Studying the local glass transition temperature Tg across a boundary, we investigate the characteristic length scales of cooperative dynamics. High molecular weight polymers have a large separation in time scales between cooperative segmental motion (α-relaxation) and chain diffusion allowing us to measure the local Tg(z) profile across a glassy-rubbery interface of polystyrene/poly(n-butyl methacrylate) using fluorescence. We find this profile in cooperative dynamics does not correlate with the 7-nm wide symmetric composition profile of the interface, but instead is very broad, spanning 350-400 nm from one bulk Tg value to another, and highly asymmetric, extending further into the glassy side.

Glassy materials reveal a rich dynamic heterogeneity, both spatially and temporally, on cooling close to the glass transition temperature (Tg).1,2 In this supercooled region, local rearrangements are thought to be facilitated through cooperative motion amongst neighboring units, such that the dynamics are coupled to their surrounding environment.3,4 Upon vitrification, the glass former enters a dynamically frustrated state where the available thermal energy is insufficient to activate this cooperative motion among multiple constituents. The length scales and mechanisms underlying these interactions that induce cooperative motion are still highly debated.1,3,5

One promising approach to understanding cooperative dynamical length scales associated with the glass transition is to introduce a local perturbation to the system, typically a boundary with faster or slower dynamics, and investigate how this perturbation alters the local dynamics. Variations on this type of approach have been treated both experimentally6–10 and theoretically,11–17 particularly investigating glass formers confined to nanoscopic dimensions.6–9,11,18,19 Because of their ubiquity and technological importance, the majority of such studies have been done on thin polymer films. One can separate the effects of a perturbing interface into two parts: (1) Molecules or segments located at the interface have their dynamics enhanced or suppressed, depending on the particular interaction. For example, at a free surface, the dynamics are typically enhanced, believed to result from the partial absence of contacts with surrounding units.13,14,20 (2) This difference in local mobility at the interface is then propagated into the material by some mechanism. For dynamics connected to the glass transition, there is evidence that this mechanism is associated with cooperative motion.10,17,21,22 This central idea suggests that the study of how a local dynamical perturbation to Tg propagates into the material can provide insight into the nature of how cooperative motion is coupled and transferred across neighboring cooperatively rearranging regions (CRRs). A single CRR, an idea originally introduced by Adam and Gibbs,23 is defined as the minimum number of units necessary to undergo a collective local rearrangement.3 Most estimates place the size of a single CRR at approximately 3 nm for typical glass formers, including polymers.2,24,25 Yet studies of glassy dynamics in nanoconfined systems suggest that dynamical perturbations of an interface persist much deeper into the material than a single CRR, several tens of nanometers in some cases before local bulk-like dynamics are recovered.8,10,26 Such studies have primarily focused on the presence of a free surface (air-material interface) as the source of dynamical perturbation to the system. However, the difficulty with determining a dynamical length scale when investigating the free surface is how to theoretically quantify the magnitude of the enhanced dynamics at the free surface.13,14,20,27,28 Some studies have estimated the free surface Tg at ∼300 K,29,30 while others have suggested that the free surface may always be liquid-like.31–33 To avoid these ambiguities of how to treat the free surface, we instead investigate here how the local Tg and associated dynamics transition from one well-defined Tg value to another.

In the present study, we take advantage of the large separation of time scales in high molecular weight polymers between cooperative segmental motion (α-relaxation) and chain diffusion. For example, at temperatures of Tg  + 15-20 K, chain diffusion occurs on time scales of many hours to days for molecular weights of 700-2000 kg/mol,34–36 while the α-relaxation time is on the order of milliseconds at such temperatures.37 This decoupling of local vs. global chain dynamics in polymers is well known and has been frequently exploited. Here, we employ this strategy to create an interface with a large 80 K step-change in Tg from one side to the other between two dissimilar polymers: polystyrene (PS) with bulk glass transition temperature T g bulk PS = 100 ° C , and poly(n-butyl methacrylate) (PnBMA) with T g bulk PnBMA = 20 ° C . We utilize a fluorescence method to measure the local Tg of a layer a fixed distance z from this interface, whose chains have been covalently tagged with pyrene dye. By assembling multilayer stacks of spin-coated polymer films with specific layer thicknesses via a water transfer process, and subsequently annealing them carefully to consolidate the sample into a single material while still limiting macroscopic diffusion of the high molecular weight chains, it is possible to create samples where the fluorescence signal originates from a 10-15 nm thick layer at a known distance z from the interface. Iterating this process to create many samples with varying values of z allows us to measure the local Tg(z) at different distances from this interface. As we vary z, the local Tg(z) measured must clearly transition from T g bulk PS = 100 ° C far from the interface on one side to T g bulk PnBMA = 20 ° C far from the interface on the other. Surprisingly, we find that the local Tg(z) takes approximately 350-400 nm to transition between these two extremes, with an asymmetric profile propagating deeper into the glassy PS side.

Figure 1 depicts the layer structures assembled for the different samples. The basic structure consists of two semi-infinite slabs (each >450 nm in thickness) to create a well-defined 7-nm interface between the high-Tg glassy polymer (PS) and low-Tg rubbery polymer (PnBMA). A 10-15 nm thick probe layer of either PS or PnBMA is inserted, where fluorescent pyrene has been covalently attached along the backbone of the chain at a level of one dye for every 70-145 monomers. On the glassy PS side, we use pyrene-labeled PS (Mw = 672 kg/mol, Mw/Mn = 1.3, with 1.4 mol. % pyrene), whereas on the rubbery PnBMA side, we use pyrene-labeled PnBMA (Mw = 1400 kg/mol, Mw/Mn = 1.7, with 0.7 mol. % pyrene). To vary the distance of the probe layer to the PS/PnBMA interface, we vary the thickness z of a neat (unlabeled) PS (Mw = 1920 kg/mol, Mw/Mn = 1.26) or PnBMA (Mw = 1210 kg/mol, Mw/Mn = 1.7) spacer layer next to the interface (see Fig. 1). Each layer is individually spin-coated to desired thickness and annealed prior to successively floating each layer atop the next. The annealing time and temperature above Tg of the multilayer stacks are carefully controlled to ensure that the PS/PnBMA interface reaches its equilibrium interfacial thickness, while still limiting the macroscopic diffusion of the pyrene-labeled layers to only a few nanometers. When measuring the low Tg PnBMA side, this necessitated assembling and annealing the multilayer structures of Fig. 1(b) in two stages as annealing the PS/PnBMA interface to equilibrium requires going above the bulk Tg of PS. (Explicit details of the sample preparation and annealing protocols for each type of sample are explained in the supplementary material.38) After annealing, this weakly immiscible system forms a stable, well-defined polymer-polymer interface of 7 nm for such high molecular weight polymers.39–43 The interfacial width w I 1 6 χ formed between two high-molecular weight polymers is determined by the unfavorable interaction parameter χ,36 which has been well characterized for PS/PnBMA in agreement with the experimentally measured interfacial width.39,42,43 Thus, this protocol creates a stable sample with an 80 K step-change in local Tg from the glassy-PS side to the rubbery-PnBMA side, whose geometry remains effectively static during the course of the experiment.40,41

FIG. 1.

Schematic illustrating sample geometries comprised of four individually spin-coated layers of either polystyrene (PS) or poly(n-butyl methacrylate) (PnBMA) assembled and annealed to form a consolidated material with a 10-15 nm thick pyrene-labeled layer located at a variable distance z from the PS/PnBMA interface (7-nm wide). High molecular weight polymers ensure that assembled morphology remains static throughout the experiment.

FIG. 1.

Schematic illustrating sample geometries comprised of four individually spin-coated layers of either polystyrene (PS) or poly(n-butyl methacrylate) (PnBMA) assembled and annealed to form a consolidated material with a 10-15 nm thick pyrene-labeled layer located at a variable distance z from the PS/PnBMA interface (7-nm wide). High molecular weight polymers ensure that assembled morphology remains static throughout the experiment.

Close modal

Use of pyrene-labeled polymers allows us to perform fluorescence measurements to determine the local Tg(z) of this layer at a given distance z from the interface. Numerous previous studies have established that the fluorescence intensity of pyrene (when covalently attached as 1-pyrenylbutyl methacrylate monomer) provides a good measure of Tg in agreement with other experimental measures such as differential scanning calorimetry and ellipsometry in thin films.8,9,40,44 Pyrene fluorescence intensity exhibits a different linear temperature dependence in the glassy and rubbery regimes such that Tg can be identified from the intersection of two linear fits as the temperature at which this change in slope occurs. It is well known that pyrene fluorescence intensity and spectral shape vary with local polarity and mobility of the material surrounding the dye.45 The sensitivity of pyrene to local Tg derives from the change in the relative amount of radiative vs. non-radiative decay of the dye as a function of temperature affected by the local density and rigidity of the surrounding polymer matrix.8,46 At higher temperatures, the higher mobility of the surrounding matrix increases the amount of non-radiative decay, showing a stronger temperature dependence in the melt region above Tg. Specific details of how we perform these fluorescence measurements have been previously reported40 and are outlined in the supplementary material.38 In short, we monitor the fluorescence intensity, every 30 s on cooling at 1 °C/min, at a single wavelength corresponding to the first peak in the pyrene spectra,40 which is known to show the most sensitivity to its local environment.45 

Figure 2 plots the normalized fluorescence intensity as a function of temperature for four different samples, each with the pyrene-labeled layer located at a different position z from the glassy-rubbery PS/PnBMA interface. Each dataset exhibits a break in the linear temperature-dependence of the fluorescence intensity identifying the local glass transition temperature Tg(z) at that distance z from the interface. The top curves were collected using a pyrene-labeled PS layer located on the glassy-PS side at a distance: z = 356 nm far from the PS/PnBMA interface reporting Tg = 99 ± 2 °C consistent with bulk PS, and z = 64 nm closer to the interface reporting a significantly reduced Tg = 63 ± 2 °C. Remarkably, we find the local Tg(z) on the PS side to be reduced by more than 35 K from its bulk value at a distance still more than 60 nm from the PS/PnBMA interface. The bottom curves were collected using a pyrene-labeled PnBMA layer located on the rubbery-PnBMA side at a distance: z = − 160 nm far from the PS/PnBMA interface reporting Tg = 20 ± 2 °C consistent with bulk PnBMA, and z = − 36 nm closer to the interface reporting Tg = 29 ± 2 °C. Each dataset displayed only a single Tg value for each location z, even when the temperature range was significantly extended. This measurement has been repeated with many different samples, varying z to map out the profile in local Tg(z) across the PS/PnBMA interface.

FIG. 2.

Fluorescence intensity as a function of temperature collected at 1 °C/min for four different samples. (Top) Pyrene-labeled PS layer located at a distance z = 356 nm away from PS/PnBMA interface reporting local Tg(z) = 99 ± 2 °C consistent with bulk PS, and at a distance z = 64 nm reporting a significantly reduced Tg(z) = 63 ± 2 °C when closer to the PS/PnBMA interface. (Bottom) Pyrene-labeled PnBMA layer located at a distance of 160 nm from the PS/PnBMA interface reporting local Tg(z) = 20 ± 2 °C consistent with bulk PnBMA, and at a distance of 36 nm reporting local Tg(z) = 29 ± 2 °C when closer to the interface.

FIG. 2.

Fluorescence intensity as a function of temperature collected at 1 °C/min for four different samples. (Top) Pyrene-labeled PS layer located at a distance z = 356 nm away from PS/PnBMA interface reporting local Tg(z) = 99 ± 2 °C consistent with bulk PS, and at a distance z = 64 nm reporting a significantly reduced Tg(z) = 63 ± 2 °C when closer to the PS/PnBMA interface. (Bottom) Pyrene-labeled PnBMA layer located at a distance of 160 nm from the PS/PnBMA interface reporting local Tg(z) = 20 ± 2 °C consistent with bulk PnBMA, and at a distance of 36 nm reporting local Tg(z) = 29 ± 2 °C when closer to the interface.

Close modal

Figure 3 graphs all the local Tg(z) values measured as a function of the labeled-layer’s position z from the interface, where positive z values denote the glassy-PS side and negative z values denote the rubbery-PnBMA side. The Tg(z) values smoothly transition from the bulk Tg of PnBMA, average Tg(z < − 150 nm) = 20.9 ± 2.0 °C, to the bulk Tg of PS, average Tg(z > 250 nm) = 100.8 ± 2.0 °C. Two features of the data are surprising: (1) the transition from one bulk Tg value to the other occurs over a very large distance 350-400 nm and (2) the Tg(z) profile is asymmetric with the perturbation in local mobility at the interface propagating much further into the glassy-PS side than the rubbery-PnBMA side. To obtain a quantitative measure of these two factors, we have fit the Tg(z) data to a hyperbolic tangent,

T g z = T g av + 1 2 Δ T g tanh 2 z γ w ,
(1)

where the difference Δ T g = T g bulk PS T g bulk PnBMA = 79 . 9 K and average T g av = 1 2 ( T g bulk PS + T g bulk PnBMA ) = 60 . 8 ° C were determined from the asymptotes of the data at large |z| and held fixed during the fit. The best fit values for the profile width w = 231 ± 5 nm and asymmetry γ = 79 ± 3 nm demonstrate that the Tg-mobility gradient is broad and strongly biased towards the (positive z) glassy-PS side. Intriguingly, a recent limited mobility model by Tito et al.15 investigating local mobility profiles near free surfaces and glassy-rubbery interfaces predicted an asymmetric mobility profile biased towards the lower-mobility (glassy) side consistent with our experimental results. The infusion of enhanced mobility at a boundary propagates a significantly larger distance into glassy material.

FIG. 3.

Measured local Tg(z) as a function of the pyrene-labeled layer’s position from the PS/PnBMA interface (positive z = glassy-PS side, negative z = rubbery-PnBMA side). Tg(z)-profile fit to hyperbolic tangent (solid-blue curve) demonstrating mobility-gradient is broad and strongly biased toward glassy-PS side. Gray-dashed curve indicates local composition profile φ(z) with 7-nm interfacial width.

FIG. 3.

Measured local Tg(z) as a function of the pyrene-labeled layer’s position from the PS/PnBMA interface (positive z = glassy-PS side, negative z = rubbery-PnBMA side). Tg(z)-profile fit to hyperbolic tangent (solid-blue curve) demonstrating mobility-gradient is broad and strongly biased toward glassy-PS side. Gray-dashed curve indicates local composition profile φ(z) with 7-nm interfacial width.

Close modal

It is important to recognize that the Tg-mobility gradient does not correlate at all with the local composition profile ϕ(z) of the two polymers across the interface. The composition profile ϕ(z) ≈ tanh(2z/wI) of PS/PnBMA interfaces has been measured by Stamm et al.39 using neutron reflectivity. For PS/PnBMA layers of comparable thickness and annealing conditions to our own work, and with a slightly smaller molecular weight to the high molecular weights used in the present study, they found the interfacial width wI = 7 nm for PS/PnBMA.39 To illustrate this composition profile, we have added a gray-dashed curve to Fig. 3 with width wI = 7 nm. If the local cooperative segmental motion associated with Tg were independent on either side of the polymer-polymer interface, the local Tg(z) would follow the local composition profile transitioning sharply at z = 0. Consider a value of z ≈ 100 nm, the local PS segments exhibit a Tg reduced by ∼30 K from its bulk T g bulk PS value, despite all PnBMA segments being over 100 nm away. These results are in strong contrast with the present views of how local Tg is expected to correlate with local composition ϕ within a small region of a few nanometers.47–49 The long length scale over which the local Tg(z) is correlated demonstrates that cooperative segmental dynamics associated with Tg can be coupled over large distances and across polymer-polymer interfaces.

A central question associated with the glass transition is the spatial extent over which cooperative dynamics are correlated. Although most studies focus on the length scale associated with a single CRR, the size over which neighboring units undergo collective rearrangements typically of order a few nanometers, we are instead interested over what larger distance are dynamics correlated across neighboring CRRs. How far away must two CRRs be to exhibit completely independent motion? The results from Figure 3 suggest that this length scale, the distance from where one region exhibits bulk dynamics to the other, is 350-400 nm. As local rearrangements occur in one CRR, its neighbor is affected by this local disturbance at its boundary altering its dynamics, which in turn influences its neighbor. This idea of there existing a dynamical length scale larger than that associated with a single CRR has been incorporated into some theoretical models of the glass transition.1,3,12,13,50 A recent unified theory of activated relaxation by Mirigian and Schweizer4,51 provides some insight into why the influence of the perturbing interface has longer range effects on the glassy side. They added a nonlocal character to the α-relaxation event incorporating a cage expansion to facilitate local hopping in highly densified systems. The free-energy barrier for this cage expansion scales with the local elastic displacement field resulting in a length scale for collective rearrangements that grows with increasing rigidity (modulus) of the system.

Various theoretical models have been applied to the study of how a dynamical perturbation at an interface propagates into glassy material.12–15,27 To keep our analysis as general as possible, we choose to focus on the facilitated kinetic Ising spin model, originally billed as the simplest model of cooperative dynamics.52 The two-spin facilitated Ising model (2SFM) by Fredrickson and Andersen52,53 contains no equilibrium phase transition and no inherent static correlation length, yet with only nearest-neighbor dynamical interactions it displays a spectrum of relaxation times and dynamic arrest as the effective temperature is reduced, characteristic of the glass transition. Butler and Harrowell12 have investigated how the 2D 2SFM behaves near a layer of pinned spins representing an interface. They found that the temperature-dependence of a dynamical correlation length ξdyn(T), defined as the distance from the boundary over which bulk behavior is recovered, matched the temperature-dependence of the mean relaxation time τ(T) in the bulk.50 This suggests that investigations of dynamical gradients near an interface can be used to learn about the dynamic length scale associated with cooperative motion. Specifically, Butler and Harrowell12 defined a surface-influence function s i n = ( τ bulk τ n ) ( τ bulk τ 1 ) with the dynamical correlation length ξdyn(T) corresponding to the distance (layer number n) when the relaxation time of the nth layer (τn) was midway between the surface value (τ1) and the bulk (τbulk), si(n = ξ) = 0.5. ξdyn(T) grew with decreasing temperature becoming much larger than a single nearest-neighbor interaction. In analogy, we can define an equivalent dynamic correlation length ξdyn(T) as the distance from the interface at which the local Tg(z) value is midway between the interface value Tg(z = 0) = 37 °C and the bulk,

s i z = ξ = T g bulk T g ( z ) T g bulk T g ( z = 0 ) = 0 . 5 .
(2)

For the glassy-PS side ξPS/PnBMA = 103 nm, while for the rubbery-PnBMA side ξPnBMA/PS = 50 nm. We suspect that the difference in these values reflects in some measure the temperature dependence of ξdyn(T) based on whether the interface has a lower or higher local Tg value than the bulk. Studies are underway to determine how this dynamic length scale depends on the nature of the interface and polymers in question.

The authors gratefully acknowledge support from National Science Foundation CAREER program (Grant No. DMR-1151646) and Emory University.

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