Glass Transition Dynamics and Surface Layer Mobility in Unentangled Polystyrene Films

See allHide authors and affiliations

Science  25 Jun 2010:
Vol. 328, Issue 5986, pp. 1676-1679
DOI: 10.1126/science.1184394


Most polymers solidify into a glassy amorphous state, accompanied by a rapid increase in the viscosity when cooled below the glass transition temperature (Tg). There is an ongoing debate on whether the Tg changes with decreasing polymer film thickness and on the origin of the changes. We measured the viscosity of unentangled, short-chain polystyrene films on silicon at different temperatures and found that the transition temperature for the viscosity decreases with decreasing film thickness, consistent with the changes in the Tg of the films observed before. By applying the hydrodynamic equations to the films, the data can be explained by the presence of a highly mobile surface liquid layer, which follows an Arrhenius dynamic and is able to dominate the flow in the thinnest films studied.

For polymers, glass transition temperature, Tg, is an important technological parameter because it is the temperature at which most polymers freeze into a solidlike glassy state (1). Studies over the past 15 years show that the Tg of nanometer polymer films can vary substantially with the film thickness, h, when h is decreased below 50 nm (28). Although the Tg of polymer films has been found to both increase (6) and decrease (48) with decreasing film thickness, the latter has drawn vastly more attention because of the often bigger size of the effect, especially in freely standing polystyrene (PS) films (8). According to conventional views, the presence of a solid substrate can decrease the configurational space available for the polymers to perform translational motions. Frictional forces between the polymer and the substrate surface can also slow down the dynamics. On the other hand, relaxation of the constraints to molecular motions at the polymer free surface can enhance the mobility. The overall dynamics of the films, and hence the Tg, should be a result of the interplay between all these effects. The most cited model for the thickness dependence Tg(h) of polymer films, namely the layer model (4, 6, 7, 9), portrays the films to consist of a highly mobile surface layer on top of a less mobile, largely bulklike inner layer. Although there have been increasing experiments supporting the existence of a surface mobile layer on polymer films, notably for PS (911), its physical properties, including how its thickness depends on temperature, T, and the polymer thickness, h, as well as the mechanism by which it alters the dynamics of the films at the glass transition, are still controversial.

Keddie et al. (7) hypothesized that the surface mobile layer exists below the bulk Tg (Tg,bulk), and when the temperature T is increased toward Tg,bulk, its thickness diverges critically, following ~(1 − Τ / Tg,bulk)−ν, where ν is a constant between 0.56 and 1 (24, 6, 7, 1214). As a result, thinner films can be filled with the surface mobile layer and melt at a lower temperature. Herminghaus et al. (12, 13) proposed that the Tg of the films is determined by the fastest surface capillary mode that can penetrate the whole film. As the film thickness decreases, the required wave vector, and hence the relaxation rate of the fastest mode, increases. So, thinner films require a lower temperature to melt. To fit the Tg(h) data, however, the model still requires the existence of a critically thickening surface mobile layer. Mechanisms that are completely independent of a free surface have also been proposed. By using molecular dynamics simulations, Varnik et al. (15) showed that an enhancement in the glass transition dynamics could be produced by confining the polymers between two repulsive, impenetrable walls. Long et al. (16) put forward another model on the premise that liquids undergoing the glass transition contain simultaneous fast and slow domains. Upon cooling, the slow domains grow at the expense of the fast ones; the glass transition occurs when the slow domains percolate through the film. Because the percolation threshold is bigger in two dimensions than in three dimensions, the Tg can be lower in thin films. Although the above views for the mechanism of Tg reduction are diversified, they can all provide a good description to the observed Tg(h) with an appropriate choice of the model parameters. Therefore, measurements additional to Tg(h) are necessary to distinguish the prevailing mechanism.

The Tg of polymer films is usually determined by the discontinuous jump in the thermal expansivity in a temperature scan (47). Only a handful of experiments have measured the viscosity (6), which provides a straightforward measure of the rapid slowing down at the glass transition. In this study, we measured the viscosity of unentangled, short-chain PS films coated on silicon by monitoring the evolution of the surface structure of the films upon annealing by atomic force microscopy (AFM) (17, 18). The PS used has a weight-average molecular weight of 2.4 kg/mol and a polydispersity index of 1.06 (19). Figure 1, A and B, shows the topographic image of an h = 4 nm film upon annealing at 328 K for 2 and 24 hours, respectively. Although the temperature used in obtaining these images is 9 K below the Tg of the bulk polymer (Tg,bulk = 337 ± 2 K, as determined by multiple thermal expansivity measurements on an h = 120 nm film upon cooling), visible features can be seen on the film surface within 2 hours of annealing, and by 24 hours, well-defined holes are developed across the film. These should be contrasted with the corresponding images obtained from an h = 79 nm film (Fig. 1, C and D), where no morphological development is discernible with the same annealing conditions. To understand the difference, we examined the evolution in the power spectral density (PSD) of the h = 4 nm films upon annealing. Figure 1E shows a sequence of PSD (open circles) obtained from one annealed at 334 K. The solid lines represent the best fit of the data to equations (S1a and S1b) derived from a model assuming the surface dynamics to be governed by thermal equilibration of the surface capillary modes. The value of the viscosity of the film, η, deduced is only ~10−5 times the bulk value. On the other hand, a low viscosity value is in keeping with the large disparity in morphological development noted between the two films.

Fig. 1

(A to D) Comparison between the AFM topographic images of PS films with h = 4 nm [(A) and (B)] and 79 nm [(C) and (D)] annealed at 328 K and annealing times of 2 and 24 hours, respectively. Scale bar, 1 μm. (E) Power spectral density of a 4-nm film upon annealing at 334 K for various times (from bottom to top): 0, 15, 30, 60, 90, 210, 600, 1080, 2100, 3840, and 7200 s (open circles). The solid lines denote the least-square fit of the data to eq. S1, as detailed in (19).

Figure 2A depicts the measured η versus T of PS films with various thicknesses (symbols). The viscosity of the thickest (h = 79 nm) film displays an excellent agreement with the published value of the bulk polymer (20) (dashed line). In contrast, the viscosity of the thinner films is significantly reduced, especially at low temperatures. The temperature dependence of the viscosity of a glass-forming liquid is conventionally related to its Tg by the Vogel-Fulcher-Tammann (VFT) relation (1).

Fig. 2

(A) Viscosity of PS films with different thicknesses (2.3 ≤ h ≤ 79 nm) plotted versus temperature (symbols). The dashed line represents the published data of the bulk polymer. The solid lines are the least-square fits of the VFT relation to the data. (B) The fitted values of TK obtained by fitting the data shown in Fig. 2A plotted as a function of the film thickness (solid circles). The solid line is the least-square fit to Eq. 2.

η(T)=η()exp(BTTK) (1)

In Eq. 1, B is a constant and TK is the Kauzmann temperature, that is, the theoretical temperature at which the configurational entropy of an ergodic, supercooled liquid vanishes (1). For our polymer in bulk, TK is 288 K, or 49 K below Tg,bulk, and B is 1620 K (20). As an initial attempt to connect our result to the Tg, we fit the VFT relation to the data in Fig. 2A and display the result by solid lines. In obtaining the fits, we varied both TK and η(∞) while keeping B equal to the bulk value. The fitted lines describe the data reasonably well. Shown in Fig. 2B are the fitted values of TK for different films. As seen, they are almost constant, close to Tg,bulk – 48 K at large h, but decrease rapidly with decreasing h for h < ~20 nm. We find that the data of TK can be fitted to the equation (solid line in Fig. 2B)TK(h)=Tg()(1+h0/h)ΔT (2)where Tg(∞), ΔT, and h0 are constant fitting parameters. The first term on the right side, which incorporates the dependence in h, coincides with the generic Tg(h) dependence found for all supported (12, 13) and freely standing films with Mw ≤ 378 kg/mol (4). The fitted value of h0, 0.6 ± 0.1 nm when Tg(∞) is fixed to Tg,bulk, matches reasonably well with that obtained by modeling the experimental Tg(h) data to Tg(h) = Tg(∞)/(1 + h0/h) (12). The fitted value of ΔT (48 ± 3 K) also agrees well with the value known for the bulk polymer. Our result thus shows that the reduction in TK(h) by viscosity measurements agrees quantitatively with the reduction in Tg(h) by thermal expansivity measurements. Notwithstanding the good agreement between the measurements and Eqs. 1 and 2, the ensuing implications are limited because the physical origins of Eqs. 1 and 2 are largely empirical and still controversial (1, 13).

Our main analysis begins when we replot the viscosity data as η/h3 versus 1/T in Fig. 3A. We find that all the data deviating from the bulk η-T curve collapse onto the straight line corresponding to the following Arrhenius dependence (solid line in Fig. 3A) 3ηh3=(165±7 Pasm-3)exp((185±3) kJ/molRT) (3)where R is the ideal gas constant. For ease of view, we have labeled all such data by filled symbols and the rest by open symbols. All the data for the 2.3 nm ≤ h ≤ 9 nm films collapse onto the Arrhenius line, whereas for the h > 9 nm films, that happens only at sufficiently low temperatures, with the onset temperature (denoted by the arrows in Fig. 3A) increasing with decreasing h. In Fig. 3B, we replot the data as η versus 1/T while preserving the same data-labeling scheme. One can see that the data that do not fall on Eq. 3 (open symbols) collapse into the bulk curve noted above. These observations suggest that the films can be separated into two regions, with one responsible for the collapse to the Arrhenius dependence seen in Fig. 3A and one for the collapse to the bulk curve seen in Fig. 3B.

Fig. 3

(A) A plot of η/h3 versus 1/T (open and solid symbols) for the same set of film thicknesses shown in Fig. 2A. The solid symbols label the data that collapse into the straight line shown by the solid line corresponding to Eq. 3. The open symbols label the rest of the data. The arrows indicate the temperatures at which the data of different thicknesses begin to depart from the solid line. (B) A plot of η versus 1/T (symbols). The same scheme used in Fig. 3A for the symbols applies here. The solid lines are generated from Eq. 6. (Inset) Schematic illustrating a simple two-layer model with homogeneous layer properties as discussed in the text. The blue curve represents the velocity profile for the fluid flow produced in the film by a uniform pressure gradient, P, predicted by Eq. 4.

We consider a bilayer film consisting of a homogeneous mobile layer at the free surface (with viscosity ηt and thickness ht) and a bulklike inner layer (with viscosity ηb and thickness hbh − ht). In this experiment, the surface topography on the film produces inhomogeneities in the Laplace and conjoining pressures and thereby pressure gradient, ∇P, parallel to the film (21). To find the flow pattern of the bilayer, we solve the Navier-Stokes equation for the system under a uniform pressure gradient, ∇P, parallel to the film, which is valid here as qh << 1. By assuming the no-slip boundary condition at both the bottom and intermediate interfaces and the interfacial tension between the two layers to be zero, we find the fluid velocity profile in the film, v(z), to be given by:v(z)=P2ηb[z22(hb+ht)z](z<hb)andv(z)=P2ηt[(zhb)22ht(zhb)ηtηbhb(2hb+ht)](z>hb) (4)

A typical solution described by Eq. 4 is shown in the inset of Fig. 3B. The total (flow) mobility of the film, defined as (∫0h v(z) dz)/(-∇P) (22), is given byMtot=hb33ηb+ht33ηt+hthb(hb+ht)ηb (5)The first two terms are simply the mobility of individual layers (denoted by Mb and Mt below) if the other layer were absent. The third is a coupling term, which under the experimental conditions is found to be always negligible. So our result shows that MtotMb + Mt. In this experiment, what we measure is the relaxation rates of the capillary modes, which are directly related to Mtot (see eq. S1b); the viscosity, on the other hand, is a derived parameter, calculated by using η = h3/(3Mtot) (21, 23). It follows that in a bilayer film, the measured viscosity is an effective one given by

ηeffh33Mtoth33(Mt+Mb) (6)

From Eq. 6, we see that the quantity being plotted in Fig. 3A is ~Mtot−1. The fact that the data for all the films with h ≤ ~9 nm collapse onto the same line means that the Mtot of those films is the same, independent of h. Independence of Mtot on h means that the thinner films transport materials as effectively as the thicker films, contrary to the convention that they should be less effective. Our result can be explained if the mobility Mtot is dominated by the top layer so that MtotMt. Any other explanation must entail the artificial relation ηeff ~ h3 and thereby more contrived arguments.

We have used Eq. 6 to calculate the expected values of ηeff by using the experimental film thickness, values of Mt−1 given by Eq. 3, and Mb = h3/(3ηbulk), where ηbulk is the viscosity of the bulk polymer (17, 20). The result, displayed by the solid lines in Fig. 3B, agrees remarkably well with the measurements without any adjustable parameters. This simple result shows that the two divergent trends (namely either Arrhenius or bulklike) found of the viscosity of the films are caused by the total mobility Mtot being almost always dominated by either Mt or Mb. Such a drastic crossover from the MtotMb to the MtotMt behavior is attributable to the vastly stronger temperature dependence of Mb(T) (VFT) than Mt(T) (Arrhenius). In the above analysis, we have assumed the mobile layer to be thin, with ht ≅ 0, whereby Mbh3/(3ηbulk). Adopting instead a small, finite ht only slightly alters the calculated values of ηeff, mostly in the limited region corresponding to T ≈ 380 K and h ≈ 11 nm, where Mb(T) and Mt(T) have similar orders of magnitude. Although we have derived the model by assuming the films to be bilayers in which individual layers have uniform properties, the result (Eq. 6), involving only the mobilities of individual layers, can be completely independent of the layer structure. In particular, Eq. 6 embraces systems with a gradient layer structure, which has been suggested recently (9, 24).

In Eq. 5, the coupling term is not symmetric with respect to ηb and ηt. By exchanging the properties of the two layers, the model cannot fit well to the data at all. This can be traced to the fact that when the bulklike layer is riding on top of the mobile layer, the coupling term would become nonnegligible, whereby Mb cannot be approximated by h3/(3ηbulk). This confirms the importance of the free surface in bringing about the reduction of the thin film Tg or enhancement in the glass transition dynamics, in disagreement with the percolation theory (16) and the possibility that the mobile layer may come from the polymer-substrate interface, as implied by simulation results (15). Our measurements are also incompatible with the models requiring the thickness of the surface mobile layer to diverge critically with temperature, as those discussed above. The fact that Mtot remains thickness independent among the h < ~11 nm films down to h = 2.3 nm (Fig. 3A, solid symbols) indicates that the surface mobile layer does not extend more than 2.3 nm from the free surface, because otherwise Mtot (≈ Mt there) should decrease with decreasing h as the surface mobile layer gets trimmed down. It can, however, be thinner if a third, dynamically dead layer (25) exists next to the substrate, for example. The coefficient, 165 ± 7 Pa⋅s⋅m−3 displayed in Eq. 3, is reliant on the total amount of material contributing to the Arrhenius transport and thus dictated by the effective surface layer thickness ht. Constancy of its value, as demonstrated by the good fit of Eq. 3 to the data (Fig. 3A, solid symbols), implies that there is at most a weak dependence of ht on T and h. This finding is in good accord with the result of Kawana et al. (25), who measured the thickness variation of the thermal expansion coefficient of PS supported by silicon and found that their data was consistent with a surface mobile layer with a constant thickness. We further note that the fluorescence-label experiment of (9) has found that the surface mobile layer extends no more than 33% into the film at T = Tg,bulk – 5 K, contrary to expectation if it had undergone critical thickening (7, 13). Nevertheless, critical thickening of a mobile layer at a water-surfactant-PS interface, differing from that studied here, is supported by a recent experiment (26). From our analysis, the surface mobile layer is able to dominate the flow in the h < 9 nm films at temperatures below ~385 K (Fig. 3A) because its viscosity at T = 385 K is at most 1/40 that of the bulklike inner layer, and the difference heightens rapidly with decreasing temperature. On the other hand, the thermal expansivity of the surface mobile layer was estimated to be only about 4 times that of the bulklike inner layer and does not change with temperature (25). Given the large discrepancies that can exist between different properties, the Tg of PS films obtained by different techniques can be different, which may explain some previous conflicting results (27).

It is remarkable that the Arrhenius dependence of Mt(T)−1 is robust and persists from 306 to 400 K and over 7 orders of magnitude (Fig. 3A). Typically, Arrhenius temperature dependence is a signature of local dynamics. Our finding suggests that the molecules close to the free surface, with reduced coordination, are free of cooperative couplings, contrary to those in the bulk. Moreover, this cooperativity-free behavior continues down to 306 K, or 31 K below Tg.bulk. The temperature dependence we found of the surface-layer mobility (Eq. 3) is consistent with those measured by lateral force microscopy (10) and surface nano-hole recovery (11). The activation energy we obtained, 185 kJ/mol, agrees within 20% with the values of 230 (10) and 150 kJ/mol (11) revealed in those experiments, although we caution that the nano-hole recovery time constant of (11) became T-independent below 303 K, outside the temperature range examined here. Although the surface dynamics discussed here do not necessarily have any direct counterpart in the bulk due to different constraints that may pertain at the free surface, we note that our activation energy roughly coincides with that of ~199 kJ/mol found in bulk PS below Tg.bulk (28) and is of similar magnitude to that for the β relaxation (~140 kJ/mol), which has been attributed to dynamics confined to a very limited extent in the polymer chain (29).

We have shown that a surface mobile layer, exhibiting Arrhenius dynamics, with a thickness of less than 2.3 nm, is responsible for the reduction in the effective viscosity and, thereby, the Tg of unentangled, short-chain polystyrene-supported films. The thickness dependence of the TK of the films found by viscosity measurement is in remarkable agreement with that of the Tg found by thermal expansivity measurement. Our result unfolds a mechanism by which the surface mobile layer can modify the overall dynamics of the films and does not require the surface layer thickness to be adjustable, thereby providing a more stringent test for the model. The other existing models, which either assume the surface mobile layer to be nonessential or to thicken critically with decreasing temperature, cannot explain our data.

Supporting Online Material

Materials and Methods

Figs. S1 and S2


References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. We thank C. A. Angell for useful discussions and G. B. McKenna for critical reading of this paper. Support from NSF (DMR-0706096 and DMR-0908651) is gratefully acknowledged.
View Abstract

Stay Connected to Science

Navigate This Article