The dielectric α-relaxation in polymer films: A comparison between experiments and atomistic simulations

The study of the glass transition in confined systems, and especially polymer films, has begun as a quest to find an inherent large length scale associated with glassy freezing [1]. Ever since Keddie et al. [2,3] reported tremendous shifts (with respect to bulk) in the glass transition temperature $(T_{\text{g}})$ of supported poly(styrene) and poly(methylmethacrylate) thin films, there has been concerted experimental efforts towards unraveling the finite-size effects of confinement [4–6]. To this end, manifold experimental techniques have been employed, with an extraordinary divergence in findings: $T_{\text{g}}$ shifts ranging from $-40\ \text{K}$ to $+15\ \text{K}$ for supported polystyrene (always studied by dielectric spectroscopy), for example, have been reported [7], and these findings were reported for situations where the upper electrode was evaporated onto the polymer film (no free surface) as well for situations with a gap between film and electrode (free surface). Looking at the temporal development of these dielectric results [7], one has to note, however, that with refinement of the experimental techniques and sample preparation procedures, there seems to be a growing agreement that, actually, there is no shift in the dielectrically determined $T_{\text{g}}$ at all upon reducing film thicknesses [8,9], even down to thicknesses around 10 nm [10], and, seemingly, even for isolated adsorbed single chains [11]. This finding is at first glance in stark contrast with the results of computer simulations of chemically realistic [12–14] as well as coarse-grained models [15–21] which have revealed from the outset that the structure and dynamics in a polymer film are modified next to a supporting substrate or at a free surface. The structural change typically extends up to 4–6 segmental thicknesses into the polymer melt, which translates into 2–3 nm for typical polymers. It is accompanied by a change in dynamics extending to a comparable distance into the film. The direction and strength of the effect were shown to depend on the nature of the surface: strongly attractive surfaces slow down relaxation in the near-surface region [17] which is even enhanced by surface corrugation [18], weakly attractive surfaces have almost no effect [17], whereas repulsive surfaces speed up relaxation in their vicinity [15,16,22]. A free surface also speeds up relaxation and more than for a repulsive wall [22]. This was in qualitative agreement with early rationalizations about the hugely varying $T_{\text{g}}$ shifts reported from different experiments [23–30]. However, independently of the choice of polymer and supporting surface [9], the results from dielectric spectroscopy in the last years tend to converge to the conclusion that the position of the segmental relaxation peak does not shift measurably with the thickness of the polymer film. To clarify the generic features of this seemingly universal result, rather than carrying out a simulation interpretation of a specific material, one has to compare typical representatives of the polymer/support systems exhibiting this phenomenon. We chose here two weakly polar polymers at neutral surfaces, where no strong specific interactions like hydrogen bonding are important. For the dielectric experiments, cis-polyisoprene (PI) supported by silica was an ideal representative of this class of systems. For the simulations 1, 4-polybutadiene (PBD) supported by graphite was the best choice, because this polymer is probably the experimentally best-studied polymer concerning the bulk glass transition (e.g., scattering, mechanical testing...) for which also quantitative, chemically realistic simulations of the bulk glass transition existed in our group [31]. Both systems are expected to show a comparable phenomenology, and this is adressed in the following.

Thin films of cis-polyisoprene $(M_{\text{w}}=44 500\ \text{g/mol})$ (Polymer Source Inc.) were prepared by spin-casting from a chloroform (Sigma-Aldrich, purity $\ge 99.9\%$) solution at a rate of 3000 rpm. Different thicknesses were obtained by varying the concentration of the solution. All thicknesses studied were prepared from semi-dilute solutions ($c>0.005\ \text{mg/ml}$, c being their mass concentration). The films were then annealed at 400 K for 24 h in an oil-free vacuum $(10^{-6}\ \text{mbar})$. Nanostructured electrodes (see [10,11,32–34]) were used to assemble the capacitors for dielectric measurements. The dielectric measurements of the bulk sample were done using a high-resolution Alpha Analyzer (Novocontrol). For all the thin films, the measurements were performed using an Andeen-Hagerling (AH) impedance bridge with an accuracy of $\le 10^{-5}$ in $\tan(\delta)$, corresponding to an error in the dielectric loss $\varepsilon^{\prime\prime}$ smaller than the symbol size in all presentations. Temperature control for both systems was regulated by a Quattro System (Novocontrol) using a jet of dry nitrogen, thereby ensuring relative and absolute errors better than 0.1 and 2 K, respectively. Due to the weak dielectric response of PI (especially of thin films), the AH bridge was used, since it has a higher accuracy than conventional dielectric spectrometers although it has the disadvantage of a limited frequency range (50–20000 Hz).

Molecular-dynamics simulations extending for up to $1\ \mu\text{s}$ in time were performed using the Gromacs [35] package with an integration time step of 1 fs. The simulations were performed for a chemically realistic melt [36] of 1, 4-polybutadiene (random copolymer of $55\%$ trans and $45\%$ cis units, 29 repeat units per chain, 720 chains) with a bulk $T_{\text{g}}=178\ \text{K}$ from DSC [37], for which the dielectric relaxation functions in the bulk had been studied by simulation before [38]. The graphite model was taken from the literature [39], and standard Lorentz-Berthelot combining rules were applied. The polymer melt was confined between two crystalline (0,0,1) graphite walls which were 10 nm ($T = 353\ \text{K}$, 323 K, 293 K, 273 K, 253 K), respectively, 20 nm ($T = 240\ \text{K}$, 225 K and 213 K) apart. The dielectric segmental relaxation was determined by a post-analysis of stored configurations calculating the dipole moment of the cis-group from its partial charges. Each cis-group was assumed to relax independently of the others, which is a reasonable approximation for 1, 4-polybutadiene of this microstructure [38]. The dielectric relaxation in the polymer film was evaluated for various layer thicknesses as measured from the graphite surface (see fig. 1). All thicknesses were multiples of the minimal layer thickness of 1.2 nm.

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The crystalline graphite support leads to a rather strong effective van der Waals attraction of the PBD segments to the surface, which is manifest not only in a segmental layering extending for about 2.5 nm into the polymer film but also in a layering in the center-of-mass density of the polymer chains [40]. Both structural changes slow down the dynamics close to the surface with respect to the center of the film, which itself exhibits bulk-like properties [40,41], indicating that the two surfaces do not interact. In addition to being heterogeneous, the dynamics is also anisotropic. Both effects are clearly shown in fig. 2, where the segmental dipolar autocorrelation function (ACF) is shown. We distinguished between the three Cartesian components of the local dipole moment (as would be observable by applying an electric field along the corresponding direction). In the center of the film, all three ACF superimpose, i.e., the relaxation is isotropic and bulk-like. Directly at the wall, the directions parallel to the supporting surface are equivalent, however, they are slowed down with respect to the center of the film. The ACF for the z-component, which is the direction perpendicular to the wall, at first superimposes upon the other two directions.

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For late times $(t>300\ \text{ps})$, it exhibits an additional slowing-down. This additional process is generated by the desorption kinetics from the attractive surface [40,41] and is predicted to be visible in scattering experiments [41] and nuclear magnetic resonance experiments [40] as well. It was not found in simulations of a coarse-grained bead-spring model at a repulsive wall [22], strengthening our argument that the long-time process seen in the dipolar relaxation is induced by the van der Waals attraction to the supporting surface. It has to be stressed, that this attraction is present for all contacts between a polymer and a solid material, albeit to a varying degree. From the relaxation $\Phi(t)$ (see fig. 2), the complex dielectric susceptibility $\varepsilon^{\ast}(\omega) = \varepsilon'(\omega) - i \varepsilon^{\prime\prime}(\omega)$ was determined using the fluctuation-dissipation relation $[\varepsilon^{\ast}(\omega) - \varepsilon_{\infty}]={\Delta \varepsilon} [1 - i \omega \hat{\Phi}(\omega)]$, to address the question as to how these clear quantitative and even qualitative changes in the segmental dynamics near an attractive supporting surface influence dielectric experiments in thin polymer films close to equilibrium. The quantities $\varepsilon_{\infty}$ and $\Delta \varepsilon$ are the relaxed dielectric permittivity and the dielectric strength, respectively [38,42], $\hat{\Phi}(\omega)$ being the one-sided Fourier transform of $\Phi(t)$. The experimentally relevant ACF is the one for the z-component, as this is the direction of the externally applied electric field. In an experiment on a thin supported polymer film the thickness of the film is varied, whereas in the simulations this was mimicked by calculating the relaxation functions as a function of layer thickness next to the confining walls (see fig. 1). From our simulations, we observe that there is a bulk-like region in the center of the film, which one also expects for films with a free surface. Thus, for the behavior next to the supporting surface, the boundary condition at the other side of the film seems not to be important, and we are able to compare our confined simulations to the experiments on the supported film with a free surface.

In fig. 2, the inset suggests that the effect of the confinement reveals itself in a broadening of the frequency spectrum at low frequencies. For the three Cartesian directions, the location of this maximum was found to be at $3.5\cdot10^{9}\ \text{Hz}$ in the core of the polymer film, whereas a shift (to $3.0\cdot10^{9}\ \text{Hz}$) was observed in the near surface layer of 1.2 nm thickness. In fig. 3, experimental findings (top part) for the film thickness dependence of the dielectric loss are compared to the corresponding simulation findings (bottom part). As explained in the methodological part, it is not possible to perform broadband dielectric spectroscopy experiments for the ultra-thin films studied here, so the experimental data are shown as a function of temperature at fixed frequency. Similarly, computational costs prohibit the simulation to be performed at a sufficient number of temperatures to allow for a plot as a function of temperature, so the simulation results are plotted at fixed temperature as a function of frequency. As the dependence of the frequency position of the maximum on temperature is monotonic, these two representations are equivalent. Focusing on the experimental results, we can clearly state that the position of the segmental peak for PI on silica does not depend on the thickness of the polymer film, in agreement with earlier findings for different polymers [8,9,33,34,43–45]. Interestingly, we find the same to be true for the simulation results. The position of the segmental relaxation peak in PBD does not depend on the thickness of the layer next to the graphite wall for which the dipolar relaxation is evaluated, except for the thinnest layer of 1.2 nm, for which a small shift was observed as discussed above. Figure 3 shows this thickness independence for a high temperature $(T=353\ \text{K})$, but it was found for all temperatures studied. Consequently, in the relaxation map in fig. 4, for both the experiments on PI on silica and the simulations of PBD on graphite, the temperature dependence of the dielectric α-process of these polymers, does not depend on the film thickness down to a thickness of a few nanometers. The data for all film thicknesses are compatible with the empirical Vogel-Fulcher-Tammann (VFT) fit for the bulk. Our simulations therefore agree with many recent dielectric spectroscopy experiments on supported films showing no change of the measured $T_{\text{g}}$ as a function of the film thickness, which is at first glance surprising in view of the shape difference between the local relaxation function near the walls and the bulk one (see fig. 2).

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The reason for this behavior is the strong effect of dihedral barriers on the relaxation properties and especially the glass transition in polymer melts [46–48]. For a melt of freely rotating PBD chains simulated at the same melt densities as the chemically realistic one, the glass transition was found to be shifted to about 60 K [48], almost 1/3 of the original value. A strong change in the density is therefore needed to overcome the influence of the dihedral barriers on the dynamics and shift $T_{\text{g}}$ appreciably, and this strong effect is only present within the first 1–2 nm next to the solid surface. A similar range over which dynamically significant density changes occur can also be expected at the free surface of a polymer film. This behavior of real polymers and chemically realistic polymer models is very different from the behavior of coarse-grained polymer models at a repulsive wall [22], for which the glass transition is mainly determined by the density effects. Correspondingly, these studies found a strong $T_{\text{g}}$ shift for bond correlation functions (which were taken as a representation of the dipole correlation functions measured in dielectric experiments) due to the density changes next to an interface. We also note that although the wall desorption process, which is clearly visible for the simulations in fig. 2, is a feature not present in the bulk dielectric relaxation, it does give rise neither to a separate peak in the dielectric loss nor to a shift of the position of the dielectric loss peak (at least for the temperature range studied), but it does give rise to a broadening of the loss peak at the low-frequency side (bottom part of fig. 3).

However, this is a small effect for all layer thicknesses when compared to the bulk, represented by the behavior in the center of the film. The experimental data (top part of fig. 3) show very few indications of a broadening even at the smallest thicknesses of 11 nm and 7 nm; however a small broadening as a function of the frequency is expected to be equivalent only to a logarithmically small broadening as a function of the temperature, so the temperature-dependent experiments are not as sensitive concerning broadening effects as frequency-dependent ones. To address the question of the presence of a broadening of the relaxation peak upon reducing the film thickness in more detail, the relaxation time distribution (RTD), $\rho(\tau)$, defined by $\Phi(t)=\int_0^\infty \rho(\tau) \text{e}^{-t/\tau} \text{d}\tau$, was derived for the experimental and the simulation data. The experimental data were fitted by the empirical Havriliak-Negami (HN) function, with the temperature dependence of the relaxation time being described by the VFT equation [42]. From this, the distribution of relaxation times is analytically given. In the regime of thick films, where the shape parameters for the HN function could be obtained from frequency-dependent dielectric experiments, it was found that they do not depend on the film thickness. Thus, for the analysis of the thin-film data, these bulk shape parameters were assumed to be applicable. This reproduced the temperature-dependent experimental spectra in fig. 3 within the experimental uncertainties. The relaxation time distribution (RTD) obtained from the experimental data is shown in the top part of fig. 5.

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For the simulations, the dipole relaxation functions of fig. 2 were fitted to a sum of two stretched exponentials, one of them being the standard Kohlrausch law generally used to model the α-process in glass-forming systems, the other one being a phenomenological fitting function for the desorption process. At an elevated temperature (e.g., $T=353\ \text{K}$ for which the relaxation function is shown in fig. 2), one obtains two well-separated peaks in the RTD. For lower temperatures, however, the time scales of the two processes visible in fig. 2 become comparable and actually cross. When this analysis is performed at the lowest simulated temperature, the result shown in the lower part of fig. 5 is obtained. For the experiment, the analysis leads to a unimodal RTD, skewed towards small-τ, where the data for all film thicknesses essentially superimpose. In the small-τ regime, a slight broadening with decreasing film thickness is visible. For the simulation, the RTD data on the large-τ side superimpose with the bulk behavior in the same way as in the experiment. On the small-τ side, the RTD is much less skewed than the experimental one, and this region seems to contain a remnant of the wall process (note that its time scale at low temperatures has become smaller than the α-relaxation time scale). However, the limited range over which the decay of the dipolar correlations was observable at this low temperature makes a quantitative discussion of this regime impossible. A comparison of the RTDs from simulation and experiment therefore suggests that for the range of film thicknesses studied, a unimodal RTD describes the segmental peak in fig. 3 (top) because the α-relaxation and the wall process have a comparable time scale which can be captured by an effective HN description. At higher temperatures, the single HN function analysis of the experimental RTD breaks down and it might be possible to separate the wall-induced processes from the α-relaxation in this temperature range.

In conclusion, a detailed comparison of experimental and simulation data on dielectric spectroscopy of ultra-thin polymer films was performed. This comparison of two different polymer-solid systems revealed strong qualitative similarities of the glass transition behavior in these systems. The physics of the glass transition in these films is determined by three influences: packing effects (or density layering), which are enhanced near the supporting wall, internal rotational barriers due to the torsion potentials, and an additional dynamic process linked to the desorption kinetics of wall attached chains in a layer of thickness $R_{\text{g}}$ next to the support. Our results suggest that the altered dynamics next to an attractive wall does not shift the dielectric glass transition temperature in films thicker than 2 nm. Only within the first 1–2 nm next to the support, the density change is large enough to overcome the influence on internal rotation barriers and induce a shift of the peak frequency of the α-process. Thus, the effect of density changes is much smaller than one would expect from coarse-grained simulations, and it occurs for film thickness which could not be studied experimentally so far. Thus, our results from simulations of a chemically realistic polymer/solid model reproduce for the first time the experimental finding that the dielectric relaxation map of a glass-forming supported film is independent of the film thickness (at least for walls at which van der Waals interactions dominate) and offer an explanation why this is so. In simulations, the spectral width of the α-process increases for thin films due to the presence of the wall desorption kinetics, and experimentally the spectrum at fixed temperature is not accessible yet for these ultra-thin films. The simulations suggest an analysis of the experimental behavior at higher temperatures, to reveal the desorption process established from recent simulations [40,41].

The authors are grateful for funding received within the Focused Research Program SPP 1369 of the German Science Foundation. The grant of CPU time at the Jülich Supercomputer Center is gratefully acknowledged.