Submitted to Journal of Chemical Physics Special Issue on: Dynamics of Polymers in Thin Films and Related Geometries

Transcription

1 Submitted to Journal of Chemical Physics Special Issue on: Dynamics of Polymers in Thin Films and Related Geometries revised version, December 2016 Influence of Chemistry, Interfacial Width and Non-Isothermal Conditions on Spatially Heterogeneous Activated Relaxation and Elasticity in Glass-Forming Free Standing Films Stephen Mirigian and Kenneth S.Schweizer* Departments of Materials Science and Chemistry, University of Illinois, 1304 West Green Street, Urbana, IL

2 ABSTRACT We employ the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory of activated relaxation to study several questions in free standing thin films of glass-forming molecular and polymer liquids. The influence of non-universal chemical aspects on dynamical confinement effects are found to be relatively weak, but with the caveat that for the systems examined the bulk ECNLE polymer theory does not predict widely varying fragilities. Allowing the film model to have a realistic vapor interfacial width significantly enhances the reduction of the film-averaged glass transition temperature, T g, in a manner that depends on whether a dynamic or pseudo-thermodynamic averaging of the spatial mobility gradient is adopted. The nature of film thickness effects on the spatial profiles of the alpha relaxation time and elastic modulus are studied under non-isothermal conditions and contrasted with the corresponding isothermal behavior. Modest differences are found if a film-thickness dependent T g is defined in a dynamical manner. However, adopting a pseudo-thermodynamic measure of T g leads to a qualitatively new form of the alpha relaxation time gradient where highly mobile layers near the film surface coexist with strongly vitrified regions in the film interior. As a consequence, the film-averaged shear modulus can increase with decreasing film thickness, despite the T g reduction and presence of a mobile surface layer. Such behavior stands in qualitative contrast to the predicted mechanical softening under isothermal conditions. Spatial gradients of the elastic modulus are studied as a function of temperature, film thickness, probing frequency and experimental protocol, and a rich behavior is found. 2

3 I. Introduction A. Background More than two decades of intense experimental investigation has revealed that geometric confinement can have large and puzzling effects on the dynamics of supercooled liquids [1-4] which are sensitive to many chemical and physical factors. For free standing thin films, a vapor surface generically speeds up dynamics which is usually attributed to enhanced mobility near the free surface somehow extending over rather large distances into the film [1-3, 5-15]. The central question is therefore the nature of the spatial gradient of mobility in the direction orthogonal to the film interface [16-18]. To date, a fundamental theoretical understanding remains elusive given the complexity of activated relaxation in bulk liquids in addition to the complications of geometric confinement, interfaces and spatial inhomogeneity [19-25]. Recently, we proposed a quantitative, force-level theory of the alpha relaxation in bulk supercooled molecular [26-28] and polymer liquids [29], the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory. Quantitative tractability for real materials is achieved based on an a priori mapping of chemical complexity [28,29] to a thermodynamic-state-dependent effective hard sphere fluid using experimental equationof-state data and (for polymers) chain conformational statistics. The structural relaxation event involves coupled cage-scale hopping and a longer range collective elastic distortion of the surrounding liquid, resulting in two inter-related, but distinct, barriers. The elastic barrier only becomes important in the deeply supercooled regime, and there it grows much faster with cooling than the local cage barrier. At the kinetic glass transition temperature the collective elastic barrier is modestly larger in absolute magnitude than its 3

4 local cage analog. The theoretical description is quasi-universal, devoid of fit parameters, has no divergences at finite temperature, and accurately captures the key features of the alpha time of molecular liquids over 14 decades [26-28]. Extension of the theory to polymer liquids is based on a disconnected Kuhn segment model [29]. Good results have been demonstrated for the glass transition temperature (T g ), dynamic fragility, and full temperature dependence of the segmental relaxation time for many polymers, including the chain length dependence of T g. However, the unusually low and high fragilities of certain long chain polymers (which are not typical of molecular liquids) are not captured. ECNLE theory has been generalized to treat free-standing films and predicts the spatial gradient of the alpha relaxation time as a function of temperature, film thickness and location in the film [30,31]. Relaxation speeds up because of both a reduction of the cage barrier near the liquid-vapor interface due to loss of neighbors, and the cutoff and dynamical softening of the longer range collective elasticity cost for hopping near the interface. These two effects are strongly coupled. Quantitative results were obtained for the mobility gradient based on a minimalist description of structure and parameters representative of a generic nonpolar van der Waals liquid. For context, our main prior main findings are as follows [30,31]. We predict temperature dependent near surface mobile layers and their many consequences are qualitatively consistent with dynamic measurements. Specifically, (a) mobile layer size and its thermal evolution [3,5-7], (b) 2-step relaxation (with very different amplitudes and non-exponential stretching exponents) of a film-averaged time-dependent correlation function [5] emerging at temperatures only modestly above the bulk T g, (c) an emergent high frequency wing in dielectric loss experiments [32], (d) a mean fragility that 4

5 decreases with film thickness, (e) large T g gradients with contributions close to and far from the interface, (f) significant film-thickness dependent dynamic T g reductions as deduced from the frequency domain response, and (g) a 4-8 decade acceleration of surface diffusion near the bulk T g [33,34]. So-called pseudo-thermodynamic experiments [1,7-11,15] were also addressed based on model ellipsometry calculations and contrasted with the results of the dynamical calculations [31]. A larger T g reduction with film thickness is predicted compared to its dynamic analog, with a functional form in accord with measurements. The dynamic and pseudo-thermodynamic T g shifts obtained from calculations of experimental observables are very well reproduced based on a purely theoretically defined T g as either when the film-averaged alpha time equals 100 seconds or as the film-average of the local T g gradient, respectively. Fundamental to all our results is that faster relaxation is not solely due to direct free surface effects which propagate inwards. Rather, the origin of accelerated dynamics arises from the coupling of physical effects associated with both near the interface and deep in the film. B. Goals and Outline The present article extends ECNLE theory to address multiple new issues. The technical theoretical aspects and treatment of chemical complexity (mapping) are nearly identical to our prior work [31]. As background, the key physical ideas and approximations are sketched in section II. Section III studies how the chemical variability of molecules and polymers influences T g -gradients and film-averaged T g reductions. Our prior thin film work [30,31] assumed an infinitely sharp (step function) liquid-vapor interface. Section IV studies the dynamical consequences of allowing the interface to have a nonzero width, and large effects are predicted. Section V examines how mobility 5

6 gradients change if different film thickness samples are compared under non-isothermal conditions, corresponding to keeping the distance from the film-thickness-dependent dynamic or pseudo-thermodynamic T g fixed. Large and novel effects (e.g., coexistence of mobile and pinned layers) are discovered which may be relevant to mechanical and other measurements. Changes of the dynamic shear modulus in thin films under both isothermal and nonisothermal conditions is studied in Section VI. The possibility of a reduction of T g and existence of near surface mobility gradients, but a stiffer filmaveraged mechanical modulus, is explored. The paper concludes in Section VII with a summary and outlook. Details of the new interface models utilized are given in the Appendix. II. ECNLE Theory, Mapping and Models Key concepts of ECNLE theory, and how it is implemented for real materials, is sketched in sections IIA and IIB for bulk liquids [26-29] and free standing films [30,31], respectively. All technical details are given elsewhere [26-31]. Physical and chemical effects not included in our present formulation are discussed in section IIC. A. Bulk Liquids ECNLE theory describes activated relaxation as a mixed spatially local-nonlocal rare event composed of a cage scale large amplitude particle jump which is coupled to a relatively long range, spontaneous, collective elastic fluctuation needed to sterically accommodate the hop [26,27]. The inset of Fig.1 shows a cartoon of the physical idea. The foundational quantity to describe the relaxation of a single spherical particle (diameter, d) in the bulk liquid (volume or packing fraction ) is an angularly-averaged 6

7 particle-displacement-dependent (r) dynamic free energy, F dyn (r) F ideal (r) F caging (r), the derivative of which determines the effective force exerted on a moving tagged particle due to its surroundings. The localizing cage contribution, F caging (r), captures the effect of interparticle interactions and local structure on the nearest neighbor length scale, and is quantified by the equilibrium pair correlation function, g(r), or Fourier space static structure factor S(k). To execute a large amplitude rearrangement over the predicted (for 0.43 ) local barrier (height, F B ) requires a small amount of extra volume be created in the surrounding fluid, which is realized via a spontaneous collective elastic fluctuation of molecules outside the cage. The corresponding radially-symmetric harmonic strain field, u(r), has a pure shear symmetry, and from a continuum linear elasticity calculation it decays as an inverse square power law: u(r) r eff r cage r 2, r r cage 3d /2 (1) where r cage is the location of the first minimum of g(r). The strain field amplitude is set by the calculated orientationally-averaged mean local cage expansion length, r eff, which is of order of (or less than) the small localization or vibrational amplitude length, r loc (minimum of the dynamic free energy): r eff 3r 2 /32r cage r loc (2) where r r B r loc d is the single particle jump distance. The activation barrier associated with the elastic strain field follows by summing over all harmonic 7

8 particle displacements outside the cage region 1 ( ) elastic V dr K0 K0 / 2 4 () 12 eff cagek0 2 (3) rcage F u r dr r u r r r where all length scales are in units of d, r is a vector (different from r in the local dynamic free energy) with origin at the center of the cage region of the local relaxation event, V is the macroscopic volume of liquid outside the cage, and K 0 is the curvature (harmonic stiffness) of the dynamic free energy at its minimum which determines the transient vibrational amplitude of particles outside the cage. This harmonic curvature constant is proportional to the emergent dynamic shear (not bulk) modulus of the glassy liquid [27,28]. The second equality in Eq.(3) is for an isotropic material, and the final expression is an accurate analytic expression which shows that F elastic is determined by the harmonic stiffness of the transient localized state and the effective jump distance, both determined by the local dynamic free energy [27,28]. The local and elastic collective barriers are therefore intimately coupled, and the total barrier for the alpha process is F total = F B + F elastic. The alpha or structural relaxation time follows from a standard Kramers calculation of the mean first passage time over a barrier [26-29]. The theory is rendered quantitatively predictive for real materials by mapping molecules [28], or disconnected Kuhn statistical segments for polymers [29], to an effective hard sphere fluid guided by the requirement that the hard sphere fluid exactly reproduce the equilibrium dimensionless density fluctuation amplitude (compressibility) of the liquid, S 0 (T ) k B T T. The latter thermodynamic quantity also sets the amplitude of nm-scale density fluctuations, and follows from the experimental equation-of-state (EOS), thereby yielding a material-specific temperature-dependent effective hard sphere 8

9 packing fraction, eff (T ) [28]. The mapping involves 4 known chemically-specific parameters: A and B (entropic and cohesive energy parameters, respectively) associated with the EOS, the number of elementary sites that define a rigid molecule, N s (e.g., N s =6 for benzene) or Kuhn segment of a polymer (N s =38.4 for polystyrene which follows from monomer structure and backbone characteristic ratio), and the effective hard sphere diameter of a molecule or Kuhn segment, d. The microscopic parameters (A, B, N s, d) at 1 atm pressure relevant to all molecules and polymers studied in this article are given elsewhere [28,29]. Knowledge of eff (T ) allows the effective hard sphere pair structure to be computed using integral equation theory [27,35] as a function of temperature and chemistry, which then determines the dynamic free energy, from which all dynamical results follow [26-29]. Figure 1 shows no adjustable parameter [26, 28] mean alpha relaxation time calculations in an Angell plot representation for 3 molecular (one nonpolar, two hydrogen bonding) and 2 polymer liquids. These systems have diverse predicted T g ( K) and fragility (m~51-96) values [28,29]. The bulk theory for nonpolar molecular liquids is in very good agreement with all aspects of experiment [32,34]. For polymers [29], the T g predictions are in good agreement with measurements, but dynamic fragilities vary only over the molecule-like range of ~80-100, thereby missing the much higher and lower values experimentally observed. B. Free Standing Films A thin film introduces spatial heterogeneity of structure and dynamics. For free standing films with vapor interfaces, we previously adopted a minimalist model based on a perfectly sharp interface and the neglect of confinement-induced spatial changes of film 9

10 density, compressibility, and packing correlations. These simplifications imply that only two generic purely dynamic mechanisms of how a free surface modifies the spatially nonlocal activated relaxation event are emphasized [30,31]: (i) a direct surface effect close to the vapor interface mainly associated with the loss of nearest neighbors and its effect on the local cage barrier, and (ii) a longer range confinement effect that is mainly associated with the cutoff of the strain field at the vapor interface which reduces the collective elastic barrier. These two effects are fundamentally coupled in a spatially heterogeneous manner via gradients of all physical properties associated with the dynamic free energy. The inset of Figure 2a schematically depicts the key physical effects. The local barrier is reduced when a particle is less than a cage radius from the surface due to a loss of nearest neighbors. At distances further from the surface than r cage, the local barrier and all dynamic free energy properties take on their bulk values, but the elastic barrier is still reduced due to the finite film volume and elastically softer material near the interface. Because of its longer ranged nature, elastic barrier reduction extends deep into the film. The above physical effects are technically quantified in the ECNLE theory framework as discussed in detail in our prior article [31]. Reduction of the number of neighbors, and hence confining forces, on the cage scale enters the dynamic free energy as [31]: F dyn (r) F ideal (r) (z)f cage (r), where F ideal (r) 3k B T ln(r / d), and (z) quantifies the missing nearest neighbors and can be analytically computed based on simple geometric considerations. At the surface (z 0) 1/2, which smoothly interpolates to the bulk behavior (z r cage ) 1. Angular averaging of the local forces on a tagged particle is performed corresponding to retaining a radially-symmetric 10

11 description of the hopping event at a fixed depth (z) in the film [31]. This simplification is adopted for both consistency with the bulk theory and tractability reasons; to a first approximation it seems reasonable given the microscopic jump event occurs on a length scale well below the particle size. The presence of (z) in the dynamic free energy implies that all dynamical properties that enter the activated hopping time are modified in a z-dependent manner. The strain field that enters the elastic barrier calculation is a function of both the distance from the cage center and the location of the relaxation event in the film, in addition to being cutoff at the vapor surface. Thus, the collective elastic barrier is reduced via three mechanisms: (i) a position-dependent strain field ur ( ; z), (ii) the position-dependent reduction in local (harmonic) stiffness K 0 (z), and (iii) the integration over a finite film volume, the cutoff effect. When a relaxation event occurs close to the surface all three effects are important. Even events far from the interface are influenced by a reduced elastic barrier due to the near surface softening (stiffness reduction) and cutoff effects. The three contributions are coupled via the z-dependent dynamic free energy and the need to integrate over film volume. Thus, elastic barrier reduction extends well beyond the cage scale. The position-dependent mean alpha time associated with barrier crossing is computed in as in the bulk [28] but all factors now depend on location in the film [31]. The main frame of Figure 2 shows a model calculation of the relaxation time gradient at the bulk T g for a range of film thicknesses using polystyrene parameters. The alpha time is massively reduced at the surface but transitions rather quickly to a more gentle reduction that extends deep into the film. For thin enough films, no bulk region 11

12 exists, emphasizing the nonlocal physics associated with how vapor interfaces modify the collective elastic barrier. From knowledge of the mobility gradient, many dynamical properties can be computed. We cite here key formulas discussed in detail previously [26-31] that are employed to study the issues of present interest. A local, depth-dependent, T g -profile is defined by when the local relaxation time reaches 100s. From this, a thickness-dependent (h), theoretically-defined pseudo-thermodynamic glass transition for the film is the average of this T g -profile: h T g h 1 h T (z)dz (4) g 0 The corresponding theoretical dynamic glass transition temperature is defined as when the film-averaged alpha time is 100 seconds: h (T g ) h 1 h (z;t ) dz 100 sec (5) g 0 A film-averaged generic relaxation function, relevant (for example) to the dye rotation experiments of Paeng and Ediger [5], is defined as: C(t) e t / (z;t;h) h (6) In agreement with experiment [5], we found at lower temperatures a double KWW function of the form C(t) a 1 e (t / fast ) 1 a 2 e (t / slow ) 2 (7) is needed to fit the theoretical calculations, while at higher temperatures a single KWW function is sufficient. The frequency domain analog (per dielectric spectroscopy[2,32]) is: 12

13 C ''( ) dt cos(t)c(t) 0 (z) 2 h 1 (z) (8) Its inverse peak frequency corresponds to a mean relaxation time, and our predictions for T g (h) based on it agree essentially exactly with the dynamic T g defined in Eq.(5). Calculations can also be done that mimic ellipsometry measurements which detect changes of the film thermal expansion coefficient by using our calculated mobile layer thickness, an effective two layer model [2,18], and the liquid and glass values of thermal expansivity [31]. From this one can deduce a pseudo-thermodynamic T g (h), which we found based on the step function interfacial density model agrees almost exactly with that computed more simply using Eq.(4). As an empirical exercise motivated by experimental and simulation studies, we previously found [31] that the theoretical results for T g (h) can be fit to the widelyemployed form [36]: T g (h) T g,bulk 1 a h (9) where a and are adjustable parameters. A fundamental theoretical basis for Eq.(9), even for the simplest case when 1, likely does not exist given the continuous spatial gradient of the activation barrier and alpha time in the film. However, since the barrier and alpha time gradients are rapidly varying near the surface (see [31] and Fig.2), but slowly varying in the film interior, an effective two layer picture of a mobile surface region and a roughly bulk-like interior can be a reasonable approximation, especially for a film-averaged scalar quantity such as T g (h). This rationalization of Eq.(9) is buttressed 13

14 by our previous finding that the predictions of Eq.(6) for C(t) can be empirically fit with a two process model per Eq.(7), in accord with experimental observations [5]. As discussed recently in the context of applications of ECNLE theory to polymer nanocomposites [37] and bulk glassy polymer melts [29], the local shear modulus (in the absence of the alpha relaxation) at a location z in a film of thickness h can be approximately computed based on a mode-coupling-like formula for intermolecular stress [29,37]: 0 2 G(z;h) k T B d dk k dk S(k) exp k 2 r 2 loc (z;h) 3S(k) (10) where r loc (z) is the localization length (minimum of dynamic free energy). A frequencydependent dynamic shear modulus is determined using a simple Maxwell [29,37] model: G(;z;h) G(z;h) (z;h) 1 (z;h) 2 (11) 2 Film-averaged analogs of Eqs.(10) and (11) follow immediately. C. Caveats Before proceeding, we acknowledge the zeroth order and quasi-universal nature of ECNLE theory, mapping and models. With regards to chemistry, the effective hard sphere model has no explicit (only implicit via the EOS and Kuhn length) information about molecular or monomer shape, hydrogen-bonding, or chain connectivity effects beyond the Kuhn length scale (~1-2 nm). With regards to equilibrium structure, beyond the lack of explicit influence of the chemical effects mentioned above on packing, there is no density layering perpendicular to the film surface, and no change of pair correlations or dimensionless compressibility in the film relative to the bulk effective hard sphere 14

15 liquid. Concerning activated dynamics, there are no effects of surface tension, no explicit effect of confinement on motion parallel to the surface, and the isotropic bulk functional form (r-dependence) of the strain field is retained and simply cutoff at the interface. Finally, the thin film theory inherits from bulk isotropic ECNLE theory [26-29] the simplifications of no intrinsic relaxation time distribution due to structural and/or dynamic heterogeneity, and no physical aging. All these simplifications are worthy of future attention, but represent difficult and open theoretical problems. Here we focus on the minimalist force-level theory which provides a no adjustable/fit parameter description and allows quantitative predictions to be made for real materials under deeply supercooled conditions. III. Local Chemistry Effects Despite its quasi-universal nature, bulk ECNLE theory carries chemical information as embedded in the material EOS, number of rigidly moving interaction sites (N s ), and the Kuhn length (for polymers) [26-29]. Our prior studies of free-standing films [30,31] employed only parameters for polystyrene (PS). Here we perform calculations for the following systems: glycerol (G), sorbitol (S), orthoterphenyl (OTP), polymethylmethacrylate (PMMA) and PS; the relevant (A,B,N s ) parameters are given elsewhere [28,31]. The Kuhn length of these two polymers is ~ 1.45 nm, and the effective diameters (d) are: 0.6, 0.7, 0.9, 1.0, and 1.2 nm for G, S, OTP, PMMA and PS, respectively [28,29]. The predicted bulk glass transition temperatures (in Kelvin) and dynamic fragilities are: (202,58), (368,51), (267,81), (425,89), (408,96), for G, S, OTP, PMMA, PS, respectively [28,29]. Except for sorbitol (a heavily hydrogen-bonding 15

16 molecule), the theoretical T g values are close to experiment (within 10%). The predicted dynamic fragilities of G and OTP are in excellent agreement with experiment [28], while the polymer theory significantly underestimates the fragilities of PS and PMMA [29]. The main frame of Figure 3 shows calculations of the T g gradient normalized to the bulk value as a function of location in a 15 nm film. They are qualitatively all of the same form, with the hydrogen-bonding systems exhibiting a larger gradient. The inset shows calculations of the corresponding pseudo-thermodynamically determined T g shifts (based on Eq.(4)) as a function of film thickness for all the systems except sorbitol. At h=10 nm, the T g reductions vary from ~10 to 22 K, though the functional form is generic. As an attempt to collapse these results, the main frame of Fig. 4 plots the calculations versus film thickness in units of molecule or Kuhn segment size, d. In this representation the results separate into two sets of curves, one for molecules and one for polymers. The larger T g depressions of the latter is likely due to their higher bulk T g values. Evans and Torkelson [38,39] recently suggested that there exists a correlation between fragility in the bulk and the amount of film T g depression (measured pseudo-thermodynamically); see [40,41] for uncertainties concerning this deduction. Motivated by this experimental work, the inset of Fig.4 plots the normalized T g (h) versus film thickness scaled by a numerical factor. Our results collapse onto a master curve with a scale factor in a small range only slightly above unity. Perhaps such a collapse is not surprising given the quasiuniversal nature of the mapping to an effectve hard sphere fluid employed in our theory. Although the theoretical collapse is based on exactly the same procedure employed experimentally by Evans and Torkelson [38], the values of required vary much more in the experimental study than in our calculations. Moreover, as one sees from the PS and 16

17 PMMA results in inset of Fig.3, we do not find larger T g shifts for larger bulk fragility materials. We caution that our conclusions about fragility-t g shift correlations in polymeric thin films is tentative given the documented inability of the present version of ECNLE theory to capture the very high fragilities of some polymers like PS and PMMA (~ ) [29]. Progress has very recently been made for this problem [42]. On the other hand, as a speculative comment, the good collapse in the inset of Fig.4 could hold for more general reasons since, to good approximation, the pseudo-thermodynamic T g depression scales as [30] ~1/h, and hence rescaling film thickness by a factor amounts to simply rescaling a length scale. However, see our comments in section IIB regarding the 1/h scaling. Thus, while empirically this rescaling may have significance in terms of an empirical effective two layer model, the theoretically continuous nature of the underlying mobility gradient means that such an interpretation likely does not have a deep theoretical basis, at least within the current theory. IV. Effect of Liquid-Vapor Interfacial Width When a tagged particle is near a vapor interface, the number of nearest neighbors is reduced. The forces exerted on a tagged particle by its neighbors enters via the local caging term in the dynamic free energy. Thus, to capture the reduction of caging constraints, which are dominated by the nearest neighbors, the dynamic free energy of NLE theory is modified as F dyn (r;) F id (r) (z)f cage (r;). Here, (z) quantifies the reduction in number of nearest neighbors and is computed as the ratio of the mean 17

18 density averaged over a spherical cage region at distance z from the surface to its analog in the bulk: (z) vcage (z) (12) v cage bulk The crucial input to determine (z) is the density profile near the surface, (z). In prior work [30,31], we adopted the simplest step function model (infinitely narrow interface). Here, we study the consequences of two model interfaces of nonzero width, as sketched in the inset of Figure 5. The double step and ramp models are taken to have a width of one particle diameter ( d ), a realistic choice. How the three interfacial density profiles determine (z) is given in Appendix A. The resultant (z) functions are shown in the main frame of Figure 5. The consequences of the finite width profiles on reducing caging constraints are very similar, both leading to the dynamical caging forces decreasing more and at greater depth into the film relative to the step function interfacial model. Where in the case of the infinitely sharp interface constraints are perturbed only to a distance r cage into the film, for the two new interfacial profiles the film density reaches its bulk value a molecular distance from the strict surface of the film, leading to perturbed dynamical constraints that extend to a larger distance r cage. This greater reduction in local caging constraints in turn leads to greater reduction in the long ranged elastic barrier far from the surface via the coupling mechanism discussed above. The inset of Figure 6 shows a representative example of the effect of interfacial structure on the alpha time profile for a PS film of h=12 nm at 400K (25 K below the theoretical bulk T g ). The two step and ramp profiles lead to almost identical predictions, a 18

19 generic finding based on other calculations (not shown). Due to the wider interface, the near surface fast mobility region is wider by about one particle diameter. Consequences of the modified mobility profiles on the film-averaged glass transition temperatures as a function of film thickness are shown in the main Figure 6. The solid symbols show the T g depression from a sharp interface, while the open symbols show the effects of a finite-width profile. Dynamic T g shifts are obtained using Eq.(5) and pseudo-thermodynamic shifts using Eq.(4). The effect of the ramp interfacial profile on the dynamic T g is modest, appearing noticeably only for the thinnest films. The reason is that the averaging inherent in the dynamic T g is weighted more towards the inner, slowest parts of the film, which are orders of magnitude slower than the faster surface layer. Because of this, the effects of a modified surface boundary condition only begin to appear when this fast surface region comprises a sufficiently large fraction of the film that there is effectively no longer an inner slow region. It is only then that the modification of dynamics in the fast region due to the ramp boundary condition becomes apparent. In contrast, the pseudo-thermodynamic averaging is more sensitive to the interfacial boundary condition because the fast and slow regions of the film contribute more equally to the average, and larger T g reductions are predicted to be observed. For example, T g is reduced by ~55 K for a 12 nm film, and significant reductions can extend to film thicknesses of the order of 100 nm or more. Our theoretical results in Fig.6 are empirically well fit by Eq.(9) with a/d = 1.56 (1.12) and = 1.07 (1.59) for the pseudo-thermodynamic (dynamic) T g -shift calculation. The pseudo-thermodynamic shifts have roughly the same simple functional form as previously found [31], varying roughly as ~1/h, in contrast to the dynamic T g shift 19

20 behavior. Thus, for the question of the shape of T g (h), the dynamic calculation is more sensitive to the interfacial boundary conditions. On the other hand, for the overall magnitude of T g -reduction, the pseudo-thermodynamic calculation is more sensitive. We note that the extracted length scale, a in Eq.(9), is of order the molecular size, which is larger than found in our prior study [31] based on the step function interface model. Experimentally, it is difficult to directly access the full mobility profile that is the central feature of our theory. One of the most direct (though still highly averaged) experimental probes is the dye reorientation experiments of Paeng and Ediger [5]. In our prior paper [31], we analyzed such a measurement based on computing C(t) in Eq.(6) using the step function interfacial model; representative results are shown in Figure 7 as the open symbols. The closed symbols show the corresponding new results for the ramp profile; we show in the inset the two relaxation times (which never intersect) extracted via a fit to Eq.(7). Although the correlation functions have noticeably different amplitudes for the two interfacial models, the extracted relaxation times are nearly indistinguishable. Moreover, we find that the KWW stretching exponents (very small for the fast process [31]) are nearly identical to what we reported previously [31] based on the step function interfacial profile. The ramp profile leads to a very short time process near 100 ps, which is far outside the experimental window [5], but is manifested as a modest reduction in the amplitude of the long time process (or single time) process in the double (or single) KWW fit. Thus, we conclude that dynamical measurements are relatively insensitive to the precise film boundary conditions for the models studied. V. Nonisothermal Conditions 20

21 All our studies to date have examined dynamic effects at fixed temperature. This isothermal condition is the simplest protocol, but some measurements, such as the bubble-inflation deduced mechanical creep studies of McKenna and coworkers [43,44], cannot be performed this way for practical reasons. Specifically, measurements are made for films of different thicknesses at a (roughly) fixed distance from the film glass transition temperature, T g (h). Since the latter varies with thickness, this is a nonisothermal measurement. More generally, studying dynamic and mechanical response in thin films based on such a protocol is of theoretical interest given the complex consequences of a mobility gradient and measurement-dependent vitrification temperatures. Here we explore this issue using (unless stated otherwise) PS parameters and the infinitely sharp step function interfacial density profile. A. Mobility Gradient and Film-Averaged Relaxation Times The alpha relaxation time gradient under isothermal conditions at the bulk T g for various film thicknesses was shown in Figure 2. Figures 8a and 8b present its nonisothermal analog at fixed (thickness-dependent) film T g determined pseudothermodynamically and dynamically, respectively. Comparing Figures 2 and 8, one sees that at/near the surface the massive degree of relaxation time speed up is very similar. On the other hand, there are curve crossings near the interface under non-isothermal conditions that are not present isothermally. The mobile layer width, as defined as the part of the film with a relaxation time less than 100 seconds, is significantly narrower under pseudo-thermodynamic compared to dynamic non-isothermal conditions. For the latter (Fig.8b), the relaxation time in the film center only very slightly exceeds 100 seconds. In qualitative contrast, the behavior in the film interior under pseudo- 21

22 thermodynamic conditions (Fig.8a) is remarkably different, with alpha times far larger than 100 seconds, and more so as the film thins. Thus, the novel picture emerges of a dynamic coexistence of a very mobile surface region layer with a highly vitrified interior region, with alpha times varying by up to 18 decades across the thinnest film. The above differences between the two non-isothermal mobility gradients reflect different averaging over gradients. One expects they will have large consequences on film-averaged mechanical response and other observable properties. An indication of this is suggested by the inset of Fig.8a which shows the film-averaged alpha relaxation time as a function of film thickness. It is, by definition, invariant to thickness based on the dynamic T g (h), per the behavior of bulk homogeneous liquids. In qualitative contrast, for the pseudo-thermodynamic protocol, starting at a very large film thickness of order several hundreds of nanometers, the film-averaged alpha time increases beyond 100 seconds, and becomes enormously larger below ~25-50 nm (e.g., ~10 6 s for h~10 nm). This behavior is consistent with the effective vitrification of the film interior, and implies a breakdown of the bulk dynamic relaxation time criterion for defining a kinetic glass transition temperature. Given the log-log format of the inset, the results roughly suggest there are two apparent power law regimes. At large film thicknesses (h~ nm), the average alpha time grows weakly (roughly as ~1/h 3/4 ) as the film thins. But for thinner films (h~4-20 nm), a much stronger apparent power law growth is predicted, roughly scaling as ~h -7. A crossover length scale can be defined by the (extrapolated) intersection of these two power laws which we find occurs at a large film thickness of h*~26 nm. All our calculations have assumed dynamic (structural) equilibration. In analogy with cooling a bulk liquid below T g, in a practical thin film experiment, if relaxation 22

23 times get too large the system will fall out of equilibrium depending on the time scale of the measurement (e.g., cooling rate). A classic consequence in a bulk cooling experiment is a rather sharp crossover from highly non-arrhenius growth of the alpha time to an apparent Arrhenius temperature dependence below T g. Equivalently, the effective activation barrier undergoes a sharp decrease below T g. Intriguingly, an abrupt decrease of the effective activation barrier in thin films beginning around h~30 nm has been recently observed by Fakhraai and coworkers based on pseudo-thermodynamic cooling rate T g and de-wetting measurements [45]. Whether the physics underlying this experimental finding is related to our theoretical results is unclear but warrants further investigation. Figure 9 shows an example of how the relaxation time gradient under pseudothermodynamic conditions changes for a 12 nm PS film at T g (h) if the interface has a finite width. Interestingly, large consequences are predicted including an enhanced spread of relaxation times, changes of the functional form of the gradient at the surface, wider mobile layer, and greatly enhanced tendency of the film interior to be frozen. However, these changes are not sensitive to whether the interface broadening is modeled as a double step or ramp. Quantitatively, these changes would tend to enhance the effects discussed above for the single step interface, but the qualitative picture remains the same. B. Evolution of Dynamic Solidity with Film Thickness It is of interest to quantify the coexistence of liquid and vitrified regions under pseudo-thermodynamic T g (h) protocol conditions based on a single parameter. We do this based on alpha time calculations like those presented in Figure 8a, both with and without including a simple mechanism for dynamic heterogeneity as previously discussed in 23

24 the context of bulk ECNLE theory [27,28,37]. Specifically, a Poisson probability distribution of relaxation times ( ) controlled by the mean alpha time is adopted: P; (T ) (T ) 2 e / (T ) (13) In the thin film, the mean alpha time depends on temperature, depth and thickness, (T;z;h), and the probability density becomes a spatially local function in the film: P; (T;z;h) (T;z;h) 2 e / (T ;z;h) (14) This is the simplest generalization of the microscopic mobility profile beyond a description that considers only the average alpha time. Explicit dynamic facilitation effects [46] are not taken into account. Given Eq.(14), the probability that a particle at an initial location in the film will have a relaxation time less than the experimental timescale is: texpt tex pt texpt / (15) 0 P( texpt ) d P( ; ) 1 1 e This function is shown in the inset of Figure 10 for a mean local alpha time of 100 sec. A tagged particle that relaxes at a time longer than the experimental timescale is deemed glassy, while if it relaxes faster it is liquid. The solid curve is the probability a particle is glassy on a given experimental timescale when the local mean alpha time is 100 seconds; the dashed step function corresponds to assuming no distribution of hopping times, a delta-function probability distribution. Thus, the solid curve is used to define the fraction of glassy segments at a given depth in the film. Integrating it over the relaxation time gradient throughout the film yields the global fraction of vitrified particles or glassy fraction. 24

25 Calculations of the glassy fraction as a function of film thickness at temperatures defined relative to their distance from the pseudo-thermodynamic T g (not isothermal) are shown in the main frame of Fig.10 for an experimental timescale of 100 sec. The solid curves include the Poisson dynamic heterogeneity, the discrete points do not. One immediately sees that for temperatures above the film pseudo-thermodynamic T g (h), a nontrivial fraction of the film is vitrified, even for fairly thick films. This is a consequence of the tail in the Poisson distribution which implies that in the slowest regions of the film, even though the average alpha time is faster than the experimental time (see, e.g., the yellow diamonds), some particles behave in a glassy manner. For temperatures below, or close to, the film T g, the glassy fraction grows monotonically towards complete vitrification (f glassy =1) in the bulk (large thickness) limit. For thinner films, the presence of an extremely fast surface layer means that an appreciable fraction of the film remains liquid-like, despite being up to 60 K below the bulk glass transition temperature. This trend is present in all the curves in Fig.10 for thinner films. When the temperature is modestly above its film T g, the glassy fraction initially increases (due to the reduced proportion of the film in the mobile surface layer) as the film thickens, but then decreases as it further thickens due to the competing effect of an increasing absolute temperature (although fixed relative to the increasing film glass temperature). Using a delta-function distribution for the local relaxation time probability leads to a sharp cutoff of the glass fraction (symbols) on a smaller length scale than found if the dynamic heterogeneity effect is included. We comment that our use of the Poisson distribution only in this section is motivated by the physical expectation that relaxation time fluctuations are especially 25

26 important for the question of at what large film thickness does a measureable glass fraction first emerge. Figure 10 shows this is the case, although for thinner films our calculations based on including the Poisson distribution or neglecting it are very similar. The latter point is relevant to the results presented earlier and subsequently in this article which focus on dynamical phenomena in films significantly thinner than the larger length scales germane to the question of the onset of a glassy fraction. For all these studies, we expect that including the Poisson distribution leads to much smaller, second order effects. C. Rubber Stiffening McKenna and coworkers have discovered a remarkable rubber stiffening of the compliance (inverse mechanical modulus) in free-standing polymer thin films [43,44]. It can be several orders of magnitude, is chemically-specific, and emerges in films of the order of hundreds of nanometers thick. An objective deduction is that, despite the major reduction of the film T g, some polymer chain dynamical modes that underlie the mechanical response on longer length scales beyond the segmental scale are somehow lost. A qualitative scenario involving sub-rouse modes and thermorheological complexity based on the phenomenological coupling model has been proposed for this effect [47]. A practical experimental complication is that the measurements are not done isothermally, but rather in a non-isothermal protocol where different films are studied roughly at a fixed (small) temperature increment above their thickness dependent T g. The latter is determined from dynamic shift factors, and hence is a measure of a film dynamic T g. Systematic studies of how the rubber stiffening behavior depends on the precise protocol (T-T g (h) calibration) apparently have not yet been done. 26

27 Here we present a speculative analysis for the loss of chain modes problem which builds on our predicted coexistence of mobile layers and vitrified regions under pseudothermodynamic conditions per Fig.8a. We are not suggesting this as a definitive explanation, but it may be part of the story for existing experiments and/or future ones. Our suggestion is that rubber stiffening could be related to the presence of vitrified segments that effectively crosslink (or more correctly, pin ) polymer chains on the timescale of the mechanical measurements. The distance along the chain between these pinned segments is crudely estimated as N x 1/ f glassy, where f glassy is per Fig.10. A rough estimate of the rubbery modulus of an entangled polymer film then follows as: 1 G rubber k B T 1 N e N x G 1 N e e N x G 1 f N e glassy e (16) where N e (G e ) is the bulk polymer melt entanglement chain length (plateau shear modulus). Given Fig.10, this effective rubbery modulus will grow beginning at a large film thickness since only a small fraction of glassy segments is required to modify the soft entanglement modulus (G e ~ MPa). As a numerical example, let N e ~100 monomers. Substituting f glassy = 0.1(0.5) in Eq.(16) gives a large enhancement of G rubber 10G e 50G e. Based on the calculations in Fig.10, at 5 degrees Kelvin above T g (h) these levels of enhancement will occur at a film thickness of ~150(50) nm. We caution that these numbers are not meant to be taken too seriously. Nonetheless, the idea that a coexistence of mobile and immobile segments in close spatial proximity in the thin film could lead to a high enough concentration of glassy or pinned segments on the experimental time scale such that large macroscopic mechanical effects emerge. 27

28 VI. Glassy Shear Modulus Diverse experiments (e.g., film buckling, AFM, capillary instability, direct mechanical measurement) have found that under isothermal conditions (often well below the bulk T g ), the mechanical stiffness (Young s modulus, shear modulus) decreases in free-standing films as the film thins [48-50]. For example, reductions have been observed for PS commencing at a film thickness ~25-30 nm with a modulus drop relative to the bulk material by a factor of 2-4 at h~10 nm [48,49]. Such mechanical softening seems natural if mobile surface layers exist. On the other hand, the non-isothermal creep measurements find the glassy modulus stiffens, by a factor ~2, with no clear variation with film thickness [43,44]. This glassy film stiffening is surprising, and was suggested as possible evidence against [43] the existence of mobile layers. But, these experiments are not isothermal, which we suggest represents a crucial difference relative to measurements that find mechanical softening [48-50] and mobile surface layers [5-7, 33,34]. Here we perform model calculations of the dynamic shear modulus of thin films using Eqs.(10) and (11). We study possible differences based on isothermal versus nonisothermal measurements and the role of measurement frequency. Based on the employed theoretical methods, it is known that in the bulk the plateau glassy shear modulus G (ignoring the long time alpha relaxation process so particles become dynamically trapped) is almost completely determined by the dynamic localization length [27,28]: Gd 3 k B T d 3 10 K d d 3 0 r loc 2, (17) 28

29 where K 0 is the harmonic curvature well of the dynamic free energy minimum. Because K 0 is a function of depth in the film, the glassy modulus is a function of z, G(z). Non-isothermal calculations of G(z) for various film thicknesses are shown in Figure 11 for PS parameters, a step function interface density model, and at the pseudothermodynamic film T g (h). The results are normalized by the dynamic shear modulus of the bulk liquid at the bulk T g. Near the surface, all the films are mechanically softer, as expected. As found for the relaxation time gradient, the film center is very glassy, and for the thinnest films shows a mechanical stiffening by a factor of ~2 relative to its bulk analog. This effect diminishes as film thickness (or temperature, not shown) increases, until for the thickest films the shear modulus in the film center is essentially that of the bulk. The inset of Fig.11 presents calculations of the local, frequency-dependent dynamic modulus. The results for a 12 nm PS film at the pseudo-thermodynamic film T g show that at higher frequencies only the fastest part of the film near the surface is probed and the film appears stiffer compared to the response at lower frequencies. In essence, the frequency sets a length scale at which the film appears glassy. Averaging the frequency-dependent modulus gradient over the film yields the net mechanical response probed in ensemble-averaged measurements. Figure 12 shows results under isothermal conditions at the bulk T g. As expected, as the film thins the modulus decreases, and more so as the probing frequency becomes smaller. For example, for h=10 nm, modulus reductions vary from ~10% to a factor 2 as the probing frequency decreases from a GHz to 1 Hz. These trends are natural given the alpha time gradient of 29

30 Fig.2 and presence of mobile layers. The mechanical softening is qualitatively consistent with isothermal experiments [56,59,60] and simulations [51,52]. The main frame of Figure 13 shows non-isothermal film-averaged shear modulus calculations at the dynamic film T g as a function of film thickness for various probing frequencies. At lower frequencies, this average shear modulus again exhibits softening as the film thins (and, consequently, is cooled under non-isothermal conditions), consistent with the presence of the mobile layer. At higher frequencies, the softening decreases, and the trend ultimately reverses and the film appears to stiffen when it becomes very thin. The latter behavior is in qualitative contrast to the isothermal calculations of Fig.12. This dramatic behavior becomes even stronger when the calculations are done at the pseudothermodynamic (colder) T g, as shown in the inset of Fig.13. Here, for the thinnest films and highest frequencies, the film averaged glassy modulus can increase by nearly a factor of two relative to the bulk. Even at the lowest frequency of 0.01 Hz, where the bulk has essentially no dynamic shear modulus (see ref.[29]), the thinnest films have an average modulus close to or higher than the bulk glassy value. Thus, independent of whether the glass transition of the film is characterized by the pseudo-thermodynamic or dynamic T g, sufficiently thin films display glassy stiffening at higher frequencies due to the extreme dynamic heterogeneity across the film, despite the presence of highly mobile near-surface layers. Thus, our zeroth order calculations provide, via the prediction that mobile and vitrified layers can coexist, a possible reconciliation of the non-isothermal creep-based deductions of glass stiffening [43,44] with the isothermal probe rotation [5], surface diffusion [33,34] and particle embedding [6,7] measurements of mobile interfacial layers. 30

31 VII. Summary and Discussion We have extended the ECNLE dynamical theory of free standing molecular and polymer films to study new issues motivated by both fundamental questions and recent experimental studies. Nonuniversal chemical effects in the present theory are relatively weak. However, given that the dynamic fragility in bulk liquids is dominated by the long range collective elastic process contribution to the activation barrier in ECNLE theory, this conclusion may change if the theory is improved [42] to capture the very low and very high fragility in glass-forming polymer materials [29]. A vapor interface of only one molecule or segment diameter wide was found to significantly increase the magnitude of T g reductions relative to our prior study [31] that adopted a step function liquid-vapor interfacial profile, and in a manner that depends on whether a dynamic or pseudothermodynamic perspective is adopted. However, it has only relatively minor effects on the functional form of the thickness dependence of T g shifts or the 2-step time dependence predicted for time domain correlation function measurements. The consequences of confinement and interfaces on mobility gradients and elastic modulus changes in thin films under non-isothermal conditions was studied and contrasted with the corresponding isothermal behavior. Modest differences were found if a film-thickness dependent T g (h) is defined in a dynamical manner. However, adopting a pseudo-thermodynamic measure of T g (h) leads to qualitatively new and large differences for the spatial form of the mobility gradient. For example, highly mobile layers near the film surface can coexist with strongly vitrified regions in the film interior. As a consequence, the film-average shear modulus can increase with decreasing film thickness, in qualitative contrast to the softening behavior predicted under isothermal 31

32 conditions. This result may reconcile differences between, for example, mechanical creep [43,44] and probe rotation [5] measurements. Gradients of the elastic modulus were studied as a function of temperature, film thickness, probing frequency and experimental protocol, and rich and varied behavior in broad agreement with observations was found. Much theoretical and model/mapping realism development remains to be done, including the topics discussed in section IIC. Perhaps the largest statistical mechanical challenge is to generalize the thin film ECNLE approach to systems with solid surfaces (supported and capped films) of variable adsorption strength and mechanical stiffness [1,2,15-18,21,36,45,53-55]. How and why substrate elasticity and degree of interfacial mixing modify the spatial gradient of the local cage and collective elastic barriers is a fascinating problem that must be addressed to understand the many puzzles recently identified by experiment [38,39,54,55] and simulation [21,56]. Our finding that a small increase of the vapor interface width leads to much greater and longer range T g reductions may be relevant to the recent striking observation by Bagley and Roth of very large and long range gradients of T g reduction in thick glassy polymer films supported on a more mobile rubbery macroscopic substrate where the degree of interfacial mixing far exceeds the Kuhn length [54]. Progress on such issues is also relevant to glassy dynamics in porous media and polymer nanocomposites. Theoretical efforts are underway in these directions and will be reported in future publications. Acknowledgements. This work was supported by the U.S. Department of Energy, Basic Energy Sciences, Materials Science Division via Oak Ridge National Laboratory. We gratefully acknowledge helpful discussions and/or correspondence with Mark Ediger, Chris Evans, Zahra Fakhraai, Rob Riggleman, Connie Roth, David Simmons, Alexei 32

33 Sokolov, Michael Vogel, Bryan Vogt and Lian Yu. A special thanks is extended to Greg McKenna for many illuminating discussions and correspondences, including his patient explanation of bubble inflation creep measurements and related topics. APPENDIX A The vapor-liquid interfacial density profile is defined as (z). In our prior work, it was taken to be a step function (z) (z) (z) 1(z h) 0 (A1) where (x) is the Heaviside function. In a real liquid, the density decreases from its bulk value inside the film to zero outside it over some length scale, typically a molecular diameter. To explore this we study the two finite interfacial width models shown in Fig.5. The two step function model is defined as 2 (z) (z) 1(z ) (z )1(z h ) (z h )1(z h) (A2) where is the height of the step function at the surface edges, here fixed at 0.5. The parameter is the step function width at the film edges and is fixed to be d, the molecular (or Kuhn segment) diameter. A perhaps more realistic form for the density profile is given by a smooth ramp over the length scale ramp (z) z (z) 1(z ) (z )1(z h ) h z This profile involves one parameter, here again fixed to be d. (z h )1(z h) (A3) 33

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38 [56] F.Klameth and M.Vogel, J.Phys.Chem.Lett., 6, 4385 (2015). Figure Captions. Figure 1. Temperature dependence of the bulk alpha relaxation time for (from least to most fragile) sorbitol (blue circles, T g =362K, m=51), glycerol (red squares, T g =202K, m=58), orthoterphenyl (yellow diamonds, T g =267K, m=81), polystyrene (green triangles, T g =425K,m=89), and polymethylmethacrylate (gray triangles, T g =408K, m=96). Inset: Schematic of the bulk ECNLE alpha relaxation process composed of a large amplitude local hopping motion coupled to a longer range, collective, harmonic elastic motion. Figure 2. Alpha relaxation time profile plotted as a function of the normalized distance from the film center at the bulk T g of PS films of thickness (from inner to outer ) 6 nm (blue), 12 nm (red), 18 nm (yellow), 36 nm (green), and 60 nm (gray). Inset: Schematic of relaxation in a free standing thin film. Figure 3. Profile of local vitrification temperatures normalized to their bulk values for films of thickness 15d as a function of distance from the film center (z) in units of particle diameter. No adjustable parameter calculations are shown for (from least to most fragile) sorbitol (blue circles), glycerol (red squares), OTP (yellow diamonds), PS (green triangles), and PMMA (gray triangles). Inset. Change of the pseudo-thermodynamic T g as a function of film thickness (in nm) for films of glycerol (blue circles), OTP (red squares), PS (yellow diamonds), and PMMA (green triangles). Figure 4. Same as the inset of Figure 3, but plotted versus film thickness in units of the particle diameter. Inset: Pseudo-thermodynamic T g, normalized to its bulk value, as a function of scaled film thickness for sorbitol (blue circles, =1.3), glycerol (red squares, =1), OTP (yellow diamonds, =1.1), PS (green triangles, =1.15), and PMMA (gray 38

39 triangles, =1.05). The scaling parameter has been chosen to best collapse all the curves (arbitrarily relative to glycerol). Figure 5. The function (z) of Eq.(12) plotted versus normalized location in the film. Colors correspond to the interfacial density profiles shown in the inset. Inset: Model interfacial density profiles. Blue (upper) curve is the single step function, red curve is a two step functional form with a thickness, and the yellow curve is a ramp profile with thickness of. Here we set d, one particle diameter. Figure 6. The film-averaged glass transition temperature as a function of film thickness in nanometers. The circles show the pseudo-thermodynamic results, the squares show the dynamic results. The closed symbols employ the prior [31] step function profile for the interfacial density, the open symbols use the ramp profile; the two step profile model gives results essentially identical to that obtained using the ramp profile. The solid curves are fits to the open symbols of corresponding color to Eq.(9); the blue curve has parameters a = 1.56 and = 1.07, and the red curve has a = 1.12 and = Inset: The alpha relaxation time profile across a PS film at 400 K using the simple step profile (blue circles), the two step profile (red squares), and the ramp profile (yellow diamonds). Figure 7. Correlation function of Eq.(6) at 420 K (blue), 425 K (red), and 430 K (yellow) for PS (bulk T g =425 K). The open symbols use the step function density profile, the closed symbols use the ramp profile. Inset: The extracted short (squares) and long (diamonds) relaxation times using Eq.(7), and the relaxation time extracted using a single KWW function (circles). Open symbols are for the step function interfacial profile, close symbols are for the ramp profile. Figure 8a. Relaxation time profile at the film pseudo-thermodynamic T g versus the normalized distance from the film center for films of thickness 3.6 nm (blue, T g =361K), 6 nm (red, T g =385K), 12 nm (yellow, T g =403K), and 36 nm (green, T g =417K). Note that each curve is at a different absolute temperature. Inset: The film-averaged relaxation time at the film T g (h) plotted versus film thickness. Blue circles use the pseudo- 39

40 thermodynamic T g, red squares use the dynamic T g. Note that each point represents a different temperature. The two dashed lines though the blue circles represent the effective power law behavior discussed in the text. Figure 8b. Relaxation time profile at the film dynamic T g as a function of reduced location in the film for films of thickness 3.6 nm (blue, T g =391K), 6 nm (red, T g =408K), 12 nm (yellow, T g =418K), and 36 nm (green, T g =422.5K). Note that each curve is at a different temperature. Figure 9. Relaxation time profile for a 12 nm thick PS film at its pseudo-thermodynamic T g using the simple step (blue circles), two step (red squares), and ramp (yellow diamonds) interfacial density profiles. The corresponding temperatures are 403 K, 383 K, and 383 K, respectively. Figure 10. The fraction of glassy segments as a function of film thickness. The points use only the local mean relaxation time to define mobility, the solid curves include the Poisson model of dynamic heterogeneity. The blue curve (circles) are at the pseudothermodynamic T g 5K, the red curve (squares) is at the pseudo-thermodynamic T g, the yellow curve (diamonds) is at the pseudo-thermodynamic T g +5K, and the green curve (triangles) is at the pseudo-thermodynamic T g + 10K. The film temperature changes with thickness. Inset: An example of the probability that a tagged segment has a relaxation time larger than (T;z;h), here taken to be 100 s. The curve employs a local Poisson distribution of relaxation times, while the dashed line corresponds to no local relaxation time distribution (delta function at the mean alpha time). Figure 11. The spatially-resolved high frequency glassy modulus at the pseudothermodynamic film T g versus the normalized location in the film (as measured from its center) for PS films of thickness 3.6 nm (blue), 6 nm (red), 12 nm (yellow), and 300 nm (green). The corresponding temperatures of the films of are 361 K, 385 K, 403 K, and 425 K, respectively. The modulus is normalized by its value in the bulk material at the bulk glass transition. Inset: Spatial gradient of the local frequency-dependent modulus of 40

41 a film of thickness 12 nm at the pseudo-thermodynamic T g =403 K for frequencies of 10-4 (blue circles), 0.01 (red squares), 1 (yellow diamonds) and 100 Hz (green triangles). Figure 12. The isothermal (at the bulk T g = 425 K) film-averaged shear modulus divided by its bulk value as a function of film thickness for frequencies of (from bottom to top) 1 (blue circles), 100 (red squares), 1000 (yellow diamonds), 10 6 (green triangles), 10 9 (gray triangles), (open circles) Hz. Figure 13. The film-averaged shear modulus at the film dynamic T g as a function of film thickness for frequencies of (from bottom to top) 0.1 (blue circles), 1 (red squares), 1000 (yellow diamonds), 10 6 (green triangles), 10 9 (gray triangles), and (open circles) Hz. Inset: Analog of the main frame but at the film pseudo-thermodynamic T g as a function of film thickness for frequencies of (from bottom to top): 0.01 (blue circles), 0.1 (red squares), 1 (yellow diamonds), 1000 (green triangles), 10 6 (gray triangles), 10 9 (open circles), and (open squares) Hz. 41

42 Figures Figure 1. Temperature dependence of the bulk alpha relaxation time for (from least to most fragile) sorbitol (blue circles, T g =362K, m=51), glycerol (red squares, T g =202K, m=58), orthoterphenyl (yellow diamonds, T g =267K, m=81), polystyrene (green triangles, T g =425K,m=89), and polymethylmethacrylate (gray triangles, T g =408K, m=96). Inset: Schematic of the bulk ECNLE alpha relaxation process composed of a large amplitude local hopping motion coupled to a longer range, collective, harmonic elastic motion. 42

43 Figure 2. Alpha relaxation time profile plotted as a function of the normalized distance from the film center at the bulk T g of PS films of thickness (from inner to outer ) 6 nm (blue), 12 nm (red), 18 nm (yellow), 36 nm (green), and 60 nm (gray). Inset: Schematic of relaxation in a free standing thin film. 43

44 Figure 3. Profile of local vitrification temperatures normalized to their bulk values for films of thickness 15d as a function of distance from the film center (z) in units of particle diameter. No adjustable parameter calculations are shown for (from least to most fragile) sorbitol (blue circles), glycerol (red squares), OTP (yellow diamonds), PS (green triangles), and PMMA (gray triangles). Inset. Change of the pseudo-thermodynamic T g as a function of film thickness (in nm) for films of glycerol (blue circles), OTP (red squares), PS (yellow diamonds), and PMMA (green triangles). 44

45 Figure 4. Same as the inset of Figure 3, but plotted versus film thickness in units of the particle diameter. Inset: Pseudo-thermodynamic T g, normalized tog(red squares, =1), OTP (yellow diamonds, =1.1), PS (green triangles, =1.15), and PMMA (gray triangles, =1.05). The scaling parameter has been chosen to best collapse all the curves (arbitrarily relative to glycerol). 45

46 Figure 5. The function (z) of Eq.(12) plotted versus normalized location in the film. Colors correspond to the interfacial density profiles shown in the inset. Inset: Model interfacial density profiles. Blue (upper) curve is the single step function, red curve is a two step functional form with a thickness, and the yellow curve is a ramp profile with thickness of. Here we set d, one particle diameter. 46

47 Figure 6. The film-averaged glass transition temperature as a function of film thickness in nanometers. The circles show the pseudo-thermodynamic results, the squares show the dynamic results. The closed symbols employ the prior [31] step function profile for the interfacial density, the open symbols use the ramp profile; the two step profile model gives results essentially identical to that obtained using the ramp profile. The solid curves are fits to the open symbols of corresponding color to Eq.(9); the blue curve has parameters a = 1.56 and = 1.07, and the red curve has a = 1.12 and = Inset: The alpha relaxation time profile across a PS film at 400 K using the simple step profile (blue circles), the two step profile (red squares), and the ramp profile (yellow diamonds). 47

48 Figure 7. Correlation function of Eq.(6) at 420 K (blue), 425 K (red), and 430 K (yellow) for PS (bulk T g =425 K). The open symbols use the step function density profile, the closed symbols use the ramp profile. Inset: The extracted short (squares) and long (diamonds) relaxation times using Eq.(7), and the relaxation time extracted using a single KWW function (circles). Open symbols are for the step function interfacial profile, close symbols are for the ramp profile. 48

49 Figure 8a. Relaxation time profile at the film pseudo-thermodynamic T g versus the normalized distance from the film center for films of thickness 3.6 nm (blue, T g =361K), 6 nm (red, T g =385K), 12 nm (yellow, T g =403K), and 36 nm (green, T g =417K). Note that each curve is at a different absolute temperature. Inset: The film-averaged relaxation time at the film T g (h) plotted versus film thickness. Blue circles use the pseudothermodynamic T g, red squares use the dynamic T g. Note that each point represents a different temperature. The two dashed lines though the blue circles represent the effective power law behavior discussed in the text. 49

50 Figure 8b. Relaxation time profile at the film dynamic T g as a function of reduced location in the film for films of thickness 3.6 nm (blue, T g =391K), 6 nm (red, T g =408K), 12 nm (yellow, T g =418K), and 36 nm (green, T g =422.5K). Note that each curve is at a different temperature. 50

51 Figure 9. Relaxation time profile for a 12 nm thick PS film at its pseudo-thermodynamic T g using the simple step (blue circles), two step (red squares), and ramp (yellow diamonds) interfacial density profiles. The corresponding temperatures are 403 K, 383 K, and 383 K, respectively. 51

52 Figure 10. The fraction of glassy segments as a function of film thickness. The points use only the local mean relaxation time to define mobility, the solid curves include the Poisson model of dynamic heterogeneity. The blue curve (circles) are at the pseudothermodynamic T g 5K, the red curve (squares) is at the pseudo-thermodynamic T g, the yellow curve (diamonds) is at the pseudo-thermodynamic T g +5K, and the green curve (triangles) is at the pseudo-thermodynamic T g + 10K. The film temperature changes with thickness. Inset: An example of the probability that a tagged segment has a relaxation time larger than (T;z;h), here taken to be 100 s. The curve employs a local Poisson distribution of relaxation times, while the dashed line corresponds to no local relaxation time distribution (delta function at the mean alpha time). 52

53 Figure 11. The spatially-resolved high frequency glassy modulus at the pseudothermodynamic film T g versus the normalized location in the film (as measured from its center) for PS films of thickness 3.6 nm (blue), 6 nm (red), 12 nm (yellow), and 300 nm (green). The corresponding temperatures of the films of are 361 K, 385 K, 403 K, and 425 K, respectively. The modulus is normalized by its value in the bulk material at the bulk glass transition. Inset: Spatial gradient of the local frequency-dependent modulus of a film of thickness 12 nm at the pseudo-thermodynamic T g =403 K for frequencies of 10-4 (blue circles), 0.01 (red squares), 1 (yellow diamonds) and 100 Hz (green triangles). 53

54 Figure 12. The isothermal (at the bulk T g = 425 K) film-averaged shear modulus divided by its bulk value as a function of film thickness for frequencies of (from bottom to top) 1 (blue circles), 100 (red squares), 1000 (yellow diamonds), 10 6 (green triangles), 10 9 (gray triangles), (open circles) Hz. 54

55 Figure 13. The film-averaged shear modulus at the film dynamic T g as a function of film thickness for frequencies of (from bottom to top) 0.1 (blue circles), 1 (red squares), 1000 (yellow diamonds), 10 6 (green triangles), 10 9 (gray triangles), and (open circles) Hz. Inset: Analog of the main frame but at the film pseudo-thermodynamic T g as a function of film thickness for frequencies of (from bottom to top): 0.01 (blue circles), 0.1 (red squares), 1 (yellow diamonds), 1000 (green triangles), 10 6 (gray triangles), 10 9 (open circles), and (open squares) Hz. 55

56 τ α (sec) T g T

57 τ α (z;h) z h