We study experimentally the temperature evolution of the thickness of the interfacial layer, Lint(T), between bulk matrices and the surface of nanoparticles in nanocomposites through broadband dielectric spectroscopy. Analyses revealed a power-law dependence between the logarithm of structural relaxation time in the interfacial layer, τint(T), and the Lint(T): lnτint(T)/τ0Lintβ(T)/T, with τ0 ∼ 10−12 s, and β index ∼0.67 at high temperatures and ∼1.7 at temperatures close to the glass transition temperature. In addition, our analysis revealed that the Lint(T) is comparable to the length scale of dynamic heterogeneity estimated from previous nonlinear dielectric measurements and the four-point NMR [ξNMR(T)], with Lint(T) ∼ ξNMR(T). These observations may suggest a direct correlation between the Lint(T) and the size of the cooperatively rearranging regions and have strong implications for understanding the dynamic heterogeneity and cooperativity in supercool liquids and their role in interfacial dynamics.

Interfaces and interfacial regions play critical roles in the macroscopic properties of nanostructured and nanocomposite materials.1–4 Any interface, introduced by phase separation or by inserting hard materials into a softer matrix, leads to a modification of the material properties on a significant distance away from the interface.5,6 Classical examples are thin polymer films2,7–10 and polymer nanocomposites (PNCs),11,12 where the critical parameters are the gradient of macroscopic properties, i.e., dynamics and modulus, at the interface with a characteristic length scale Lint.5,13–16 In the case of thin polymer films, the estimates of Lint scatter from several nanometers to more than 100 nm,8,17–19 and the problem remains highly controversial.18–20 

Extensive work has been done in the past to investigate the Lint and its relationship with the dynamics of the interfacial layer. For instance, Stevenson and Wolynes proposed in 2008 that Lint(T) can be related to a characteristic size of the cooperatively rearranging regions (CRR), ξ(T), which is considered to be an intrinsic property of any glass-forming system, including polymers.21 Based on the Random First-Order Transition (RFOT) theory, they argued that Lint ∼ 10 − 20 ξ.21 Because the size of CRR is expected to increase upon approaching the glass transition temperature (Tg), Lint(T) should also increase upon cooling.22 Moreover, the RFOT suggests a quantitative relationship between Lint(T) and the structural relaxation time τα(T): logταTξTβ/TLintβ(T)/T, where β is an exponent reflecting the fractal dimension of the CRR structure.22 

Recently, molecular dynamics simulations provided estimates of the interfacial layer in thin polymer films and analyzed its temperature evolution.14,15,23 These studies suggest that the thickness of the interfacial layer is not very large, Lint(T) ∼ 2–5 segment sizes.15,23–26 Moreover, the computer simulations demonstrated an increase in Lint(T) upon cooling.14,23,24 Analyses of the simulation results revealed a linear relationship between the log[τα(T)] and the Lint(T): log[τα(T)] ∝ Lint(T)/T,14,23,24 which was ascribed to string-like structures in Refs. 14 and 24. However, these results were limited to very high temperatures and covered only ∼3–4 order changes in structural relaxation time far above Tg,14,15,23–25 while the region close to Tg is the most interesting from both the fundamental and applied aspects.

Very recently, Phan et al.27 and Diaz-Vela et al.28 demonstrated that the generic features of the interfacial dynamics can be understood through a pure interface induced change in the activation energy barrier at the interfacial layer. According to Schweizer and Simmons,29 the activation energy at the polymer–vacuum interface can be factorized into a position dependent component and a temperature dependent component, leading to a double-exponential dynamic profile at the interface. The apparent increase in the Lint(T) upon cooling is mainly due to the increase in the temperature dependent activation energy barrier, while the position dependent energy barrier has a characteristic penetration length, ζ(T), which saturates upon super-cooling.28,30,31 Specifically, the Phan–Schweizer theory for polymer–vacuum interface predicts an almost linear decrease in Lint(T) with an increase in temperature.32 Moreover, the theory suggests that the temperature dependence of the interfacial layer thickness is already encoded in the temperature dependence of structural relaxation time of the τα(T) of the bulk material:29,32Lint(T) ∝ ln[ln(τα(T)/τ0)], where τ0 is the structural relaxation time at infinitely high temperature. In addition, the slope of the lnlnταTτ0vs LintT offers a direct estimation of the 1/ζ(T).

In contrast to the rich theoretical work and computer simulations, experimental quantifications on Lint(T) and τint(T) are extremely challenging, especially in thin polymer films due to the very small volume of the interfacial region and the required high accuracy in analysis. Alternatively, bulk nanocomposite samples with a large surface-to-volume ratio possess sufficient fractions of the interfacial regions, offering an ideal platform for quantifying the thickness and the dynamics of the interfacial layer and their temperature evolution.33–36 Recent experimental studies revealed that in PNC, Lint(T) is ∼2–6 nm, which increases with an increase in chain rigidity and upon cooling.37 It was also suggested that Lint(T) can be related to the length scale of dynamic heterogeneity, which is often connected to the size of CRR.37 

The current work extends previous studies on the Lint(T) of PNCs37–39 to a detailed analysis on the correlation between the Lint(T) and the τint(T) at the interfacial layer in several nanocomposite materials based on the results of broadband dielectric spectroscopy (BDS). The presented study covers a large (∼7 orders) range of structural relaxation time close to Tg and reveals a power-law dependence lnτint(T)/τ0Lintβ(T)/T with β ∼ 0.67–1.7 and the Lint(T) comparable to the dynamic heterogeneity length scale of the bulk matrix, ξ(T).

Nanocomposites with well-dispersed silica nanoparticles in the small molecular liquid (glycerol) and three polymers with different chain rigidities were included in this study: poly(vinyl acetate) [PVAc, molecular weight (MW) of 40 kg/mol, polydispersity index, PDI = 1.76, and the characteristic ratio, C = 9.4], poly(propylene glycol) (PPG, MW = 4 kg/mol, PDI = 1.07, and C = 5.1), and poly(2-vinylpyridine) (P2VP, MW = 101 kg/mol, PDI = 1.07, and C = 10.0). Silica nanoparticles with a radius of RNP = 12.5 nm were used for nanocomposites of glycerol, PVAc, and PPG, and RNP = 15 nm for nanocomposites of P2VP. The silica nanoparticles were synthesized through a modified Stöber’s method in ethanol at a concentration of 15 mg/ml.40,41 According to our recent small angle x-ray scattering,42 the sizes of the nanoparticles follow a log-normal distribution with polydispersity σ = 0.17. The nanoparticle loadings of glycerol nanocomposite, PVAc nanocomposite, PPG nanocomposite, and P2VP nanocomposite are φNP = 23.6 vol. %, 20.5 vol. %, 18 vol. %, and 26 vol. %, respectively. The characteristic interparticle surface-to-surface distance dIPS=16πφNP1/32RNP=9.8 nm,11.5 nm,13.1 nm, and 10.4 nm for the glycerol nanocomposite, the PVAc nanocomposite, the PPG nanocomposite, and the P2VP nanocomposite, respectively.37,38 The detailed sample preparations, characterizations for the nanoparticle loadings and dispersion states, and the BDS measurements as well as the detailed analysis of the BDS spectra of PNCs can be found in previous publications.37–39 

Figure 1(a) shows a comparison of representative dielectric spectra (the loss spectra, ε″(ω), in the mainframe and the derivative spectra, εder(ω), in the inset) of the neat PVAc (red circles) and PVAc/SiO2 nanocomposite (PVAc-20.5) (blue squares) at T = 363 K. An interfacial layer relaxation process shows up as a low-frequency broadening of the main α-relaxation peak in both the ε″(ω) and εder(ω) spectra [Fig. 1(a)]. Detailed analyses of the spectra of PNC have been published elsewhere.37–39 Important parameters such as the structural relaxation time of the interfacial layer, τint(T) [Fig. 1(b)], the structural relaxation time of the bulk matrix, τα(T) [Fig. 1(b)], and the volume fraction of the interfacial layer, φint(T) [see the inset of Fig. 1(c)], can be identified at each T. From the spherical geometry of the nanoparticle, the average thickness of the interfacial layer can be estimated by LintT=φNP+φint(T)φNP1/31RNP [Fig. 1(c)]. Similar analyses had been performed for all four nanocomposites in our earlier papers.37,38Figure 1(d) presents the summary of the Lint(T) [see the mainframe of Fig. 1(d)] and τint(T) [see the inset of Fig. 1(d)] at temperatures close to Tg. Note that the mainframe of Fig. 1(d) is essentially a replot of the mainframe of Fig. 5(d) of Ref. 37, which serves as a starting point for the in-depth analysis presented in the current work. It shows that Lint(T) increases almost linearly upon cooling in all systems, in agreement with the recent theoretical predictions,29,32 and its temperature variations and absolute values differ between different materials. Moreover, the ratios of τint(T)/τα(T) remain almost constant for nanocomposites, as shown in the inset of Fig. 1(b), which is also predicted by the recent theoretical efforts.32 

FIG. 1.

(a) Dielectric loss spectra (mainframe), ε″(ω), and derivative spectra (inset), εder(ω), of neat poly(vinyl acetate) (PVAc) (red circles) and PVAc-20.5 nanocomposites (blue squares). The solid red lines are the fit to the Havriliak–Negami functions for the neat PVAc. The blue lines are the fit to the interfacial layer model.39 The alpha process and the interfacial layer process can be clearly identified in the PVAc-20.5 spectra. Details of the analysis are published in Refs. 37–39. (b) The alpha relaxation time of the neat PVAc, the bulk PVAc of the PVAc-20.5, and the interfacial layer of the PVAc-20.5. The inset shows the ratio of τint(T)/τα(T) of PVAc-20.5. (c) The interfacial layer thickness, Lint(T), of the PVAc-20.5 nanocomposite from the dielectric spectroscopy measurements. The inset shows the temperature dependence of the interfacial layer volume fraction, φint(T). (d) The Lint(T) (mainframe) and the τint(T) for the glycerol-23.6, the PPG-18, the PVAc-20.5, and the P2VP-26 nanocomposites. The data in the mainframe of Fig. 1(d) are a replot of Fig. 5(d) in Ref. 37.

FIG. 1.

(a) Dielectric loss spectra (mainframe), ε″(ω), and derivative spectra (inset), εder(ω), of neat poly(vinyl acetate) (PVAc) (red circles) and PVAc-20.5 nanocomposites (blue squares). The solid red lines are the fit to the Havriliak–Negami functions for the neat PVAc. The blue lines are the fit to the interfacial layer model.39 The alpha process and the interfacial layer process can be clearly identified in the PVAc-20.5 spectra. Details of the analysis are published in Refs. 37–39. (b) The alpha relaxation time of the neat PVAc, the bulk PVAc of the PVAc-20.5, and the interfacial layer of the PVAc-20.5. The inset shows the ratio of τint(T)/τα(T) of PVAc-20.5. (c) The interfacial layer thickness, Lint(T), of the PVAc-20.5 nanocomposite from the dielectric spectroscopy measurements. The inset shows the temperature dependence of the interfacial layer volume fraction, φint(T). (d) The Lint(T) (mainframe) and the τint(T) for the glycerol-23.6, the PPG-18, the PVAc-20.5, and the P2VP-26 nanocomposites. The data in the mainframe of Fig. 1(d) are a replot of Fig. 5(d) in Ref. 37.

Close modal

To investigate the correlation between the dynamics at the interface and the thickness of the interfacial layer and to follow the theoretical ideas and recent computer simulations, we compare the structural relaxation time of the interfacial layer log(τint(T)) to Lint(T)/T [Fig. 2(a)], and ln[ln(τα(T)/τ0)] to Lint(T) [Fig. 2(b)]. In Fig. 2(a), we normalized the values of Lint(T)/T with respect to the corresponding values of each PNC at T = 1.05Tg, LN,int(T)/T = [Lint(T)/T]/[Lint(T = 1.05Tg)/(1.05Tg)]. In order to see more clearly the common features between the log(τint(T)) and the LN,int(T)/T, a small vertical shift of the log(τint(T)) has been applied for glycerol-23.6 and PPG-18, as shown in the inset of Fig. 2(a). Interestingly, the values of the shifted log(τint(T)) for all four different PNCs fall onto a master curve with respect to LN,int(T)/T, indicating a rather universal relationship between the log(τint(T)) and the LN,int(T)/T. Moreover, all four nanocomposites show an obvious upturn of log(τint(T)) with respect to LN,int(T)/T at temperatures approaching Tg, indicating an increase in the fractal dimension of CRR upon cooling in the framework of the RFOT approach. Although previous computer simulations predicted a linear relationship between log(τint(T)) and LN,int(T)/T at high temperatures, the upturn at temperatures close to Tg has not been observed before.

FIG. 2.

(a) log(τint(T)) as a function of LN,int(T)/T, where LN,int(T)/T = [Lint(T)/T]/[Lint(T = 1.05Tg)/(1.05Tg)] for all four nanocomposites under study. With a slightly vertical shift of log(τint(T)), a master-like curve can be constructed, as shown in the inset. The vertical shift factors are −1.25, −0.04, 0, and 0 for glycerol-23.6, PPG-18, PVAc-20.5, and P2VP-26, respectively. The yellow dashed line in the inset is drawn to guide the eyes to show the deviation in the slopes of the shifted log(τint(T)) vs LN,int(T)/T at the high and the low temperatures. (b) ln[ln(τα(T)/τ0)] as a function of Lint(T). The four nanocomposites share almost the same slope with 1/ζ(T) ∼ 1/(2.7 nm), indicating a saturation of the position dependent energy barrier in the super-cooled region of the experiments.

FIG. 2.

(a) log(τint(T)) as a function of LN,int(T)/T, where LN,int(T)/T = [Lint(T)/T]/[Lint(T = 1.05Tg)/(1.05Tg)] for all four nanocomposites under study. With a slightly vertical shift of log(τint(T)), a master-like curve can be constructed, as shown in the inset. The vertical shift factors are −1.25, −0.04, 0, and 0 for glycerol-23.6, PPG-18, PVAc-20.5, and P2VP-26, respectively. The yellow dashed line in the inset is drawn to guide the eyes to show the deviation in the slopes of the shifted log(τint(T)) vs LN,int(T)/T at the high and the low temperatures. (b) ln[ln(τα(T)/τ0)] as a function of Lint(T). The four nanocomposites share almost the same slope with 1/ζ(T) ∼ 1/(2.7 nm), indicating a saturation of the position dependent energy barrier in the super-cooled region of the experiments.

Close modal

Interestingly, our analysis [Fig. 2(b)] also reveals that ln[ln(τα(T)/τ0)] varies linearly with Lint(T) for all four nanocomposites over the entire temperature range of the measurements. We used τ0 = 10−12 s in Fig. 2(b), but choosing the value of the τ0 = 10−13–10−11 s will not change the following discussion. Remarkably, according to the theory,32 the slope of the ln[ln(τα(T)/τ0)] ∝ Lint(T) leads to an experimental quantification of the characteristic length of the position dependent barrier. The analysis estimates ζ(T) ∼ 2.7 nm for all nanocomposites studied here. Although the almost constant ζ(T) ∼ 2.7 nm for all four nanocomposites with a different chemistry can hardly be rationalized in other theories, it actually agrees quantitatively with the proposal of the positional dependent energy barrier idea at the interface that saturates at low temperature and is controlled by the surface chemistry.29 

It is intriguing to see that the same set of experimental data can be described by the two different viewpoints. Naïvely, τα(T) = τ0 exp(Ea(T)/kT) with Ea(T) being the absolute activation energy and k being the Boltzmann constant. Thus, Ea(T) = kT ln(τα(T)/τ0). According to the Adam–Gibbs theory,43 the activation energy Ea(T) = NCRR(T)E = (T)βE, where NCRR(T) is the number of relaxers in the CRR, A is the pre-factor, β is the fractal dimension of the CRR, and E is the activation energy at infinitely high temperatures. Thus, a plot of ln(Ea(T)) vs ln(ξ(T)) could provide the β parameter, i.e., the fractal dimension of the CRR. The inset of Fig. 3 shows a plot of ln(Ea(T)) with respect to ln(Lint(T)/Lint(T = 1.05Tg)), where Ea(T) = kT ln(τint(T)/τ0) and τ0 = 10−12 s. In addition, in this case, the choice of τ0 between 10−11 s and 10−13 s will only slightly influence the absolute values of β and will not change the discussions below. Figure 3 presents the ln(Ea(T)) with a slight vertical shift of the data presented in the inset. The shifting factors for Glycerol-23.6, PPG-18, PVAc-20.5, and P2VP-26 are −0.04, 0, −0.36, and −0.54, respectively. Since the vertical shift will not change the slope, i.e., the β parameter, the following discussion will be based on the shifted ln(Ea(T)) vs ln(Lint(T)/Lint(T = 1.05Tg)) curves that form an interesting master curve, reflecting some possible generic features of the relationship between the thickness of the interfacial layer and its characteristic relaxation times (Fig. 3). It starts with the slope β ∼ 0.67 and increases to ∼1.7 at temperatures close to Tg. Interestingly, recent computer simulations revealed the fractal dimension of the string-like CRR having a value of β = 0.7–1.3 at high and intermediate temperatures.24,44–46 The same string-like structure at high temperature as well as a more compact structure of the CRR with fractal dimension below 3 upon approaching Tg has also been predicted at high temperatures by the RFOT theory.22 Therefore, the experimental observations here seem to be consistent with the predictions of previous computer simulations14,15,23,24 and agree qualitatively with the predictions from RFOT where the fractal dimension of the CRR increases with cooling toward Tg.22 

FIG. 3.

A plot of the shifted ln(Ea(T)) with respect to the ln(Lint(T)/Lint(T = 1.05Tg)), the slope of which provides a direct estimate of the shape of the CRR. The shift factors are −0.04, 0, −0.36, and −0.54 for glycerol-23.6, PPG-18, PVAc-20.5, and P2VP-26, respectively. A slope of 0.67 has been observed at high temperatures, and an upturn to a slope of around 1.7 has been found for temperatures close to Tg. The inset presents the raw data of ln(Ea(T)) of the four samples before shift.

FIG. 3.

A plot of the shifted ln(Ea(T)) with respect to the ln(Lint(T)/Lint(T = 1.05Tg)), the slope of which provides a direct estimate of the shape of the CRR. The shift factors are −0.04, 0, −0.36, and −0.54 for glycerol-23.6, PPG-18, PVAc-20.5, and P2VP-26, respectively. A slope of 0.67 has been observed at high temperatures, and an upturn to a slope of around 1.7 has been found for temperatures close to Tg. The inset presents the raw data of ln(Ea(T)) of the four samples before shift.

Close modal

We note that our analysis above is based on the assumption that Lint(T) ∝ ξ(T). In order to analyze a possible relationship between Lint(T) and ξ(T), we first compare the interfacial layer volume fraction φint(T) to the number of correlated structural units Ncorr(T) estimated for glycerol using linear and non-linear dielectric spectroscopies. Three types of Ncorr(T) studies of neat glycerol have been published previously: the dielectric derivative measurements,47,48 the third-order susceptibility measurements, χ3(ω),49–51 and the fifth-order susceptibility measurements, χ5(ω).52 Direct comparison of the normalized φint(T), φN,int(T) = φint(T)/φint(T = 1.05Tg) for the glycerol nanocomposite to the normalized Ncorr(T), NN,corr(T) = Ncorr(T)/Ncorr(T = 1.05Tg) revealed striking similarity in their temperature variations over the temperature range of T/Tg from 1.0 to 1.2 [Fig. 4(a)]. This good agreement of the temperature dependence of the φN,int(T) and the NN,corr(T) over a wide temperature range presents a first experimental support to the proposed correlation between the interfacial layer thickness and the dynamic heterogeneity length.

FIG. 4.

(a) Temperature dependence of the normalized interfacial volume fraction in glycerol-23.6 (the red circles), φN,int(T), and normalized Ncorr, NN,corr(T), from nonlinear dielectric measurements of glycerol [the blue squares (from Ref. 50), the olive line (from Ref. 48), and the orange triangles (from Ref. 51)]. (b) The comparison in absolute values between the Lint(T) of nanocomposites and the ξNMR(T) from the four-point NMR measurements of neat glycerol and neat PVAc. The ξNMR(T) are adopted from Refs. 53 and 54.

FIG. 4.

(a) Temperature dependence of the normalized interfacial volume fraction in glycerol-23.6 (the red circles), φN,int(T), and normalized Ncorr, NN,corr(T), from nonlinear dielectric measurements of glycerol [the blue squares (from Ref. 50), the olive line (from Ref. 48), and the orange triangles (from Ref. 51)]. (b) The comparison in absolute values between the Lint(T) of nanocomposites and the ξNMR(T) from the four-point NMR measurements of neat glycerol and neat PVAc. The ξNMR(T) are adopted from Refs. 53 and 54.

Close modal

To be more quantitative, we want to further compare the values of Lint(T) with the dynamic heterogeneity length scale measured by four-point NMR, ξNMR(T), which is considered to be the most direct experimental way of measuring the dynamic heterogeneity length. Due to the challenge in the multidimensional NMR measurements, only limited data exist in the literature.53,54 As shown in Fig. 4(b), the absolute values of the Lint(T) of both PVAc and glycerol nanocomposites are larger than the ξNMR(T) of neat PVAc and neat glycerol with Lint(T) ∼ 1.7 − 3.5 ξNMR(T). The comparison of the absolute values of Lint(T) and ξNMR(T) presents a further support to the recent proposal of Lint(T) ∝ ξ(T).14,23 However, it is Lint(T) ∼ 1.7 − 3.5 ξNMR(T), i.e., much smaller than the predicted by RFOT.21 This is also consistent with the results of simulations that estimate Lint to be just a few segments in length.15,23–26

At the same time, our results are consistent with the recent theoretical predictions23,28,29,32 for the thickness of the polymer–vacuum interface Lint(T) [Fig. 2(b)]. In that case, Lint(T) is defined by the bulk behavior of the relaxation time and should vary linear with ln[ln(τα(T)/τ0)], without involvement of any dynamic heterogeneity length scale or the size of the CRR.55,56 Yet, the estimate from the theory Lint(T) is just a few segments,32 consistent with our experimental data. Thus, the presented analysis cannot currently differentiate these two different approaches. However, we emphasize that both approaches relate Lint(T) to the bulk material properties, either to the length scale of dynamic heterogeneity or to the temperature variations of its structural relaxation time.

In conclusion, the presented analyses revealed a clear correlation between the temperature dependence of the characteristic relaxation time, ln(τα(T)/τ0), and the interfacial layer thickness, Lint(T), in nanocomposite materials. In particular, we present the first experimental confirmation of the relationship ln(τα(T)/τ0) ∝ Lintβ(T)/T proposed in the RFOT theory and found in simulations of thin polymer films and polymer nanocomposites.14,15,23 Moreover, our analysis revealed an increase in the exponent β upon approaching Tg. Furthermore, the quantitative analyses demonstrated that the interfacial layer thickness in the nanocomposites is comparable to the dynamic heterogeneity length, ξ(T), with Lint(T) ∼ 1.7 − 3.5 ξ(T). It is not obvious whether the observed correlations between the Lint(T) and the ξ(T), and the behavior of the exponent β are characteristic just for the interfacial regions or will be also valid for bulk glass-forming liquids and polymers. If the latter is correct, it might open an interesting way of studies of CRR and dynamic heterogeneities by analyzing interfacial regions in composite materials and thin films.

It is puzzling that the same set of experimental data can also be rationalized within the recently published theoretical approach27–29,32,57 based on the interface induced dynamic gradient, predicting Lint(T) ∝ ln[ln(τα(T)/τ0)]. This approach does not involve dynamic heterogeneity. Unfortunately, current experiments cannot measure accurately the dynamic gradient at the interface that is critical to distinguish the Schweizer–Simmons approach29 and the dynamic heterogeneity approach.15,21,24,26 From this perspective, an experimental mapping of the dynamic gradient at the interface at temperature close to Tg is critical for testing various theoretical approaches.

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Science, Materials Science and Engineering Division. S.C. acknowledges the financial support from Michigan State University.

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