Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers and Other Glass Forming Systems by Dielectric Spectroscopy and Calorimetric Techniques Daniele Cangialosi Abstract Glassy dynamics under nanoscale confinement is currently a topic under intense debate in soft matter physics. The reason is that this kind of studies may deliver important insight on the glassy dynamics in general. Furthermore, from a technological point of view, there exists a rising interest in the understanding of how properties are modified at the nanoscale in comparison to the corresponding bulk system. Within this context, this chapter critically discusses the experimental findings in the field. The vast majority of results concerns thin polymer films. However, other geometries of confinement, such as polymer nanocomposites and nanospheres, are considered as well. Special attention is devoted to the kind of information achieved by a specific technique. Within this context, the ability of dielectric and calorimetric techniques is highlighted. Particular attention is devoted to the determination of the different aspects of glassy dynamics in confinement, that is, the equilibrium dynamics in terms of the rate of spontaneous fluctuations as probed by experiments where a perturbation in the linear regime is applied, on the one hand, and the outof-equilibrium dynamics in terms of thermal glass transition temperature (Tg ) and the physical aging on the other. In the latter case, the application of a temperature ramp for Tg measurements and the recovery of equilibrium in physical aging imply the application of large perturbations, in particular with amplitude well beyond that of spontaneous fluctuations. It is demonstrated how, in view of numerous experimental results, the two aspects are not one-to-one related in confinement. Specifically, the reduction in Tg and the acceleration of equilibrium recovery in the aging regime does not imply a concomitant speed-up of the rate of spontaneous fluctuations, which is in several cases found to be unaltered in comparison to the bulk. Finally, a description of suitable frameworks to describe such phenomenology is presented with special attention to the free volume hole diffusion (FVHD) model. This is shown to quantitatively D. Cangialosi (B) Centro de Física de Materiales (CSIC-UPV/EHU), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain e-mail: swxcacad@sw.ehu.es F. Kremer (ed.), Dynamics in Geometrical Confinement, Advances in Dielectrics, 339 DOI: 10.1007/978-3-319-06100-9_13, © Springer International Publishing Switzerland 2014 340 D. Cangialosi catch the acceleration of physical aging and the Tg depression with no need to assume any acceleration on the intrinsic molecular mobility of the glass former. Keywords Glass transition -of-Equilibrium dynamics · Molecular mobility · Linear response · Out Abbreviations AG theory CD CRR FVHD NEXAFS PALS PMMA RFOT SMFM VFT Adam-Gibbs Theory Capacitive Dilatometry Cooperative Rearranging Region Free Volume Hole Diffusion Model Near-Edge X-Ray Absorption Fine Structure Positron Annihilation Lifetime Spectroscopy Poly(methyl methacrylate) Random First-order Theory Shear Modulation Force Microscopy Vogel-Fulcher-Tammann 1 Introduction Modern technology often requires the employment of materials confined at the nanoscale, that is, with typical dimensions in the submicron range. In such configuration, properties can be dramatically affected due to the dominant role of those portions of the material close to the interface. Therefore, the knowledge of how the interface interferes with the overall material performance is crucial. Among those properties that can be deeply affected under nanoscale confinement, those related to glass transition phenomena have been intensively investigated in the last years. The reason is that glass forming materials are a widely employed class of materials since the beginning of civilization (their use among phoenicians dating back to the third millennium a.C. is reported). Among them a subclass is represented by glass forming polymers and most of the studies of the glass transition in confinement involve these systems. The one-dimensional confinement, namely that based on polymer thin films, is by far the most investigated, although in recent years much effort has been devoted to the investigation of confined glass formers in other geometries. In particular polymer nanocomposites and nanospheres, and glass formers in nanopores have recently received considerable attention. In this chapter, we first briefly introduce the main aspects of the phenomenology of the glass transition. In particular, the existence of a typical length scale associ- Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 341 ated with density fluctuations relevant for glassy dynamics will be introduced. This, among the different facets of the phenomenology of the glass transition, has historically represented one of the main rationale to study glass forming systems confined at length scales approaching those relevant for glassy dynamics. After introducing these aspects, we provide a review on the debate regarding the effect of confinement on such phenomena. This discussion involves the apparently controversial views based on different experimental observations, where the glassy dynamics is seen to either speed-up, slow-down, or remain equal to that of the bulk system. In doing so, the following aspects will be emphasized: (i) the conceptual difference among the different kind of determinations delivering information on nonequivalent aspects of the glass transition phenomenon; (ii) how—given a certain confinement length scale, e.g., the thickness in thin polymer films—the applied cooling rate, the kind of substrate, and the preparation conditions affect the deviation of glass transition phenomena from the bulk behavior. In this context, the numerous experimental contributions will be reviewed. Particular emphasis will be provided to the study of glassy dynamics by dielectric relaxation and calorimetric techniques. These allow the determination of the different aspects of glassy dynamics, in some cases in a single experiment. The critical analysis emphasizing the different factors affecting the glass transition in confinement of such contributions can in principle provide a suitable framework to understand what kind of deviations should be expected in given confining conditions and experimental protocols. Finally, some possible theoretical frameworks, recently introduced into the scientific debate, to account for the different aspects of glassy dynamics in confinement will be reported. 2 Glassy Dynamics: Established Facts From a thermodynamic point of view, the fate of a liquid cooled down below the melting temperature is that of transforming to the most stable crystalline phase. However, for kinetic reasons, a considerable number of liquids can be supercooled, that is, they remain amorphous over large time scales [29, 110]. Among glass forming systems, polymers represent an important subclass, due the fact that crystallization is often hindered by chain connectivity and, in some cases, conformational irregularities. Further temperature reduction of a supercooled liquid at a given rate leads to the formation of a glass, that is, a disordered system with the mechanical properties of a solid. The temperature marking the liquid to glass transformation is commonly addressed as the glass transition temperature (Tg ). At such temperature, a jump in thermodynamic coefficients (the specific heat, the compressibility, the coefficient of thermal expansion, etc.) occurs. This is reminiscent of a second-order thermodynamic transition in the Ehrenfest classification [110]. However, several experimental aspects of the glass transition indicates that this is actually not the case. Among them, probably the most evident is the fact that the Tg depends on the cooling rate, that is, the timescale of the experiment. Nowadays, it is well accepted that the glass transition is intimately linked to the dramatic slowing down of density fluctuations associated to 342 D. Cangialosi the glassy dynamics, the so-called α relaxation, with decreasing temperature occurring in the supercooled state. This is often described by the Vogel-Fulcher-Tammann (VFT) equation: τ = τ0 exp(B/(T − T0 )). Here, τ is the relaxation time relevant for spontaneous fluctuation, τ0 a pre-exponential factor, and B and T0 the Vogel activation energy and temperature, respectively. For bulk glass formers, it has been shown that, not only there exists a relation between the time scale of spontaneous fluctuations τ and the Tg , but that the two magnitudes are unequivocally related to each other [34, 125]. This is reflected in the dependence of the Tg with the cooling rate, exhibiting the same VFT behavior as that of τ . Hence, cooling down a supercooled melt below its Tg implies that, in the timescale of the experiment, dictated by the applied cooling rate, the rate of spontaneous fluctuations is too small to maintain equilibrium. However, once a glass is brought into the glassy state, it spontaneously evolves toward equilibrium, an aspect of glassy dynamics commonly known as physical aging. As for the Tg , in bulk glass formers, it has been shown that the rate of approach to equilibrium is exclusively determined by that of spontaneous fluctuations [55, 66]. Nevertheless, it has to be remarked that, from a conceptual point of view, measuring τ , on the one hand, and the Tg or the time to equilibrate in the physical aging regime, on the other, are two separate aspects of the dynamics of the glass transition. In particular, measuring τ requires the application of a perturbation in the linear regime, that is, with amplitude smaller than those of spontaneous fluctuations, in order to fulfill the fluctuation-dissipation theorem [17, 91]. Conversely, measuring the Tg at a given rate or the recovery of equilibrium in the physical aging regime entails the application of perturbations beyond the linear regime (e.g., a change of temperature with a certain cooling ramp in Tg measurements). As will be seen in the next sections of the chapter, this conceptual difference constitutes an important ingredient in the understanding of experimental results on glassy dynamics in confinement. A long-debated aspect of the drastic slowing down of the dynamics of the glass transition when decreasing the temperature is the possible existence of a concomitant growing length scale, that is, the spatial extent of the relaxation. This has been long ago put forward by Adam-Gibbs (AG) [1], who theorized that the magnitude controlling the slowing down of the α relaxation with decreasing temperature is the configurational entropy (Sc ). According to the AG theory, the α relaxation occurs via cooperative rearrangement of several basic structural units. The AG theory suggests that the number of units (z ∗ ) involved in cooperatively rearranging regions (CRR) increases with decreasing temperature and such temperature variation is also related to the configurational entropy: z ∗ ∼ Sc−1 . Since the AG theory, numerous approaches have been presented in the search for the relevant length scale of the α relaxation. As the AG theory, some of them, such as that proposed by Donth [33, 56] and the random first-order theory (RFOT) [76], rely on thermodynamics. Conversely others, such as that based on string-like motion [32] and the four point dynamic susceptibility [8], are based on the estimation of dynamically correlated structural units. Among the different approaches seeking for the relevant length scale of the α process, within the context of this chapter, it is worth considering that based on the self-concentration [73]. Such concept was introduced Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 343 to explain the presence of two glass transitions in miscible polymer blends. It relies on the fact, that due to the limited size of CRR(s), the effective concentration within the cooperative volume differs from the macroscopic one [18, 19, 25]. In particular, it is richer in the component of the target unit at the center of the CRR. As such the self-concentration approach provides information on how far the dynamics of a given structural unit is affected by the surrounding. This is especially relevant once the effect of an interface in nanoscale confinement is considered. In all approaches providing an estimation of the relevant length of the α relaxation, this is generally found to be in the order of several nanometers or, for some polymer and within some approaches, even smaller than 1 nm [5, 18, 19]. 3 Glassy Dynamics Under Nanoscale Confinement: Experimental Observations 3.1 Thermal Glass Transition In the previous section, among the different aspects of glassy dynamics, those related to the presence of a typical length scale have been emphasized. As a consequence, if such a length scale really exists, it is possible to speculate a- priori that once the dimensions of the glass former approach those of such length scale, a modification of glassy dynamics must be expected. In this context, pioneering studies of McKenna and co-workers [60] on several low molecular weight glass formers confined in nanopores with diameters as low as 4 nm showed clear depression of the calorimetric Tg . The first report of confinement effects in glass forming polymers was presented by Keddie et al. [62]. They employed ellipsometry to determine the temperature dependence of the thickness of thin polystyrene (PS) films supported on silicon wafers. They found significant Tg depression in films thinner than 50 nm. Since the work of Keddie et al. [62], a huge amount of work has been presented on the glass transition of thin polymer films in different configurations, included supported [4, 30, 35, 37, 38, 42, 44, 47, 51, 52, 107, 111, 117, 123, 124], capped [16, 46, 77, 98] and freestanding films [4, 10, 16, 41, 63, 64, 84, 102, 126]. Most of the results generally indicate that nanoscale confinement induces Tg depression. As in the work of Keddie et al. [62], effects on the Tg are visible at thicknesses smaller than 50 nm and the largest depression is of the order of 30 K for films thinner than 10 nm. Apart from this, it is important to remark that Tg determinations deliver considerable scattering of data. This implies that other factors, beyond the film thickness, are of importance in determining the magnitude of Tg depression in thin polymer films. This becomes immediately clear once Tg data of freestanding polymer films are considered [4, 10, 16, 41, 63, 64, 84, 102, 126]. In such a case, decreases of Tg in comparison to the corresponding bulk polymer as large as 70 K are observed and effects are visible already for thicknesses larger than 100 nm [10]. Furthermore, while for thin PS films the vast majority 344 D. Cangialosi 90 nm 24 nm 11 nm 6 nm 5 nm 3 2 1 -1 ln(q [s]) ∼ ln(τ eq [s]) 4 0 -1 2.65 2.70 2.75 2.80 2.85 2.90 2.95 -1 1000/Tg (K ) Fig. 1 Natural logarithm of the inverse cooling rate, q −1 , versus reciprocal glass transition temperature, 1000/Tg , for supported thin PS films on platinum-coated silicon nitride with different films thicknesses. Continuous lines are the fits of the FVHD model to experimental data (Reprinted with permission from Ref. [12]) 385 380 375 on Tg (K) 370 365 360 355 -1 350 q c = 20 K.min 345 qc = 0.2 K.min -1 10 100 1000 10000 Bulk Film thickness, h (nm) Fig. 2 Tg as a function of films thickness and applied cooling rate (qc ) for freestanding thin PS films. Continuous and dashed lines are the fits of the FVHD model to experimental data via two different approaches. For details regarding the difference between the two approaches see Ref. [10]. (Reprinted with permission from Ref. [10]) of studies generally provides evidence for Tg depression, there exists a significant number of studies on thin poly(methyl methacrylate) (PMMA) films supported on silicon-based substrates [52, 107], where an increase of the Tg in comparison to the bulk polymers is observed. These observations suggest that the nature of the interface is an important parameter affecting the magnitude of Tg depression. An additional parameter that has been considered in the study of thin polymer films is the dependence of the magnitude of Tg depression on the applied cooling Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 345 rate. This aspect has been considered after the introduction into the scientific debate of the study by fast calorimetry by Efremov et al. [35]. Studying thin PS films, they showed little effect, if any, of the film thickness down to several nanometers on the Tg measured at 9 × 104 K min−1 cooling rate. A systematic study on the cooling rate dependence of the Tg in thin PS films was later conducted by Fakhraai and Forrest [38] by ellipsometry, a technique delivering information of the temperature dependence of the film thickness. They showed pronounced thickness dependence of the Tg at low cooling rates, as in the vast majority of experiments. However, when the cooling rate is increased, the magnitude of the Tg depression reduces. This is shown in Fig. 1, where data of Fakhraai and Forrest [38] are summarized. Similar investigations have been performed by calorimetry at two cooling rates in freestanding thin PS films [10]. These results are presented in Fig. 2. With regard to supported or capped thin films, very recently, the effect of polymer adsorption on the substrate has been investigated. In particular, it has been shown that extended annealing at temperatures substantially larger than the Tg of the bulk polymer (453 K for PS) induced a reduction of confinement effects on the Tg [87, 89, 106, 113, 129]. Furthermore, the annealing time dependence of such a reduction was shown to be significantly dependent on the polymer’s molecular weight, as shown in Fig. 3. In particular, for films with molecular weight larger than 160 Kg mol−1 no shift in Tg were observed for annealing times as large as 105 s. Conversely, at similar annealing times lower molecular weights films could recover the bulk Tg . This indicates that the kinetic of adsorption is somehow related to chain dynamics. Further investigation on the impact of high temperature annealing on Tg ’s deviations revealed that, rather than the thickness of the adsorbed layer, the crucial parameter determining the magnitude of Tg depression is the amount of free interface [87]. Beside the huge scientific activity on thin polymer films, several studies on the effect on the thermal Tg in different kind of confinement have been carried out. These involve polymer nanocomposites and nanospheres, and glass formers confined in nanopores. With regard to polymer nanocomposites, those systems exhibiting weak interactions with the nanoparticles’ surface generally exhibit Tg depression [7, 11, 15, 22, 105]. This result suggests a general analogy with thin polymer films [7]. Conversely, in those polymer nanocomposites where strong interactions, such as hydrogen bonding, are allowed, a Tg increase is observed [75, 100, 105]. Recent scientific activity has been devoted to the study of the thermal glass transition in polymer nanospheres. Here, results appears to be somewhat scattered even for the same type of interface. In particular, some calorimetric studies report no Tg dependence with the nanosphere diameter [48, 120]. Conversely, several works show that freestanding nanospheres with diameter of the order of several tens of nanometers generally exhibit increased Tg in comparison to the bulk [80, 83]. However, once the nanospheres diameter approaches 100 nm or larger a reduction in Tg is observed [31, 131, 132]. This nonmonotonic Tg dependence on the nanospheres diameter has been explained according to entropic arguments by Martinez-Tong et al. [80]. They pointed out that, when the nanospheres radius of curvature approaches the typical size of the macromolecules (e.g., the radius of gyration), a decrease of the number of configurational degrees of freedom occurs and, as a consequence, 346 D. Cangialosi 21 nm PS97 22 nm PS160 35 nm PS97 44 nm PS97 300 nm PS97 Tg [K] 370 360 1000 10000 100000 annealing time [s] Fig. 3 Tg as a function of annealing time at 453 K for Al-capped thin PS films. PS97 and PS160 stand for polystyrenes with molecular weight 97 and 160 Kg mol−1 , respectively (Re-adapted from Ref. [89]) the molecular motion associated to the α relaxation is slowed down. As in the case of polymer thin films, an additional ingredient in determining the magnitude of Tg deviations from bulk behavior is the nature of the interface. Recent studies showed that PS nanospheres exhibit either decrease or no-change in Tg depending on the presence of surfactants and its nature at the interface [40] or the presence of silica capped on the surface [132]. Regarding the increase of Tg in polymer nanospheres with diameter in the sub-100 nm regime [80, 83], whatever the explanation, it is obvious that there exists a marked difference with thin polymer films with the same (equivalent) size. Conversely for nanospheres with diameter larger than 100 nm, analogy with thin polymer films can be put forward [131]. Hence, the difference between polymer nanospheres and thin films must originate from the curvature at the interface in the former geometry and its effect on the polymer conformation at such interface. This conclusion is corroborated by experiments in low molecular weight glass formers confined in spherical nanopores, which exhibit decrease [60] or no change in Tg . 3.2 Out-of-Equilibrium Dynamics The glass transition marks the crossover from the equilibrium melt state to the outof-equilibrium glass. Physical aging, that is the recovery of equilibrium of the glass, Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 347 Fig. 4 Recovered enthalpy, expressed in terms of distance from equilibrium, for freestanding thin PS films and bulk PS at 358 K (Reprinted with permission from Ref. [10]) in nanoscale confinement has been deeply investigated in recent years [23, 96]. This phenomenon is intimately linked to the thermal glass transition. A Tg decrease indicates the ability of the glass to equilibrate more efficiently. Hence, if confinementdependent Tg is observed, it is obvious to expect deep effects also in the physical aging behavior. This is actually found in several works in polymer thin films [10, 65], nanocomposites [3, 11, 14, 15, 22, 24] and nanospheres [53], where the entire enthalpy or volume recovery function is obtained in calorimetric and dilatometric experiments, respectively. All these studies show faster achievement of equilibrium in the nanostructured glass in comparison to the bulk counterpart. As an example, the enthalpy recovery during physical aging of freestanding thin PS films at 358 K, taken from Ref. [10], is presented in Fig. 4. Importantly, deviations from the bulk physical aging behavior can be clearly observed at thicknesses as large as several microns. Other studies reported the aging time dependence of a given observable in a relatively limited time window. In such a case, an aging rate, defined as the slope of the decay in the observable as a function of the logarithm of the aging time [57], is determined. The studies delivering such information are based on the employment of ellipsometry [6, 43, 61, 101, 103, 104], fluorescent spectroscopy [97, 99, 100], dielectric methods [13, 26, 45, 100], permeability measurements [85, 108], dilatometry [24], and Positron annihilation lifetime spectroscopy (PALS) [109]. All of these studies point toward significant effects of confinement on the physical aging behavior. Interestingly, when the depth profile of the physical aging pattern is determined [97, 109], achievement of thermodynamic equilibrium appears to be faster the closer the free interface of the film is. Other factors, such as the presence of mechanical stress [103], the type of interface [104] and the chain architecture [43], have been shown to be of importance in determining the magnitude of the aging rate. 348 D. Cangialosi Fig. 5 Loss part of the dielectric permittivity as a function of temperature for PMMA at 12 kHz with thickness 57 nm before and after annealing for 12 h at 400 K in pure nitrogen (Reprinted with permission from Ref. [113]) 3.3 Dynamic Glass Transition Besides the investigation of the thermal glass transition, numerous studies have documented the effect of confinement on the rate of spontaneous fluctuations associated with the α process. With regard to polymer thin films, early experiments showed acceleration of the α relaxation with decreasing thickness. The first important contribution in the field was provided by Fukao and Miyamoto [46]. They showed that the molecular dynamics—probed by broadband dielectric spectroscopy (BDS)—of Al-capped PS films thinner than 20 nm was accelerated in comparison to the bulk. Similar results, also by BDS, were later reported by other authors [54, 116]. In subsequent studies, Kremer and co-workers emphasized that preparation and experimental conditions may have significant effect on BDS results [113]. In particular, they showed that annealing above Tg to remove the solvent of spin-coated thin polymer films and the environmental conditions of the experiments (nitrogen ver us air) are key factors in determining the location of the most probable frequency of relaxation in BDS experiments. This is shown in Fig. 5, where the dielectric response as a function of the temperature at 12 kHz for a thin PMMA film with thickness 57 nm is presented. In this figure, it is shown how the typical temperature of the α relaxation shifts by more than 30 K once the films are annealed over extended time well above Tg . Beside these experiments, Perlich et al. [94] showed that solvent removal in supported thin PS films was considerably more difficult than in the bulk polymer. The important consequence of these observations was that, once measured in inert atmosphere (e.g., nitrogen) and prepared under drastic conditions for solvent removal, thin films exhibited identical molecular dynamics as the bulk counterpart. This result has been found measuring the α relaxation by different techniques, included BDS [16, 67, 79, 88, 89, 112, 121, 122, 128, 130], AC-calorimetry [58, 79, 121], shear modulation force microscopy (SMFM) [49], Near-edge X-ray absorption fine structure (NEXAFS) [72] and optical photobleaching [93]. In the latter case, pronounced bulk-like dynamics was observed for freestanding PS films as thin as 10 nm, as shown in Fig. 6. Apart from thin polymer films, bulk-like dynamics has been observed in other type confinement. In particular, several polymer nanocomposites [11, 13, 15, 22] Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 349 Fig. 6 Temperature dependence of the molecular relaxation time for 17 nm freestanding thin PS film (filled symbols) and bulk PS (stars) (Reprinted with permission from Ref. [93]) and nanospheres [131] exhibit filler size and diameter independent molecular dynamics, respectively. For polymer nanocomposites exhibiting strong interaction at the interface polymer/nanofillers, a slowing down of the segmental dynamics is rather observed [68, 100]. Despite the presence of bulk-like dynamics in different confining geometries, it is important to mention that several studies suggest that such systems display a rather complex relaxational behavior. This is generally found in confined systems with at least one free surface [39, 90, 92, 93, 131] and, therefore, is attributed to the existence of relatively fast dynamics in proximity of such surface. Within this explanation, the thickness of such layer is estimated to be of the order of nanometers [93]. 4 Decoupling Between Equilibrium and Out-of-Equilibrium Dynamics In the previous section of the chapter, the importance of sample preparation and the environmental conditions employed to measure glassy dynamics in thin polymer films have been discussed [113]. The important outcome within this context is that confinement effects on the α relaxation might be a consequence of the experimental conditions employed to determine such dynamics. These experimental facts generate an important question: is the observed thermal Tg depression (and beside it the acceleration of equilibrium recovery) a true confinement effect or does this originate from the employed experimental procedure? To answer this question it is vital to recall those studies where both the intrinsic molecular mobility, on the one hand, and the Tg and recovery of equilibrium in the physical aging regime, on the other, 350 D. Cangialosi are probed in sample prepared under identical conditions and, possibly, in the same measurement. In this sense, dielectric and calorimetric methods offer a unique possibility to probe the rate of spontaneous fluctuations and the out-of-equilibrium dynamics. The first study where simultaneous measurements of these two aspects of glassy dynamics were performed is that of Lupascu et al. [78]. In samples prepared under identical conditions, they measured the molecular dynamics in the linear regime and the thermal Tg simultaneously. The latter determination is based on the so-called capacitive dilatometry (CD) method. This consists in measuring the high frequency real part of the dielectric permittivity, where no relaxational contributions are present, as a function of the temperature. The value of such permittivity is connected to the density of the glass formers via the refractive index. In doing so, for Al-capped thin PS films, Lupascu et al. [78] found a weak speed-up of the molecular dynamics in comparison to the bulk. Conversely, the Tg of thin PS films exhibited slightly more pronounced thickness dependence. A systematic investigation on the molecular dynamics and thermal Tg of thin PS films has been recently performed by Boucher et al. [16] employing BDS and calorimetry in samples subjected to the same preparation procedures and measured under identical environmental conditions. In doing so, they found a marked decoupling between the rate of spontaneous fluctuations and the thermal Tg . The former was found to be independent of the thickness and identical to that of bulk PS. Contrariwise, the thermal Tg exhibited clear depression, increasing with decreasing film thickness, and being more pronounced for freestanding films. In the case of BDS, exploiting the ability of this technique to achieve information on both aspects of glassy dynamics, this result was found in the same measurement. This result unequivocally indicates that Tg depression is a real feature of glassy dynamics in confinement and that this can be present in glass formers in confinement exhibiting bulk-like dynamics. The main outcome of this study is presented in Fig. 7, where the temperature dependence of the molecular relaxation time (τ ) (upper panel) and the thermal Tg as a function of the thickness (lower panel) are reported. Apart from the study of Boucher et al. [16], other works report deviating results depending on the information delivered in thin polymer films [27, 59, 114, 115]. In ways analogous to thin polymer films, there exists a number of recent studies by BDS and calorimetry reporting apparently contrasting results regarding the intrinsic molecular dynamics and the thermal Tg in other type of confinement. In the Sects. 3.1 and 3.2, several examples of polymer nanocomposites exhibiting Tg depression have been reviewed [7, 11, 15, 22, 105]. In analogy to thin polymer films, in those nanocomposites where the molecular dynamics have been probed, identical rate of spontaneous fluctuations as those of the bulk polymer has been found [11, 15, 22]. This result also applies to PS nanospheres, where the Tg from calorimetry [132] and CD [131] have been found to be depressed with decreasing nanospheres diameters, whereas no shifts in the intrinsic molecular mobility were detected [131]. Regarding the simultaneous measurement, or at least in samples prepared under identical conditions, of the intrinsic molecular mobility and the rate of equilibrium recovery in the physical aging regime, this has been performed in several Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 351 1 10 Bulk 0 10 -1 10 -2 10 τ (s) -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 Al-capped, BDS 1200 nm 500 nm 130 nm 30 nm 15 nm AC-calorimetry 3000 nm 280 nm 52 nm 18 nm Freestanding, BDS 1000 nm 500 nm 200 nm 300 nm 2.1 2.2 2.3 2.4 2.5 2.6 -1 1000/T (K ) 0 Tg - T g(bulk) (K) -5 -10 -15 DSC, M w = 1400 k, freestanding -20 DSC, M w = 550 k, freestanding CD, M w = 1400 k, freestanding -25 CD, M w = 1400 k, Al-capped -30 1 10 10 2 10 3 10 4 10 5 h (nm) Fig. 7 Upper panel Temperature dependence of the molecular relaxation time of thin PS films in different confinement conditions; Lower panel Thickness dependence of Tg for the same systems of the upper panel (Reprinted with permission from Ref. [16]) confining geometries, including thin polymer films [10], polymer nanocomposites [11, 13, 15, 22] and nanospheres [53, 131]. In all cases, accelerated equilibrium recovery is found despite the unaltered molecular dynamics. To understand these results, one way could be that the former is generally measured at temperatures somewhat larger than those relevant for the determination of the Tg and the recovery of equilibrium. Hence, one could hypothesize that, at lower temperatures, a drastic variation of the temperature dependence of the typical relaxation time occurs. However, there exists a number of studies which appear to contradict such scenario. First of all, the molecular dynamics of freestanding thin PS films have been shown to possess pronounced bulk-like dynamics even for temperatures of the order of the calorimetric Tg [93]. Furthermore, the molecular relaxation time monitored during the course of aging in poly(vinyl acetate) (PVAc)/silica nanocomposites 352 D. Cangialosi has been shown to increase more rapidly in systems with larger nanofillers content. This implies that the instantaneous relaxation time, that is, the one at a given aging time, is larger in nanocomposites than in the pure polymer despite the faster evolution toward equilibrium [11]. Molecular dynamics simulations on a coarse-grained polymer nanocomposites show a significant acceleration of physical aging close to the polymer/filler interface accompanied by a slight speed-up of the molecular mobility seemingly insufficient to justify the increase of the aging rate [71]. Finally, some crucial experimental observations pointing toward the infeasibility of arguments exclusively based on the molecular mobility to describe the out-of-equilibrium dynamics are those reporting acceleration of physical aging in thin films [10, 81, 85, 95, 108] and polymer nanocomposites [9, 11, 13, 15, 22], with typical confinement length scale of the order of microns. Whatever the approach employed for the description of glassy dynamics, it is unphysical to attribute the observed acceleration to a modification of the rate of spontaneous fluctuation for systems exhibiting confinement length scale in the microns range. According to the previous observations, it is possible to conclude that, differently from bulk glass formers [34, 55, 66, 125], the rate of spontaneous fluctuations and the out-of-equilibrium dynamics, that is that monitored after the application of a perturbation in the nonlinear regime are decoupled [16, 23]. In other words, the latter aspect of glassy dynamics in confinement is not exclusively determined by the molecular mobility and some confinement-specific features must be included to fully account for it. 5 Theoretical Frameworks In the previous section of the chapter, the need to seek for an explanation to the peculiarities of glassy dynamics in confinement has been evidenced. In particular, in confinement the out-of-equilibrium dynamics, which could be expressed by an equilibration time τeq , must be connected to the molecular relaxation time τ plus an additional factor. This must depend exclusively on a confinement length. In particular, one can express the equilibration time as a function of the molecular relaxation time and a function only depending on such length: τeq = g(h)τ . Here h is the con f inement length scale, which, as will be described, in the most trivial case of freestanding films is the film thickness. Given this very general approach to the problem, the main challenge is the search for the physics behind the function g(h). In other words, whatever the approach employed to describe glassy dynamics in confinement, this must account for the experimental evidence that the rate of spontaneous fluctuations is not one-to-one related to the out-of-equilibrium dynamics. Among the numerous theoretical approaches, most of them describe the Tg depression in terms of altered intrinsic molecular mobility [50, 70]. Hence, they do not account for the entire phenomenology of the glass transition in confinement. Others, such as those based on percolation arguments [69, 74], also rely on the effect of altered molecular mobility on the Tg . However, this kind of approach is based on Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 353 the change of dimensionality of percolation under confinement, a purely geometric argument that could in principle be adapted to catch experimental observations. The free volume hole diffusion (FVHD) model potentially represents a suitable candidate for the description of glassy dynamics in confinement. This model, rather than describing the different aspects of the relaxation in glass forming liquids, exclusively attempts to describe the way a glass recovers equilibrium in the physical aging regime (or try to maintain it on cooling). Alfrey et al. [2], who first tried to develop this idea, proposed that diffusion of free volume holes and their annihilation at the external surface of the sample could be responsible for the rapidity of achievement of equilibrium in the out-of-equilibrium glass. An important interference to the development of such idea is due to Braun and Kovacs [66]. They compared the volume recovery of milled PS, with typical size of several microns, with that of the bulk polymer and found no differences, at odds with the qualitative scenario expected from the FVHD model. To overcome the apparent size-independent evolution of out-ofequilibrium glasses, Curro et al. [28] assumed the presence of an internal length scale, where free volume holes can annihilate. In doing so, they could describe several aspects of PVAc volume recovery. More recently, the model has been revitalized [21, 26, 82, 118, 119] after the finding of accelerated physical aging in numerous nanostructured glasses with typical confinement length scale of several microns or shorter [9–11, 13, 15, 22, 81, 85, 95, 108]. The basic equations to apply the model are: (i) the second equation of Fick: ∂ f v (r, t) = ∇(D∇ f v (r, t)) ∂t (1) where f v is the fractional free volume and D is the diffusion coefficient of free volume holes; and (ii) and the equation expressing the mean square displacement (MSD) (x 2 ) as a function of time for one-dimensional linear diffusion1 : x 2  = 2Dt (2) The former equation can be applied to fit the evolution of magnitudes related to the free volume holes fraction during physical aging. The description of the Tg depression can be performed employing Eq. (2). This because, within the FVHD model, the glass transition occurs when x 2  in the observation time t ∼ q −1 , where q is the cooling rate of the experiment, is of the order of the square of half the film thickness: (h/2)2 = x 2  = 4Dq −1 . A crucial point of the model is that, to describe accelerated physical aging and Tg depression, it does not requires any acceleration of the glass former molecular mobility, as suggested by several experiments. The lines in Figs. 1 and 2 are the fits of the FVHD model to Tg data of thin PS films supported [12] and freestanding [10, 12], respectively. Interestingly, the FVHD model is able to account for both the thickness and cooling rate depen1 The assumption of one-dimensional confinement is obviously true is thin films. For polymer nanocomposites and nanospheres it is approximately valid if the radius of curvature of nanoparticles and nanospheres, respectively, is considerably larger than the size of free volume holes. 354 D. Cangialosi dence of the Tg . In particular, the weak thickness dependence at high cooling rates [36, 38] is due to the VFT behavior of D in the temperature range relevant for such rates [10, 12, 15]. Conversely, at lower temperatures, in particular those relevant for determinations of the Tg at relatively low cooling rates, the diffusion coefficient exhibits weak Arrhenius temperature dependence [10, 12, 15]. This gives rise to large variations of the Tg with the film thickness. Successful fits of the model to the physical aging of freestanding thin PS films [10, 82, 118, 119] and several polymer nanocomposites [13, 15, 22] were also achieved, as shown by the lines in Fig. 2 for the former systems as a showcase. With regard to supported or capped films, it is worth pointing out that only a portion of the polymer surface is available for elimination of free volume holes. This is due to irreversible chain adsorption [87, 89, 106, 129]. In this case, adsorbed chains constitute an infinitely high potential energy barrier to be overcome by free volume holes. Hence, these will be eliminated only at the free interface. A recent experimental study actually showed that the magnitude of Tg depression in capped thin PS films scales with the amount of free interface, in qualitative agreement with the FVHD model. Subsequently, such agreement was quantitatively tested and confirmed [86]. As such, the FVHD model, in combination with the experimental evidence reporting on chain adsorption at the interface, is able to account for the relatively limited Tg depression observed in supported and capped films in comparison to freestanding films. In relation to the presence of some energetic barrier at the interface, the FVHD model may provide an explanation to the contrasting results obtained for the deviation from bulk behavior of the Tg of polymer nanospheres exhibiting different kind of surfactants (if any) [40] or silica capped [132] at the polymer interface. Further investigation is required in this sense. 6 Conclusions and Perspectives This chapter has emphasized the recent advancements in the understanding of the overall phenomenology of glassy dynamics in confinement. In view of recent results, it has been shown that, in the search for an explanation for the physics behind the observed Tg depression and accelerated recovery of equilibrium in confinement, arguments exclusively based on the alteration of the intrinsic molecular mobility are not sufficient. Therefore, any theoretical effort to describe glassy dynamics in confinement must account for this fact. In this context, the FVHD model potentially constitutes a suitable candidate to describe the physics of Tg depression and accelerated recovery of equilibrium, without need to invoke any change in the intrinsic molecular mobility. Obviously, such model needs to be tested for a variety of confinement configurations. Apart from the search for a suitable framework to describe experimental results of glassy dynamics in confinement, the ability of nanostructured glasses to maintain equilibrium at temperatures lower than for bulk glass formers may open the perspective to so-far unexplored temperature ranges. This in view of the fact that in Equilibrium and Out-of-Equilibrium Dynamics in Confined Polymers 355 confinement the general features of the intrinsic molecular mobility, as discussed in this chapter, as well as the thermodynamics (at least for freestanding films thicker than 30 nm) [127] are essentially unaltered in comparison to the corresponding bulk glass formers. This implies that it is in principle possible to achieve information on the properties of equilibrium glasses down in the energy landscape. Within the context of the dynamics and thermodynamics of glass forming liquids, this implies that insight on the alleged divergence of the relaxation time and the vanishing of the configurational entropy at a finite temperature, that is, the Vogel and the Kauzmann temperatures respectively, can be obtained. Apart from this, a recent study on the enthalpy recovery of several polymer glasses in bulk over prolonged aging times (more than 1 year) revealed a complex scenario of both dynamics and thermodynamics [20]. In particular, it was found that recovery of equilibrium occurs in two stages, with partial and complete enthalpy recovery. 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