Article pubs.acs.org/JPCB
Fast Surface Diffusion of Amorphous o‑Terphenyl and Its Competition with Viscous Flow in Surface Evolution Wei Zhang,† Caleb W. Brian,‡ and Lian Yu*,†,‡ †
School of Pharmacy and ‡Department of Chemistry, University of WisconsinMadison, Madison, Wisconsin 53705, United States ABSTRACT: Surface self-diffusion coefficients have been measured for the model molecular glass o-terphenyl (OTP) through surface-grating decay driven by capillarity. The decay mechanism transitions from viscous flow at high temperatures to surface diffusion at low temperatures; for 1000 nm wavelength gratings, the transition occurs at Tg + 11 K. The surface diffusion of OTP is 108 times faster than bulk diffusion at Tg and even faster at lower temperatures because of its weaker temperature dependence. At Tg, OTP has approximately the same bulk diffusivity as the previously studied molecular liquid indomethacin, but its surface diffusion is 100 times faster. While the molecular glass-formers exhibit transitions from viscous flow to surface diffusion as the mechanism of capillarity-driven surface flattening, polystyrenes and silicates show no such transition under comparable conditions, suggesting slower surface diffusion on these materials and a general dependence of surface diffusion on intermolecular forces. The velocity of surface crystal growth on molecular glasses is proportional to surface diffusivity, indicating a common kinetic barrier for both processes for temperatures below Tg.
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INTRODUCTION Surface mobility influences many processes in science and technology, including catalysis, sintering, and crystal growth. Among its different measures, the surface diffusion coefficient Ds characterizes the in-plane translation of atoms or molecules on the surface. Although this property has been studied extensively for crystalline metals and semiconductors,1,2 only recently have there been reports of Ds for molecular solids.3−5 These studies show that similar to atomic solids, molecular solids can have vastly faster diffusion at the free surface. These results help develop a broad perspective of surface mobility across classes of materials, and are relevant for understanding several phenomena of current interest: fast surface crystal growth on molecular glasses,6,7 formation of stable glasses by vapor deposition,8,9 and the surface mobility of polymer glasses.10,11 A surface contour can evolve under the driving force of surface tension in several mechanismsviscous flow, evaporation−condensation, bulk diffusion, and surface diffusion.12 Mullins showed how surface diffusion can be distinguished from other mechanisms by direct evaluation and wavelength dependence. Surface diffusion flattens a sinusoidal surface at a rate proportional to q4, where q is the spatial frequency, while the other mechanisms do so at rates that have weaker dependence on q. This way of identifying and measuring surface diffusion has been applied to crystalline metals and silicon13,14 and to molecular glasses.3,4 By simulation, Malshe et al. found that surface diffusivity obtained from surface-grating decay agrees with that from the growth of mean square displacement of single particles.15 © 2015 American Chemical Society
Recent studies of glass-forming molecular liquids found that the mechanism of surface flattening driven by surface tension changes from viscous flow at high temperatures to surface diffusion at low temperatures.3,4 For a sinusoidal surface of 1000 nm wavelength on indomethacin and nifedipine (IMC and NIF; see Figure 2 for their structures), this transition occurs at 12 K above the glass transition temperature Tg at a relatively low viscosity (107.5 Pa s). In contrast, similar studies on the liquids of polystyrenes16−18 and silicates19 observed that viscous flow remains the mechanism of surface-grating decay up to the highest viscosities studied (1010 Pa s), with no sign of transition to another decay mechanism. This difference suggests that surface diffusion is faster on the glasses of small organic molecules than on polymeric and silicate glasses, and motivates inquiries into the material dependence of surface diffusion. As temperature decreases toward Tg, viscosity increases sharply and viscous relaxation slows. We anticipate that for a glassformer with faster surface diffusion, viscous flow loses to surface diffusion as the mechanism of capillarity-driven surface flattening at a lower viscosity. This study investigated the surface mobility of the model molecular glass former o-terphenyl (OTP). OTP has weaker intermolecular forces than the previously studied IMC and NIF,3,4 a consequence of its smaller size and lack of polar groups, and together these systems enable a test of the material dependence of surface diffusion. We report that the surface diffusion of OTP is vastly faster than its bulk diffusion. Received: December 21, 2014 Revised: February 20, 2015 Published: March 24, 2015 5071
DOI: 10.1021/jp5127464 J. Phys. Chem. B 2015, 119, 5071−5078
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The Journal of Physical Chemistry B Compared to IMC, OTP has faster surface diffusion at the same bulk diffusivity, and the mechanism of its surface evolution changes from viscous flow to surface diffusion at a lower viscosity. We analyze these results in reference to the surface mobility of polystyrene and silicate liquids and theoretical predictions.20,21 Molecular glasses grow crystals much faster on the surface than in the interior7 and we report that the velocity of surface crystal growth is approximately proportional to the surface diffusion coefficient, indicating a common kinetic barrier for the two processes.
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MATERIALS AND METHODS o-Terphenyl (OTP, 99% pure) was purchased from SigmaAldrich and used as received. Surface gratings were embossed unto liquid OTP at 273 K (27 K above Tg) in flowing N2 with master gratings of different wavelengths (740−8200 nm). The masters were purchased from Rainbow Symphony (1000 and 2000 nm), separated from data storage discs (CD for 1500 nm, DVD for 740 nm), or replicated from glass gratings through a Norland UV-curing optical adhesive (3300 and 8200 nm). The masters were gold-coated to avoid transfer of contaminants. The masters were confirmed by atomic force microscopy to be sinusoidal, except for those of 3300 and 8200 nm wavelengths, which were blazed (“sawtooth”). Upon cooling to 243 K, the master was detached from OTP, yielding a glass film with a corrugated surface. The OTP films thus prepared were ca. 150 μm thick, much thicker than any grating wavelength used so that the rate of viscous relaxation did not depend on film thickness. The decay of surface grating was monitored in flowing N2 at a constant temperature maintained with a Linkam THMS 600E stage. A helium−neon laser (632.8 nm) passed normally through the sample and the first-order diffraction was measured in transmission with a photodiode (Si Transimpedence Amplified Photodetector from Thorlabs) interfaced with an oscilloscope or a National Instruments NI-DAQ data acquisition card. The square root of the diffraction intensity was taken to be proportional to the grating amplitude.
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RESULTS Figure 1a shows typical decay curves for OTP surface gratings I/I0 vs t, where I/I0 is the normalized first-order diffraction intensity and t is time. The decay at each temperature is fitted with an exponential function or a stretched exponential function, I/I0 = exp[−(KIt)β], with β close to 1. In the latter case, we regard KI as characterizing the average decay kinetics during the measurement time. Figure 1b plots the decay constant K vs temperature T for 1000 nm wavelength gratings; here K is the grating-amplitude decay constant, and half of KI, because of the relation I ∝ h2. The solid symbols are data directly measured with 1000 nm wavelength gratings; the open symbols are data measured at other wavelengths and converted to λ = 1000 nm, after confirming the wavelength dependence at the respective temperatures (Figure 1c). By doing so we obtained the values of K at higher temperatures at which the grating decay was too fast to be measured by our method at λ = 1000 nm. Together, the measured decay constants span 5 orders of magnitude. The upper limit of this range was determined by grating decay being faster than temperature equilibration, and the lower limit by the fracture of OTP glasses upon cooling on a substrate (silicate) of lower thermal expansion coefficient.
Figure 1. (a) Representative decay curves of 1000 nm wavelength OTP gratings at different temperatures. (b) Decay constant K vs T for 1000 nm wavelength OTP gratings. (c) Wavelength dependence of K at different temperatures.
To help identify the mechanism of surface evolution, we measured the wavelength dependence of K (Figure 1c). At 263 K (Tg + 17 K), we find K ∝ λ−1 for λ = 1000−8200 nm; the slope of the log K vs log λ plot is −0.97. At 228 K (Tg − 18 K) and 223 K (Tg − 23 K), we find K ∝ λ−4 for λ = 740−2000 nm; the slope of the log K vs log λ plot is −4.1. As we discuss later, these wavelength dependences indicate surface evolution by viscous flow and surface diffusion, respectively. Mullins12 showed that for a sinusoidal grating, the grating amplitude h decays exponentially: h = h0 exp( −Kt )
(1)
where the decay constant K is given by K = Fq + Aq2 + Dq3 + Bq 4 5072
(2) DOI: 10.1021/jp5127464 J. Phys. Chem. B 2015, 119, 5071−5078
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The Journal of Physical Chemistry B
effect, the bulk-diffusion term never becomes large enough under our experimental conditions to be the mechanism of surface grating decay. This conclusion is confirmed by direct evaluation using the known bulk diffusivity and by the wavelength dependence of K: should bulk diffusion be the decay mechanism, K would be proportional to λ−3, which was not observed. After identifying surface diffusion as the mechanism for grating decay at low temperatures, we calculate the surface diffusion coefficients Ds using eq 2. For this calculation, we use ν = Ω−2/3 = 2 nm−2. The results (Figure 2a) are compared with
In eq 2, q = 2π /λ
F=
A=
γ 2η p0 γ Ω2 (2πm)1/2 (kT )3/2
D = A′ + C =
B=
ρ0 DGγ Ω2 kT
+
Dv γ Ω kT
Dsγ Ω2ν kT
Here, λ is the grating wavelength, γ the surface tension, η the viscosity, p0 the equilibrium vapor pressure, Ω the molecular volume, m the molecular mass, ρ0 the equilibrium vapor density, DG the diffusion coefficient of evaporated molecules in the inert atmosphere, Dv the bulk diffusion coefficient, Ds the surface diffusion coefficient, and ν the number of molecules per unit area of surface. The terms in eq 2 correspond to different mechanisms of surface evolution: viscous flow (F), evaporation−condensation (A and A′), bulk diffusion (C), and surface diffusion (B). These different terms have distinct wavelength dependence, with the surface-diffusion term having the strongest dependence (K ∝ λ−4). It is worth noting that the F term is for a liquid film whose thickness is much greater than the grating wavelength, h ≫ λ, a condition met by our films (h ≈ 150 μm). For OTP, the evaporation−condensation and bulk diffusion terms are at least 600 times smaller than the observed K. For this calculation, we obtain γ by extrapolating the literature data22,23 to lower temperatures, and the values of p0, ρ0 and Dv from the work of Mapes et al.24 DG is taken to be 0.1 cm2/s, the typical value for organic molecules diffusing under ambient pressure.25 Ω = 0.3 nm3 is calculated from the specific volume of the OTP glass (0.89 cm3/g or 205 cm3/mol);26 this calculation assumes that the diffusing unit is an OTP molecule. The viscous flow term reproduces the observed K almost exactly at high temperatures (above Tg + 11 K). For this calculation, we obtain the bulk viscosity of OTP from ref 27: log η (Pa s) = −13.889 + 4003.3/[2.303(θ + 101.3)], where θ is temperature in °C. In Figure 1b, the curve running through the high-temperature data is the predicted K for viscous flow (Fq), with no adjustment. This excellent agreement indicates that viscous flow is the mechanism for grating decay above Tg + 11 K. This conclusion is further supported by the wavelength dependence K ∝ λ−1 (Figure 1c), which is expected for surface evolution by viscous flow. Whereas viscous flow accounts for the high-temperature decay of OTP surface gratings, it fails to do so at lower temperatures (Figure 1b). By elimination, surface diffusion is the only mechanism for surface smoothing at low temperatures. This assignment is verified by the wavelength dependence of K. According to eq 2, if surface diffusion is the mechanism for grating decay, K is proportional to λ−4. This wavelength dependence was indeed observed at low temperatures (Figure 1c). It is noteworthy that near Tg, the viscosity and diffusivity of liquid OTP decouple (deviate from the Stokes−Einstein relation),24 implying that the ratio of the bulk-diffusion term to the viscous-flow term increases with cooling. Despite this
Figure 2. (a) Surface and bulk diffusion coefficients, Ds and Dv, of amorphous OTP. (b) Ds and Dv of glass-forming molecular liquids vs Tg-scaled temperature (Tg = 246 K for OTP, 315 K for IMC and NIF, and 347 K for TNB; Tg is the onset temperature during heating at 10 K/min of a glass prepared by cooling at 10 K/min). 5073
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The Journal of Physical Chemistry B the bulk diffusion coefficients Dv.24 Surface diffusion is 108 times faster than bulk diffusion at Tg and even faster with cooling, because of its weaker temperature dependence. At the temperatures of our study, the temperature dependence of Ds is Arrhenius, with an activation energy Ea of 71 kJ/mol. Figure 2b extends the comparison of Ds and Dv to include other systems.3,4,24,28,29 The temperature is scaled by the Tg measured by differential scanning calorimetry. In this format, the Dv data of the different liquids nearly collapse to one master curve. The Dv data for tris-naphthylbenzene (TNB) were included here to verify the validity of the master curve. Figure 2b shows that while the bulk diffusivities of the molecular liquids have similar values at the same Tg-scaled temperature, their surface diffusivities can vary substantially. The three molecular glasses all show much faster diffusion at the free surface than in the bulk, with Ds/Dv = 106−108 at Tg. The ratio increases with cooling, since Dv decreases faster than Ds. The Ds of TNB has not been measured by surface-grating decay, but its value estimated from the embedding of nanoparticles is in broad agreement with those of the other systems.5 Note that at the same Tg/T (approximately the same Dv), surface diffusion is the fastest on OTP, followed by NIF, and then by IMC. This order correlates with the relative strengths of intermolecular forces in the three systems, a point we will discuss later. In constructing Figure 2b, we have revised slightly the published Ds values of IMC3 and NIF4 so that the molecular volumes Ω are all calculated consistently according to Mullins’s model:12 Ω = molar volume/Avogadro’s number. In previous work, Ω is taken to be the volume of a sphere of the molecular diameter. This correction changes Ω from 0.3 to 0.4 nm3.
observed to exhibit such transitions: IMC,3 NIF,4 and OTP (this work). In Figure 3, we characterize this transition by
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DISCUSSION This study has determined the surface self-diffusion coefficients of the model molecular glass OTP. Similar to the previously studied IMC and NIF,3,4 surface diffusion is vastly faster than bulk diffusion. For these systems, bulk diffusivities have nearly the same value at the same Tg-scaled temperature (Figure 2b), but surface diffusivities differ significantly, with OTP having the fastest surface diffusion. As they are cooled to become glasses, all three molecular liquids show a transition from viscous flow to surface diffusion as the mechanism of surface evolution. The transition viscosity (106.5−107.5 Pa s) is low relative to the viscosity expected at Tg, with the value for OTP being the lowest. In this section, we discuss these results in reference to the surface mobility of polystyrenes and silicates and theoretical predictions. For IMC, NIF, and OTP, surface diffusivity differs significantly at the same Tg-scaled temperature (Figure 2b). This difference correlates with the strength of intermolecular forces. OTP is a smaller molecule than NIF and IMC (Figure 2); OTP is incapable of hydrogen bonding, while NIF and IMC are. Between the latter two, IMC forms stronger hydrogen bonds because of its carboxyl group. Thus, the strength of intermolecular forces follows the order OTP < NIF < IMC. The trend revealed by these molecular liquids is that at the same Tg-scaled temperature, increasing the strength of intermolecular forces significantly slows down surface diffusion, but the effect is minor on bulk diffusion. It would be valuable to test the generality of this conclusion. Transition from Viscous Flow to Surface Diffusion as Mechanism of Surface Evolution. Among liquids of different compositions, only molecular liquids have been
Figure 3. Comparison of grating decay constants K and liquid dynamics for OTP, IMC and NIF (a) K vs structural relaxation time τα. (b) K vs viscosity η. At high fluidity, the relations K ∝ τα−1 and K ∝ η−1 hold (straight lines), indicating grating decay occurs by viscous flow.
plotting the decay constant K for 1000 nm wavelength surface gratings against two measures of fluidity: structural relaxation time measured by dielectric spectroscopy (τα)30−32 and viscosity.27,33 In these plots, temperature (and fluidity) decreases left to right. The viscosity of NIF, unknown at the temperatures of this study, is estimated from the data at higher temperatures34 and its τα,32 assuming η ∝ τα. Figure 3 shows that for all three systems, K is proportional to τα−1 and η−1 at high temperatures (straight lines). These two proportionalities anticipate each other, given the generally valid relation η ∝ τα, and both indicate surface smoothing by viscous flow at high temperatures. This mode of surface evolution, however, is lost at sufficiently large τα and η. As we have demonstrated here for OTP (Figure 1c) and previously for IMC3 and NIF,4 the new mechanism for surface evolution is surface diffusion. This assignment relies on the direct evaluation of the contributions from various mechanisms (eq 2) and the wavelength dependence of the grating decay rate. The observed relation K ∝ λ−4 identifies surface diffusion as the operative mechanism. This relation also rules out bulk diffusion as the responsible mechanism despite the known enhancement of bulk diffusion relative to viscous flow.24,28,29 5074
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The Journal of Physical Chemistry B From Figure 3, one obtains the level of fluidity at which the transition from viscous flow to surface diffusion occurs: ταt ≈ 0.1 s and ηt ≈ 107.5 Pa s for IMC and NIF, and ταt ≈ 0.01 s and ηt ≈ 106.5 Pa s for OTP (the subscript t denotes “transition”). These values are well below those expected for liquids at Tg (τα ≈ 100 s and η ≈ 1012 Pa s). That the transition occurs well above Tg indicates that it is unrelated to the bulk glass transition. The lower transition viscosity of OTP is explained by its faster surface diffusion, which controls surface-grating decay at higher fluidity. To further study the material dependence of surface diffusion, we compare in Figure 4 glass-forming liquids composed of small organic molecules, polystyrenes (molecular weights 5.6 kg/mol to 1.6 Mg/mol),16−18,35,36 and silicates.19,37,38 Compared to small organic molecules, the interaction between polymer molecules is stronger because of
their larger sizes and their entanglement, and the interaction between the atomic units in silicates is stronger because of their covalent and ionic bonding. Together these materials can probe the dependence of surface diffusion on intermolecular forces. In Figure 4, we plot ηs against ηb, where ηs is the viscosity calculated from the capillarity-driven decay of surface waves and ηb is the viscosity from such bulk measurements as torsional creep. For the decay of a sinusoidal surface on a thick liquid, ηs = πγ/λK, where all the parameters are defined in the context of eq 2; for thin films (whose thickness is smaller than the wavelength λ), the substrate effect on viscous relaxation must be accounted for.17,35,36 In this format, we expect an agreement between ηs and ηb at low viscosities, but a disagreement at high viscosities if viscous flow yields to surface diffusion as the mechanism of surface evolution. Of course this comparison is fair only at a common wavelength, since longer wavelength favors viscous flow to dominate surface evolution (eq 2). In Figure 4, the standard wavelength for comparing the different materials is 1000 nm; this is the wavelength at which the smallmolecule systems were studied and the wavelength bracketed by the data of polystyrenes and silicates. Note that in this analysis, ηs is the apparent viscosity calculated from surfaceevolution rates, not intended to represent the viscosity of the hypothetical mobile surface layer.11 Figure 4 shows a remarkable difference between the various materials. For the small-molecule systems (Figure 4a) ηs and ηb agree at low viscosities, but disagree at high viscosities. This result is already anticipated by Figure 3b and signals the transition to surface diffusion as the mechanism of surface flattening. In contrast, polystyrenes (Figure 4b) and silicates (Figure 4c) show good agreement between ηs and ηb up to the highest viscosities studied (1010 Pa s). These results argue that at the same bulk viscosity, surface diffusion is slower on polystyrene and silicate glasses than on the glasses of small organic molecules. Given the greater cohesive forces in polystyrenes and silicates, these results suggest that surface diffusion slows down in general with increasing intermolecular interactions. The data in Figures 4a and 4b, viewed together, suggest that low-molecular-weight polystyrenes might exhibit a transition from viscous flow to surface diffusion as the mechanism of surface evolution. This speculation is consistent with the observation of Chai et al. that ηs calculated from the evolution of a stepped surface of a 3 kg/mol PS matches ηb above Tg, but deviates from ηb below Tg (ηb is estimated to be 1010 Pa s at Tg).11 Their data are not plotted in Figure 4b because unlike those displayed, their experiment was not performed at a known or constant wavelength (a Fourier transform of a stepped surface returns multiple wavelengths). It would be valuable to extend their study with surface gratings of known wavelengths. A peculiarity revealed by Figure 4 is that some ηs values are vertically raised above the line ηs = ηb, by as much as one decade. This feature is likely a result of the combined experimental errors in measuring ηs and ηb; for example, ηb near Tg has such a strong temperature dependence that a small temperature error can cause a sizable disagreement. Surface Diffusion and Surface Crystal Growth on Molecular Glasses. Molecular glasses can grow crystals much faster on the free surface than in the interior.6,7 These crystals rise upward as they grow laterally, without penetrating deep into the bulk. For IMC and NIF, the velocity of surface crystal growth us is proportional to surface diffusivity Ds, suggesting
Figure 4. Comparison of the apparent viscosity calculated from surface-evolution rates and the bulk viscosity for glass-forming liquids composed of (a) small organic molecules, (b) polystyrenes (PS), and (c) silica-rich oxides. The PS data are from Hamdorf et al.16 (Mw = 5.6 kg/mol), Peng et al.18 (15, 41, 220, and 1571 kg/mol), Rognin et al.35 (30 and 130 kg/mol), Kim et al.17 (123 kg/mol), and Wang et al.36 (37 kg/mol). They are shown in two groups, as originally reported, one around the ηs = ηb line, the other slightly above it. The silicate data are from Wang et al.,19 Cassidy and Gjostein,37 and Kishi et al.,38 each group having studied multiple materials. 5075
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The Journal of Physical Chemistry B that surface crystal growth is supported by surface diffusion.4 There have been other explanations of fast surface crystallization; for example, the easier release of crystallizationinduced tension at the free surface.39 To better understand the phenomenon, in Figure 5, we extend the previous comparison
bulk crystal growth is received through a two-dimensional interface. Surface crystals have the height of many molecules and their growth velocity refers to the lateral velocity measured in top view. Because of its height, a surface crystal can grow one molecular length only after acquiring n times more molecules, where n is the crystal height in molecular diameter, in analogy with the building of a stone fence, whose extension by one stone takes 10 stones for a fence 10-stones high. In bulk crystal growth, every molecule joining the crystal advances the growth front by one molecular length. In this sense, bulk crystal growth is more “efficient”. Thus, even though surface diffusion vastly outpaces bulk diffusion, surface crystal growth is faster than bulk crystal growth by a smaller factor.7 Comparison of Experimental Results with Theoretical Models. Two theoretical models have been proposed to predict the degree to which mobility is enhanced at a glass surface: τα /τsurf = (τα /τ0)0.5
(3)
τα /τsurf = (τα /tc)n
(4)
In these equations, τα and τsurf are bulk and surface relaxation times, respectively. Equation 3 originates from the Random First Order Transition (RFOT) theory, where τ0 = 1 ps,20 and eq 4 from the Coupling Model (CM), where tc ≈ 2 ps and n is obtained from the Kohlrausch−Williams−Watt (KWW) exponent for fitting the bulk relaxation kinetics: Φ = exp[−(t/τα)1−n].21 For OTP, n = 0.5 (ref 30), and eqs 3 and 4 are nearly the same, both predicting τα/τsurf = 107 for τα = 100 s. This ratio approximately matches the observed ratio Ds/Dv = 108 for OTP at Tg. (Note that the two ratios need not match exactly given the known decoupling between diffusion and α relaxation.24,28,29) The theories also correctly predict the weaker temperature dependence of surface mobility. Concerning the material dependence of surface mobility, RFOT predicts the same τα/τsurf for all liquids at the same τα, which is at odds with the different surface diffusivities of various molecular liquids (Figure 2b). CM does allow different τα/τsurf ratios for different systems, but the literature values of n are too varied to assess its performance. Overall, the agreement between experiment and theory is encouraging while further work is needed to understand the material dependence of surface mobility.
Figure 5. Correlation between the surface crystal growth rate us and surface diffusion coefficient Ds for OTP, IMC, NIF, and amorphous silicon. α and γ refer to two polymorphs of IMC crystals.
of us and Ds to include OTP, whose surface crystal growth was recently reported,7 and amorphous silicon, whose us was measured for the surface growth of hemispherical grains40 and Ds from the evolution of surface grooves around them.41 We find that in general, faster surface crystal growth correlates with faster surface diffusion. The data for molecular glasses are approximately described by us ≈ Ds/(3 μm), while the a-Si point deviates slightly from the relation. This simple relation is surprising given the differences between these systems in Tg, molecular structure, crystal-growth velocity, and in the case of IMC, polymorphism (α IMC grows segregated needles while γ IMC compact domains). This relation suggests that among the many factors expected to influence surface crystal growth, surface diffusion is the rate-limiting step. If the trend in Figure 5 is general, we expect the slower surface diffusion of polymers and silicates to be associated with slower surface crystal growth. It would be of interest to test this idea. At present, reports differ on whether crystal growth is faster on the surface of silicate glasses than in the interior.42−45 The relation us ≈ Ds/ (3 μm) (Figure 5) is similar to the relation observed for bulk crystal growth in liquid OTP and IMC, uv ≈ Dv/ (50 nm), where uv is the velocity of bulk crystal growth and Dv bulk diffusivity.46 The latter relation describes diffusion-limited crystal growth in a bulk liquid above Tg. The similarity between the two relations suggests the soundness of the classical notion that crystal growth and diffusion have a common kinetic barrier, so long as the right diffusivity is employed. We note that the classical model is formulated to treat crystal growth in the interior of a liquid, not for crystal growth on the surface of a glass. It is of interest that the 3 μm in the relation us ≈ Ds/(3 μm) is longer than the 50 nm in the relation uv ≈ Dv/(50 nm), implying that at the same diffusivity, crystal growth is faster in the bulk than at the surface. This difference exists because the flux supporting surface crystal growth is received through a one-dimensional contact line between the crystal and the glass, whereas the flux supporting
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CONCLUSION We have measured the surface self-diffusion coefficients of oterphenyl glasses by surface grating decay. Similar to the previously studied IMC and NIF, the flattening of 1000 nm wavelength gratings on OTP occurs by viscous flow at high temperatures, but by surface diffusion at lower temperatures. Surface diffusion is 108 times faster than bulk diffusion at Tg and even faster at lower temperatures. At the same bulk diffusivity, molecular glasses can have different surface diffusion coefficients, in correlation with the strength of intermolecular interactions. All three molecular glass-formers (IMC, NIF, and OTP) exhibit a transition from viscous flow to surface diffusion as the mechanism of surface evolution, whereas polystyrenes and silicates undergo no such transition under the same conditions, indicating faster surface diffusion on molecular glasses. The proportionality between the rates of surface crystal growth and surface diffusion supports the view that surface diffusion is the rate-limiting step for surface crystal growth. RFOT and CM successfully predict the large surface enhance5076
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ment of mobility, while further work is needed to understand its material dependence. This study suggests that stronger intermolecular interactions lead to slower surface diffusion. This trend is observed within the group of molecular glasses, and between the molecular glasses and the glasses of polystyrenes and silicates, with the latter having stronger intermolecular forces and slower surface diffusion. To further test this relation, it may be fruitful to extend the current study to low-molecular-weight polymers. With decreasing molecular weights, a transition might be observed from viscous flow to surface diffusion as the mechanism of surface evolution. Such data would allow a fuller understanding of the material dependence of surface mobility. It is also of interest to study amorphous solids of different compositions to test the generality of the relation between surface crystal growth and surface diffusion.
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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, R. Burnette, and B. Bending for helpful discussions and assistance with instrumentation.
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REFERENCES
(1) Ehrlich, G.; Stolt, K. Surface Diffusion. Annu. Rev. Phys. Chem. 1980, 31, 603−637. (2) Seebauer, E. G.; Allen, C. E. Estimating Surface Diffusion Coefficients. Prog. Surf. Sci. 1995, 49, 265−330. (3) Zhu, L.; Brian, C. W.; Swallen, S. F.; Straus, P. T.; Ediger, M. D.; Yu, L. Surface Self-Diffusion of an Organic Glass. Phys. Rev. Lett. 2011, 106, 256103. (4) Brian, C. W.; Yu, L. Surface Self-Diffusion of Organic Glasses. J. Phys. Chem. A 2013, 117, 13303−13309. (5) Daley, C. R.; Fakhraai, Z.; Ediger, M. D.; Forrest, J. A. Comparing Surface and Bulk Flow of a Molecular Glass Former. Soft Matter 2012, 8, 2206−2212. (6) Sun, Y.; Zhu, L.; Kearns, K. L.; Ediger, M. D.; Yu, L. Glasses Crystallize Rapidly at Free Surfaces by Growing Crystals Upward. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 5990−5995. (7) Hasebe, M.; Musumeci, D.; Powell, C. T.; Cai, T.; Gunn, E.; Zhu, L.; Yu, L. Fast Surface Crystal Growth on Molecular Glasses and Its Termination by the Onset of Fluidity. J. Phys. Chem. B 2014, 118, 7638−7646. (8) Swallen, S. F.; Kearns, K. L.; Mapes, M. K.; Kim, Y. S.; McMahon, R. J.; Ediger, M. D.; Wu, T.; Yu, L.; Satija, S. Organic Glasses with Exceptional Thermodynamic and Kinetic Stability. Science 2007, 315, 353−356. (9) Zhu, L.; Yu, L. Generality of Forming Stable Organic Glasses by Vapor Deposition. Chem. Phys. Lett. 2010, 499, 62−65. (10) Yang, Z.; Fujii, Y.; Lee, F. K.; Lam, C.; Tsui, O. K. C. Glass Transition Dynamics and Surface Layer Mobility in Unentangled Polystyrene Films. Science 2010, 328, 1676−1679. (11) Chai, Y.; Salez, T.; McGraw, J. D.; Benzaquen, M.; DalnokiVeress, K.; Raphaël, E.; Forrest, J. A. A Direct Quantitative Measure of Surface Mobility in a Glassy Polymer. Science 2014, 343, 994−999. (12) Mullins, W. W. Flattening of a Nearly Plane Solid Surface due to Capillarity. J. Appl. Phys. 1959, 30, 77−83. (13) Bonzel, H. P.; Latta, E. E. Surface Self-Diffusion on Ni(110): Temperature Dependence and Directional Anisotropy. Surf. Sci. 1978, 76, 275−295. 5077
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Fast Surface Diffusion of Amorphous o‑Terphenyl and Its Competition with Viscous Flow in Surface Evolution Wei Zhang,† Caleb W. Brian,‡ and Lian Yu*,†,‡ †
School of Pharmacy and ‡Department of Chemistry, University of WisconsinMadison, Madison, Wisconsin 53705, United States ABSTRACT: Surface self-diffusion coefficients have been measured for the model molecular glass o-terphenyl (OTP) through surface-grating decay driven by capillarity. The decay mechanism transitions from viscous flow at high temperatures to surface diffusion at low temperatures; for 1000 nm wavelength gratings, the transition occurs at Tg + 11 K. The surface diffusion of OTP is 108 times faster than bulk diffusion at Tg and even faster at lower temperatures because of its weaker temperature dependence. At Tg, OTP has approximately the same bulk diffusivity as the previously studied molecular liquid indomethacin, but its surface diffusion is 100 times faster. While the molecular glass-formers exhibit transitions from viscous flow to surface diffusion as the mechanism of capillarity-driven surface flattening, polystyrenes and silicates show no such transition under comparable conditions, suggesting slower surface diffusion on these materials and a general dependence of surface diffusion on intermolecular forces. The velocity of surface crystal growth on molecular glasses is proportional to surface diffusivity, indicating a common kinetic barrier for both processes for temperatures below Tg.
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INTRODUCTION Surface mobility influences many processes in science and technology, including catalysis, sintering, and crystal growth. Among its different measures, the surface diffusion coefficient Ds characterizes the in-plane translation of atoms or molecules on the surface. Although this property has been studied extensively for crystalline metals and semiconductors,1,2 only recently have there been reports of Ds for molecular solids.3−5 These studies show that similar to atomic solids, molecular solids can have vastly faster diffusion at the free surface. These results help develop a broad perspective of surface mobility across classes of materials, and are relevant for understanding several phenomena of current interest: fast surface crystal growth on molecular glasses,6,7 formation of stable glasses by vapor deposition,8,9 and the surface mobility of polymer glasses.10,11 A surface contour can evolve under the driving force of surface tension in several mechanismsviscous flow, evaporation−condensation, bulk diffusion, and surface diffusion.12 Mullins showed how surface diffusion can be distinguished from other mechanisms by direct evaluation and wavelength dependence. Surface diffusion flattens a sinusoidal surface at a rate proportional to q4, where q is the spatial frequency, while the other mechanisms do so at rates that have weaker dependence on q. This way of identifying and measuring surface diffusion has been applied to crystalline metals and silicon13,14 and to molecular glasses.3,4 By simulation, Malshe et al. found that surface diffusivity obtained from surface-grating decay agrees with that from the growth of mean square displacement of single particles.15 © 2015 American Chemical Society
Recent studies of glass-forming molecular liquids found that the mechanism of surface flattening driven by surface tension changes from viscous flow at high temperatures to surface diffusion at low temperatures.3,4 For a sinusoidal surface of 1000 nm wavelength on indomethacin and nifedipine (IMC and NIF; see Figure 2 for their structures), this transition occurs at 12 K above the glass transition temperature Tg at a relatively low viscosity (107.5 Pa s). In contrast, similar studies on the liquids of polystyrenes16−18 and silicates19 observed that viscous flow remains the mechanism of surface-grating decay up to the highest viscosities studied (1010 Pa s), with no sign of transition to another decay mechanism. This difference suggests that surface diffusion is faster on the glasses of small organic molecules than on polymeric and silicate glasses, and motivates inquiries into the material dependence of surface diffusion. As temperature decreases toward Tg, viscosity increases sharply and viscous relaxation slows. We anticipate that for a glassformer with faster surface diffusion, viscous flow loses to surface diffusion as the mechanism of capillarity-driven surface flattening at a lower viscosity. This study investigated the surface mobility of the model molecular glass former o-terphenyl (OTP). OTP has weaker intermolecular forces than the previously studied IMC and NIF,3,4 a consequence of its smaller size and lack of polar groups, and together these systems enable a test of the material dependence of surface diffusion. We report that the surface diffusion of OTP is vastly faster than its bulk diffusion. Received: December 21, 2014 Revised: February 20, 2015 Published: March 24, 2015 5071
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The Journal of Physical Chemistry B Compared to IMC, OTP has faster surface diffusion at the same bulk diffusivity, and the mechanism of its surface evolution changes from viscous flow to surface diffusion at a lower viscosity. We analyze these results in reference to the surface mobility of polystyrene and silicate liquids and theoretical predictions.20,21 Molecular glasses grow crystals much faster on the surface than in the interior7 and we report that the velocity of surface crystal growth is approximately proportional to the surface diffusion coefficient, indicating a common kinetic barrier for the two processes.
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MATERIALS AND METHODS o-Terphenyl (OTP, 99% pure) was purchased from SigmaAldrich and used as received. Surface gratings were embossed unto liquid OTP at 273 K (27 K above Tg) in flowing N2 with master gratings of different wavelengths (740−8200 nm). The masters were purchased from Rainbow Symphony (1000 and 2000 nm), separated from data storage discs (CD for 1500 nm, DVD for 740 nm), or replicated from glass gratings through a Norland UV-curing optical adhesive (3300 and 8200 nm). The masters were gold-coated to avoid transfer of contaminants. The masters were confirmed by atomic force microscopy to be sinusoidal, except for those of 3300 and 8200 nm wavelengths, which were blazed (“sawtooth”). Upon cooling to 243 K, the master was detached from OTP, yielding a glass film with a corrugated surface. The OTP films thus prepared were ca. 150 μm thick, much thicker than any grating wavelength used so that the rate of viscous relaxation did not depend on film thickness. The decay of surface grating was monitored in flowing N2 at a constant temperature maintained with a Linkam THMS 600E stage. A helium−neon laser (632.8 nm) passed normally through the sample and the first-order diffraction was measured in transmission with a photodiode (Si Transimpedence Amplified Photodetector from Thorlabs) interfaced with an oscilloscope or a National Instruments NI-DAQ data acquisition card. The square root of the diffraction intensity was taken to be proportional to the grating amplitude.
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RESULTS Figure 1a shows typical decay curves for OTP surface gratings I/I0 vs t, where I/I0 is the normalized first-order diffraction intensity and t is time. The decay at each temperature is fitted with an exponential function or a stretched exponential function, I/I0 = exp[−(KIt)β], with β close to 1. In the latter case, we regard KI as characterizing the average decay kinetics during the measurement time. Figure 1b plots the decay constant K vs temperature T for 1000 nm wavelength gratings; here K is the grating-amplitude decay constant, and half of KI, because of the relation I ∝ h2. The solid symbols are data directly measured with 1000 nm wavelength gratings; the open symbols are data measured at other wavelengths and converted to λ = 1000 nm, after confirming the wavelength dependence at the respective temperatures (Figure 1c). By doing so we obtained the values of K at higher temperatures at which the grating decay was too fast to be measured by our method at λ = 1000 nm. Together, the measured decay constants span 5 orders of magnitude. The upper limit of this range was determined by grating decay being faster than temperature equilibration, and the lower limit by the fracture of OTP glasses upon cooling on a substrate (silicate) of lower thermal expansion coefficient.
Figure 1. (a) Representative decay curves of 1000 nm wavelength OTP gratings at different temperatures. (b) Decay constant K vs T for 1000 nm wavelength OTP gratings. (c) Wavelength dependence of K at different temperatures.
To help identify the mechanism of surface evolution, we measured the wavelength dependence of K (Figure 1c). At 263 K (Tg + 17 K), we find K ∝ λ−1 for λ = 1000−8200 nm; the slope of the log K vs log λ plot is −0.97. At 228 K (Tg − 18 K) and 223 K (Tg − 23 K), we find K ∝ λ−4 for λ = 740−2000 nm; the slope of the log K vs log λ plot is −4.1. As we discuss later, these wavelength dependences indicate surface evolution by viscous flow and surface diffusion, respectively. Mullins12 showed that for a sinusoidal grating, the grating amplitude h decays exponentially: h = h0 exp( −Kt )
(1)
where the decay constant K is given by K = Fq + Aq2 + Dq3 + Bq 4 5072
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The Journal of Physical Chemistry B
effect, the bulk-diffusion term never becomes large enough under our experimental conditions to be the mechanism of surface grating decay. This conclusion is confirmed by direct evaluation using the known bulk diffusivity and by the wavelength dependence of K: should bulk diffusion be the decay mechanism, K would be proportional to λ−3, which was not observed. After identifying surface diffusion as the mechanism for grating decay at low temperatures, we calculate the surface diffusion coefficients Ds using eq 2. For this calculation, we use ν = Ω−2/3 = 2 nm−2. The results (Figure 2a) are compared with
In eq 2, q = 2π /λ
F=
A=
γ 2η p0 γ Ω2 (2πm)1/2 (kT )3/2
D = A′ + C =
B=
ρ0 DGγ Ω2 kT
+
Dv γ Ω kT
Dsγ Ω2ν kT
Here, λ is the grating wavelength, γ the surface tension, η the viscosity, p0 the equilibrium vapor pressure, Ω the molecular volume, m the molecular mass, ρ0 the equilibrium vapor density, DG the diffusion coefficient of evaporated molecules in the inert atmosphere, Dv the bulk diffusion coefficient, Ds the surface diffusion coefficient, and ν the number of molecules per unit area of surface. The terms in eq 2 correspond to different mechanisms of surface evolution: viscous flow (F), evaporation−condensation (A and A′), bulk diffusion (C), and surface diffusion (B). These different terms have distinct wavelength dependence, with the surface-diffusion term having the strongest dependence (K ∝ λ−4). It is worth noting that the F term is for a liquid film whose thickness is much greater than the grating wavelength, h ≫ λ, a condition met by our films (h ≈ 150 μm). For OTP, the evaporation−condensation and bulk diffusion terms are at least 600 times smaller than the observed K. For this calculation, we obtain γ by extrapolating the literature data22,23 to lower temperatures, and the values of p0, ρ0 and Dv from the work of Mapes et al.24 DG is taken to be 0.1 cm2/s, the typical value for organic molecules diffusing under ambient pressure.25 Ω = 0.3 nm3 is calculated from the specific volume of the OTP glass (0.89 cm3/g or 205 cm3/mol);26 this calculation assumes that the diffusing unit is an OTP molecule. The viscous flow term reproduces the observed K almost exactly at high temperatures (above Tg + 11 K). For this calculation, we obtain the bulk viscosity of OTP from ref 27: log η (Pa s) = −13.889 + 4003.3/[2.303(θ + 101.3)], where θ is temperature in °C. In Figure 1b, the curve running through the high-temperature data is the predicted K for viscous flow (Fq), with no adjustment. This excellent agreement indicates that viscous flow is the mechanism for grating decay above Tg + 11 K. This conclusion is further supported by the wavelength dependence K ∝ λ−1 (Figure 1c), which is expected for surface evolution by viscous flow. Whereas viscous flow accounts for the high-temperature decay of OTP surface gratings, it fails to do so at lower temperatures (Figure 1b). By elimination, surface diffusion is the only mechanism for surface smoothing at low temperatures. This assignment is verified by the wavelength dependence of K. According to eq 2, if surface diffusion is the mechanism for grating decay, K is proportional to λ−4. This wavelength dependence was indeed observed at low temperatures (Figure 1c). It is noteworthy that near Tg, the viscosity and diffusivity of liquid OTP decouple (deviate from the Stokes−Einstein relation),24 implying that the ratio of the bulk-diffusion term to the viscous-flow term increases with cooling. Despite this
Figure 2. (a) Surface and bulk diffusion coefficients, Ds and Dv, of amorphous OTP. (b) Ds and Dv of glass-forming molecular liquids vs Tg-scaled temperature (Tg = 246 K for OTP, 315 K for IMC and NIF, and 347 K for TNB; Tg is the onset temperature during heating at 10 K/min of a glass prepared by cooling at 10 K/min). 5073
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The Journal of Physical Chemistry B the bulk diffusion coefficients Dv.24 Surface diffusion is 108 times faster than bulk diffusion at Tg and even faster with cooling, because of its weaker temperature dependence. At the temperatures of our study, the temperature dependence of Ds is Arrhenius, with an activation energy Ea of 71 kJ/mol. Figure 2b extends the comparison of Ds and Dv to include other systems.3,4,24,28,29 The temperature is scaled by the Tg measured by differential scanning calorimetry. In this format, the Dv data of the different liquids nearly collapse to one master curve. The Dv data for tris-naphthylbenzene (TNB) were included here to verify the validity of the master curve. Figure 2b shows that while the bulk diffusivities of the molecular liquids have similar values at the same Tg-scaled temperature, their surface diffusivities can vary substantially. The three molecular glasses all show much faster diffusion at the free surface than in the bulk, with Ds/Dv = 106−108 at Tg. The ratio increases with cooling, since Dv decreases faster than Ds. The Ds of TNB has not been measured by surface-grating decay, but its value estimated from the embedding of nanoparticles is in broad agreement with those of the other systems.5 Note that at the same Tg/T (approximately the same Dv), surface diffusion is the fastest on OTP, followed by NIF, and then by IMC. This order correlates with the relative strengths of intermolecular forces in the three systems, a point we will discuss later. In constructing Figure 2b, we have revised slightly the published Ds values of IMC3 and NIF4 so that the molecular volumes Ω are all calculated consistently according to Mullins’s model:12 Ω = molar volume/Avogadro’s number. In previous work, Ω is taken to be the volume of a sphere of the molecular diameter. This correction changes Ω from 0.3 to 0.4 nm3.
observed to exhibit such transitions: IMC,3 NIF,4 and OTP (this work). In Figure 3, we characterize this transition by
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DISCUSSION This study has determined the surface self-diffusion coefficients of the model molecular glass OTP. Similar to the previously studied IMC and NIF,3,4 surface diffusion is vastly faster than bulk diffusion. For these systems, bulk diffusivities have nearly the same value at the same Tg-scaled temperature (Figure 2b), but surface diffusivities differ significantly, with OTP having the fastest surface diffusion. As they are cooled to become glasses, all three molecular liquids show a transition from viscous flow to surface diffusion as the mechanism of surface evolution. The transition viscosity (106.5−107.5 Pa s) is low relative to the viscosity expected at Tg, with the value for OTP being the lowest. In this section, we discuss these results in reference to the surface mobility of polystyrenes and silicates and theoretical predictions. For IMC, NIF, and OTP, surface diffusivity differs significantly at the same Tg-scaled temperature (Figure 2b). This difference correlates with the strength of intermolecular forces. OTP is a smaller molecule than NIF and IMC (Figure 2); OTP is incapable of hydrogen bonding, while NIF and IMC are. Between the latter two, IMC forms stronger hydrogen bonds because of its carboxyl group. Thus, the strength of intermolecular forces follows the order OTP < NIF < IMC. The trend revealed by these molecular liquids is that at the same Tg-scaled temperature, increasing the strength of intermolecular forces significantly slows down surface diffusion, but the effect is minor on bulk diffusion. It would be valuable to test the generality of this conclusion. Transition from Viscous Flow to Surface Diffusion as Mechanism of Surface Evolution. Among liquids of different compositions, only molecular liquids have been
Figure 3. Comparison of grating decay constants K and liquid dynamics for OTP, IMC and NIF (a) K vs structural relaxation time τα. (b) K vs viscosity η. At high fluidity, the relations K ∝ τα−1 and K ∝ η−1 hold (straight lines), indicating grating decay occurs by viscous flow.
plotting the decay constant K for 1000 nm wavelength surface gratings against two measures of fluidity: structural relaxation time measured by dielectric spectroscopy (τα)30−32 and viscosity.27,33 In these plots, temperature (and fluidity) decreases left to right. The viscosity of NIF, unknown at the temperatures of this study, is estimated from the data at higher temperatures34 and its τα,32 assuming η ∝ τα. Figure 3 shows that for all three systems, K is proportional to τα−1 and η−1 at high temperatures (straight lines). These two proportionalities anticipate each other, given the generally valid relation η ∝ τα, and both indicate surface smoothing by viscous flow at high temperatures. This mode of surface evolution, however, is lost at sufficiently large τα and η. As we have demonstrated here for OTP (Figure 1c) and previously for IMC3 and NIF,4 the new mechanism for surface evolution is surface diffusion. This assignment relies on the direct evaluation of the contributions from various mechanisms (eq 2) and the wavelength dependence of the grating decay rate. The observed relation K ∝ λ−4 identifies surface diffusion as the operative mechanism. This relation also rules out bulk diffusion as the responsible mechanism despite the known enhancement of bulk diffusion relative to viscous flow.24,28,29 5074
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The Journal of Physical Chemistry B From Figure 3, one obtains the level of fluidity at which the transition from viscous flow to surface diffusion occurs: ταt ≈ 0.1 s and ηt ≈ 107.5 Pa s for IMC and NIF, and ταt ≈ 0.01 s and ηt ≈ 106.5 Pa s for OTP (the subscript t denotes “transition”). These values are well below those expected for liquids at Tg (τα ≈ 100 s and η ≈ 1012 Pa s). That the transition occurs well above Tg indicates that it is unrelated to the bulk glass transition. The lower transition viscosity of OTP is explained by its faster surface diffusion, which controls surface-grating decay at higher fluidity. To further study the material dependence of surface diffusion, we compare in Figure 4 glass-forming liquids composed of small organic molecules, polystyrenes (molecular weights 5.6 kg/mol to 1.6 Mg/mol),16−18,35,36 and silicates.19,37,38 Compared to small organic molecules, the interaction between polymer molecules is stronger because of
their larger sizes and their entanglement, and the interaction between the atomic units in silicates is stronger because of their covalent and ionic bonding. Together these materials can probe the dependence of surface diffusion on intermolecular forces. In Figure 4, we plot ηs against ηb, where ηs is the viscosity calculated from the capillarity-driven decay of surface waves and ηb is the viscosity from such bulk measurements as torsional creep. For the decay of a sinusoidal surface on a thick liquid, ηs = πγ/λK, where all the parameters are defined in the context of eq 2; for thin films (whose thickness is smaller than the wavelength λ), the substrate effect on viscous relaxation must be accounted for.17,35,36 In this format, we expect an agreement between ηs and ηb at low viscosities, but a disagreement at high viscosities if viscous flow yields to surface diffusion as the mechanism of surface evolution. Of course this comparison is fair only at a common wavelength, since longer wavelength favors viscous flow to dominate surface evolution (eq 2). In Figure 4, the standard wavelength for comparing the different materials is 1000 nm; this is the wavelength at which the smallmolecule systems were studied and the wavelength bracketed by the data of polystyrenes and silicates. Note that in this analysis, ηs is the apparent viscosity calculated from surfaceevolution rates, not intended to represent the viscosity of the hypothetical mobile surface layer.11 Figure 4 shows a remarkable difference between the various materials. For the small-molecule systems (Figure 4a) ηs and ηb agree at low viscosities, but disagree at high viscosities. This result is already anticipated by Figure 3b and signals the transition to surface diffusion as the mechanism of surface flattening. In contrast, polystyrenes (Figure 4b) and silicates (Figure 4c) show good agreement between ηs and ηb up to the highest viscosities studied (1010 Pa s). These results argue that at the same bulk viscosity, surface diffusion is slower on polystyrene and silicate glasses than on the glasses of small organic molecules. Given the greater cohesive forces in polystyrenes and silicates, these results suggest that surface diffusion slows down in general with increasing intermolecular interactions. The data in Figures 4a and 4b, viewed together, suggest that low-molecular-weight polystyrenes might exhibit a transition from viscous flow to surface diffusion as the mechanism of surface evolution. This speculation is consistent with the observation of Chai et al. that ηs calculated from the evolution of a stepped surface of a 3 kg/mol PS matches ηb above Tg, but deviates from ηb below Tg (ηb is estimated to be 1010 Pa s at Tg).11 Their data are not plotted in Figure 4b because unlike those displayed, their experiment was not performed at a known or constant wavelength (a Fourier transform of a stepped surface returns multiple wavelengths). It would be valuable to extend their study with surface gratings of known wavelengths. A peculiarity revealed by Figure 4 is that some ηs values are vertically raised above the line ηs = ηb, by as much as one decade. This feature is likely a result of the combined experimental errors in measuring ηs and ηb; for example, ηb near Tg has such a strong temperature dependence that a small temperature error can cause a sizable disagreement. Surface Diffusion and Surface Crystal Growth on Molecular Glasses. Molecular glasses can grow crystals much faster on the free surface than in the interior.6,7 These crystals rise upward as they grow laterally, without penetrating deep into the bulk. For IMC and NIF, the velocity of surface crystal growth us is proportional to surface diffusivity Ds, suggesting
Figure 4. Comparison of the apparent viscosity calculated from surface-evolution rates and the bulk viscosity for glass-forming liquids composed of (a) small organic molecules, (b) polystyrenes (PS), and (c) silica-rich oxides. The PS data are from Hamdorf et al.16 (Mw = 5.6 kg/mol), Peng et al.18 (15, 41, 220, and 1571 kg/mol), Rognin et al.35 (30 and 130 kg/mol), Kim et al.17 (123 kg/mol), and Wang et al.36 (37 kg/mol). They are shown in two groups, as originally reported, one around the ηs = ηb line, the other slightly above it. The silicate data are from Wang et al.,19 Cassidy and Gjostein,37 and Kishi et al.,38 each group having studied multiple materials. 5075
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The Journal of Physical Chemistry B that surface crystal growth is supported by surface diffusion.4 There have been other explanations of fast surface crystallization; for example, the easier release of crystallizationinduced tension at the free surface.39 To better understand the phenomenon, in Figure 5, we extend the previous comparison
bulk crystal growth is received through a two-dimensional interface. Surface crystals have the height of many molecules and their growth velocity refers to the lateral velocity measured in top view. Because of its height, a surface crystal can grow one molecular length only after acquiring n times more molecules, where n is the crystal height in molecular diameter, in analogy with the building of a stone fence, whose extension by one stone takes 10 stones for a fence 10-stones high. In bulk crystal growth, every molecule joining the crystal advances the growth front by one molecular length. In this sense, bulk crystal growth is more “efficient”. Thus, even though surface diffusion vastly outpaces bulk diffusion, surface crystal growth is faster than bulk crystal growth by a smaller factor.7 Comparison of Experimental Results with Theoretical Models. Two theoretical models have been proposed to predict the degree to which mobility is enhanced at a glass surface: τα /τsurf = (τα /τ0)0.5
(3)
τα /τsurf = (τα /tc)n
(4)
In these equations, τα and τsurf are bulk and surface relaxation times, respectively. Equation 3 originates from the Random First Order Transition (RFOT) theory, where τ0 = 1 ps,20 and eq 4 from the Coupling Model (CM), where tc ≈ 2 ps and n is obtained from the Kohlrausch−Williams−Watt (KWW) exponent for fitting the bulk relaxation kinetics: Φ = exp[−(t/τα)1−n].21 For OTP, n = 0.5 (ref 30), and eqs 3 and 4 are nearly the same, both predicting τα/τsurf = 107 for τα = 100 s. This ratio approximately matches the observed ratio Ds/Dv = 108 for OTP at Tg. (Note that the two ratios need not match exactly given the known decoupling between diffusion and α relaxation.24,28,29) The theories also correctly predict the weaker temperature dependence of surface mobility. Concerning the material dependence of surface mobility, RFOT predicts the same τα/τsurf for all liquids at the same τα, which is at odds with the different surface diffusivities of various molecular liquids (Figure 2b). CM does allow different τα/τsurf ratios for different systems, but the literature values of n are too varied to assess its performance. Overall, the agreement between experiment and theory is encouraging while further work is needed to understand the material dependence of surface mobility.
Figure 5. Correlation between the surface crystal growth rate us and surface diffusion coefficient Ds for OTP, IMC, NIF, and amorphous silicon. α and γ refer to two polymorphs of IMC crystals.
of us and Ds to include OTP, whose surface crystal growth was recently reported,7 and amorphous silicon, whose us was measured for the surface growth of hemispherical grains40 and Ds from the evolution of surface grooves around them.41 We find that in general, faster surface crystal growth correlates with faster surface diffusion. The data for molecular glasses are approximately described by us ≈ Ds/(3 μm), while the a-Si point deviates slightly from the relation. This simple relation is surprising given the differences between these systems in Tg, molecular structure, crystal-growth velocity, and in the case of IMC, polymorphism (α IMC grows segregated needles while γ IMC compact domains). This relation suggests that among the many factors expected to influence surface crystal growth, surface diffusion is the rate-limiting step. If the trend in Figure 5 is general, we expect the slower surface diffusion of polymers and silicates to be associated with slower surface crystal growth. It would be of interest to test this idea. At present, reports differ on whether crystal growth is faster on the surface of silicate glasses than in the interior.42−45 The relation us ≈ Ds/ (3 μm) (Figure 5) is similar to the relation observed for bulk crystal growth in liquid OTP and IMC, uv ≈ Dv/ (50 nm), where uv is the velocity of bulk crystal growth and Dv bulk diffusivity.46 The latter relation describes diffusion-limited crystal growth in a bulk liquid above Tg. The similarity between the two relations suggests the soundness of the classical notion that crystal growth and diffusion have a common kinetic barrier, so long as the right diffusivity is employed. We note that the classical model is formulated to treat crystal growth in the interior of a liquid, not for crystal growth on the surface of a glass. It is of interest that the 3 μm in the relation us ≈ Ds/(3 μm) is longer than the 50 nm in the relation uv ≈ Dv/(50 nm), implying that at the same diffusivity, crystal growth is faster in the bulk than at the surface. This difference exists because the flux supporting surface crystal growth is received through a one-dimensional contact line between the crystal and the glass, whereas the flux supporting
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CONCLUSION We have measured the surface self-diffusion coefficients of oterphenyl glasses by surface grating decay. Similar to the previously studied IMC and NIF, the flattening of 1000 nm wavelength gratings on OTP occurs by viscous flow at high temperatures, but by surface diffusion at lower temperatures. Surface diffusion is 108 times faster than bulk diffusion at Tg and even faster at lower temperatures. At the same bulk diffusivity, molecular glasses can have different surface diffusion coefficients, in correlation with the strength of intermolecular interactions. All three molecular glass-formers (IMC, NIF, and OTP) exhibit a transition from viscous flow to surface diffusion as the mechanism of surface evolution, whereas polystyrenes and silicates undergo no such transition under the same conditions, indicating faster surface diffusion on molecular glasses. The proportionality between the rates of surface crystal growth and surface diffusion supports the view that surface diffusion is the rate-limiting step for surface crystal growth. RFOT and CM successfully predict the large surface enhance5076
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The Journal of Physical Chemistry B
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ment of mobility, while further work is needed to understand its material dependence. This study suggests that stronger intermolecular interactions lead to slower surface diffusion. This trend is observed within the group of molecular glasses, and between the molecular glasses and the glasses of polystyrenes and silicates, with the latter having stronger intermolecular forces and slower surface diffusion. To further test this relation, it may be fruitful to extend the current study to low-molecular-weight polymers. With decreasing molecular weights, a transition might be observed from viscous flow to surface diffusion as the mechanism of surface evolution. Such data would allow a fuller understanding of the material dependence of surface mobility. It is also of interest to study amorphous solids of different compositions to test the generality of the relation between surface crystal growth and surface diffusion.
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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, R. Burnette, and B. Bending for helpful discussions and assistance with instrumentation.
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