Effect of bulk aging on surface diffusion of glasses Caleb W. Brian, Lei Zhu, and Lian Yu
Citation: J. Chem. Phys. 140, 054509 (2014); doi: 10.1063/1.4863556 View online: http://dx.doi.org/10.1063/1.4863556 View Table of Contents: http://aip.scitation.org/toc/jcp/140/5 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 140, 054509 (2014)
Effect of bulk aging on surface diffusion of glasses Caleb W. Brian, Lei Zhu, and Lian Yua) Department of Chemistry and School of Pharmacy, University of Wisconsin–Madison, Madison, Wisconsin 53705, USA
(Received 11 October 2013; accepted 16 January 2014; published online 5 February 2014) The effect of physical aging on surface diffusion has been determined for two organic glasses, Indomethacin and Nifedipine. The two systems exhibit similar aging kinetics typical of organic glasses. Surface diffusivity remains unchanged despite significant bulk aging that nearly equilibrates the systems and increases the bulk relaxation time by orders of magnitude. The finding is relevant for understanding the stability of amorphous materials and the formation of low-energy glasses by vapor deposition. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863556] INTRODUCTION
The mobility of molecules at the free surface, in reference to those in the bulk, is important in condensed-phase science. Although much is known about the surface mobility of metals, semiconductors, and polymers,1–3 little data exist on organic solids of relatively low molecular weights.4, 5 Such data are necessary for developing organic materials for many applications and for understanding surface mobility across classes of materials. Characterizing surface mobility of organic solids is particularly relevant for two areas of current research: fast surface crystal growth on organic glasses6–8 and formation of low-energy glasses by physical vapor deposition.9–11 Both phenomena are attributed to the high mobility of surface molecules. Recently, theories have been proposed for the surface mobility of glasses,12, 13 further stimulating work in this area. Glasses form by cooling liquids, condensing vapors, and evaporating solutions while avoiding crystallization. An important property of glasses is their relaxation or aging over time toward the equilibrium liquid state, leading to higher density, lower energy, and slower molecular motions.14–17 Despite extensive studies of this phenomenon, however, little is known about the effect of glass aging on surface mobility. There have been no measurements of surface diffusivity of glasses that have been aged systematically. In their current stage of development, the theoretical models12, 13 offer no predictions of the effect of glass aging on surface mobility. The notion that there exists a mobile surface layer atop a solid glass18 suggests that surface molecules can remain equilibrated as the bulk ages, with their mobility adjusting quickly to the bulk state. The surface response to bulk aging would be more complex if the surface layer itself undergoes the glass transition.19 To assess the interplay between surface and bulk dynamics, we studied the effect of physical aging on the surface diffusivity of two organic glasses, Indomethacin and Nifedipine (Scheme 1). We found that the two systems have similar aging kinetics typical of organic glasses. Surface diffusion is unafa) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. Telephone: (608) 263 2263. 0021-9606/2014/140(5)/054509/6/$30.00
fected by significant bulk aging that nearly equilibrates the systems and increases the bulk structural relaxation times by orders of magnitude. The finding is consistent with the notion that the mobile surface layer of organic glasses is weakly coupled to bulk molecules. METHOD
Indomethacin [1-(p-chlorobenzoyl)-5-methoxy-2-methylindole-3-acetic acid (IMC), γ polymorph] was obtained from Sigma (St. Louis, MO). Nifedipine [1,4-Dihydro2,6-dimethyl-4-(2-nitrophenyl)-3,5-pyridinedicarboxylic acid dimethyl ester (NIF), α polymorph] was obtained from FWD Chem (99.8% purity). Differential Scanning Calorimetry (DSC) was performed with a TA Instruments Q2000 unit to measure enthalpy recovery during glass aging. Each sample was 3–6 mg and was placed in a crimped Al DSC pan, leaving no head space, so that the condition of aging approximately matched that of the samples in the surface-diffusion measurements. For shorter aging (up to several hours), the sample always stayed in the DSC cell; it was melted at 20 K above the crystal melting point, cooled to an aging temperature, and held for a desired time. For subsequent analysis, the sample was cooled and reheated at 10 K/min to determine the change of the glass’s enthalpy and its fictive temperature. For longer aging (>10 h), the sample was stored outside the DSC cell in a desiccator at the laboratory temperature, and the other steps were the same. To prepare surface gratings, plastic 1000 nm gratings (Edmond Scientific) were gold coated to serve as masters. The masters were placed over an IMC or NIF liquid at 353 K, 45 K above the glass transition temperature Tg (315 K) for 1 min. Samples were cooled to laboratory temperature (295 ± 1 K) and aged in a desiccator for allotted times, with the masters intact. Freshly made control samples were prepared the day the aged samples were measured. To start a measurement of surface grating decay, the master was detached to expose a surface grating and its smoothing over time was monitored. The embossed surface grating was nearly sinusoidal as confirmed by atomic force microscopy (AFM).
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(a)
SCHEME 1. Molecular structures of Nifedipine and Indomethacin.
The smoothing of surface gratings was monitored in dry N2 with an atomic force microscope (Veeco Multimode IV) or an optical microscope (Nikon Optiphot 2). Tapping-mode AFM scans were performed perpendicular to the grating direction. The height profile was Fourier-transformed to obtain the grating amplitude. Optical diffraction was recorded in transmission with the aid of a Bertrand lens. The white light was filtered with a 530 nm band-pass filter. The square root of the first order diffraction intensity was taken to be proportional to the grating amplitude. A 1000 nm grating on IMC decayed in ca. 2 weeks at 295 K; the sample was periodically removed from a desiccator, measured in dry N2 , and returned to the desiccator for further decay. A 1000 nm grating on NIF flattened in ca. one day at 295 K; the sample was followed in the same way as an IMC sample or was placed in a N2 -purged hot stage (Linkam THMS600) and measured real-time with more precise control of temperature. RESULTS Kinetics of glass aging
This study compared the surface mobility of freshly made and aged glasses of IMC and NIF. We used the enthalpy decrease and the enthalpic fictive temperature Tf to characterize the progress of glass aging. Figure 1(a) shows the typical DSC data collected for this purpose. To calculate the enthalpy decrease from a freshly made glass to an aged glass, their heat capacities were integrated from the glassy state to a common temperature above Tg , at which the two glasses are identical (Figure 1(b)). The Tf of a glass is the equivalent temperature of an equilibrated liquid that, if vitrified at Tf , has the same enthalpy as the glass.20 A freshly made glass has a Tf that is approximately its Tg measured by DSC, and a glass aged to equilibrium has a Tf that is identical to the temperature of aging. Figures 2(a) and 2(b) show the enthalpy decrease and the Tf of an IMC glass as a function of the aging time ta . The data from this work are compared with those of Kearns et al. (open circles);21 the two sets are in good agreement. The longest aging time is determined by the crystallization of IMC glasses, and can be as long as several months if no free surface is exposed during aging.6–8 At our aging temperature (Ta = 295 K), the equilibrated glass would have a Tf of 295 K and an enthalpy decrease of −8.2 J/g. The latter value is calculated from H∞ = Cpg (Ta − Tg ), where Cpg is the heat-capacity change upon the glass transition. For IMC,
(b)
FIG. 1. DSC analysis of glass aging. (a) Heat capacity curves for a freshly made and aged IMC glass. (b) Enthalpy decrease on aging is calculated from the integrated Cp curves of freshly made and aged glasses. The fictive temperature Tf is found by extrapolating the liquid line to intersect the glass line.
Cpg = 0.41 J/g K22 and is assumed to be constant in the temperature range of interest. Owing to faster crystallization, the aging kinetics of NIF glasses cannot be followed as long as IMC glasses. For the overlapping periods, however, the two glasses aged at similar rates. As Figure 2(c) shows, the Tf values of the two glasses decrease at the same rate within experimental error. This similarity is expected given the similar dynamics of structural relaxation of IMC23 and NIF24 liquids. Because of this similarity, we assume that for a partially crystallized NIF glass, the uncrystallized portion reaches the same stage of aging as an IMC glass under the same condition. At the “steady state” of aging, the enthalpy and the Tf of an IMC glass decrease at 295 K according to dH/d log ta = −1.6 J/g and dTf /d log ta = −3.5 K. These rates are typical for organic glasses under similar conditions; see Table I for a comparison of IMC with other systems. For later comparison with surface mobility, we calculate the bulk mobility of the glasses during aging. The decreasing enthalpy and fictive temperature suggest slowing molecular motions with glass age. Following Kovacs,14 we calculate the effective relaxation time τ eff as a function of glass age. τ eff is calculated from the Tf vs. ta data (Figure 2(b)) and the definition: τeff = −1/d ln(Tf − Ta )/dt.
(1)
Figure 3 shows that log τ eff increases approximately linearly with log ta from minutes to months. The increase of τ α with time follows Struik’s law:15 τ α ∝ tμ , with μ ≈ 0.9. By
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(a)
(b)
FIG. 3. Bulk relaxation time τ α vs. the age of an IMC glass at 295 K. τ eff : effective relaxation time of Kovacs (Eq. (1)). τ AGV : relaxation time from the AGV model (Eq. (2)). The two methods yield similar results. The dotted curve indicates the speculated approach to equilibrium.
gives a reasonably accurate description for the decrease of bulk mobility of IMC glasses upon aging. Note also that the Tf used in this calculation is the enthalpic Tf , not the entropic Tf implicit in the AGV model.
(c)
Effect of glass aging on surface diffusion
FIG. 2. Aging of an IMC glass at 295 K characterized by the decrease of (a) enthalpy and (b) Tf vs. the aging time ta . The data are from this work and Ref. 21 (open circles). The dotted curves indicate the speculated approach to equilibrium. (c) Tf vs. ta data for IMC and NIF glasses at 298 K (Tg – 17 K). The two glasses have similar aging kinetics.
this measure, the bulk mobility of an IMC glass slows by orders of magnitude in the course of aging. In Figure 3, we also show the relaxation time predicted from Tf and the AdamGibbs-Vogel (AGV) equation:15 ln τa = ln τ0 + DT0 /[T (1 − T0 /Tf )],
(2)
where T = 295 K, τ 0 = 4.4 × 10−20 s, D = 17, T0 = 234 K for IMC.23 Note the good agreement between τ eff and τ α from the AGV model. This agreement suggests that the AGV model
Figure 4 shows the decay curves of 1000 nm surface gratings of IMC and NIF glasses, both freshly made and aged. For the data from optical diffraction, the fraction decayed is calculated as φ = I/I0 , where I0 and I are diffraction intensities at the initial time and time t. For the data from AFM, the fraction decayed is calculated as φ = (h/h0 )2 , where h0 and h are the amplitudes of the surface grating at time zero and t. (Because I ∝ h2 , it is necessary to square the ratio h/h0 to compare with I/I0 .) Within experimental error, the two methods report the same decay kinetics. Note that for NIF, partial crystallization could occur in long aging. Crystallization began from the edge of the glass film that was not in contact with the master grating. Nonetheless, there were regions of uncrystallized glass where surface-grating decay could be measured, and we assume that these regions reached the same stage of aging as an IMC glass under the same condition. Figure 4 shows that the decay curves for glasses of different ages overlap each other, showing no systematic variation with glass age. Each decay curve was fitted with an exponential function, h = h0 exp(−Kt), or a stretched exponential, h = h0 exp[−(Kt)β ], with β close to unity, to obtain the grating decay constant K. The average decay constant for 1000 nm
TABLE I. Kinetics of aging of organic glasses.a Glass
M (g/mole)
Tg (K)
Cpg (J/g K)
Ta (K)
dH/d log ta (J/g)
dTf /d log ta (K)
IMC ROY OTP PS
357.79 259.29 230.30 2M
315 259 246 377
0.41 0.43 0.49 0.28
295 248 233 362
−1.5 −1.4 −1.1 −0.6
−3.5 −3.2 −2.5 −2.3
a
Reference This work, 22 25 26 27
M: molecular weight. Cpg : heat capacity change at Tg . Ta : aging temperature. ROY is 5-methyl-2-[(2-nitrophenyl)amino]-3thiophenecarbonitrile. OTP is o-terphenyl. PS is polystyrene.
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FIG. 5. Bulk relaxation time τ α and surface diffusion coefficient Ds at 295 K as functions of glass age for IMC and NIF. With aging τ α increases by orders of magnitude but Ds remains approximately constant. The dotted curve indicates the speculated approach to equilibrium. The horizontal bars indicate the times for measuring surface grating decay, during which significant aging can occur for a freshly made glass. For already aged glasses, additional aging during measurement is insignificant. FIG. 4. Decay kinetics at 295 K of 1000 nm surface gratings on freshly made and aged glasses of IMC and NIF. The legends indicate the times of aging. The aging temperature was 295 K (Tg – 20 K). For IMC, both AFM and light microscopy (LM) were used to follow surface smoothing; for NIF, only LM was used.
gratings is log K (1/s) = −5.9 for IMC and −5.0 for NIF, consistent with the previous reports for freshly made glasses.4, 5 Previous studies have established that under the conditions of our study, surface smoothing occurs by surface diffusion.4, 5 This assignment relies on Mullins’ model of surface evolution by various mechanisms: viscous flow, evaporation-condensation, bulk diffusion, and surface diffusion. Mullins described how each mechanism can be identified through numerical evaluation and wavelength dependence. Following this procedure, we showed that below Tg , surface diffusion dominates other mechanisms for the decay of 1000 nm gratings on the glasses of IMC4 and NIF.5 Figure 5 shows the surface diffusion coefficient Ds as a function of glass age for the two systems. We use Mullins’ model to calculate Ds from the grating decay constant K:28 K = (Ds γ 2 ν/kT )(2π/λ)4 .
(3)
In Eq. (3), λ is the wavelength of a sinusoidal surface grating, γ is the surface free energy, the molecular volume, Ds the surface diffusion coefficient, and ν the number of molecules per unit area of surface. For our systems, we assume γ = 0.05 N/m, ≈ 3 × 10−28 m3 , and ν = 1.6 × 1018 molecules/m2 .4, 5 In this model, surface diffusion refers to lateral diffusion in the top layer of molecules. Figure 5 shows that there is no significant change of Ds up to the longest aging time (2 months). This insensitivity of surface mobility to glass age is in contrast to the strong effect of aging on bulk mobility, also displayed in Figure 5. Two months of aging at 295 K decreases the glass enthalpy by 6 J/g and its Tf by 15 K (Figure 2), nearly equilibrating the system; this process is expected to increase the structural relaxation time by four orders of magnitude. The significant bulk aging, however, leads to no observable change in surface diffusivity.
It is worth noting that in a surface-smoothing measurement, a freshly made glass can age significantly. The horizontal bar in Figure 5 marks the measurement time for 1000 nm gratings; the faster-decaying NIF grating took shorter time to measure. During this time, a freshly made glass ages rapidly and its τ α is expected to increase by 2 or 3 orders of magnitude. There is no corresponding change, however, in the surface-smoothing kinetics: the decay curve of a freshly made glass is identical to that of a well-aged glass (Figure 4). This constancy, again, argues for the insensitivity of surface diffusion to glass age. (In contrast to freshly made glasses, well-aged glasses undergo minimal aging during measurement; in fact, if horizontal bars were added to cover the times of measurement for aged glasses they would be barely visible on the log time scale in Figure 5.) The foregoing argument becomes even stronger on considering the data collected at different grating wavelengths λ.4, 5 Because the measurement time scales with λ4 (Eq. (3)), measurements at different λ sample different glass ages; however, our data collected at λ = 330–2000 nm yielded the same surface diffusion coefficients. As a control, we investigated the possibility that the act of removing the master grating off an organic glass could somehow perturb its surface, creating a physical state that would relax at the same rate, regardless of the age of the underlying bulk glass. To test this possibility, we heated a fresh glass to 323 K (Tg + 8 K) for 200 s and returned it to 303 K to measure the rate of surface grating decay. The time of heating corresponds to many structural relaxation times of the liquid at 323 K and should suffice to relax any mechanical disturbance from removing the master grating, but is short enough that the surface grating was largely intact after the treatment. We observed no significant effect as a result of refreshing the sample. As another control, we varied the glass thickness to test its effect on the kinetics of surface grating decay, hypothesizing that glass thickness could influence its rate of aging. We simultaneously measured the decay of surface gratings on
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NIF glasses 40 and 400 μm thick at 303 K. No significant difference was observed between the two samples. DISCUSSION
This study found that bulk aging has no significant effect on the surface diffusion of the organic glasses IMC and NIF. Despite almost complete relaxation of the glasses to equilibrium at Tg − 20 K, we observed no change of surface diffusivity within experimental error. IMC and NIF exhibit similar aging kinetics typical for organic glasses and we speculate that the insensitivity of surface diffusion to bulk age is a general phenomenon for these materials under similar conditions. Figure 6 shows a physical picture of the near-surface mobility of an IMC glass that is consistent with the existing data. Zhu et al. have used surface-grating decay to measure the surface diffusivity Ds .4 Swallen and Ediger used isotope labeling and secondary-ion mass spectroscopy to determine the bulk diffusivity Dv .29 The gray zone in Figure 6 indicates the region through which Ds transitions to Dv – the “mobile surface layer.” At 315 K (Tg ), Ds is one million times larger than Dv . At 295 K (Tg − 20 K), we estimate Dv = 10−24 m2 /s for an equilibrated liquid, by extrapolating the higher-temperature data.29 At this temperature, a freshly made glass undergoes aging (Figure 2) and its Dv is expected to decrease towards its equilibrium value (arrow). Despite the bulk aging effect, this work has found that Ds remains constant over time at 295 K. The thickness of the mobile surface layer (δ) through which Ds transitions to Dv has not been firmly established for organic glasses. Based on measurements of near-surface ion mobility, Bell et al. argue that δ is 3 nm for a glass of 3-methylpentane.30 Stevenson and Wolynes suggest a more gradual transition from surface to bulk mobility for an IMC glass, with δ = 20 nm.12 In our study, the combination of surface-smoothing rates and Mullins’ model yields the lateral diffusivity for the top layer of molecules. Although this work does not explicitly assess the mobility of sub-surface molecules, an estimate can be made of its upper bound, as follows. For surface diffusion to dominate over bulk diffusion as the mechanism of surface smoothing, Dv /Ds < νq ≈ aq, where Dv is the diffusivity of molecules immediately
FIG. 6. Diffusivities at the free surface and in the bulk of an IMC glass (Tg = 315 K).
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beneath the surface, a is the molecular diameter, and q = 2π /λ is the spatial frequency.4 At λ = 1000 nm, we obtain Dv /Ds < 0.006; that is, diffusivity must decrease rapidly from the top layer of molecules to those immediately below. Further studies are warranted to establish the details of the transition from surface to bulk mobility. The mobility of a surface molecule is expected to depend on its special environment: its lack of molecular neighbors above the surface (in the free space), its interaction with interior molecules under the surface, and its interaction with other molecules within the surface layer. In this picture, surface mobility should reflect at least to some extent the state of the bulk material. The insensitivity of surface diffusivity to the age of the bulk glass therefore suggests that the surface layer is sufficiently independent so that large changes in the bulk are not felt at the surface. It is also worth noting that the unchanging surface diffusivity with bulk aging could reflect the fact that our method probes the diffusion of only those molecules exposed to the free space; should other methods be used that simultaneously sample many layers of molecules, a greater effect of bulk aging might be observed on the (average) surface mobility. The surface diffusivity observed in this study is equivalent to the bulk diffusivity of a moderately viscous liquid: the Ds of IMC at Tg − 20 K (10−15 m2 /s) is similar to its Dv at Tg + 40 K. At this level of diffusivity, surface molecules are expected to be equilibrated as the bulk ages. The possibility exists in theory that with sufficient cooling, a near-surface layer vitrifies, allowing the observation of an aging effect.19 By extrapolation, we estimate that IMC and NIF glasses must be cooled to ∼0.7 Tg for Ds to reach 10−20 m2 /s, the typical bulk diffusivity at Tg . At this temperature, bulk aging is expected to be slow, possibly allowing the observation of a surface aging effect. Surface mobility has been used to explain fast crystal growth on the free surface of organic glasses.6–8 Our finding that surface diffusion does not slow substantially with glass age agrees with the observation that surface crystal growth can occur on well-aged glasses. Surface mobility is also invoked to explain the formation of low-energy, high-density glasses by vapor deposition, including those of IMC and NIF.9–11 These glasses have such low energies and high densities that liquid-cooled glasses reach the same levels only after impractically long aging (estimated to be a million years). In this sense, these vapor-deposited glasses are liquid-cooled glasses of extremely old ages. The formation of stable glasses by vapor deposition relies on the efficient equilibration of molecules at the free surface before they are buried and locked in place by later-depositing molecules. The fact that such equilibration can occur on the surface of a “super-aged” glass implies that low bulk mobility need not mean low surface mobility. It would be of interest to directly measure surface diffusion on stable glasses prepared by vapor deposition. The weak dependence of surface mobility on glass age is consistent with the observation that aging has no significant effect on the smoothing kinetics of the rubbed surfaces of polystyrene.31 Note that in the polystyrene experiments, aging occurred well below Tg but surface-smoothing was monitored near Tg , at which temperature the glasses may be
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partially rejuvenated. In our study, the two temperatures are the same. A further difference between the two studies is that our measurement probes the surface diffusion of relatively small organic molecules, whereas the relaxation of rubbed polystyrene surfaces may involve both conformational and translational motions of polymer chains. Two models are available that relate the surface and bulk mobility of glasses.12, 13 Stevenson and Wolynes extended the Random First Order Transition (RFOT) treatment of bulk particles to surface particles, and proposed Eq. (4) for calculating the surface relaxation time τ surf from the bulk relaxation time τ α :12 τsurf = (τ0 τα )0.5 ,
(4)
where τ 0 ≈ 1 ps. Capaccioli et al. associated the relaxation of surface particles with the “primitive” motion of bulk particles envisioned by the coupling model, and proposed Eq. (5):13 τsurf = tc n τα 1−n .
(5)
In Eq. (5), tc ≈ 2 ps for molecular liquids and (1 − n) is the exponent obtained from fitting the bulk relaxation kinetics to the Kohlrausch-Williams-Watt (KWW) function: φ = exp[−(t/τ α )1−n ]. Equations (4) and (5) are nearly equivalent if n = 0.5 is entered in Eq. (5). At this stage of development, neither model makes specific predictions of the effect of bulk aging on surface mobility. Entering in Eqs. (4) and (5) the bulk relaxation time that reflects the glass age implies a significant slowdown of surface relaxation. For the systems of this study, the orders-of-magnitude increase in τ α (Figure 3) would lead to smaller but still substantial increase in τ surf . It is worth noting here the different physical pictures of surface mobility envisioned in the various studies. The surface-smoothing kinetics as interpreted by Mullins’ model yield the surface diffusivity of the top layer of molecules (those exposed to the free space). By contrast, the surface relaxation time calculated by Eq. (4) refers to a near-surface domain several particles deep. Such differences could lead to different dependence of a chosen measure of surface mobility on glass age. CONCLUSION
We have studied the effect of physical aging on the surface diffusion of two small-molecule organic glasses, Indomethacin and Nifedipine. At 20 K below Tg , these two glasses exhibit significant aging at rates typical of organic glasses. Surface diffusivity remains unchanged, however, despite significant bulk aging that nearly equilibrates the systems and increases the bulk relaxation time by orders of magnitude. Our finding is consistent with the persistence of surface crystal growth on aged glasses and the ability to form low-energy, high-stability glasses by vapor deposition. Despite low mobility in the bulk, surface molecules can remain mobile to support these surface-mediated processes. This finding is also consistent with the observation that physical aging has little effect on the surface-smoothing kinetics of rubbed polystyrene glasses.31
Our finding is relevant for understanding and controlling the stability of amorphous materials. Physical aging is known to lower the energy of glasses, reduce their volumes, and decrease the mobility of molecules. This work shows, however, that surface diffusion is unaffected by bulk relaxation. If physicochemical changes occur through surface molecules, e.g., surface crystallization,8 the rate will be insensitive to glass age and bulk mobility. Further progress in this area could benefit from studies of low-energy vapor-deposited glasses to assess the effect of even higher glass density on surface mobility. It may be of interest to investigate the surface mobility of organic glasses at lower temperatures for the existence of near-surface glass transitions. Complementary methods could be developed that help determine the thickness of the surface mobile layer.
ACKNOWLEDGMENTS
We thank the NSF (DMR-1206724) for supporting this work. 1 G.
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Citation: J. Chem. Phys. 140, 054509 (2014); doi: 10.1063/1.4863556 View online: http://dx.doi.org/10.1063/1.4863556 View Table of Contents: http://aip.scitation.org/toc/jcp/140/5 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 140, 054509 (2014)
Effect of bulk aging on surface diffusion of glasses Caleb W. Brian, Lei Zhu, and Lian Yua) Department of Chemistry and School of Pharmacy, University of Wisconsin–Madison, Madison, Wisconsin 53705, USA
(Received 11 October 2013; accepted 16 January 2014; published online 5 February 2014) The effect of physical aging on surface diffusion has been determined for two organic glasses, Indomethacin and Nifedipine. The two systems exhibit similar aging kinetics typical of organic glasses. Surface diffusivity remains unchanged despite significant bulk aging that nearly equilibrates the systems and increases the bulk relaxation time by orders of magnitude. The finding is relevant for understanding the stability of amorphous materials and the formation of low-energy glasses by vapor deposition. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863556] INTRODUCTION
The mobility of molecules at the free surface, in reference to those in the bulk, is important in condensed-phase science. Although much is known about the surface mobility of metals, semiconductors, and polymers,1–3 little data exist on organic solids of relatively low molecular weights.4, 5 Such data are necessary for developing organic materials for many applications and for understanding surface mobility across classes of materials. Characterizing surface mobility of organic solids is particularly relevant for two areas of current research: fast surface crystal growth on organic glasses6–8 and formation of low-energy glasses by physical vapor deposition.9–11 Both phenomena are attributed to the high mobility of surface molecules. Recently, theories have been proposed for the surface mobility of glasses,12, 13 further stimulating work in this area. Glasses form by cooling liquids, condensing vapors, and evaporating solutions while avoiding crystallization. An important property of glasses is their relaxation or aging over time toward the equilibrium liquid state, leading to higher density, lower energy, and slower molecular motions.14–17 Despite extensive studies of this phenomenon, however, little is known about the effect of glass aging on surface mobility. There have been no measurements of surface diffusivity of glasses that have been aged systematically. In their current stage of development, the theoretical models12, 13 offer no predictions of the effect of glass aging on surface mobility. The notion that there exists a mobile surface layer atop a solid glass18 suggests that surface molecules can remain equilibrated as the bulk ages, with their mobility adjusting quickly to the bulk state. The surface response to bulk aging would be more complex if the surface layer itself undergoes the glass transition.19 To assess the interplay between surface and bulk dynamics, we studied the effect of physical aging on the surface diffusivity of two organic glasses, Indomethacin and Nifedipine (Scheme 1). We found that the two systems have similar aging kinetics typical of organic glasses. Surface diffusion is unafa) Author to whom correspondence should be addressed. Electronic mail:
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fected by significant bulk aging that nearly equilibrates the systems and increases the bulk structural relaxation times by orders of magnitude. The finding is consistent with the notion that the mobile surface layer of organic glasses is weakly coupled to bulk molecules. METHOD
Indomethacin [1-(p-chlorobenzoyl)-5-methoxy-2-methylindole-3-acetic acid (IMC), γ polymorph] was obtained from Sigma (St. Louis, MO). Nifedipine [1,4-Dihydro2,6-dimethyl-4-(2-nitrophenyl)-3,5-pyridinedicarboxylic acid dimethyl ester (NIF), α polymorph] was obtained from FWD Chem (99.8% purity). Differential Scanning Calorimetry (DSC) was performed with a TA Instruments Q2000 unit to measure enthalpy recovery during glass aging. Each sample was 3–6 mg and was placed in a crimped Al DSC pan, leaving no head space, so that the condition of aging approximately matched that of the samples in the surface-diffusion measurements. For shorter aging (up to several hours), the sample always stayed in the DSC cell; it was melted at 20 K above the crystal melting point, cooled to an aging temperature, and held for a desired time. For subsequent analysis, the sample was cooled and reheated at 10 K/min to determine the change of the glass’s enthalpy and its fictive temperature. For longer aging (>10 h), the sample was stored outside the DSC cell in a desiccator at the laboratory temperature, and the other steps were the same. To prepare surface gratings, plastic 1000 nm gratings (Edmond Scientific) were gold coated to serve as masters. The masters were placed over an IMC or NIF liquid at 353 K, 45 K above the glass transition temperature Tg (315 K) for 1 min. Samples were cooled to laboratory temperature (295 ± 1 K) and aged in a desiccator for allotted times, with the masters intact. Freshly made control samples were prepared the day the aged samples were measured. To start a measurement of surface grating decay, the master was detached to expose a surface grating and its smoothing over time was monitored. The embossed surface grating was nearly sinusoidal as confirmed by atomic force microscopy (AFM).
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(a)
SCHEME 1. Molecular structures of Nifedipine and Indomethacin.
The smoothing of surface gratings was monitored in dry N2 with an atomic force microscope (Veeco Multimode IV) or an optical microscope (Nikon Optiphot 2). Tapping-mode AFM scans were performed perpendicular to the grating direction. The height profile was Fourier-transformed to obtain the grating amplitude. Optical diffraction was recorded in transmission with the aid of a Bertrand lens. The white light was filtered with a 530 nm band-pass filter. The square root of the first order diffraction intensity was taken to be proportional to the grating amplitude. A 1000 nm grating on IMC decayed in ca. 2 weeks at 295 K; the sample was periodically removed from a desiccator, measured in dry N2 , and returned to the desiccator for further decay. A 1000 nm grating on NIF flattened in ca. one day at 295 K; the sample was followed in the same way as an IMC sample or was placed in a N2 -purged hot stage (Linkam THMS600) and measured real-time with more precise control of temperature. RESULTS Kinetics of glass aging
This study compared the surface mobility of freshly made and aged glasses of IMC and NIF. We used the enthalpy decrease and the enthalpic fictive temperature Tf to characterize the progress of glass aging. Figure 1(a) shows the typical DSC data collected for this purpose. To calculate the enthalpy decrease from a freshly made glass to an aged glass, their heat capacities were integrated from the glassy state to a common temperature above Tg , at which the two glasses are identical (Figure 1(b)). The Tf of a glass is the equivalent temperature of an equilibrated liquid that, if vitrified at Tf , has the same enthalpy as the glass.20 A freshly made glass has a Tf that is approximately its Tg measured by DSC, and a glass aged to equilibrium has a Tf that is identical to the temperature of aging. Figures 2(a) and 2(b) show the enthalpy decrease and the Tf of an IMC glass as a function of the aging time ta . The data from this work are compared with those of Kearns et al. (open circles);21 the two sets are in good agreement. The longest aging time is determined by the crystallization of IMC glasses, and can be as long as several months if no free surface is exposed during aging.6–8 At our aging temperature (Ta = 295 K), the equilibrated glass would have a Tf of 295 K and an enthalpy decrease of −8.2 J/g. The latter value is calculated from H∞ = Cpg (Ta − Tg ), where Cpg is the heat-capacity change upon the glass transition. For IMC,
(b)
FIG. 1. DSC analysis of glass aging. (a) Heat capacity curves for a freshly made and aged IMC glass. (b) Enthalpy decrease on aging is calculated from the integrated Cp curves of freshly made and aged glasses. The fictive temperature Tf is found by extrapolating the liquid line to intersect the glass line.
Cpg = 0.41 J/g K22 and is assumed to be constant in the temperature range of interest. Owing to faster crystallization, the aging kinetics of NIF glasses cannot be followed as long as IMC glasses. For the overlapping periods, however, the two glasses aged at similar rates. As Figure 2(c) shows, the Tf values of the two glasses decrease at the same rate within experimental error. This similarity is expected given the similar dynamics of structural relaxation of IMC23 and NIF24 liquids. Because of this similarity, we assume that for a partially crystallized NIF glass, the uncrystallized portion reaches the same stage of aging as an IMC glass under the same condition. At the “steady state” of aging, the enthalpy and the Tf of an IMC glass decrease at 295 K according to dH/d log ta = −1.6 J/g and dTf /d log ta = −3.5 K. These rates are typical for organic glasses under similar conditions; see Table I for a comparison of IMC with other systems. For later comparison with surface mobility, we calculate the bulk mobility of the glasses during aging. The decreasing enthalpy and fictive temperature suggest slowing molecular motions with glass age. Following Kovacs,14 we calculate the effective relaxation time τ eff as a function of glass age. τ eff is calculated from the Tf vs. ta data (Figure 2(b)) and the definition: τeff = −1/d ln(Tf − Ta )/dt.
(1)
Figure 3 shows that log τ eff increases approximately linearly with log ta from minutes to months. The increase of τ α with time follows Struik’s law:15 τ α ∝ tμ , with μ ≈ 0.9. By
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(a)
(b)
FIG. 3. Bulk relaxation time τ α vs. the age of an IMC glass at 295 K. τ eff : effective relaxation time of Kovacs (Eq. (1)). τ AGV : relaxation time from the AGV model (Eq. (2)). The two methods yield similar results. The dotted curve indicates the speculated approach to equilibrium.
gives a reasonably accurate description for the decrease of bulk mobility of IMC glasses upon aging. Note also that the Tf used in this calculation is the enthalpic Tf , not the entropic Tf implicit in the AGV model.
(c)
Effect of glass aging on surface diffusion
FIG. 2. Aging of an IMC glass at 295 K characterized by the decrease of (a) enthalpy and (b) Tf vs. the aging time ta . The data are from this work and Ref. 21 (open circles). The dotted curves indicate the speculated approach to equilibrium. (c) Tf vs. ta data for IMC and NIF glasses at 298 K (Tg – 17 K). The two glasses have similar aging kinetics.
this measure, the bulk mobility of an IMC glass slows by orders of magnitude in the course of aging. In Figure 3, we also show the relaxation time predicted from Tf and the AdamGibbs-Vogel (AGV) equation:15 ln τa = ln τ0 + DT0 /[T (1 − T0 /Tf )],
(2)
where T = 295 K, τ 0 = 4.4 × 10−20 s, D = 17, T0 = 234 K for IMC.23 Note the good agreement between τ eff and τ α from the AGV model. This agreement suggests that the AGV model
Figure 4 shows the decay curves of 1000 nm surface gratings of IMC and NIF glasses, both freshly made and aged. For the data from optical diffraction, the fraction decayed is calculated as φ = I/I0 , where I0 and I are diffraction intensities at the initial time and time t. For the data from AFM, the fraction decayed is calculated as φ = (h/h0 )2 , where h0 and h are the amplitudes of the surface grating at time zero and t. (Because I ∝ h2 , it is necessary to square the ratio h/h0 to compare with I/I0 .) Within experimental error, the two methods report the same decay kinetics. Note that for NIF, partial crystallization could occur in long aging. Crystallization began from the edge of the glass film that was not in contact with the master grating. Nonetheless, there were regions of uncrystallized glass where surface-grating decay could be measured, and we assume that these regions reached the same stage of aging as an IMC glass under the same condition. Figure 4 shows that the decay curves for glasses of different ages overlap each other, showing no systematic variation with glass age. Each decay curve was fitted with an exponential function, h = h0 exp(−Kt), or a stretched exponential, h = h0 exp[−(Kt)β ], with β close to unity, to obtain the grating decay constant K. The average decay constant for 1000 nm
TABLE I. Kinetics of aging of organic glasses.a Glass
M (g/mole)
Tg (K)
Cpg (J/g K)
Ta (K)
dH/d log ta (J/g)
dTf /d log ta (K)
IMC ROY OTP PS
357.79 259.29 230.30 2M
315 259 246 377
0.41 0.43 0.49 0.28
295 248 233 362
−1.5 −1.4 −1.1 −0.6
−3.5 −3.2 −2.5 −2.3
a
Reference This work, 22 25 26 27
M: molecular weight. Cpg : heat capacity change at Tg . Ta : aging temperature. ROY is 5-methyl-2-[(2-nitrophenyl)amino]-3thiophenecarbonitrile. OTP is o-terphenyl. PS is polystyrene.
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FIG. 5. Bulk relaxation time τ α and surface diffusion coefficient Ds at 295 K as functions of glass age for IMC and NIF. With aging τ α increases by orders of magnitude but Ds remains approximately constant. The dotted curve indicates the speculated approach to equilibrium. The horizontal bars indicate the times for measuring surface grating decay, during which significant aging can occur for a freshly made glass. For already aged glasses, additional aging during measurement is insignificant. FIG. 4. Decay kinetics at 295 K of 1000 nm surface gratings on freshly made and aged glasses of IMC and NIF. The legends indicate the times of aging. The aging temperature was 295 K (Tg – 20 K). For IMC, both AFM and light microscopy (LM) were used to follow surface smoothing; for NIF, only LM was used.
gratings is log K (1/s) = −5.9 for IMC and −5.0 for NIF, consistent with the previous reports for freshly made glasses.4, 5 Previous studies have established that under the conditions of our study, surface smoothing occurs by surface diffusion.4, 5 This assignment relies on Mullins’ model of surface evolution by various mechanisms: viscous flow, evaporation-condensation, bulk diffusion, and surface diffusion. Mullins described how each mechanism can be identified through numerical evaluation and wavelength dependence. Following this procedure, we showed that below Tg , surface diffusion dominates other mechanisms for the decay of 1000 nm gratings on the glasses of IMC4 and NIF.5 Figure 5 shows the surface diffusion coefficient Ds as a function of glass age for the two systems. We use Mullins’ model to calculate Ds from the grating decay constant K:28 K = (Ds γ 2 ν/kT )(2π/λ)4 .
(3)
In Eq. (3), λ is the wavelength of a sinusoidal surface grating, γ is the surface free energy, the molecular volume, Ds the surface diffusion coefficient, and ν the number of molecules per unit area of surface. For our systems, we assume γ = 0.05 N/m, ≈ 3 × 10−28 m3 , and ν = 1.6 × 1018 molecules/m2 .4, 5 In this model, surface diffusion refers to lateral diffusion in the top layer of molecules. Figure 5 shows that there is no significant change of Ds up to the longest aging time (2 months). This insensitivity of surface mobility to glass age is in contrast to the strong effect of aging on bulk mobility, also displayed in Figure 5. Two months of aging at 295 K decreases the glass enthalpy by 6 J/g and its Tf by 15 K (Figure 2), nearly equilibrating the system; this process is expected to increase the structural relaxation time by four orders of magnitude. The significant bulk aging, however, leads to no observable change in surface diffusivity.
It is worth noting that in a surface-smoothing measurement, a freshly made glass can age significantly. The horizontal bar in Figure 5 marks the measurement time for 1000 nm gratings; the faster-decaying NIF grating took shorter time to measure. During this time, a freshly made glass ages rapidly and its τ α is expected to increase by 2 or 3 orders of magnitude. There is no corresponding change, however, in the surface-smoothing kinetics: the decay curve of a freshly made glass is identical to that of a well-aged glass (Figure 4). This constancy, again, argues for the insensitivity of surface diffusion to glass age. (In contrast to freshly made glasses, well-aged glasses undergo minimal aging during measurement; in fact, if horizontal bars were added to cover the times of measurement for aged glasses they would be barely visible on the log time scale in Figure 5.) The foregoing argument becomes even stronger on considering the data collected at different grating wavelengths λ.4, 5 Because the measurement time scales with λ4 (Eq. (3)), measurements at different λ sample different glass ages; however, our data collected at λ = 330–2000 nm yielded the same surface diffusion coefficients. As a control, we investigated the possibility that the act of removing the master grating off an organic glass could somehow perturb its surface, creating a physical state that would relax at the same rate, regardless of the age of the underlying bulk glass. To test this possibility, we heated a fresh glass to 323 K (Tg + 8 K) for 200 s and returned it to 303 K to measure the rate of surface grating decay. The time of heating corresponds to many structural relaxation times of the liquid at 323 K and should suffice to relax any mechanical disturbance from removing the master grating, but is short enough that the surface grating was largely intact after the treatment. We observed no significant effect as a result of refreshing the sample. As another control, we varied the glass thickness to test its effect on the kinetics of surface grating decay, hypothesizing that glass thickness could influence its rate of aging. We simultaneously measured the decay of surface gratings on
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NIF glasses 40 and 400 μm thick at 303 K. No significant difference was observed between the two samples. DISCUSSION
This study found that bulk aging has no significant effect on the surface diffusion of the organic glasses IMC and NIF. Despite almost complete relaxation of the glasses to equilibrium at Tg − 20 K, we observed no change of surface diffusivity within experimental error. IMC and NIF exhibit similar aging kinetics typical for organic glasses and we speculate that the insensitivity of surface diffusion to bulk age is a general phenomenon for these materials under similar conditions. Figure 6 shows a physical picture of the near-surface mobility of an IMC glass that is consistent with the existing data. Zhu et al. have used surface-grating decay to measure the surface diffusivity Ds .4 Swallen and Ediger used isotope labeling and secondary-ion mass spectroscopy to determine the bulk diffusivity Dv .29 The gray zone in Figure 6 indicates the region through which Ds transitions to Dv – the “mobile surface layer.” At 315 K (Tg ), Ds is one million times larger than Dv . At 295 K (Tg − 20 K), we estimate Dv = 10−24 m2 /s for an equilibrated liquid, by extrapolating the higher-temperature data.29 At this temperature, a freshly made glass undergoes aging (Figure 2) and its Dv is expected to decrease towards its equilibrium value (arrow). Despite the bulk aging effect, this work has found that Ds remains constant over time at 295 K. The thickness of the mobile surface layer (δ) through which Ds transitions to Dv has not been firmly established for organic glasses. Based on measurements of near-surface ion mobility, Bell et al. argue that δ is 3 nm for a glass of 3-methylpentane.30 Stevenson and Wolynes suggest a more gradual transition from surface to bulk mobility for an IMC glass, with δ = 20 nm.12 In our study, the combination of surface-smoothing rates and Mullins’ model yields the lateral diffusivity for the top layer of molecules. Although this work does not explicitly assess the mobility of sub-surface molecules, an estimate can be made of its upper bound, as follows. For surface diffusion to dominate over bulk diffusion as the mechanism of surface smoothing, Dv /Ds < νq ≈ aq, where Dv is the diffusivity of molecules immediately
FIG. 6. Diffusivities at the free surface and in the bulk of an IMC glass (Tg = 315 K).
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beneath the surface, a is the molecular diameter, and q = 2π /λ is the spatial frequency.4 At λ = 1000 nm, we obtain Dv /Ds < 0.006; that is, diffusivity must decrease rapidly from the top layer of molecules to those immediately below. Further studies are warranted to establish the details of the transition from surface to bulk mobility. The mobility of a surface molecule is expected to depend on its special environment: its lack of molecular neighbors above the surface (in the free space), its interaction with interior molecules under the surface, and its interaction with other molecules within the surface layer. In this picture, surface mobility should reflect at least to some extent the state of the bulk material. The insensitivity of surface diffusivity to the age of the bulk glass therefore suggests that the surface layer is sufficiently independent so that large changes in the bulk are not felt at the surface. It is also worth noting that the unchanging surface diffusivity with bulk aging could reflect the fact that our method probes the diffusion of only those molecules exposed to the free space; should other methods be used that simultaneously sample many layers of molecules, a greater effect of bulk aging might be observed on the (average) surface mobility. The surface diffusivity observed in this study is equivalent to the bulk diffusivity of a moderately viscous liquid: the Ds of IMC at Tg − 20 K (10−15 m2 /s) is similar to its Dv at Tg + 40 K. At this level of diffusivity, surface molecules are expected to be equilibrated as the bulk ages. The possibility exists in theory that with sufficient cooling, a near-surface layer vitrifies, allowing the observation of an aging effect.19 By extrapolation, we estimate that IMC and NIF glasses must be cooled to ∼0.7 Tg for Ds to reach 10−20 m2 /s, the typical bulk diffusivity at Tg . At this temperature, bulk aging is expected to be slow, possibly allowing the observation of a surface aging effect. Surface mobility has been used to explain fast crystal growth on the free surface of organic glasses.6–8 Our finding that surface diffusion does not slow substantially with glass age agrees with the observation that surface crystal growth can occur on well-aged glasses. Surface mobility is also invoked to explain the formation of low-energy, high-density glasses by vapor deposition, including those of IMC and NIF.9–11 These glasses have such low energies and high densities that liquid-cooled glasses reach the same levels only after impractically long aging (estimated to be a million years). In this sense, these vapor-deposited glasses are liquid-cooled glasses of extremely old ages. The formation of stable glasses by vapor deposition relies on the efficient equilibration of molecules at the free surface before they are buried and locked in place by later-depositing molecules. The fact that such equilibration can occur on the surface of a “super-aged” glass implies that low bulk mobility need not mean low surface mobility. It would be of interest to directly measure surface diffusion on stable glasses prepared by vapor deposition. The weak dependence of surface mobility on glass age is consistent with the observation that aging has no significant effect on the smoothing kinetics of the rubbed surfaces of polystyrene.31 Note that in the polystyrene experiments, aging occurred well below Tg but surface-smoothing was monitored near Tg , at which temperature the glasses may be
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partially rejuvenated. In our study, the two temperatures are the same. A further difference between the two studies is that our measurement probes the surface diffusion of relatively small organic molecules, whereas the relaxation of rubbed polystyrene surfaces may involve both conformational and translational motions of polymer chains. Two models are available that relate the surface and bulk mobility of glasses.12, 13 Stevenson and Wolynes extended the Random First Order Transition (RFOT) treatment of bulk particles to surface particles, and proposed Eq. (4) for calculating the surface relaxation time τ surf from the bulk relaxation time τ α :12 τsurf = (τ0 τα )0.5 ,
(4)
where τ 0 ≈ 1 ps. Capaccioli et al. associated the relaxation of surface particles with the “primitive” motion of bulk particles envisioned by the coupling model, and proposed Eq. (5):13 τsurf = tc n τα 1−n .
(5)
In Eq. (5), tc ≈ 2 ps for molecular liquids and (1 − n) is the exponent obtained from fitting the bulk relaxation kinetics to the Kohlrausch-Williams-Watt (KWW) function: φ = exp[−(t/τ α )1−n ]. Equations (4) and (5) are nearly equivalent if n = 0.5 is entered in Eq. (5). At this stage of development, neither model makes specific predictions of the effect of bulk aging on surface mobility. Entering in Eqs. (4) and (5) the bulk relaxation time that reflects the glass age implies a significant slowdown of surface relaxation. For the systems of this study, the orders-of-magnitude increase in τ α (Figure 3) would lead to smaller but still substantial increase in τ surf . It is worth noting here the different physical pictures of surface mobility envisioned in the various studies. The surface-smoothing kinetics as interpreted by Mullins’ model yield the surface diffusivity of the top layer of molecules (those exposed to the free space). By contrast, the surface relaxation time calculated by Eq. (4) refers to a near-surface domain several particles deep. Such differences could lead to different dependence of a chosen measure of surface mobility on glass age. CONCLUSION
We have studied the effect of physical aging on the surface diffusion of two small-molecule organic glasses, Indomethacin and Nifedipine. At 20 K below Tg , these two glasses exhibit significant aging at rates typical of organic glasses. Surface diffusivity remains unchanged, however, despite significant bulk aging that nearly equilibrates the systems and increases the bulk relaxation time by orders of magnitude. Our finding is consistent with the persistence of surface crystal growth on aged glasses and the ability to form low-energy, high-stability glasses by vapor deposition. Despite low mobility in the bulk, surface molecules can remain mobile to support these surface-mediated processes. This finding is also consistent with the observation that physical aging has little effect on the surface-smoothing kinetics of rubbed polystyrene glasses.31
Our finding is relevant for understanding and controlling the stability of amorphous materials. Physical aging is known to lower the energy of glasses, reduce their volumes, and decrease the mobility of molecules. This work shows, however, that surface diffusion is unaffected by bulk relaxation. If physicochemical changes occur through surface molecules, e.g., surface crystallization,8 the rate will be insensitive to glass age and bulk mobility. Further progress in this area could benefit from studies of low-energy vapor-deposited glasses to assess the effect of even higher glass density on surface mobility. It may be of interest to investigate the surface mobility of organic glasses at lower temperatures for the existence of near-surface glass transitions. Complementary methods could be developed that help determine the thickness of the surface mobile layer.
ACKNOWLEDGMENTS
We thank the NSF (DMR-1206724) for supporting this work. 1 G.
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