Note pubs.acs.org/Macromolecules
Surface Diffusion of Polymer Glasses Wei Zhang† and Lian Yu*,†,‡ †
School of Pharmacy and ‡Department of Chemistry, University of WisconsinMadison, Madison, Wisconsin 53705, United States
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the method has been validated by simulations to yield accurate surface diffusivity for glasses.21 For glass-forming liquids, an important mechanism of surface evolution is viscous flow (collective movement of a liquid driven by a pressure gradient). This mechanism differs from surface diffusion and must be carefully distinguished. For a Newtonian liquid, viscous flow decays a surface wave at the rate K = Fq = (γ/2η)q, where η is the viscosity. Thus, viscous flow can be distinguished from surface diffusion by direct calculation of the decay rate and from its q dependence (K ∝ q vs K ∝ q4). Notice that the result K = Fq holds for deep liquids, defined as qd ≫ 1, where d is the depth of the liquid to a substrate; for shallow liquids, the decay rate is influenced by the substrate.22 In this study, the deep-liquid condition prevails. A sinusoidal surface grating (λ = 290−8200 nm) was embossed onto a thick glass film of polystyrene (PS); upon removing the master, the flattening of the surface was monitored over time by AFM or laser diffraction in flowing N2 (Figure 1).14,15 AFM was always the preferred method;
he surface mobility of polymer glasses has been a subject of active research and continuing interest.1−3 There is growing evidence for a mobile surface layer on the order of several nanometers thick. The nature of this mobile layer, however, remains inadequately understood; for example, theories4,5 and simulations6−12 predict a steep mobility gradient beneath a free surface on the length scale of a flexible segment (∼1 nm), while experiments have not tested this picture at the necessary resolution.13 It is also noteworthy that despite the extensive studies, there has been no report of the lateral surface diffusion coefficients for polymer glasses. This property is a fundamental measure of surface mobility and closely related to the surface-mobility gradient through chain dimensions. Here we report the first measurement of surface diffusivity for polystyrene glasses. The results show a 5 orders of magnitude enhancement of diffusion at the free surface. We also find that the enhancement effect is weaker than that for small-molecule glasses,14,15 a difference attributed to a steep mobility gradient and deeper penetration of polymer chains into the bulk. The strong material dependence of surface diffusion suggests large system-to-system variations of surface-facilitated processes, for example, surface crystal growth16 and formation of stable glasses by vapor deposition.17 Mullins derived the equation of surface evolution by the lateral diffusion of surface particles:18
rn = B∇2 κ In this equation, rn is the velocity of the surface element along its normal with respect to the bulk material, κ is the curvature of the surface, and B is a mobility constant given by γDs Ω2ν/kT, where γ is the surface tension, Ds is the surface diffusion coefficient, Ω is the molecular volume, and ν is the areal density of surface molecules. The equation is derived under the following assumptions: (1) isotropy of surface properties, (2) the chemical potential of a surface particle changes by Δμ = γΩκ from a point of zero curvature to a point of curvature κ, and (3) a surface particle in a chemical-potential gradient ∇μ drifts at an average velocity given by v = −(Ds/kT)∇μ (the Nernst−Einstein relation). Assuming small surface slopes and variation of the surface height W only in one dimension, x, Mullins’s equation is reduced to ∂W /∂t + B ∂ 4W /∂x 4 = 0
(1)
For a sinusoidal surface grating of wavelength λ, eq 1 predicts an exponential decrease of the grating amplitude with a decay constant K = Bq4, where q = 2π/λ is the spatial frequency. The very strong q dependence K ∝ q4 is a signature of surface diffusion and useful for distinguishing it from other mechanisms of surface evolution.18 Mullins’s method has been used to determine the surface diffusivity of crystalline metals19 and silicon20 and of small-molecule glasses14,15 and is here applied to polymer glasses. Though often used to study crystal surfaces, © XXXX American Chemical Society
Figure 1. Representative decay kinetics for PS surface gratings. (A) PS1.1k at 293 K measured by AFM. h: grating amplitude. Inset: AFM images at two decay times. (B) PS1.1k at 328 K measured by laser diffraction. I/I0: normalized diffraction intensity. Inset: structure of PS. The curves are exponential fits. Received: October 19, 2015 Revised: December 10, 2015
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molecular-weight PS. Excellent agreement is obtained with no adjustment for PS1.7k and PS3.7k and after a vertical shift by 0.45 decade for PS1.1k. (The shift necessary for PS1.1k is likely a result of the residual error of the master curve, especially at low molecular weights.) As Figure 2B shows, the viscous-flow mechanism is further confirmed by the wavelength dependence K ∝ λ−1 that is expected for this mechanism.18 Although viscous flow controls surface flattening at high temperatures, this control is lost at low temperatures for PS1.1k and PS1.7k; the observed K greatly exceeds Fq and scales as λ−4 (Figure 2B). These results indicate a change of decay mechanism from viscous flow to surface diffusion. This mechanistic change is a general feature for small-molecule glasses14,15 and is consistent with the observation with a stepped PS surface (Mw = 3 kg/mol), which evolves by wholefilm flow at high temperatures and through a surface mobile layer at low temperatures.2 From the second y-axis of Figure 2A, note that the viscosity at which the mechanistic transition occurs is much higher for PS1.1k and PS1.7k (∼1010 Pa s) than for OTP (106.5 Pa s), consistent with slower surface diffusion for PS. The third fraction studied, PS3.7k, showed no change of decay mechanism down to the lowest temperature tested, at which η ≈ 1011 Pa s, suggesting even slower surface diffusion; we made no attempt to measure its surface diffusion. Figure 2C shows the surface diffusion coefficients calculated from the decay constants K. For this calculation, we assume that any center-of-mass translation of a polymer chain on the surface displaces a volume equal to the average volume occupied by a chain in the bulk: Ω = Vm/NA = Mn/ρNA, where Vm is the molar volume (volume per mole of polymer chains), NA is Avogadro’s number, ρ is the density (1.03 g/cm3 for PS),25 and ν = Ω−2/3. In Figure 3, we compare the surface diffusivity of the PS fractions studied and the bulk diffusivity of a similar PS fraction (1.9 kg/mol)26 as well as the corresponding data on smallmolecule glasses.14,15,27−30 The temperature has been scaled by Tg. In this format, the Dv values of various systems roughly cluster together, serving as a reference for evaluating the Ds values. Note that for the PS studied (1.1 and 1.7 kg/mol) Ds at Tg is roughly 105 times larger than Dv, indicating highly mobile surface molecules. Note also that surface diffusion in PS is much slower than in the small-molecule glasses; for OTP, Ds/ Dv = 108 at Tg, much larger than the ratio for the PS fractions studied. The smaller Ds of PS is associated with a steeper temperature slope, indicating a higher kinetic barrier. The overall effect of increasing molecular size is to bring Ds closer to Dv, both in absolute value and in temperature slope, an effect we shall discuss later. It is relevant to compare this study with two previous studies,2,3 all aimed to identify a surface-transport process that differs from ordinary liquid flow. This study measured the decay of surface gratings using thick films; the previous workers measured the spontaneous roughening3 or the evolution of a stepped surface2 using thin supported films. We measured the decay of surface waves at one spatial frequency q at a time; they simultaneously followed surface evolution at multiple spatial frequencies. Our experiments used thick films, characterized by qd ≫ 1, while the previous work used thin films (qd ≪ 1). This last distinction is important since it is a source of confusion. In a thick film, viscous flow decays a surface wave at the rate (γ/ 2η)q, whereas in a thin film, the decay rate is (γd3/3η)q4 under the no-slip condition.22 In the latter case, the equation of surface evolution at small surface slopes has the same form as
diffraction was used to measure fast decays at high temperatures. The expected relation I ∝ h2 was verified between diffraction intensity and grating amplitude. All diffraction experiments were performed real-time, with the sample residing on a Linkam temperature stage. AFM experiments shorter than 1 day were conducted real-time using an AFM temperature stage; for longer experiments, a sample was annealed offline at a chosen temperature and returned to the AFM for analysis at room temperature, during which surface evolution was negligible. An important feature of our experiment is that the film thickness (d ≈ 150 μm) greatly exceeds the grating wavelength so that the condition qd ≫ 1 is met and the substrate has no effect on relaxation near the surface. We studied three PS fractions: “PS1.1k” [Mw = 1110 g/mol, Mw/ Mn = 1.12, Tg = 307 K (onset)/310 K (inflection)]; “PS1.7k” (Mw = 1700 g/mol, Mw/Mn = 1.06, Tg = 319 K/323 K); and “PS3.7k” (Mw = 3680 g/mol, Mw/Mn = 1.08, Tg = 350 K/352 K). We measured Tg by differential scanning calorimetry (DSC) during heating at 10 K/min after cooling at 10 K/min. Figure 2A shows the grating decay constant K as a function of temperature for the PS studied. We included data on the
Figure 2. (A) Decay constant K vs T at λ = 1000 nm for three PS fractions and OTP (Tg = 246 K).15 Decay is by viscous flow (Fq) at high temperatures and by surface diffusion (Bq4) at low temperatures. (B) K vs λ data for identifying the mechanism for surface evolution. K ∝ λ−1 holds at high temperatures (viscous flow) and K ∝ λ−4 at low temperatures (surface diffusion). (C) Calculated surface diffusion coefficients Ds. The lines are guide to the eye. Tg is the DSC value measured on heating at the onset of the transition.
small molecule o-terphenyl (OTP) for comparison.15 For all these systems, high-temperature decay occurs by viscous flow. This conclusion follows from the nearly perfect agreement between the K values observed and calculated (dotted curves labeled Fq); for this calculation, γ = 0.035 N/m (ref 23) and η is from the viscosity master curve reported in ref 24 for lowB
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4 compares the values of B from the three studies. We find an approximate agreement between these studies on similar PS
Figure 4. Comparison of the surface-mobility constants B from this work (PS1.1k, Tg = 307 K; PS1.7k, Tg = 319 K), ref 3 (PS2.4k, Tg = 337 K), and ref 2 (PS3k, Tg = 343 K).
fractions (1−3 kg/mol): B falls in the range 10−32−10−33 m4/s at Tg and appears to have stronger temperature dependence with increasing molecular weight. For these PS fractions, the calculated Ds values are approximately 10−16 m2/s at Tg (Figure 5). This agreement is encouraging given the significant difference between the experiments and suggests the potential for thin-film experiments to report the surface dynamics of bulk glasses. It is important to note that while the different studies yielded roughly consistent mobility constants B, they provide different interpretations.2,3 Using Mullins’s model, we interpret B as γDsΩ2ν/kT, which characterizes the lateral diffusion of molecules in the top surface layer (exposed to free space). This interpretation does not imply that the mobile surface layer is one-molecule thick; it is fully consistent with the existence of a mobility gradient suggested by theories and simulations.4−12 This interpretation offers no direct information about the mobility field beneath the free surface, other than requiring for internal consistency that local diffusion of molecules beneath the top layer be sufficiently slow so that surface diffusion, not bulk diffusion, dominates the observed surface evolution.14 In contrast, Chai et al. describe the surface-transport process conforming to eq 1 as viscous f low in a thin surface layer of unknown thickness hm that has a uniform viscosity ηm and no slip from the bulk material underneath. In this model, B = γhm3/(3ηm). These two models are mathematically equivalent for modeling experimental data. We prefer the surface diffusion picture as a basis for comparing the surface transport of smallmolecule glasses and short-chain polymer glasses and for comparing experimental results and theoretical predictions.5 Which interpretation is more accurate can be tested by simulations that compare the relative contributions of surface and subsurface molecules to the evolution of a glass surface. We now discuss the slower surface diffusion of PS relative to small-molecule glasses (Figure 3). To focus this discussion, Figure 5A shows the surface and bulk diffusion coefficients, Ds and Dv, evaluated at Tg for various systems plotted against the molecular weight M. Note the large decrease of Ds with M compared with the smaller decrease of Dv. Simulations have shown that the end-to-end vectors Ree of PS chains are almost
Figure 3. Surface and bulk diffusion coefficients (Ds and Dv) vs Tg/T. The Dv values are clustered while Ds shows stronger molecular dependence. The temperature has been scaled by the DSC Tg (onset): 307 K for PS1.1k, 319 K for PS1.7k, 246 K for OTP, 315 K for IMC and NIF, 347 K for TNB. For IMC, Dv above 10−15 m2/s is obtained by extending low-temperature data29 by viscosity and the Stokes− Einstein relation.
eq 1, but the mobility constant B is replaced by γd3/3η.2 Thus, in a thin-film experiment, the q4 dependence is the signature of both surface transport (eq 1) and whole-f ilm viscous f low under the no-slip condition, as observed by Chai et al.2 In our thick-film experiments, in contrast, the two mechanisms are readily distinguished by their different q dependence: q4 for surface transport; q for viscous flow. This should be considered an advantage of thick-film experiments. To distinguish the two mechanisms in thin-film studies, previous workers relied on the thickness dependence of surface evolution: viscous flow requires the observed rate to vary as d3, with the associated viscosity matching the known bulk value, while surface transport requires that film thickness has no effect on the mobility constant3 or the surface profile that develops in a selfsimilar manner.2 There are other issues concerning the use of nanometer-thin films. Given that a solid substrate alters the structure and dynamics of polymer chains within several nanometers,7,10 it is unclear whether the free surface of a supported nanometer-thin film accurately represents that of a bulk glass. While a no-slip condition is generally assumed for a supported thin film, Chen et al. recently reported the possibility of slippage during annealing.31 Since these issues do not arise in thick-film experiments, they provide important control for thinfilm experiments. It is significant that all three studies identified a surfacetransport process on polystyrene, which differs from viscous flow and whose rate varies as Bq4 (eq 1). The “primary data” from all three studies are the mobility constant B (B corresponds to γMt in ref 3 and γhm3/(3ηm) in ref 2). Figure C
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in rough agreement with the prediction of Mirigian and Schweizer.5 This agreement is encouraging given the steep mobility gradient being studied and our crude analysis. The current data provide no information on mobility deeper into the bulk. The picture proposed here qualitatively explains the gradual approach of surface diffusivity to bulk diffusivity with increasing molecular size (Figure 3). Although we have studied only short PS chains, we expect longer chains to penetrate deeper into the bulk and their surface diffusion to be even slower, eventually approaching the bulk rate. The slower diffusion as a result of polymer chains spanning a spatial gradient of mobility has analogy with the observation in the bulk that diffusion is enhanced relative to the Stokes− Einstein prediction near Tg for small molecules such as OTP, but not for a 17-mer of PS.26 This bulk effect is attributed to the existence of spatial domains 1−4 nm in size with fast and slow dynamics.36 For molecules small enough to fit in each domain, the observed diffusivity is biased by fast domains, but the effect vanishes for molecules so large that they span one or more domains. This analogy between the surface and bulk situations is broadly consistent with the notion that the length scale of heterogeneous dynamics in the bulk is comparable to that of the mobility field beneath the surface.6,10 To summarize, we have observed that surface diffusion dominates lateral surface mass transport for short-chain polystyrene glasses at micrometer to nanometer length scales. At higher temperatures, viscous flow is the mechanism for surface evolution. Surface diffusion is 105 times faster than bulk diffusion at Tg, indicating highly mobile surface molecules. This surface enhancement of diffusion, however, is smaller than that observed for small-molecule glasses. The difference is likely a result of the steep mobility gradient and significant penetration of polymer chains into the bulk. We expect the effect to be even stronger for longer chains and anticipate a similar effect for molecules that associate so strongly (e.g., by hydrogen bonds) that they diffuse as larger units. A strong material dependence of surface diffusion implies significant system-to-system variation for processes that rely on surface mobility, for example, surface crystal growth16 and formation of stable glasses by vapor deposition.17 In these processes, polymers may behave qualitatively differently from small molecules.37
randomly oriented at a melt surface (with a slight preference for parallel orientation and an average angle of 60° from the surface normal).32,33 These simulations also find that chain ends are enriched at the surface and phenyl rings tend to point outward. We depict this situation in Figure 5B. This structure makes it unlikely that surface diffusion is a process of moving chains that
Figure 5. (A) Surface and bulk diffusivity at Tg vs molecular weight. Ds values are from this work (solid symbols) and calculated from data in refs 2 and 3 (open symbols). (B) Comparison of the small molecule OTP and a 16-mer of PS at the glass surface. Polymer chains penetrate deeper and have slower center-of-mass diffusion.
are completely on the surface, as in the case of n-alkanes diffusing on metals.34 The structure implies that PS chains penetrate significantly into the bulk; for a 16-mer (Ree ≈ 2.7 nm),35 the penetration is about 1.5 nm. We explain the slower surface diffusion of PS on the basis of this penetration and the steep mobility gradient beneath a free surface predicted by theories and simulations.5,6,8,9,11 We define the mobility gradient as k = −d log τ/dz, where τ is the local structural relaxation time and z the depth (k is evaluated at the free surface where z = 0). For a liquid of Lennard-Jones spheres, k is about 1 decade per particle diameter at the lowest temperature accessible by simulations (still well above the laboratory Tg) and is expected to increase with cooling.6 Similar k is predicted for LJ spheres joined by springs to simulate polymer chains;8,9 for a PMMA model, Xia et al. find k ≈ 1 decade/nm when τbulk = 1 ns.11 At our DSC Tg (τbulk ≈ 10 s), Mirigian and Schweizer predict a much larger k of 6 decades/nm.5 We speculate that the segments that penetrate into the bulk move so slowly that they limit the whole-chain diffusion as measured by our experiments, even though the segments at the surface are more mobile. In this picture, the ratio Ds/Dv = 105 for PS oligomers at Tg would reflect the local mobility roughly 1.5 nm beneath the surface. The chain ends at the surface are expected to move faster, perhaps at a similar rate as the small molecule OTP (108 times the bulk mobility at Tg). A simple aromatic hydrocarbon, OTP, is a fair representation of a PS segment. In this picture, the surface mobility gradient is estimated to be 3 decades/nm at Tg,
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected] (L.Y.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank the NSF (DMR-1206724) for supporting this work and M. D. Ediger and Kenneth S. Schweizer for helpful discussions.
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REFERENCES
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