.5 nm

^{−2}) for C

_{18}grafts. This effect is considerably stronger than the difference in alkyl mass might suggest. Another remarkable result is that the volume fraction of the interfacial layer stays approximately constant for all grafting densities and the graft length.

_{8}grafting density of about 1.5 nm

^{−2}. Moreover, the three-dimensional representations of the dispersion state available through the RMC simulations allow for a precise determination of the interfacial layer thickness, in agreement with the corresponding volume fraction measured by BDS. It is thus the original combination of BDS, SAXS, and RMC which enables the determination of the nanometric interfacial layer thickness. Finally, the evolution of the interfacial layer volume fraction with (hypothetical) thickness for different dispersions shows that this function is also a valuable tool for the analysis of particle dispersions.

_{g}and the overall segmental dynamics of the material, and thus on the mechanical properties of the material. In parallel, modifying the particle dispersion, with aggregation or percolation, also influences the mechanical properties like moduli and resistance to rupture, or, in the case of carbon black fillers, conductivity. We believe that this study, which combines several techniques for a precise determination of interfacial thicknesses, will open the way to new investigations and hopefully control of macroscopic material properties by molecular design of interfacial layer properties.

## Supplementary Materials

**a**) and of the C

_{8}surface-modified NPs (

**b**) diluted in ethanol (Φ

_{NP}= 0.3%v). (

**c**) SAXS intensities (symbols) of P2VP nanocomposites filled with bare and surface-modified NPs at low volume fraction (Φ

_{NP}= 2.0, 1.9, 1.9, 1.5, and 0.3%v, for bare and C

_{8}-NPs with grafting density from 0.8 up to 2.9 nm

^{−2}). The line is the NP form factor in P2VP/silica contrast for comparison with the PNC data; Figure S3: Dielectric loss spectra of P2VP PNCs with different surface modifications of the silica NPs for the series with Φ

_{NP}= 15 (

**a**) and 20%v (

**b**) at 423 K. The solid lines represent the ILM fit with additional MWS and conductivity terms at low frequency and the β-process at high frequency; Table S1: ILM fit parameters of silica-P2VP PNCs using eqs (S1–S3); Figure S4: Frequency dependence of the imaginary part of the dielectric permittivity (squares) in 30%v‐PNCs at T = 423 K for (

**a**) bare NPs and (

**b**) C

_{8}1.3 nm

^{−2}grafted NPs. The solid lines represent the ILM fits as discussed in the main text, including dc-conductivity, MWS and secondary β processes (dashed and dotted lines as indicated); Figures S5: Top row: SAXS scattered intensities of P2VP-silica PNCs of different surface modifications for (

**a**) 15%v-series, (

**b**) 20%v, and (

**c**) 30%v. The particle form factor is superimposed (black line). Bottom row: corresponding apparent structure factors with RMC fits (solid lines) for the series at (

**d**) 15%v, (

**e**) 20%v, and (

**f**) 30%v; Figure S6: Comparison of the SAXS intensities of two similar P2VP-nanocomposites with C

_{8}surface-modified NPs (grafting density = 1.3 nm

^{−2}). The arrow indicates the position of the repulsive peak; Figure S7: Center-to-center distance d associated with the peak position in Figure S4, d = 2π/q0, as function of the grafting density (

**a**) and the silica volume fraction (

**b**). Log-log scale in (

**b**). The solid line is a fit to a power law with exponent equal to −1/3; Figure S8: (

**a**) Structure factor of the 15%v-PNC at 1.3 nm

^{−2}considering a shifted I(q) by ±2% (red and blue data). Solid lines are the corresponding RMC fits. (

**b**) IPS vs. surface-to-surface distance, normed to the same quantity in a hard-sphere gas of same parameters, using the data in (

**a**); Figure S9: Pair-correlation function for the 15%v‐PNCs with different grafting density of C

_{8}-silane as indicated in the legend; Figure S10: Volume fraction of interfacial layer Φ

_{IL}

^{PNC}as a function of interfacial layer thickness with respect to the entire sample Φ

_{IL}

^{PNC}+ Φ

_{NP}+ Φ

_{bulk}= 1, for 30%v-PNCs with different grafting density. Circles are bare NPs (Φ

_{NP}= 30.7%v) and squares are high silane grafting (C

_{8}2.9 nm

^{−2}, Φ

_{NP}= 25.2%v). The dashed lines represent the silica volume fraction of each sample as determined by the same algorithm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**TEM micrographs of P2VP PNCs at nominal NP volume fraction of 2%v (

**left**) and 20%v (

**right**) with C

_{8}-grafted silica. The grafting densities (0–2.4 nm

^{−2}) vary from top to bottom as indicated in the legend. The exact volume fractions of all samples are given in Table 1.

**Figure 2.**Comparison of the dielectric loss spectra of neat P2VP (black crosses, data normalized to the weight polymer fraction, 1 − Φ

_{w}) and PNCs with different surface modifications of the silica NPs for the series with nominal Φ

_{NP}= 30%v at T = 423 K. Arrows indicate the position of the maximum of the loss peak in neat polymer (black) and PNC with bare NPs (blue). The black dashed line is a fit with eq 1 for the α-process, plus the β-process and a conductivity term, while the colored lines represent the ILM fit with β- and MWS processes, plus conductivity.

**Figure 3.**(

**a**) Dielectric loss of a hypothetical system made of 100% of interfacial layer for P2VP PNCs (Φ

_{NP}= 30%v, T = 423 K, silane grafting as indicated in the legend). (

**b**) Volume fraction of interfacial layer for PNC series at 15%v, 20%v, and 30%v volume fraction, as a function of C

_{8}-grafting density. (

**c**) Segmental relaxation time of the interfacial layer, relative to the neat, at 423 K for the same series as in (

**b**).

**Figure 4.**(

**a**) SAXS results of P2VP PNCs with 30%v of silica: scattered intensity as a function of wave vector, for different grafting densities as given in the legend. (

**b**) Apparent structure factors obtained by dividing by the average form factor of the NPs.

**Figure 5.**Interparticle spacing function (IPS) vs. surface-to-surface distance, normed to the same quantity in a hard-sphere fluid of same parameters (concentration, size distribution), for different grafting densities as indicated in the legend. The nominal silica volume fractions of the PNCs are (

**a**) 15%v, (

**b**) 20%v, and (

**c**) 30%v.

**Figure 6.**(

**a**) Evolution of the normalized contact values with NP volume fraction (15, 20, and 30%v). (

**b**) Average second-particle surface-to-surface distances determined by integration of the raw IPS up to a sum of two neighboring NPs. (

**c**) Integrated raw IPS from surface to twice the silane length using 2L = 2.5 nm for C

_{8}(resp. 5.1 nm for C

_{18}). All plots are represented as a function of surface modification, for the three volume fraction series. Results obtained for the 15%v-series with C

_{18}surface modification [12] are included in grey for comparison.

**Figure 7.**Volume fraction of interfacial layer Φ

_{IL}

^{PNC}as a function of interfacial layer thickness with respect to the entire sample (Φ

_{IL}

^{PNC}+ Φ

_{NP}+ Φ

_{bulk}= 1), for different NP dispersions. The dashed lines represent the silica volume fraction of each sample as determined by the same algorithm. (

**a**) 15%v-PNCs, circles are bare NPs (Φ

_{NP}= 15.3%v), and diamonds are high silane grafting (C

_{8}2.9 nm

^{−2}, Φ

_{NP}= 12.6%v). (

**b**) 30%v-PNC with bare NPs (30.7%v) compared to the prediction of an (here inappropriate) cubic model with overlap correction [17].

**Figure 8.**Thickness of the interfacial layer determined by a combination of BDS, SAXS, and RMC, as a function of C

_{8}grafting density. The average value over all volume fractions is represented by a solid line. Dashed line: average value considering a cubic NP arrangement with overlap.

**Table 1.**NP volume fractions in PNCs, and C

_{8}-grafting densities on NPs suspended in solvent, both determined by TGA.

Bare | C_{8} 0.8/nm^{2} | C_{8} 1.3/nm^{2} | C_{8} 2.4/nm^{2} | C_{8} 2.9/nm^{2} | |
---|---|---|---|---|---|

2%v-series | 2.0% | 1.9% | 1.9% | 1.5% | 0.3% |

15%v-series | 15.3% | 13.3% | 12.4% | 11.9% | 12.6% |

20%v-series | 22.4% | 21.1% | 19.5% | 18.5% | 19.0% |

30%v-series | 30.7% | 28.1% | 26.5% 26.1% | 25.2% |

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**MDPI and ACS Style**

Genix, A.-C.; Bocharova, V.; Carroll, B.; Dieudonné-George, P.; Chauveau, E.; Sokolov, A.P.; Oberdisse, J.
Influence of the Graft Length on Nanocomposite Structure and Interfacial Dynamics. *Nanomaterials* **2023**, *13*, 748.
https://doi.org/10.3390/nano13040748

**AMA Style**

Genix A-C, Bocharova V, Carroll B, Dieudonné-George P, Chauveau E, Sokolov AP, Oberdisse J.
Influence of the Graft Length on Nanocomposite Structure and Interfacial Dynamics. *Nanomaterials*. 2023; 13(4):748.
https://doi.org/10.3390/nano13040748

**Chicago/Turabian Style**

Genix, Anne-Caroline, Vera Bocharova, Bobby Carroll, Philippe Dieudonné-George, Edouard Chauveau, Alexei P. Sokolov, and Julian Oberdisse.
2023. "Influence of the Graft Length on Nanocomposite Structure and Interfacial Dynamics" *Nanomaterials* 13, no. 4: 748.
https://doi.org/10.3390/nano13040748