Effect of nanostructure on the thermal glass transition and physical aging in polymer materials

Progress in Polymer Science 54–55 (2016) 128–147

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Progress in Polymer Science journal homepage: www.elsevier.com/locate/ppolysci

Effect of nanostructure on the thermal glass transition and physical aging in polymer materials Daniele Cangialosi a , Angel Alegría a,c , Juan Colmenero a,b,c,∗ a b c

Centro de Física de Materiales (CSIC-UPV/EHU), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain Departamento de Física de Materiales, Universidad del País Vasco (UPV/EHU), Apartado 1072, 20080 San Sebastián, Spain

a r t i c l e

i n f o

Article history: Available online 23 November 2015 Keywords: Glass transition Segmental mobility Enthalpy relaxation Physical aging Confinement Diffusion model

a b s t r a c t We review the recent activity aiming to clarify glassy dynamics in nanostructured polymer glasses, in particular thin films, nanocomposites and nanospheres. Special emphasis is devoted to recent results on the out-of-equilibrium dynamics, that is, the way the system leaves equilibrium when cooling, marking the thermal glass transition, or recovers it once in the glassy state – the so-called physical aging. Apart from those systems exhibiting strong interactions at the interface, we show that a huge number of studies probing glassy dynamics in nanostructured glasses finds negative deviations from bulk Tg and accelerated physical aging. Analysis of the dependence of the rate of spontaneous fluctuations – namely the linear dynamics – on nanostructuring indicates that there exists a significant component exhibiting bulk-like dynamics. This is the case even in the most extreme case of nanostructuring, that is, semi-isolated polymer chains and freestanding thin film with thickness ∼10 nm. In the latter case this is found at temperatures around the bulk glass transition temperature (Tg ), that is, in the range where deviations of the out-of-equilibrium dynamics are normally observed. All together these results indicate that the linear dynamics alone cannot provide an exhaustive description of the out-of-equilibrium dynamics in nanostructured systems. In this case, purely geometric factors must be included. We discuss recent approaches aiming to capture the phenomenology of glassy dynamics in nanostructured glasses. Special attention is dedicated to the free volume hole diffusion (FVHD) model. © 2015 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Phenomenology of the glass transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.1. Equilibrium dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.2. Out-of-equilibrium dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.1. Glass transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.1.1. Thin polymer films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.1.2. Polymer nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.1.3. Polymer nanospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

∗ Corresponding author at: Centro de Física de Materiales (CSIC-UPV/EHU), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain. E-mail address: [email protected] (J. Colmenero). http://dx.doi.org/10.1016/j.progpolymsci.2015.10.005 0079-6700/© 2015 Elsevier Ltd. All rights reserved.

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3.2.

4. 5. 6. 7.

Physical aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.2.1. Thin polymer films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.2.2. Polymer nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.2.3. Polymer nanospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Connection to polymer segmental dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Factors controlling the out-of-equilibrium dynamics in nanostructured glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

1. Introduction Recent developments in modern technology have motivated increasing interest in the study of amorphous polymers with typical dimensions in the nanometers range. These nanostructured systems can be employed in a wide range of technologically relevant applications. For instance, submicron thick polymer films are exploited as templates in microelectronics [1], non-biofouling protective coatings [2], membranes in separation technologies [3], active layers in photovoltaic cells [4]. Polymer nanocomposites are extensively employed in tire technologies and in those application where mechanical reinforcement, enhancement of barrier properties, flame resistance, electro-optical, and bactericidal properties, are required [5]. Polymer nanospheres have received increasing interest in the last years in applications such as vehicles in drug delivery [6], components in fluorescent imaging [7], performance reinforcing additives [8], and components in photonic structures [9]. Beside the technological interest of nanostructured polymers, the reduction of the typical size of the material has raised numerous fundamental concerns regarding the physics behind the alteration of properties at the nanoscale. Among the topics currently subject of investigation, the alteration of glassy dynamics in nanostructured glasses has been one of the most intensely debated since the finding in the early 90s of depressed glass transition temperature (Tg ) in thin polymer films [10,11]. Since then, a huge number of studies regarding glassy dynamics in nanostructured polymers have been reported. After a brief introduction to the main aspects of the glass transition, we focus on the recent activity on the glassy dynamics in the non-linear regime of nano-structured polymers, that is, the efficiency of maintaining equilibrium when cooled down from the melt (providing the Tg ) or recovering it once in the glassy state. In doing so, we critically review those studies where the effect of nanostructure on the Tg and the recovery of equilibrium in the out-of-equilibrium glass is investigated. We show that, in these systems, the non-linear dynamics may not be exclusively related to the rate of spontaneous fluctuations, that is, to the glassy dynamics in the linear regime. In view of this finding, we emphasize the role of the typical length scale of nanostructuring, an argument based on geometric aspects, in determining the efficiency of equilibration in the glassy state. Finally, we recall the theoretical activity aiming to elucidate the connection

of the non-linear to the linear dynamics via geometric arguments. 2. Phenomenology of the glass transition 2.1. Equilibrium dynamics Liquids differ from crystalline solids not only for the absence of structural order, but also because they exhibit molecular motion beyond atomic vibrations. The time scale at which such motion occurs provides the rate of spontaneous thermal fluctuations in the liquid. The temperature dependence of such fluctuations is Arrhenius-like for standard liquids. However, for those liquids that can be supercooled below their melting temperature, the socalled glass-forming liquids, such temperature dependence drastically becomes more pronounced. This is shown in Fig. 1 (left panel) where the typical time scale of spontaneous fluctuations (), represented as the frequency corresponding to the most probable rate in experiments based on oscillatory fields (ωmax =  −1 ), is shown for polystyrene (PS) [12]. Within more or less limited temperature intervals, such strong temperature dependence is often described by the Vogel–Fulcher–Tammann (VFT) equation [13–15]: ωmax = ω0 exp[−B/(T − T0 )]

(1)

with ω0 , B and T0 the pre-exponential factor, the Vogel activation parameter and the Vogel temperature respectively. As can been seen in Fig. 1 (continuous line, left axis), the VFT equation provides a suitable description of experimental data. In the case of polymers, the process exhibiting VFT behavior is called the segmental or the ˛ relaxation. In the rest of the Review, we will refer indifferently to rate of spontaneous fluctuations, linear dynamics or segmental (or ˛) relaxation. This behavior is believed to be associated to the cooperative rearrangement of several structural units. Numerous attempts to estimate the length associated to such rearrangement have been presented since this idea was first introduced by Adam and Gibbs (AG) [16]. Their theory relies on the connection of dynamics of glass forming liquids to their thermodynamics via the configurational entropy (Sc ). According to the theory the size (z) of the region involved in glassy dynamics increases with decreasing temperature following the relation: z ∼ S−1 . Since the AG theory numerous approaches have

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Fig. 1. Relaxation time of spontaneous fluctuations as a function of the inverse temperature (left ordinate axis and small symbols) and cooling rate as a function of the inverse Tg (right ordinate axis and large symbols). The lines are the fits to the VFT equation. Reproduced with permission from [12], Copyright 2014, the American Institute of Physics.

been presented providing a typical length scale of glassy dynamics. Some of them, in ways analogous to the AG approach, are based on thermodynamics. Among them we recall the random first order transition (RFOT) theory [17] and the approach of Donth [18]. Other approaches are those based on string-like motion [19], the four point dynamic susceptibility [20] and the self-concentration [21,22]. In all cases, estimations of the typical length scale provided values of the order of 1–3 nanometers at Tg . In the context of the present Review, the idea of a typical length underlying glassy dynamics has constituted a formidable encouragement to study glassy dynamics in nanostructured systems. If a length scale really exists, once the size of the sample becomes of the order of such length scale, effects on the glassy dynamics should be observed. An important aspect of the dynamics of glass forming liquids is the nonexponential nature of its relaxation function. In particular, the time decay function (ϕ(t)) associated to spontaneous fluctuations follows the so-called Kohlrausch–Williams–Watts (KWW) equation [23,24]:



(t) = exp −

t 

ˇKWW (2)

Apart from the fluctuations related to the segmental relaxation, polymers exhibit a variety of specific relaxation processes [25,26]. This, among glass-forming liquids, makes them especially complex systems. The motion of the overall polymer chain is one of them though is not directly relevant to glassy dynamics. Conversely, as will be discussed later, chain dynamics is, for instance, of utmost importance in determining the degree of adsorption on a substrate of confined polymers, a factor with deep influence on the glass transition in nanostructured glasses [27]. Finally, similarly to other glass-forming liquids, amorphous polymers exhibit localized motion generally faster than the segmental relaxation [28]. In polymers, these are generally related to some internal degrees of freedom [25]. To close this section, it is important to emphasize that determining experimentally the characteristics of the rate of spontaneous fluctuations imposes the application of linear perturbations, that is, those exhibiting amplitude smaller than those corresponding to spontaneous

fluctuations. In particular, the experimental procedure must fulfill the fluctuation dissipation theorem (FDT) [29,30]. 2.2. Out-of-equilibrium dynamics In experiments where the temperature is decreased at a finite rate, there exists a temperature, that is, the experimental Tg , below which the supercooled liquid transforms into a glass. Apart from the dramatic effects on the properties of the glass former, the transformation into a glass implies a gradual step-like change in thermodynamic coefficients (specific heat, coefficient of thermal expansion, compressibility etc.) [31]. Though this kind of behavior is reminiscent of an Ehrenfest second order transition, the glass transition cannot be classified as a thermodynamic transition. First of all it does not fulfill the criterion of the Prigogine–Defay ratio [32]: (VT)−1 Cp k/˛2 = 1. Furthermore, different from thermodynamic transitions, the Tg depends on the applied cooling rate. In particular, the Tg increases with the cooling rate (ˇ), as shown in Fig. 1 (right axis) [12]. The explanation to this phenomenology is that the glass transition is a kinetic event mediated by the molecular mobility. In particular, the transformation of the supercooled liquid into a glass occurs when the typical time scale of spontaneous fluctuations is such that – in the time scale of the experiment, proportional to the reciprocal of the applied cooling rate – equilibrium cannot be maintained and a glass is formed. As a proof of the intimate link between the molecular mobility and the glass transition, the cooling rate dependence of the Tg can also be described by the VFT equation: ˇ = ˇ0 exp[−B/(Tg − T0 )]

(3)

In the wide range of temperature shown in Fig. 1 and for bulk glass formers, the parameters B and T0 able to fit the Tg dependence on ˇ are the same as those describing the temperature dependence of the time scale of spontaneous fluctuations. This can be seen in Fig. 1 and the connection between these two aspects of glassy dynamics in bulk glass formers has been investigated in detail in several studies [12,33,34].

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Another manifestation of the kinetic nature of the glass transition is the spontaneous evolution toward the relative thermodynamic equilibrium represented by the supercooled melt. This process, known as physical aging, is ubiquitous in glasses due to their intrinsic out-ofequilibrium nature. Similarly to the glass transition, an intimate link between the rate of recovery of equilibrium and that of spontaneous fluctuations has been established [35–38]. In particular, it has been shown that recovery of equilibrium can be described accounting for the nonexponential nature of glassy dynamics (see Eq. (2)). The stretching exponent needed to describe recovery of equilibrium data is generally identical to that fitting the decay function of spontaneous fluctuations. Once the non-linear behavior of the equilibrium recovery – arising from the aging time dependence of the typical relaxation time of spontaneous fluctuations () – is incorporated into the non-exponential behavior of glassy dynamics a satisfactory description of aging data is achieved, at least not too far from Tg . This has been shown in the past by several approaches [39–44] where the non-exponential and nonlinear nature of equilibrium recovery in the physical aging regime are variously described. For the interested reader, several monographs on the phenomenology of physical aging can be consulted [35,36,45–48]. Despite the numerous evidences showing a one-to-one connection between the equilibrium, in terms of the rate of spontaneous fluctuations, and the out-of-equilibrium dynamics, that is the Tg and the physical aging, it must be remarked that the two aspects are conceptually different. The rate of spontaneous fluctuations is an intrinsic property of the glass former. As such, in the experimental practice it can only be determined performing measurements in the linear regime, that is, fulfilling the FDT [29,30], as discussed in the previous section. In experiments probing the rate of spontaneous fluctuations, the so-called linear structural relaxation is monitored. Conversely, measurements based on the application of a cooling ramp to determine the Tg or those where the (non-linear) structural recovery of equilibrium is monitored over a given time scale, are based on the application of perturbations beyond the linear regime. This conceptual difference implies that the connection between the equilibrium and out-of-equilibrium dynamics not necessarily must hold in all glass-forming systems. Within the context of the present Review, we emphasize how numerous experimental evidences show that, in nanostructured glasses and under certain conditions, arguments based on the rate of spontaneous fluctuations are not sufficient to capture the overall phenomenology of out-of-equilibrium glassy dynamics. In this case, apart from the rate of spontaneous fluctuations, purely geometric factors determine the magnitude of deviation of the Tg and the rate of recovery of equilibrium from bulk behavior. 3. Experimental results Investigations on glassy dynamics in nanostructured polymer glasses include a huge number of experimental results. In this section of the Review, rather than mentioning all of them, we recall those studies that, according to us, have constituted major advancements

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in the understanding of glassy dynamics in confinement. In doing so, we distinguish among the different kind of nanostructured polymer glasses ranging from polymer thin films, nanocomposites and nanospheres. For each kind of nanostructured system, we will emphasize their peculiarities and the analogy among each other. 3.1. Glass transition 3.1.1. Thin polymer films Among nanostructured polymer glasses, thin films are certainly the most investigated systems. Research activity in this topic has been promoted by the seminal work of Keddie et al. [11]. They performed direct measurements of the Tg on thin PS films supported on silicon wafers by means of ellipsometry, a technique delivering the temperature dependent thickness of films. The main outcome of the study was that the Tg deviates from the bulk values for films thinner than 30–40 nm. Since then a huge number of experiments was performed on a variety of configurations, including supported, capped and freestanding films. Supported and, to a less extent, capped thin films are by far the most investigated film configurations and PS is the most explored polymer. These two film configurations have in common the presence of a solid substrate at least on one interface. The vast majority of experiments in these systems show a Tg depression in comparison to the bulk. Some years after the seminal study of Keddie et al. [11], the numerous results on the Tg of thin PS films were summarized in a review [49]. These results showed a clear trend toward Tg depression, showing up at thicknesses below 40 nm and being as large as 30–40 K below 10 nm. However, the scattering of such results was so large that, at that time, it was already evident that other factors, beyond the film thickness, were playing a role in determining the magnitude of Tg depression. Among them, the role of the applied cooling rate was unequivocally shown to be a crucial factor. Studies in this sense were promoted by Efremov et al. [50], performing specific heat measurements on thin PS films supported on platinum at cooling rates as large as thousands of Ks−1 . Surprisingly, in that work no discernible deviations of the Tg from bulk behavior were found. In subsequent studies, Fakhraai and Forrest [51], and Glor and Fakhraai [52] explored the cooling rate dependence of thin PS films by means of ellipsometry. They found that, for all thicknesses, the Tg deviation from bulk behavior systematically decreases with the cooling rate increase. This is shown in Fig. 2 where the cooling rate is presented as a function of Tg −1 (data taken from Ref. [51] and rearranged in Ref. [53]). Similar results were obtained by Gao et al. [54] and Shamim et al. [55] for thin PS and PC films, respectively. In this case, deviations from bulk Tg were more pronounced when thin films were deposited on a Krytox oil, whereas films deposited directly on the chip for FSC exhibited deviations from bulk Tg only below 30 nm. Fakhraai and Forrest [51], Glor and Fakhraai [52], Gao et al. [54] and Shamim et al. [55] interpreted their data showing Tg depression as resulting from enhanced segmental dynamics of the films in comparison to bulk PS. However, as will be described in detail in a subsequent section, alternative explanations based on non-relaxation arguments can be provided.

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Fig. 2. 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 free volume holes diffusion model (FVHD) model to experimental data. Reproduced with permission from [53], Copyright 2014, Elsevier Ltd.

21 nm PS97 22 nm PS160 35 nm PS97 44 nm PS97 300 nmPS97

370

Tg [K]

Within the context of Tg deviations in supported and capped polymers the kind of interface is certainly another important factor. In this sense, numerous studies emphasize the importance of the interfacial energy. This was first shown by Keddie et al. [56]. They studied the Tg behavior of thin poly(mehtyl methacrylate) (PMMA) films supported on gold and native silicon oxide wafers by ellipsometry. For PMMA supported on the former substrate a reduction in Tg was found. Conversely, thin PMMA films supported on native silicon oxide wafers exhibited enhanced Tg in comparison to bulk PMMA. In this case it was speculated that the increased interfacial energy due to hydrogen bonding of PMMA with silicone oxide was responsible for the observed Tg increase. The importance of the substrate was later shown by van Zanten et al. [57] for thin PS films supported on hydrogen-terminated silicon substrates. In contrast to the vast majority of experiments on PS, they found positive Tg deviations. More recent studies systematically investigated how the interface can be opportunely tuned to produce increase, decreases or no change in Tg in both experiments [58–64] and simulations [65,66]. Beyond the role of interfacial energy in determining the magnitude of Tg deviations from bulk, recently it has been shown how such deviations can be varied by modifying the degree of adsorption [67] of thin polymer films at the interface with their support [27,68–70]. This depends on the kind of substrate and the annealing time well above the Tg . In Fig. 3 the former effect is emphasized. In particular, the Tg as a function the annealing time at 453 K, that is at ∼ Tg (bulk) + 80 K, is shown for thin PS films capped between aluminum. Here, the Tg is measured by capacitive dilatometry, a technique sensitive to the material density [71,72]. As can be observed, the Tg is depressed in comparison to the bulk before annealing. However, a progressive Tg increase occurs during the course of annealing and the bulk value is achieved in some cases. The bulk Tg recovery occurs faster for lower molecular weights. This suggests that chain adsorption, driven by the overall chain dynamics, is intimately associated with the Tg variation with annealing

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. Adapted with permission from [27], Copyright 2011, the Nature Group.

time. That irreversible chain adsorption occurs during annealing above well Tg was corroborated showing that there exists a layer, the so-called Guiselin brush [73], at the interface that cannot be removed even employing good solvents [27]. Furthermore, Napolitano et al. [68] showed how the essential parameter determining the distance from bulk Tg was the amount of free interface. In Section 6, we will show how such experimental evidence can be adequately represented by the theoretical framework based on free volume holes diffusion. Additional parameters – which have been shown to play a role in determining Tg deviations in supported thin polymer films – are the chain architecture and the presence of mechanical stress induced by spin coating and the unavoidable cooling ramp employed to obtain the Tg itself. Regarding the former, it was shown that thin films of star

D. Cangialosi et al. / Progress in Polymer Science 54–55 (2016) 128–147

Fig. 4. Tg as a function of films thickness and applied cooling rate (qc ) for freestanding-like 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. [86]. Reproduced with permission from [86], Copyright 2012, the American Chemical Society.

polymers exhibit suppression, enhancement or no change in Tg , depending on the number or arms and their molecular weight [74]. Furthermore, oligomer and dendrimers exhibited larger deviations than linear high molecular weight thin PS films [75]. Regarding the role of residual stress due to rapid solvent evaporation in spin coated samples, this have been emphasized in the past [76]. However, quantifying how this effect can influence the Tg is currently matter of debate [77]. A very recent study suggests that the release of stress in the melt state does not substantially affect the Tg of thin PS films [78]. Freestanding thin polymer films have been somewhat less investigated than supported or capped films. Nevertheless, there exists a considerable number of studies in this configuration [79–86]. The main outcome of these studies is that freestanding polymer films exhibit large Tg depression, if compared to supported and capped. Such depression can be as large as 70 K for ∼ 30 nm thick PS films, as shown in the seminal study of Forrest et al. [79]. Furthermore effects can be detected for films larger than 100 nm [86]. In this case, the absence of any interfacial interaction, including chain adsorption with a substrate, makes the determination of the Tg less amenable to the scattering observed in supported and capped films. In the case of freestanding films, the magnitude of deviations depends on the applied cooling rate as shown by Boucher et al. [86]. In that study, it was found that lower cooling rates give rise to more pronounced Tg depression. This is shown in Fig. 4 where the Tg of freestanding thin PS films is shown as a function of thickness at two cooling rates. As can be seen, the effect of reducing the film thickness on the Tg is considerably more pronounced when a cooling rate of 0.2 K min−1 is applied. 3.1.2. Polymer nanocomposites The Tg of polymer nanocomposites has been investigated in recent years for a variety of polymer/nanofiller couples. The results are qualitatively analogous to those of supported and capped films [87]. However, two main dif-

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ferences need to be emphasized: (i) the magnitude of Tg deviations from bulk behavior are generally milder than those of thin films; (ii) for small nanofiller sizes (smaller then ∼ 10 nm), their radius of curvature approaches the molecular dimensions. The latter argument implies that, for nanocomposites with small nanofiller size, the dimensionality of nanostructuring is different from that of thin films [88]. However, in analogy to supported and capped films, the Tg of polymer nanocomposites can exhibit positive or negative deviations as well as no difference with the bulk counterpart. Tg increase has been found for polymer/nanofillers with strong interfacial interactions. In several cases this is associated to the presence of hydrogen bonding at the interface. Lee and Lichtenhan [89] observed an 8 K increase in Tg in epoxy/POSS nanocomposites with 10 wt.% POSS. Similar results were obtained by Lu and Nutt [90] investigating epoxy nanocomposites with montmorillonite (MMT). They found a 7 K increase in Tg for the nanocomposite with the largest filler content (10 wt.%). By means of fluorescent spectroscopy, Torkelson and co-workers [91,92] found a considerable Tg increase for poly(2-vinylpyridine) (P2VP) in nanocomposites with alumina and silica; 17 and 10 K respectively. Similar results were obtained by Rittigstein et al. [92], and Priestley et al. [93] in PMMA nanocomposites with silica (diameter of the nanofiller d = 10–15 nm), though with a more moderate increase in Tg with respect to the previously mentioned systems (5 K for the nanocomposites with the largest nanofiller content). Poly(dimethylsiloxane)/silica nanocomposites were shown to exhibit Tg increase of 2–3 K in comparison to the bulk polymer [94]. Several examples of polymer nanocomposites with single-walled carbon nanotubes (SWCNT) can be found, where Tg increase is observed. Among them, those with PS [95], PMMA [96] and poly(llactide) [97] all exhibit increase in Tg . Finally a slight Tg increase (if any), never exceeding 2 K, was found for nanocomposites of styrene-butadiene rubber (SBR) with alumina and silica [98,201]. Several examples exist in the literature where Tg is unaltered or depressed in comparison to the bulk polymer. This can be found in systems very similar to those where a Tg increase is observed. This is the case of PMMA/silica nanocomposites where several studies indicate no change or decrease, for low [99,100] and high [101] area of nanofiller to volume of polymer ratios respectively. This appears to be in contradiction with the results of Priestley et al. [93] that found a Tg increase instead for the same nanocomposites. The main difference between the two sets of studies is that in Refs. [99–101] silanized silica nanoparticles are employed, whereas Priestley et al. [93] produced PMMA nanocomposites with untreated silica. As a result, hydrogen bonding between PMMA and the silica nanoparticles is of prominent importance in the latter study. Conversely, the presence of silanized silica significantly reduces the interfacial interaction. No detectable change in Tg was found for other nanocomposites. Poly(vinyl acetate)/silica [102], poly(ether imide)/carbon fiber nanocomposites [103] belong to this category. Among nanocomposites exhibiting depressed Tg in comparison to the bulk, apart from PMMA/(silanized) silica nanocom-

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posites [99–101], those of PS [101] and PVAc [104] with (silanized) silica, those of PS with gold [105–107] and silver [108], and those of poly(dimethyl siloxane) (PDMS) with silica and silica/titania [109] can be recalled. In all cases the decrease in Tg never exceeds 10 K. Apart from those studying where monotonous deviations from bulk behavior are observed, there exist at least two studies where the Tg decreases for low filler content and increases or remain constant above a certain threshold. This is the case for PS in nanocomposites with silver [108] and silica [101], respectively. As mentioned at the beginning of the section, Tg deviations from bulk behavior are milder than those observed in thin polymer films (never larger than 15–20 K). This, apart from the ubiquitous presence of a substrate (the nanofiller) at the interface, is likely due to the unlikeness of achieving area of interface to volume of polymer ratios as large as those of nanometric films, due to the tendency of aggregation of nanoparticles at large filler contents. 3.1.3. Polymer nanospheres The first study on the glass transition of polymer nanospheres dates back the early 80s. Employing differential scanning calorimetry (DSC), Gaur and Wunderlich [110] found no discernible Tg deviations from the bulk on PS nanospheres with diameters as small as 85 nm. However, in comparison to bulk PS, they observed significantly different traces of the specific heat as a function of temperature with highly broadened transition from melt to glass and decreased specific heat jump, that is, the difference between the melt and glass specific heats. Similar results were found by Ming et al. [111], and Sasaki et al. [112] by DSC in PS nanospheres in aqueous suspension. More recent studies showed apparently contrasting results on the glass transition behavior of PS nanospheres exposed to air or in general to a fluid with weak interfacial interactions [113,114]. Here the Tg was shown to exhibit pronounced negative deviations from bulk behavior. These were as large as 50 K for the smallest nanospheres (∼ 100 nm) and visible for nanospheres with diameter of several hundreds nanometers. These results were proved by two different techniques, that is, DSC and CD delivering the specific heat and volume respectively and are shown in Fig. 5. Here pronounced Tg depression was visible at diameters as large as 380 nm and, for the lowest diameter (∼ 100 nm), a reduction of 60 K was observed. However, in Ref. [113], it was also shown that, once PS nanospheres are covered by a silica shell, no detectable deviations from bulk behavior exist. The importance of the nature of the interface around polymer nanospheres was highlighted by Feng et al. [115,116]. They showed how the Tg of PS [115] and PMMA [116] in aqueous dispersion can be tuned depending on the kind of surfactant (if any) at the polymer/water interface. In particular, surfactant free PS nanospheres, and to less extent those covered by a nonionic surfactant, exhibit significant Tg depression. This can be as large as 70 K for surfactant free nanospheres with diameter ∼ 40 nm. Conversely, for PS nanospheres covered by a nonionic surfactant no dependence of the Tg on the diameter was observed with values comparable to those of bulk PS. A similar behavior

Fig. 5. Tg as a function of diameter measured by CD (red triangles) and DSC (blue circles, green diamonds are for the “dynamic” Tg measured by BDS). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Reproduced with permission from [114], Copyright 2013, John Wiley & Sons Inc.

was found for PMMA nanospheres, although Tg depression was found in the case of nonionic surfactant, whereas those nanospheres recovered with anaionic surfactant exhibited bulk-like Tg . Pronounced Tg depression was found f or surfactant free PMMA nanospheres, though the extent of such reduction was somewhat lower than for PS. Beside those studies showing negative Tg deviations from bulk behavior, it is worth mentioning that there exists examples of Tg increase. This has been shown for poly(ethyl acrylate) (PEMA) nanospheres [117,118]. This result has been found even for surfactant free nanospheres, a result that seems to be at odds with those on surfactant free PS [115] and PMMA [116] with similar diameters. 3.2. Physical aging Equilibrium recovery during the course of physical aging after cooling from the melt into the glassy state is a phenomenon intimately related to the glass transition. In particular, the Tg is by definition the temperature below which the glass former enters the physical aging regime. As such, results on physical aging must be interpreted in connection to the glass transition behavior. In particular, the distance from the Tg of the sample determines the driving force of physical aging, i.e. the distance of a thermodynamic magnitude from its equilibrium. Hence, a preliminary characterization of the Tg of the nanostructured glass is propaedeutic for an accurate understanding of the aging behavior. In reviewing the available literature on the physical aging of nanostructured polymers, we can distinguish among those studies reporting an increase, decrease or no change of the polymer Tg . Before reviewing the recent activity on the physical aging in nanostructured materials, it is important to point out that the rate of physical aging can be defined in different manners. In particular, some authors employ a definition based on the slope of a portion of the recovery function. Others define the aging rate as the time to reach the plateau in this function, that is, the equilibration time. Thus, the former definition applies to those aging experiments where

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only a portion of the recovery function is monitored. In this case the aging rate is defined as the slope of that portion of the recovery function [92,119,120]. Conversely, when the entire recovery function is characterized [100,121,122], the often employed definition of the rapidity of physical aging relies on the time of equilibration. 3.2.1. Thin polymer films In comparison to Tg determinations, there exists considerably less reports where the physical aging of thin polymer films is studied. The first investigations in this sense involved polymer membranes with thickness in the micrometer range. The employment of these systems in gas separation made this kind of study appealing from a technological point of view. Due to the way membranes are employed, an important detail of these investigations is that physical aging was always monitored in films in the freestanding configuration or freestanding-like, that is, deposited on a non-adsorbing substrate. The first report in the topic was that of Pfromm and Koros [123], who monitored the evolution of gas oxygen and nitrogen permeabilities in glassy polysulfone (PSF) and polyimide (PI). They showed that the decrease of gas permeability resulting from the densification occurring during equilibrium recovery was proceeding significantly faster in 0.5 ␮m thin films. Since that publication the intense activity showed that acceleration of physical aging is a common feature of freestanding films of polymer membranes [122,124–127]. In several cases such acceleration is visible at thicknesses larger than microns. For thickness below 100 nm, the physical aging process was so rapid that when data collection began after cooling below the Tg , the gas permeability had already decreased significantly and the evolution toward equilibrium appeared to be slowed down. This is shown in Fig. 6, where the evolution of the oxygen permeability of PSF films was monitored during the course of aging [122]. An attempt to quantitatively interpret these results was carried out by McCaig, Paul and Barlow [128] within the approach based on free volume holes diffusion. This will be detailed in Section 6. Consistent with this interpretation, in this context a recent study by Rowe et al. [129], by means of variable energy positron annihilation lifetime spectroscopy (PALS), a technique able to provide a depth profile of the free volume. They showed that acceleration of physical aging is more pronounced near the polymer interfaces in comparison to the interior of the film. Although in none of the studies on micrometer thick polymer membranes the Tg was measured, the accelerated recovery of equilibrium suggests that negative Tg deviations from the bulk behavior must be expected. A model developed by Dorkenoo and Pfromm [124] connected the rate of achievement of equilibrium and Tg . This was experimentally found in micrometers thick PS films investigated by Boucher et al. [86], who measured the enthalpy recovery and the Tg of freestanding-like PS films ranging from several microns to 30 nm. In this work acceleration of physical aging, that is, reduction of the time scale of equilibration, well agrees with the Tg depression, observable at thicknesses as large as several microns. Similar results in freestanding-like thin PS films, showing accelerated physical aging if considered at a given aging temperature, were

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obtained by Koh et al. [130] and Koh and Simon [121]. Indications of accelerated recovery of equilibrium are provided by molecular dynamics simulations [131]. Here an increase of the rate of equilibrium recovery, taken as the slope of the recovery function, is observed for moderately thin films (down to ∼ 10 nm). Thinner films exhibit reduced aging rate, indicative of nearly achieved equilibrium. For what concerns supported and capped polymer films, the first report was that of Kawana and Jones [132]. They monitored volume recovery by means of ellipsometry of thin PS films supported on silica wafers and previously shown to exhibit Tg depression [133]. In doing so, they noticed that in films thinner than 18 nm, no overshoot in the coefficient of thermal expansion is observed. This result indicates that very thin films already equilibrate after cooling from the melt state. Hence, similarly to freestanding films, these results points toward faster equilibration times in thin films in comparison to the bulk. Faster evolution than the bulk counterpart was also found in capped polycarbonate films following the change of the real part of the dielectric permittivity during physical aging [134,135]. Indications of faster equilibration of thin PS films supported on silicon wafers were provided by Vignaud et al. [136]. Employing ellipsometry and X-ray reflectivity they showed that thin PS films exhibit coefficient of thermal expansion in the glassy state larger than bulk. This implies that in the glassy state, for a given thermal history (in this case standard cooling from above Tg ), thin films are denser than bulk PS and, hence, closer to equilibrium. A considerable number of studies based the characterization of physical aging on monitoring the slope of a portion of the recovery function. To this aim, techniques such as ellipsometry [137–141], fluorescent spectroscopy [93,119,120] and dielectric methods [142] have been recently employed. The general outcome of these studies is that reducing the film thickness alters the rate of physical aging. However, care must be taken to interpret these results. A reduced slope of a portion of the recovery function requires the knowledge of the stage of physical aging corresponding to such a portion. In particular, this can originate from the slowness of the equilibration process; in this case only the very beginning of the equilibration process is monitored. Otherwise the reason for the reduced slope may be due to the fact that the glass is already close to the thermodynamic state corresponding to the equilibrated supercooled melt. This was highlighted by Priestley et al. [120], who selectively labeled thin PMMA layers with fluorescent probes and followed the evolution of the fluorescent intensity during the course of aging. They found the largest aging rate in the middle of the film and reduced aging rates at both the free and supported interfaces. However, in the former case this was due to the vicinity of the free surface to its Tg , whereas in the case of the supported interface this was related to the sluggishness of the approach to equilibrium due to the reduced mobility at the interface with silica. This is shown in Fig. 7 where the aging time evolution of the fluorescent intensity corresponding to each layer is presented. Apart from the work where the physical aging is directly monitored, it is worth mentioning that studies where the coefficient of thermal expansion in the glassy state is stud-

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Fig. 6. Oxygen permeability as a function of the aging time at 308 K and 2 atm for PSF films with different thicknesses. Reproduced with permission from [129], Copyright 2009, John Wiley & Sons Inc.

ied for thin PS films [133,136,144,145]. These results show that thin films exhibit densities in the glassy state larger than that of the bulk polymer [136]. This can attributed to more efficient equilibration of thin films. To close this subsection, it is worth mentioning the efficiency of equilibration appears to be a universal feature of glass forming thin films, that is, including non-polymeric ones. This was recently shown by Leon-Gutierrez et al. [146], who investigated the enthalpy recovery in thin toluene films. Similarly to thin polymer films, they found faster approach to equilibrium than bulk. This is especially evident in films thinner than 10 nm. These exhibit decrease of the Tf larger than 15 K in only 30 min at annealing temperature 3 K below the nominal Tg . This constitutes a rather spectacular decrease in Tf , considering that bulk toluene Tf only decreased by about 5 K in the same annealing conditions.

3.2.2. Polymer nanocomposites Among studies on nanocomposites exhibiting decrease or no change in Tg , there exist somewhere the physical aging is investigated too. As expected from the sign of Tg deviation from bulk behavior, all these studies point toward faster equilibrium achievement. This has been found in PMMA/(silanized)silica [99–101], PS/(silanized)silica [101], PS/gold [105], PVAc/silica [102,104] and poly(ether imide)(PEI)/carbon-fiber [103] nanocomposites. In other studies where Tg depression is observed too, the rate of physical aging, within the definition based on the slope of a portion of the recovery function, is decreased in comparison to that of the bulk. As discussed in the previous subsection, this is easily explained considering that the recovery function has almost completely decayed, due to the tiny departure from equilibrium as a result of the small

Fig. 7. Evolution of the normalized fluorescent intensity at different positions of supported PMMA film with total thickness of 1000 nm. The thickness of each labeled layer is 25 nm. Reproduced with permission from [143], Copyright 2009, the Royal Society of Chemistry.

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Tg –Ta , where Ta is the aging temperature. This is found in PS/silica nanocomposites [91]. In the case of polymer nanocomposites for which the Tg increases with the filler fraction, there exists unanimous consensus that the aging process, whatever its definition, slows down. This has been tested in a number of nanocomposites including epoxy in nanocomposites with POSS [89] and MMT [147], P2VP in nanocomposites of alumina and silica [91] and PMMA/(non-silanized) silica nanocomposites [92,93]. Among these nanocomposites, those based on SWCNT constitutes an important category of those systems exhibiting slow-down of equilibrium recovery [95–97]. 3.2.3. Polymer nanospheres Physical aging studies on polymer nanospheres are scarce and, to the best of our knowledge, there exists only one study in the topic [113]. Here, the enthalpy recovery of silica capped and PS nanospheres in aqueous solution was investigated. In the former case, the time scale to reach equilibrium was substantially unmodified in comparison to bulk PS. This result is compatible with the lack of dependence of the Tg on the nanospheres diameter [148]. Conversely, nanospheres in aqueous solution were found to age considerably faster than the bulk at a given temperature, a result visible for nanospheres with diameter as large as 400 nm. This result well agrees with the depressed Tg found in freestanding PS nanospheres [114,148]. Although it does not involve polymers, it is worth mentioning a physical aging study on the low molecular weight glass-former o-terphenyl (OTP) confined in nanopores [149]. In analogy to polymer nanospheres, this system exhibits 3-dimensional confinement. Furthermore, enthalpy recovery was found to reach equilibrium faster in OTP confined in the nanopores than in bulk, a result analogous to that found for PS nanospheres in aqueous solution. However, it is worth mentioning that, in this case, the possible role of negative pressure effects, arising from the contraction inside the nanopores, was considered to explain the accelerated aging. 4. Connection to polymer segmental dynamics As remarked in Section 1, the way a glass former leaves equilibrium when cooled down through the Tg at a given rate or recovers it when annealed isothermally below Tg – the so-called out-of-equilibrium dynamics, depends on the rate of spontaneous fluctuations in the glass itself. Given the conceptual difference between equilibrium and out-ofequilibrium dynamics, the former depending exclusively on the real temperature, while the latter being related to both the real and the fictive temperature Tf the question is: does the rate of spontaneous fluctuations alone determine the out-of-equilibrium dynamics of a glass? In bulk glass formers there are several indications that one can reasonably describe the out-of-equilibrium dynamics on the basis of exclusively the rate of spontaneous fluctuations (see Fig. 1). In this section, we review the most important results regarding the characterization of the rate of spontaneous fluctuations in nanostructured glass-forming polymers. To assess such a rate, measurements of the molecular mobil-

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ity by means of techniques working in the linear regime are considered. Pioneering studies of the molecular mobility in thin polymer films by Fukao and Miyamoto [150] employed broadband dielectric spectroscopy (BDS) to investigate the molecular mobility of thin PS films capped between aluminum. They found that the segmental dynamics was substantially unaltered down to 20 nm, whereas significant acceleration was observed for lower thicknesses. Similar results were later obtained by others [151,152]. More recently, Serghei and Kremer [153] emphasized the role of preparation conditions in the determination of the typical time scale of spontaneous fluctuations. In particular, they showed that thin films annealed during prolonged time at temperatures considerably above Tg slowly recovered bulk-like behavior. They attributed the accelerated segmental dynamics observed at short annealing times to the presence of solvent trapped at the polymer/substrate interface. Perlich et al. [154] showed this to be a possibility. Subsequent studies showed how supported and capped thin polymer films generally exhibit bulk-like molecular mobility even for thickness of a few nanometers [85,155–164]. A recent study on (semi)isolated polymer coils, that is the smallest achievable polymer system, demonstrated that the segmental mobility remains bulk-like even for these systems [165]. In Fig. 8, the segmental relaxation time (), taken from Ref. [85], is shown as a function of the inverse temperature for thin PS films in a wide range of thicknesses and measured by different linear techniques, that is, BDS and alternating current AC-calorimetry. As can be observed, bulk-like thickness independent segmental mobility is found. A pronounced component exhibiting bulk-like dynamics was found in free-standing thin PS films as thin as ∼ 10 nm following the reorientation of dye molecules [166]. In this case, however, it has to be remarked that, beyond the bulk-like behavior, a fraction of dye molecules exhibits a fast component. This was attributed by the authors to the presence of accelerated segmental dynamics in proximity of the free interfaces. Furthermore, the presence of a fasterthan-bulk component in the dynamics of freestanding films has been found by several authors [114,166–170]. Concerning polymer nanocomposites and nanospheres exhibiting Tg depression and accelerated physical aging, results analogous to those of supported and capped thin films have been presented. This is the case of PVAc [104], and PMMA and PS [101] in nanocomposites with silica and PS/gold nanocomposites [105]. In all cases the segmental dynamics, determined by BDS, was found to be independent of the filler content and bulk-like. Concerning PS nanospheres, there exist two studies, one by BDS [114] and the other following the closure of voids between nanospheres by small angle neutron scattering (SANS) [171], where the linear dynamics exhibit bulk-like relaxation. Apart from the bulk-like relaxation, PS nanospheres exhibit a relatively fast relaxation process [114]. This behavior, at least qualitatively, mimics that of freestanding thin polymer films [166]. A review of the results on the linear dynamics of different nanostructured systems, showing the presence

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Fig. 8. Segmental relaxation time as a function of the inverse temperature for thin PS films of different thicknesses and configurations. Reproduced with permission from [85], Copyright 2012, the Royal Society of Chemistry.

of predominant bulk-like linear dynamics even in freestanding systems appears to be in contradiction with the enhanced out-of-equilibrium dynamics, that is, Tg depression and accelerated physical aging, found in the same systems. This has been found in experiments on exactly the same samples where the linear and non-linear dynamics were probed in identical samples and in some cases simultaneously [85,101,105,109,114,172,173]. These results suggest that the linear and the non-linear glassy dynamics are not unequivocally related in nanostructured glasses. The most striking indication of the absence of a one-to-one correlation between the two aspects is that films as thin as a few nanometers (including freestanding films [166]), or even (semi)isolated polymer coils [165], exhibit dominant bulk-like dynamics, whereas enhanced out-of-equilibrium dynamics is observed even for nanostructured glasses with typical length scale of the order of microns. This is found in films and nanocomposites exhibiting depressed Tg [85,101,104,105,108] and accelerated physical aging [122,124–127]. One objection to the lack of full correlation between equilibrium and out-of-equilibrium dynamics could be that the former is normally determined at temperatures somewhat higher than those relevant for the melt to glass transition and equilibrium recovery. Hence, one could hypothesize that, upon temperature reduction, the equilibrium dynamics strongly deviates from bulk behavior and speed up in ways consistent with the acceleration of the out-of-equilibrium dynamics [52]. In Fig. 9 we illustrate how such scenario is at odds with the experimental evidence. In the upper panel of the figure, results on the linear dynamics of freestanding thin PS films, taken from Ref. [166], are presented. The middle and lower panels shows the calorimetric traces of freestanding-like thin PS films, taken from Ref. [85], and the temperature dependent thickness measured by ellipsometry for a 33 nm thick freestanding PS film, taken from Ref. [83]. Stacked films are investigated in Ref. [85], however, these behave similar to a single freestanding film, provided that very high molecular weights polymers are employed. As can be observed,

in the temperature range relevant for the bulk Tg , freestanding thin PS films with thickness as low as 14 nm exhibit a significant relaxation component with bulk-like dynamics. Conversely the calorimetric and ellipsometric plots in the middle and lower panels of Fig. 9 show no indication of decrease of the specific heat and linear coefficient of thermal expansion in the temperature range of the bulk Tg . This is certainly the case for films thinner than ∼100 nm, which exhibit melt specific heat at temperatures around the bulk Tg , and for the 33 nm thick freestanding film measured by ellipsometry exhibiting no hint of a thermal transition in the same temperature range. These results all together indicate no thermal glass transition signature in a temperature range where the segmental dynamics exhibits bulk-like relaxation time of the order of 1000 s. Further evidence of the impossibility of describing the out-of-equilibrium dynamics through arguments based on the rate of spontaneous fluctuations comes from equilibrium recovery experiments in PVAc/silica nanocomposites [104] where the segmental relaxation time is monitored during physical aging by dielectric techniques. This has been shown to increase more rapidly in systems with larger nanofiller content, as shown in Fig. 10, where the instantaneous relaxation time , probed by thermally stimulated depolarization current, is shown as a function of the aging time for nanocomposites with different filler contents. This implies that the instantaneous segmental relaxation time, that is, the relaxation time at a given aging time, is larger in nanocomposites than in the pure polymer despite the faster evolution toward equilibrium (see values of  in Fig. 10 at any aging time). 5. Factors controlling the out-of-equilibrium dynamics in nanostructured glasses The review of the results on glassy dynamics in nanostructured materials suggests that this is affected by numerous factors. Hence, finding a rationale to such results constitutes a major challenge in the topic.

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Fig. 9. Segmental relaxation time (upper panel); specific heat (middle panel); and thickness (lower panel) as a function of temperature for freestanding thin PS films. The arrow indicates the location of PS bulk Tg . Reproduced with permission from [166], Copyright 2011, the American Chemical Society; [85], Copyright 2012, the Royal Society of Chemistry; [83], Copyright 2011, the American Physical Society.

In the previous section, it has been shown how arguments exclusively based on the rate of spontaneous fluctuations are not sufficient to completely account for the phenomenology of the out-of-equilibrium dynamics. Nevertheless, it is worth emphasizing that in those cases where

a Tg increase and a decrease of the rate of physical aging is observed, the nanostructured polymer exhibits concomitant slow-down of the rate of spontaneous fluctuations. At least, this was found when both the equilibrium and the

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Fig. 10. Segmental relaxation time as a function of the aging time for pure PVAc, and nanocomposites with silica with the following filler content: 8 (V8) and 25% (V25). Reproduced with permission from [104], Copyright 2012, the American Physical Society.

out-of-equilibrium dynamics were measured in the same system [61,89,92,174]. Concerning nanostructured polymer glasses exhibiting Tg depression, a factor that has been emphasized is the applied cooling rate. Several studies in both supported [51,175] and freestanding films [86,176], and polymer nanocomposites [105] indicate that Tg deviations are more pronounced at relatively low cooling rates. This means that part of the scattering in the determination of the Tg depends on the applied cooling rate. From the point of view of the sample configuration, a crucial role in determining the deviation of the out-of-equilibrium dynamics from bulk behavior is the amount of free interface. This is straightforward if one compares the deviations of the out-of-equilibrium dynamics in freestanding, on the one hand, and supported and capped films, on the other. In the former case, these are considerably more pronounced than in the latter, with Tg depression that can be as large as 80 K. For what concern nanostructured glasses in the presence of a substrate, reviewing the available literature indicates that the out-of-equilibrium dynamics may undergo deep variations depending on the nature of the substrate [177]. Furthermore, even for the same kind of substrate Tg deviations can be changed by varying the thermal history of the film. As commented in Section 3.1.1, this has been proved for thin PS films annealed between aluminum layers at Tg + 60 K by Napolitano and ¨ Wubbenhorst [27] (see Fig. 3). Briefly recalling the outcome of this study, the degree of irreversible adsorption (formation of Guiselin brushes [73], with effects that can extend over several polymer radii of gyrations [178]) was related to a Tg depression, and that the bulk value is restored with complete chain adsorption. In a subsequent study, the same authors showed how the essential parameter determining the deviation from bulk Tg was the amount of free interface between the polymer and the substrate [68].

Napolitano et al. [68] provided the only systematic study showing the importance of irreversible chain adsorption on the Tg . However, this allows rationalizing the wide scattering of results on the Tg of nanostructured glasses. This is immediate for what concerns the different magnitude of Tg deviations in freestanding nanostructured glasses, on the one hand, and supported and capped, on the other. In the former case the amount of free interface includes the whole surface of the nanostructured glass. In the case of supported and capped systems, the amount of free interface depends on: (i) the kind of substrate; (ii) the thermal history followed to produce the film. Apart from Napolitano and coworkers [27,68] the latter factor has been scarcely investigated. Several reports show that different substrates can give rise to a broad distribution of deviations from bulk Tg . Instructive examples are provided by the behavior observed with PS [115] and PMMA [116] nanospheres. In both cases, as expected according to arguments involving the amount of free interface, bare nanospheres exhibit the largest Tg depression. Conversely, the Tg of PS nanospheres significantly deviates from bulk behavior in systems covered with ionic surfactant, whereas it remains bulk-like for those exhibiting a non-ionic surfactant on the surface. The opposite occurs in PMMA nanospheres. Within the interpretation of Tg depending on the amount of free interface, it is possible to speculate that the nature of the surfactant induces various degrees of adsorption as a result of different Van der Waals interactions between polymer and surfactant. Obviously, these arguments are based on the assumption that these nanospheres exhibit prevalent bulklike rate of spontaneous fluctuations. This has been shown to be the case in PS nanospheres in contact with air [114]. However, it is worth remarking that in all cases where deviations of the out-of-equilibrium from bulk-like behavior are observed, it is highly desirable to have information on the linear dynamics too. Arguments based on the amount of free interface can be invoked to explain the range of Tg deviations in thin PS

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films supported on layers of different thickness of silicon wafers [63]. This modifies the Van der Waals interaction between polymer and substrate that in turn affects the level of adsorption. This implies different amount of free interface depending on the thickness of the silicon wafer layer. 6. Theoretical framework To explain the overall phenomenology of out-ofequilibrium glassy dynamics in nanostructured polymers, one should be able to account for all experimental observations. This is a challenging task, considering the numerous factors playing a role once the typical dimension of the glass is reduced to the nanometer or even micrometer length scale. In discussing the theoretical activity, it should be remarked that there exist several approaches attempting to explain the deviations in glassy dynamics in nanostructured materials on the basis of purely relaxation arguments. One such description of the dynamic Tg corresponds to a particular time scale of spontaneous fluctuations (e.g. 100 s) [179–182]. For freestanding thin polymer films or with one free surface, it has been recently shown that the segmental dynamics is affected by the presence of a mobile layer at the free interface(s) [163,166]. As a result, the thickness averaged dynamic Tg deviates from bulk behavior. However, given the presence of pronounced bulk-like relaxation even for freestanding films as thin as 14 nm [166], theoretical approaches based on relaxation arguments to describe the thermal Tg depression in thin polymer films present serious flaws. As discussed in Section 5, within the relaxation approaches to the Tg of films, deviations from melt thermodynamic coefficients (heat capacity, coefficient of thermal expansion etc.) would be observed in proximity of the bulk Tg . That is not observed as shown in Fig. 9. In view of the previous discussion, a suitable theoretical approach to describe the overall phenomenology of outof-equilibrium glassy dynamics in nanostructured glasses requires to account for effects beyond the role of the rate of spontaneous fluctuations. In other words, it is possible to define an equilibration time  eq , that is, the inverse of the cooling rate corresponding to a given Tg or the time scale to reach equilibrium in the physical aging regime. This must depend on the time scale of spontaneous fluctuations  together with a function exclusively related to the length of nanostructuring h:  eq =  g(h). Within this context, the major challenges are: (i) seeking for the physics behind the function g(h); (ii) defining the length scale h of nanostructuring. Among the approaches aiming to describe glassy dynamics in nanostructured systems, those based on percolation have been developed on the base of altered linear dynamics [183,184]. However, given the fact that these approaches are based on geometric arguments, that is, the change of dimensionality of percolation upon nanostructuring, they could also be developed to account for the enhancement of out-of-equilibrium dynamics maintaining a pronounced bulk-like behavior of the rate of spontaneous fluctuations.

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The free volume hole diffusion (FVHD) model has been extensively applied to describe Tg depression and acceleration of equilibrium recovery in nanostructured glasses exhibiting pronounced bulk-like linear dynamics. This was first introduced in the early 80s by Simha and co-workers [185], who described volume recovery of bulk glassy PVAc [35]. A similar approach was adopted by Perez [186,187], who also developed a defect diffusion model to describe the out-of-equilibrium dynamics of glassy PVAc. The purpose of these models was to describe the way equilibrium recovery is achieved. As such they must be distinguished from those models aiming to describe the rate of spontaneous fluctuations employing free volume arguments [188,189]. According to the model of Simha and co-workers [185], diffusion of free volume holes toward an interface is responsible for maintenance of equilibrium when cooling down a melt, or recovering it once in the glassy state. In applying the model to volume recovery of bulk glassy PVAc, Simha and co-workers [185] achieved satisfactory description of data. However, to overcome the fact that macroscopic samples do not exhibit any dependence of equilibrium recovery on the size of the specimen, they had to assume the existence of an ill-defined internal length scale, of the order of several hundreds nanometers, where free volume holes would annihilate after diffusion. This was evidently a limitation of the model and for several years it did not receive much attention. However, we notice by passing that this limitation could be removed in the case of nanostructured polymers were a natural length scale does exist. The finding of acceleration of physical aging in polymer membranes as thick as several microns, dating back the mid 90s [123], provided a significant revitalization of the FVHD model [86,99,100,105,125,134,190]. This is essentially based on the following two equations: (i) the second equation of Fick: ∂fv (˜r , t) = ∇ (D∇ fv (˜r , t)) ∂t

(4)

where f v(˜r , t) is the fractional free volume at position r and time t, and D is the diffusion coefficient of free volume holes; and (ii) and the equation expressing the mean square displacement (MSD) x2 (t) as a function of time for one-dimensional linear diffusion. (The assumption of onedimensional confinement is obviously true in 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.): x2 (t) = 2Dt

(5)

The second equation of Fick can be employed to determine the spatio-temporal evolution of the free volume during physical aging. The description of the Tg can be performed through Eq. (5). As a result of its definition, at the Tg the system is out of equilibrium after a temperature quench. Within the FVHD model, this implies that at Tg the furthest free volume holes from the interface, that is, the ones located at h/2, half the typical length scale of nanostructuring, are just unable to diffuse out of the interface, in the observation time scale, proportional to the inverse of the

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cooling rate ˇ. This can be easily translated by the following equation: (h/2) = 2D(Tg )ˇ−1 2

(6)

A crucial point of the model is that to predict enhanced out-of-equilibrium dynamics, it does not require any alteration of the equilibrium dynamics of the glass. This is easily seen in Eq. (6). For instance, one can assume a thickness independent diffusion coefficient (bulk-like). For thicker samples larger values of D(Tg ) must be employed in Eq. (6). Considering that the diffusion coefficient increases with the temperature, this implies larger Tg for thicker films, as found in experiments. In the application of the FVHD model the assumption of a bulk-like diffusion coefficient was imposed. This was a reasonable assumption considering that, even in the most extreme case of nanostructuring [165,166], pronounced bulk equilibrium dynamics is observed. However, one could in principle refine the model to account for the modification of the linear dynamics in proximity of an interface. An important point for the correct application of the model arises from the fact that macroscopic samples, that is, those with typical nanostructure dimension larger than microns, exhibit no dependence of the out-of-equilibrium dynamics on such a dimension. This implies that with increasing this dimension the mechanism of equilibration based on diffusion of free volume holes must be gradually replaced by a “bulk” mechanism, that is, the one normally employed to describe out-of-equilibrium dynamics in bulk glasses. This problem has been approached either by assuming the presence of an internal length scale, as originally proposed by Simha and co-workers [185], or employing the most general formalism for the bulk mechanism [86]. In the latter case, for the description of the out-of-equilibrium dynamics in nanostructured systems there is no need to make any assumption regarding the mechanism of equilibration in bulk glasses. The application of the model is straightforward in freestanding films, given the availability of the entire interface to diffusion of free volume. In such a case, the length scale of nanostructure is simply the film thickness [86,128]. In Figs. 4 and 11, it can be seen how the entire phenomenology of the out-of-equilibrium dynamics in freestanding PS films is accurately described by the FVHD model. This is true for both the description of enthalpy recovery data (Fig. 11) and the Tg at two different cooling rates (Fig. 4). Both approaches based on the presence of an internal length scale (continuous lines) and that based on the most general formalism for the bulk mechanism (dashed lines) provide accurate descriptions of the thickness dependence of the Tg at both investigated cooling rates. In the latter case, this would imply that there exist two mechanisms of recovery of equilibrium: one relevant for bulk glasses; and the other, based on free volume holes diffusion, exhibiting increasing impact once the typical length scale of the nanostructuring is decreased. In the case of supported and capped films or any nanostructured system with an interacting interface, the application of the model requires considering the fraction of free interface. This is due to the fact that elimination of free volume holes cannot occur in the presence of an

Fig. 11. Evolution of the enthalpy with the aging time for freestandinglike thin PS films. Reproduced with permission from [86], Copyright 2012, the American Chemical Society.

energetic barrier too large to overcome. That is the case for the portion of the interface where irreversible chain adsorption has occurred during preparation of the nanostructured material, or the subsequent thermal treatment. Once the FVHD model is employed for aluminum capped films where both the amount of free interface and the magnitude of Tg deviation are characterized [68], accurate description of Tg data obtained by capacitive dilatomery is achieved [191]. In particular, in Ref. [191] it is shown that the amount of free interface predicted by the FVHD model on the basis of the magnitude of Tg depression scales with that obtained experimentally [68]. The amount of free interface predicted by the model is obtained via Eq. (6) as Afree = V/heff , where the effective thickness heff , is the thickness of the freestanding films exhibiting the same Tg depression as the capped film under consideration. Similar arguments, based on the FVHD model, have been recently employed to describe the dependence of the Tg of supported PS films subjected to different degrees of high temperature annealing and exposed to an upper interface [192]. Although the amount of free interface is rarely characterized for nanostructured systems obtained in identical conditions, the FVHD model can be applied considering that the amount of free interface is constant and, therefore, its ratio with the sample volume scales with the film thickness. This is done for the set of data of Fakhraai and Forrest [51] and shown in Fig. 2 [53]. As can be observed, the continuous lines, which are the fits of the model to experimental data, accurately catch both the thickness and the cooling rate dependence of the Tg . Application of FVHD in a simplified version has been proposed to describe the acceleration of physical aging of polycarbonate (PC) films capped between gold [134]. The outcome of this study was that experimental results, showing acceleration of physical aging with decreasing film thickness, are at least compatible with the FVHD model. Similar considerations can be used when applying the FVHD model to polymer nanocomposites. In this case, as for supported and capped films, the nanostructure length scale depends not only on the fraction of nanofiller and the

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degree of aggregation, but also on the degree of adsorption at polymer/nanofiller interface. The latter determines the amount of free interface per nanoparticle. Having clarified this, similarly to supported thin films, it is possible to apply the FVHD model for a single set of nanocomposites, prepared in identical experimental conditions. In such a way the amount of free interface per nanoparticle is maintained constant and the only difference among samples is the fraction of nanofiller. Successful fits of enthalpy recovery results to the FVHD model have been reported in several polymer nanocomposites including PMMA/silica [99,101,190], PS/gold [105] and PS/silica [101]. A qualitative explanation on the scattering of results obtained on the Tg deviation in polymer nanospheres [113–116] can be provided within the idea that the free interface determines the efficiency of free volume transport from and into the glass. One can speculate that depending on the chemistry at the interface (presence of an inorganic layer or different types of surfactant), it is possible to tune the Tg of the sample by simply changing the amount of free interface. Hence, on a qualitative basis, the FVHD model provides an explanation on the different magnitude of Tg deviation in polymer nanospheres. Additional indications of the suitability of the FVHD model to describe the out-of-equilibrium dynamics of nanostructured glasses comes from experiments where the depth profile of the evolution of the aging rate [120], the free volume during equilibrium recovery [129] and the Tg obtained by x-rays photoelectron spectroscopy (XPS) [193] are monitored. In all cases, the spatio-temporal profile in aging experiments and the Tg distribution are paraboliclike, in qualitative agreement with the description based on Eq. (4). 7. Conclusions and perspectives We have reviewed the activity of the last decade on the topic of glassy dynamics in nanostructured polymer glasses, that is, thin films, nanocomposites and nanospheres. Special attention has been devoted to the review of the out-of-equilibrium dynamics in terms of deviations of the Tg and the physical aging from bulklike behavior. In view of several recent findings, we show how the out-of-equilibrium dynamics cannot be exclusively related to the equilibrium dynamics, that is, the rate of spontaneous fluctuations measured by techniques working in the linear regime. This conclusion indicates that glassy dynamics in nanostructured glasses exhibit a behavior markedly different from that of bulk glass formers. For the latter system a one-to-one relation between out-of-equilibrium and equilibrium dynamics was previously demonstrated. To understand these results, we have emphasized the conceptual difference between linear and non-linear determinations of glassy dynamics. Furthermore we have discussed recent developments aiming to explaining the entire phenomenology of glassy dynamics in nanostructured glasses, with particular emphasis on the FVHD model. This model is able to provide a suitable description of experimental data and is consistent with the faster out-of-equilibrium dynamics in nanostructured glasses exhibiting large free interface.

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An important consequence of the previous analysis is that nanostructured glasses with large free interface are able to maintain equilibrium more efficiently than bulk glass formers. Hence, they are able to explore the energy landscape down to regions not easily accessible in bulk glass former. This while maintaining prominent bulk-like equilibrium dynamics, as discussed in this review, as well as the thermodynamics (at least for freestanding films thicker than about 30 nm [194,195]). In the context of the knowledge of the equilibrium dynamics and thermodynamics deep in the glassy state, information regarding the divergence of the relaxation time at the Vogel temperature and the vanishing configurational entropy at a finite temperature, that is, the so-called Kauzmann temperature [196] can be achieved. This problem has been faced characterizing glasses exhibiting low Tf [197–199]. Furthermore, studies of nanostructured systems deep in the landscape can provide insight on the recent finding on the complex behavior of the dynamics and thermodynamics below the bulk Tg [38,200]. It was found that recovery of equilibrium occurs in two stages, with partial and complete enthalpy recovery. This implies that the existence of multiple recovery mechanisms, each of them leading the glass into a relative minimum in the energy. Together with the idea that equilibrium recovery of nanostructured glasses is achieved via two mechanisms, one relevant for the bulk and the other for nanostructured glasses, this indicates a complex nature of the kinetics of glasses equilibration. Unfortunately, in bulk glass formers, the time scale involved to monitor all recovery stages are often very long (larger than one year in Ref. [38]). Exploring nanostructured glasses with enhanced out-of-equilibrium dynamics can provide new insight on the topic with time scales amenable to the normal practice of the experimental work. In this sense, it is worth recalling a recent work on freestanding thin PS films where two discontinuities in the coefficient of thermal expansion were detected by ellipsometry on cooling at rate of several K min−1 [83]. Such finding, in relation to the double equilibration mechanism found in the enthalpy recovery of bulk polymers, opens new perspective on the exploration of glasses low in the energy landscape exploiting the mentioned peculiarities of nanostructured glasses.

Acknowledgements The author acknowledges the University of the Basque Country and Basque Country Government (Ref. No. IT-654-13 (GV)), Depto. Educaci´ıon, Universidades e Investigaci´ıon; and Spanish Government (Grant No. MAT2012-31088) for their financial support.

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