Three-Layer Model for the Emergence of Ultrastable Glasses from the

Article pubs.acs.org/JPCB

Three-Layer Model for the Emergence of Ultrastable Glasses from the Surfaces of Supercooled Liquids Jayachandra Hari Mangalara, Michael D. Marvin, and David S. Simmons* Department of Polymer Engineering, The University of Akron, 250 South Forge Street, Akron, Ohio 44325-0301, United States ABSTRACT: Ultrastable glasses produced by vapor deposition exhibit properties consistent with glasses that have been aged for thousands of years or more. These materials’ properties are believed to emerge from the presence of a mobile layer at the surface of supercooled liquids that allows access to lower-energy states. However, the precise mechanism by which this enhanced mobility is translated into ultrastable glass behavior remains incompletely understood. Here we show that enhanced densities and stabilities consistent with ultrastable glasses specifically can emerge as a result of a mismatch in the length scales of thermodynamic and dynamic gradients at the surfaces of equilibrium supercooled liquids. In particular, ultrastable glass properties can be understood within a three-layer model of the interface in which a “facilitated layer” intermediate between the surface and bulk exhibits bulk-like liquid-state density but suppressed Tg. This mismatch in length-scale has previously been correlated with the scale of cooperative rearrangements in the supercooled state, suggesting that ultrastable glasses may be a direct consequence of the cooperative nature of dynamics in equilibrium supercooled liquids.

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formulation of free volume theory 21 suggests that ⎡ qv ⎤ τ = τ0 exp⎣⎢ v −0v ⎦⎥, where τ is segmental relaxation time, τ0 ̅ 0 and q are constants, v0 is some minimal occupied or excluded volume, and v ̅ is the specific volume. Within a free volume perspective of near-surface dynamics, this (or some other similar) relationship is expected to hold for dynamics near the free surface, with the enhanced mobility simply reflecting a reduced local density. Since these models preserve the bulkstate relationship between relaxation time and density or free volume, they suggest that enhanced surface mobility should not lead to an enhanced glass-state density, but simply to trapping in the bulk-state glassy density at a lower temperature. We therefore highlight two key open questions regarding the origin of ultrastable glasses: (1) Why does the surface region of enhanced stability allow access to higher densities, rather than simply freezing in the bulk glassy density at a lower temperature? (2) Do these materials leverage an already-existing region of enhanced density present at the surfaces of quiescent glassy films or are the high densities and stabilities found in these materials created by the deposition process itself? Here we suggest a more detailed mechanistic understanding of the origin of the extraordinarily high thermodynamic and kinetic stabilities observed in ultrastable glasses. This understanding indicates that a layer of enhanced glass-state density exists near the surface of quiescent liquids as a result of a mismatch in the range over which thermodynamic properties

INTRODUCTION The last several years have seen the emergence of a new class of glassy materials“ultrastable” glasses produced by vapor deposition. These materials exhibit unusual kinetic and thermodynamic stability, with enhanced densities, enhanced moduli,1,2 suppressed energies, and increased softening temperatures. Estimates suggest that these enhancements would require tens to thousands of years to obtain via a typical temperature quench into the glass state. Because of these extraordinary properties and speculation that they may closely approach a posited “ideal glass” state,3,4 these materials have been the focus of great attention.2,3,5−8 Experimental and simulation9,10 studies have attributed ultrastable glass properties to the presence of a surface layer of enhanced mobility. This view is commensurate with longstanding “two-layer” models for the surface behavior of glassforming liquids, wherein a thin layer of enhanced mobility can be found immediately at the surface before giving way to bulklike material.11 This mobile surface layer is posited to allow access to configurations lower on the material’s potential energy landscape, enabling access to higher densities and lower energies than are accessible via normal melt quenches.3,12,13 However, the precise mechanism by which this surface mobility is translated into high-stability glasses remains unsettled. In particular, the enhancement in glassy density found in ultrastable glasses is not consistent with some of the most influential models of enhanced surface mobility during glass formation, which view the shift in near-surface dynamics as directly emerging from an enhancement in local free (or specific) volume14−17 or segmental rattle space18,19 near the interface. These perspectives are conceptually similar to the idea of the surface layer as possessing an enhanced “rheological temperature” relative to the bulk.20 For example, the Doolittle © XXXX American Chemical Society

Received: May 10, 2016

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higher density in this region than in the bulk, with the density enhancement given by

and mobility are altered near a free surface in supercooled liquids, as described below. This mismatch allows the system to locally circumvent the usual relationship between density and mobility to achieve glass-state densities greater than bulk. The observation of this layer of enhanced density at the surface of a model quiescent film supports the view that vapor deposition processes leverage the equilibrium behavior of surfaces in supercooled liquids and that deposition-specific effects, such as induced molecular orientation,22,23 are not the central mechanism driving the high stabilities of these materials.

⎛ ∂ρ ρ(T ) − ρB (T ) = ⎜⎜ ⎝ ∂T f

− liquid

∂ρ ∂T

⎞ ⎟[Tgf − Tgb] ⎟ glass ⎠

(1) b

where Tg is a mean Tg of the facilitated layer and Tg is the mean Tg in the bulk. This enhancement in density will naturally be tracked by a reduction in energy following an analogous equation in the specific heat, consistent with observations in ultrastable glasses. This view is consistent with recent experimental work indicating that vapor-deposited glasses modestly below Tg indeed exhibit thermodynamic properties consistent with the supercooled liquid state.40,41

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PROPOSED MODEL Several decades of evidence suggest that Tg and segmental relaxation time vary over a range on the order of 10 nm or more near the surface of many glass-forming liquids.11,24−31 Evidence to this effect includes shifts in overall film Tg, direct observations of gradients in the glass transition temperature32−34 and relaxation dynamics,35,36 and predictions of long-range gradients from computer simulation.37−39 By contrast, gradients in density ρ at liquid/gas interfaces far from the vapor−liquid critical point are generally much shorterranged, on the order of 1 nm.38,39 An apparently unrecognized implication of this mismatch in dynamic and thermodynamic interfacial length scales, reflecting a breakdown of the free-volume-layer view,39 is that the standard two-layer model of film dynamics outlined above is inadequate to describe the surfaces of supercooled liquids. Instead, such a material is minimally divided into three layers, as shown in Figure 1: a layer immediately at the surface with

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COMPARISON TO SIMULATION To illustrate this effect more clearly, we employ data from previously published simulations of a freestanding film of a linear bead−spring polymer.42 The methodological details of this simulation can be found in our earlier publication. In summary, the simulations consist of an approximately 20 nm thick film (employing a common conversion of one LennardJones (LJ) distance unit to approximately one nanometer43−45) of unentangled polymer chains, subject to a quench toward the glassy state. To test the three-layer model proposed above, we quantify dynamic and structural properties of the film as a function of distance from the interface, defined here as the point in the surface density gradient at which the density is half that at the center of the film. As illustrated by Figure 2, the system behaves as expected in the equilibrium liquid state: the surface layer exhibits a strong

Figure 1. Schematic of three-layer model for near-free-surface dynamics in glass-forming liquids, as described in the text. ρl and ρg denote liquid- and glass-state densities, respectively.

suppressed Tg and liquid-state ρ, a layer far from the interface possessing bulk Tg and liquid-state ρ, and an additional intermediate layer with bulk-like liquid-state ρ but suppressed Tg. We denote this as the “facilitated” layer, because, as discussed below, its liquid-state dynamics are facilitated by proximity to the free surface without it exhibiting a corresponding density suppression. In other words, this layer locally violates the freevolume-layer model expectation that local dynamics should follow local density. This facilitated layer has a unique property that is critical to the understanding of ultrastable glasses produced by vapor deposition. In the liquid state, the facilitated layer essentially follows the bulk-state density equation of state. However, its Tg is lower than that in the bulk. Therefore, at temperatures below the bulk Tg, the density of this layer will continue to increase according to the liquid-state thermal expansion coefficient, while the bulk thermal expansion coefficient drops to its glassy value. This naturally leads to a

Figure 2. Mean density (solid lines, averaged across two interfaces each for four trials) as a function of temperature for regions 0.875 LJ distance units thick, at positions z from the interface, located in the surface layer, intermediate layer, and bulk layer of the simulated free interface. Points denote the ρ data point closest to the glass transition temperature of each region (with a resolution of 0.02 in LJ temperature units). The dashed lines are extrapolations of this data from the glass transition temperature of each region, following the temperature dependence of density in the bulk glass.

suppression in density, while the facilitated layer exhibits density essentially equal to the bulk layer. However, Tg in the facilitated layer, defined for these simulations based on a common convention as the temperature at which the simulations begin to fall out of equilibrium on a time scale of 103 LJ time units, exhibits a suppression from bulk. For this reason, the facilitated layer continues to increase in density after B

DOI: 10.1021/acs.jpcb.6b04736 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B the bulk has fallen into the glassy state. We can assess the implications of this for the glass-state density by assuming that the glass-state density follows the bulk glass-state temperature dependence (local density data in the glassy state tends to be overwhelmed by noise in simulations of this size). As shown by this figure, this local failure of the bulk relationship between mobility and density naturally leads to an enhanced glass-state density, relative to bulk, of the facilitated layer. This outcome is emphasized by Figure 3 (rendered in part using the Visual Molecular Dynamics Software Package46),

The key properties of ultrastable glasses can therefore be explained by a model in which vapor-deposition methods grow the facilitated layer in a layer-by-layer fashion without exposing it to high enough temperatures to perturb its anomalously highdensity glassy state. Once this layer is buried in the bulk, it will exhibit a fictive temperature of roughly Tgfthe Tg of the facilitated domain. Calculations of the equivalent quench or annealing time will then indicate that the material corresponds to a bulk glass formed via a temperature quench to yield Tg = Tgf. Measurements of Tg in highly confined materials suggest that Tg in this domain can be suppressed by up to 50−80 K in many cases,48 making reports of fictive temperature suppressions of 30 K or more (and corresponding equivalent aging times in the tens to thousands of years) in ultrastable glasses12 quite consistent with this mechanism. Finally, we note that this more detailed mechanism is also qualitatively consistent with the experimentally observed temperature and rate dependences of ultrastable glass formation, wherein excessively high rates of vapor deposition fail to produce ultrastable glasses and the optimum temperature for deposition is commonly found to be several tens of Kelvin below the bulk Tg. Specifically, previous work has found that the dynamic gradient that is central to this model becomes smaller at high rates of quench into the glassy state.20,36,38,39,49−53 Since the effective thermal quench rate of the deposited material depends on the temperature of the substrate and the deposition rate, excessively low deposition temperatures or high deposition rates likely probe an effective high-quench-rate regime in which the facilitated layer disappears. Conversely, excessively high substrate temperatures do not fully leverage the Tg difference between the bulk and facilitated layers, effectively reducing the temperature difference in eq 1 and thereby muting the magnitude of the overall effect. This model suggests, in fact, that the optimum deposition temperature should be near the Tg of the facilitated layer, since lower temperatures will rapidly vitrify the facilitated layer without benefiting from any further liquid-like densification. Evidence on the Tg of this layer is sparse, but experiments in very thin polymer films suggest that a Tg suppression on the order of 50 K is likely in this region. This is reasonably consistent with experimental optimum temperatures of deposition on the order of 15% below Tg1. For example, in the case of trinaphthylbenzene, the optimum deposition temperature is reported to be near Tg − 50 K, relative to a bulk Tg of 347 K.12

Figure 3. Surface gradients with respect to distance z from the interface of the anomaly from bulk of Tg, liquid-state density, at local Tg,

ρ(z , T = 1.35Tgb) ρbulk (T = 1.35Tgb)

ρ(z , T = Tg(z)) ρbulk (T = Tgb)

Tg(z) Tgb

− 1, (red triangles);

− 1, (blue data points); and density

− 1, (black data points). Distance is in LJ

units, with 1 unit corresponding approximately to one nanometer. Vertical dashed lines denote approximate boundaries between surface (S), facilitated (F), and bulk-like (B) layers described in Figure 1 and the text. Error bars are standard deviations determined based on 8 interfaces (2 each for 4 trials). The background image shows a section of the interface with beads colored according to the mean local density at their local Tg, based on a smoothed fit of position to the black data points, with white indicating bulk-like density, red indicating reduced density, and blue indicating enhanced density.

which compares spatially resolved near-surface gradients in three properties: the local anomaly from bulk of Tg,

Tg(z) Tgb

− 1,

the local anomaly from bulk of density at a fixed temperature in the liquid state,

ρ(z , T = 1.35Tgb) ρbulk (T = 1.35Tgb)

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− 1, and the anomaly of the local

CONCLUSIONS We have proposed and tested versus simulation an expanded three-layer model for the surfaces of supercooled liquids. This model suggests the following answers to the open questions highlighted above regarding the origin of ultrastable glasses produced by vapor deposition. Why does the surface region of enhanced stability allow access to higher densities, rather than simply f reezing in the bulk glassy density a lower temperature? This model suggests that interfacial mobility enhancement is a necessary but not sufficient condition for achieving ultrastable glasses. A surface mobility gradient that simply reflected a local density gradient (as in free-volume models of near-surface dynamics14−17) would indeed yield a lower local Tg without enhancing the glassstate density, consistent with the “enhanced rheological temperature” picture that has sometimes been presented for supercooled liquid surfaces in the literature.20 The key

density at the local Tg as compared to that of the bulk system at its Tg,

ρ(z , T = Tg(z)) ρbulk (T = Tgb)

− 1. This figure confirms the three-layer

picture illustrated schematically in Figure 1. The facilitated layer exhibits nearly bulk-like density in the liquid state combined with a suppressed Tg, leading to a peak density enhancement of 1.5% at the layer Tg relative to the bulk density at the bulk Tg. Notably, the magnitude of this density enhancement is consistent with that observed in experimental ultrastable glasses.12 In these simulations, the breadth of this layer is of order 4 nm. Prior work has indicated that the range of the dynamic gradient driving this region continues to grow on further cooling, with extrapolations to the experimental-time scale Tg suggesting a thickness on the order of 10 nm.38 A thickness in this range of order four to ten nm is generally consistent with experimental measurements of the range of near-interface gradients in Tg and near-Tg dynamics.32,47 C

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by vapor deposition should be interpreted as a new piece of evidence for the presence of medium-ranged dynamic correlations57 in supercooled liquids.

ingredient in circumventing this very general density/mobility coupling (which is a key underpinning of the common empirical success of free volume models of glass formation in bulk polymers54) is the presence of a nanoscale “facilitated” region in which mobility is enhanced but density is essentially bulk-like. From this perspective, the very existence of ultrastable glasses produced by vapor deposition may provide a new piece of evidence supporting the existence of long-ranged gradients in dynamics at the surfaces of glass-forming liquids, corresponding to a local failure of free-volume relationships between density and mobility. Do these materials leverage an already-existing region of enhanced stability present in ultrastable glasses, or are the high densities and stabilities found in these materials created by the deposition process itself? These simulation data and reasoning suggest that an enhancement in density, suppression in energy, and enhancement in kinetic stability consistent with ultrastable glasses can emerge as a natural consequence of an equilibrium phenomenona mismatch in the range of thermodynamic and dynamic gradients at free surfaces of quiescent supercooled liquids. While deposition-specific effects such as induced molecular orientation may play a role in the details of a given ultrastable glass, these details are evidently not necessary to realize the enhanced kinetic and thermodynamic stability that most essentially characterize ultrastable glasses in general. This finding is consistent with prior simulation55 and experimental22,23,56 work finding that ultrastable glass formation does not universally correlate with formation of a particular anisotropic molecular orientation. If correct, these findings also suggest that an enhancement in density and reduction in specific energy consistent with those observed in ultrastable glasses should be present in very thin (order 10 nm or less) freestanding films well below their Tg. Metrology of such thin films remains a major challenge; however, these results suggest that these types of measurements could be of considerable value in providing new understanding of near-interface dynamics in supercooled liquids. These findings naturally raise an additional question: how can the facilitated layer exhibit bulk-like density but dramatically accelerated dynamics? Previously published evidence suggests that surface mobility enhancements may propagate into the material beyond the surface density gradient via cooperative rearrangements that are predicted by the theory of Adam and Gibbs57 to universally underpin dynamics in nonArrhenius supercooled liquids.31,38,53,58 Recent simulation results in model systems comparable to the one considered here have provided strong evidence for this picture, indicating that the range of the dynamic interface is linearly related to the size of these cooperatively rearranging regions.31,38,53,58 In this view, dynamic cooperativity enables the facilitated layer to benefit from the mobility of the free surface well beyond the range of the surface density suppression. A similar conclusion (albeit grounded in a very different physical model) is suggested by the recently developed elastically collective nonlinear Langevin equation theory for dynamics in thin films, which suggests that the absence of an elastic medium beyond the film surfaces lowers a long-ranged elastic barrier to collective relaxation over a range exceeding that of interfacial density alterations.59,60 The present results indicate that the decoupling of local density/mobility relationships suggested by both of these theories can account for the high densities and stabilities realized in ultrastable glasses. Viewed another way, these results indicate that the realization of ultrastable glasses

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. DMR1310433. The authors thank Bryan Vogt and Mark Ediger for helpful discussions and editorial advice.

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