J. Phys. Chem. B 2010, 114, 2635–2643
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Transformation of Stable Glasses into Supercooled Liquids: Growth Fronts and Anomalously Fast Liquid Diffusion Stephen F. Swallen, Katherine Windsor, Robert J. McMahon, and M. D. Ediger* Department of Chemistry, UniVersity of Wisconsin-Madison, Madison, Wisconsin 53706
Thomas E. Mates Materials Department, UniVersity of California-Santa Barbara, Santa Barbara, California 93106 ReceiVed: NoVember 11, 2009; ReVised Manuscript ReceiVed: December 30, 2009
Physical vapor deposition onto substrates near 0.85Tg can prepare organic glasses with low enthalpy, high density, and high thermal stability. Isotopically labeled multilayer films of tris(naphthyl)benzene and indomethacin stable glasses were prepared and secondary ion mass spectrometry was used to study the evolution of these materials upon heating above Tg. In contrast to ordinary glasses, when stable glasses are held above Tg they transform to a liquid via a growth front mechanism. In these experiments, growth fronts are initiated at the free surface of the glass and in some cases at the glass/substrate interface or an internal interface in the glass. For tris(naphthyl)benzene, the velocity of this growth front is observed to be nearly independent of the stability of the glass. Diffusion in the liquid that results from the growth front is initially 2-5 times faster than for the equilibrium supercooled liquid at the same temperature; the nature of this liquid is unclear. Under some circumstances, the slow evolution of this unusually mobile liquid into the equilibrium supercooled liquid can be observed. I. Introduction If a single-component liquid avoids crystallization when cooled somewhat below its melting point, the supercooled liquid is produced with properties that can be well described by extrapolations from the high-temperature liquid. At lower temperatures, when the cooling rate exceeds the molecular reorganization time, a glass is formed. Although this laboratory process does not involve a thermodynamic phase transition, properties such as heat capacity and thermal expansion coefficient change at the glass transition temperature Tg. The nonequilibrium glass can relax toward the metastable liquid structure, but this aging becomes a very slow process only a few degrees kelvin below Tg. We have recently shown that glasses prepared by physical vapor deposition can be highly stable. In comparison to ordinary glasses prepared by cooling the supercooled liquid, these glasses can have lower enthalpy, higher density, higher mechanical moduli, higher resistance to vapor uptake, and better thermal stability.1-7 The degree of stabilization relative to the ordinary glass can be experimentally controlled by the substrate temperature and the rate of deposition.4,5,8 In analogy to ordinary glasses, these stable glasses can be regarded as “superaged.” We estimate that their properties are similar to those that would be obtained by aging an ordinary glass for times in the range of 102-1012 years.8 When a single-component glass is heated above Tg, the supercooled liquid is eventually recovered (if crystallization does not intervene). If a supercooled liquid is cooled into the glass and then promptly heated, the sample will follow nearly the same volume or enthalpy curve for both cooling and heating; i.e., the sample will return to equilibrium at about the same temperature as it left equilibrium. In contrast, an ordinary glass * To whom correspondence should be addressed.
that has been aged not too far below Tg will show a delayed recovery to the equilibrium liquid structure upon heating. In differential scanning calorimetry, this gives rise to a prominent peak in the apparent heat capacity (the “enthalpy overshoot”).9 Qualitatively, the delayed response occurs because aging allows the glass to become slightly better packed and better packing leads to slower dynamics. When very efficiently packed glasses (e.g., stable glasses prepared by vapor deposition) are heated above Tg, their response is much slower than an aged sample and thus even larger enthalpy overshoots result.1,2,4,5,8 We have recently shown that transformation of a stable glass into the supercooled liquid upon heating above Tg occurs by an unprecedented growth front mechanism.10 For ordinary or aged glasses, the transformation to the liquid is almost always described by models similar to that of Tool, Narayanaswamy, and Moynihan (TNM).11-13 In this standard approach, relaxation rates depend only on temperature and structure, with structural evolution occurring everywhere in the sample at the same rate. This conventional view is clearly incompatible with our observations of a growth front that propagates into the stable glass until a liquid exists everywhere. Growth fronts naturally arise in highly nonlinear diffusion problems.14-17 For highly stable glasses, the molecular packing is so tight that the glass is taken apart from the outside, i.e., by liquid regions eating away at its surfaces. Very recent theoretical work by Wolynes, stimulated in part by our observations, predicts growth front behavior for highly stable glasses.18 This paper expands upon our initial report10 of propagating growth fronts in stable glasses prepared by vapor deposition. In particular, we investigate the following questions. For what range of deposition conditions are growth fronts observed in the transformation of the glass to the liquid phase? How does the growth front velocity depend upon the stability of initial glass? What is the nature of the mobile liquid behind the front,
10.1021/jp9107359 2010 American Chemical Society Published on Web 02/08/2010
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and are its properties the same as those of the ordinary supercooled liquid? Are growth fronts a general feature of stable organic glasses? We use secondary ion mass spectrometry (SIMS) to study molecular mobility in ordinary and stable glasses of two organic glass formers: tris(naphthyl)benzene (TNB) and indomethacin (IMC). The concentration profiles of isotopically labeled multilayer films are measured before and after annealing at temperatures around Tg. Comparison of the annealing profiles at various times provides direct access to spatially resolved relaxation rates in the samples. Our results were analyzed using a model that incorporates a growth front and Fickian diffusion behind the growth front. We find that samples vapor-deposited onto substrates held very near Tg behave as ordinary glasses; upon annealing, the concentration profile evolves promptly throughout the sample as the glass transforms into the supercooled liquid. In these cases, molecular motion is diffusive and is very well described by Fick’s law. Deposition at substrate temperatures near 0.85Tg gives very stable glasses, which evolve to the liquid by surfaceinitiated growth fronts. Similar growth fronts are seen for both TNB and IMC glasses. Quantitative analyses of the concentration profiles for stable glasses reveal that the liquid transformed from the stable glass is unexpectedly mobile, with diffusion coefficients 2-5 times larger than those of the ordinary supercooled liquid. Even more surprising, this enhanced-mobility liquid persists for times that can exceed several hundred τR, where τR is the structural relaxation time of the supercooled liquid. Under some circumstances, the slow evolution of this unusually mobile liquid into the equilibrium supercooled liquid can be observed. These results are discussed in the context of current theoretical models. We speculate that the unexpectedly mobile liquid is either metastable with respect to the known supercooled liquid (polyamorphic) or contains a nonequilibrium distribution of regions of varying mobility. II. Experimental Technique and Data Analysis 1,3-bis(1-naphthyl)-5-(2-naphthyl)benzene, commonly known as tris(naphthyl)benzene or TNB, was synthesized as described previously.19 A partially deuterio isomer was also prepared, in which the 14 sites on the 1-naphthyl groups were isotopically substituted.19 The protio and deuterio versions of TNB will be referred to as h-TNB and d-TNB, respectively. A representative 3-dimensional structure is shown as the inset to Figure 4. The glass transition temperature Tg has been found by differential scanning calorimetry (DSC) to be 347 K.8 The melting point temperature of the vapor-deposited material was found to be identical to the starting material, indicating that no chemical degradation occurred during the deposition process. Tg for the vapor-deposited TNB also matches that of the as-synthesized material for samples with the same thermal history, i.e., cooled from the supercooled liquid and then heated. Indomethacin (1-(4-chlorobenzoyl)-5-methoxy-2-methyl-3indoleacetic acid), with purity greater than 99%, was purchased from Sigma (St. Louis, MO) and was used as received. This material is referred to as IMC and is schematically shown as an inset to Figure 5. The IMC was completely crystalline and consisted primarily as the γ polymorph. Melting point (Tm ) 432.8 K) comparisons of the as-received and vapor-deposited material4 agreed with literature data for the γ polymorph. Tg was measured by DSC to be 315 K. Partially deuterated IMC (d4) was purchased from CDM Isotopes (Canada) with a stated 98.8 atom % deuteration and was used as received. Isotopically labeled multilayer films of TNB and IMC were prepared by physical vapor deposition, as described previously.20
Swallen et al. Glasses that quickly evolved into the ordinary supercooled liquid were prepared for both materials by depositing onto the silicon substrate at a temperature Tsubstrate ≈ Tg - 7 K and a deposition rate of 0.2 nm/s. Stable glasses were prepared at Tsubstrate ≈ 0.85Tg (295 K for TNB, 265 K for IMC) at 0.2 nm/s. TNB glasses that are presumed to have even greater stability were made by depositing more slowly (0.02 nm/s) at 295 K; previous work on IMC has shown that slower deposition rates yield more stable glasses.8 TNB glasses that are somewhat less stable were prepared by depositing at 0.2 nm/s at Tsubstrate ) 310 K ≈ 0.9Tg; the kinetic stability of TNB glasses deposited at different temperatures is compared in ref 21. Following deposition, the silicon substrates were broken into ∼0.5 cm2 pieces to provide numerous identical samples for subsequent annealing. Individual films were annealed in an oven that was temperature controlled to better than 0.1 K, for times ranging from 50 to 62 000 s. To prevent water absorption, the IMC samples were stored in a desiccator and annealed in a dry nitrogen atmosphere. Protio/deuterio concentration profiles were measured by secondary ion mass spectrometry (SIMS) at the University of California Santa Barbara Materials Research Center. Other experimental details and analysis techniques have been discussed previously.22 Simulations of Diffusion. In a procedure similar to that used in ref 23, calculations of the concentration profile evolution due to annealing were done with a one-dimensional random walk on a lattice. For these calculations, the experimental concentration profile at the preceding time step was utilized as the starting profile unless otherwise noted; the experimental profile was interpolated to give evenly spaced points. Diffusion was modeled by the transfer of concentration between neighboring sites at a rate determined by the diffusion coefficient. Concentration at each lattice site is identified by a value between 0 and 1, with 0 representing purely deuterio material and 1 being purely protio material. At each time step in the simulation, concentration is moved from each site such that a fraction “p” of the original concentration is removed and split equally between the two adjacent sites. The simulation utilizes reflecting boundary conditions at the ends of the lattice that mimic the boundaries of the thin film. These lattice calculations were directly compared to the measured SIMS profiles. The parameter p in the simulation was varied to produce the best fit to the experimental data and is related to the self-diffusion coefficient by
D)
( 2p )( t
tsim anneal
)( ) hreal hsim
2
(1)
Here the film thickness is hreal, and hsim is the number of lattice sites in the simulation. The experimental annealing time in seconds is tanneal, and the number of simulation time steps is tsim. The simulations were tested by comparing resulting concentration profiles to analytical solutions of Fickian diffusion in a regime where boundary effects were negligible. In all comparable cases, the numerical solutions precisely matched the analytical results. The two procedures agreed for all tested values of D and time, and for arbitrary starting concentration profiles. The lattice diffusion simulations provide a mechanism for modeling the combined effect of a growth front and Fickian diffusion. In growth front simulations, the value of p at each time and lattice site was specified as the value for the liquid (p) or for the glass (p ) 0). The boundary between the liquid and
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the glass regions in the simulation was moved at a constant velocity to match the experimentally observed change in front position between the starting and ending profiles of the simulations. Special care was taken to preserve microscopic reversibility at all points, including at growth front boundaries where the mobilities are unequal for neighboring lattice points. The transfer probabilities p between two such adjacent sites were set equal to the average of the values initially calculated at each site based on the local diffusion coefficient. III. Results Ordinary TNB Glasses. A typical route for preparing a glass is cooling the melt at a slow rate until the molecular relaxation time exceeds the time allowed during cooling. Recent experiments have shown that physical vapor deposition can also be used to produce ordinary glasses of low molecular weight organic compounds when the substrate is held within a few kelvin of Tg.4,5 Although the initial state of such vapor-deposited glasses may not be exactly the same as the glass formed by cooling the supercooled liquid, both types of glasses quickly evolve into the supercooled liquid when the temperature is increased above Tg. Supercooled liquids formed in these two ways show very similar calorimetric5,8 and structural properties,24 indicating that vapor deposition is a reliable means of preparing ordinary glasses and supercooled liquids. We have used SIMS to measure concentration profiles of isotopically labeled multilayer thin films of supercooled TNB vapor-deposited onto substrates very near Tg.22 Because this previously reported work sets the context for the work reported here on stable glasses, we briefly review it. Depth-dependent proton and deuteron intensities were measured for the asdeposited sample, and after annealing for a series of times and temperatures around Tg. Molecular motion on the length scale of ∼5 nm was measured by following the broadening of the hTNB/dTNB layer interfaces with time. Figure 1 shows the as-deposited concentration profile of an ordinary TNB glass (Tsubstrate ≈ Tg - 7 K, deposition rate of 0.2 nm/s) and subsequent concentration profiles after annealing at 345 K (black curves). The blue squares in Figure 1 are the concentration profiles predicted from evolution of the t ) 0 data set using D ) 3.5 × 10-17 cm2/s. The agreement between the Fickian diffusion prediction and the experimental data is excellent. This indicates that all h/d interfaces broaden in a Gaussian manner, that the same D value applies throughout the sample, and that the mean square displacement is linear with the annealing time. Experiments performed on ordinary glass samples with other initial concentration profiles and other thicknesses yielded essentially identical D values when annealed at 345 K. As discussed elsewhere, Fickian diffusion provides an excellent description of SIMS results for ordinary TNB glasses annealed near or above Tg (338-365 K).22 Stable TNB Glasses. Glasses that are highly stable can be produced by physical vapor deposition if the substrate temperature and deposition rate are controlled appropriately; these materials have high density,3,5,20,21 low enthalpy,5,8 higher mechanical moduli,7 lower vapor uptake,6 and high thermal stability.1,2,21,25 If the substrate is held near 0.85Tg and the deposition is slow, high mobility at the free surface during deposition allows rapid configurational sampling and results in a very stable structure that is low on the energy landscape. This has been demonstrated for several low molecular weight organic molecules.1-4 The reported properties of other systems, such as lead germinate26 and Tb-Fe glasses27 suggest that it is also possible to create stable glasses of many other types of materials.
Figure 1. Homogeneous evolution of d-TNB/h-TNB concentration profiles for an ordinary glass during annealing at 345 K (black). Alternating vapor deposition of h-TNB and d-TNB at 340 K produced the d-TNB concentration profile shown at the top, as measured by SIMS. The two lower samples were annealed 6000 and 26 600 s. Fickian diffusion fits of the concentration profiles with D ) 3.5 × 10-17 cm2/s are shown as blue squares; the as-deposited data (t ) 0 s) was used as the starting profile for the diffusion calculations.
Figure 2. Inhomogeneous evolution of d-TNB/h-TNB concentration profiles for a stable glass during annealing. These samples were deposited at 295 K at 0.2 nm/s. Thick lines are concentration profiles after annealing at 352 K for the times indicated. Thin lines are the t ) 0 curve overlaid for comparison, illustrating the progression of growth fronts entering the sample from the free surface and the glass/substrate interface.
One surprising feature of these stable glasses is the inhomogeneous nature of the transformation to the supercooled liquid state.10,21 Figure 2 demonstrates that transformation occurs via a planar growth front, with a highly mobile region propagating into the solid. The stable TNB glass samples shown in Figure 2 were vapor-deposited at 0.85Tg () 295 K) at a rate of 0.2 nm/s, and then annealed at 352 K. As shown for the three data sets at the bottom of Figure 2, the concentration profile in the center of the film in not affected by annealing (the initial
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concentration profile is shown using a thin line). All changes in the concentration profile propagate into the film from either the free surface or the glass/substrate interface. We speculate that these growth fronts are generated by enhanced mobility at the free surface and substrate interface.10 Molecular packing in the stable glass is so efficient that “unraveling” the glass from the outside is the only pathway for transformation for these thin films. This is likely a molecular scale effect: individual or small groups of molecules at the boundary between solid and mobile regions enter the liquid phase at a rate determined by the local mobility of the liquid.18 Since the front is initially generated uniformly at the film surfaces, it is easy to observe in these SIMS experiments. Comparison of Figures 1 and 2 illustrates the fundamental difference in transformation mechanisms from the glass to the supercooled liquid for ordinary glasses and stable glasses. It is important to emphasize that the only relevant difference between these two sets of TNB samples is the temperature of the substrate during vapor deposition; all other variables such as deposition rate, sample storage, and annealing protocols were identical. The stable glass relaxation mechanism is inconsistent with models currently used to describe the kinetics of the glass to liquid transformation. The Tool-Narayanaswamy-Moynihan (TNM) model11-13 gives relaxation rates that depend upon the temperature T and the instantaneous structure of the material, described by the fictive temperature Tf. In the model neither of these depends upon spatial position within the sample, so transformation of the glass into the liquid occurs at the same rate throughout the sample. The data shown in Figure 2 require a new description for the transformation of stable glasses to a liquid. During the transformation, the samples must be considered to have at least two relaxation rate distributions, with mobility in the liquid phase being at least several orders of magnitude larger than in the glass. Indeed, the solid ahead of the growth front is so kinetically stable that no molecular motion on the length scale measured by SIMS is observed even at temperatures well above Tg for very long annealing times. At 352 K, the center portion of the sample in Figure 2 showed no structural relaxation even at times exceeding 10 000 s. At this temperature, the structural relaxation time of the supercooled liquid τR is 2 s.28 All other depositions of TNB at 295 K and 0.2 nm/s also showed that any increase in mobility upon annealing occurred only with passage of the growth front through the sample. Stable TNB Glasses Formed at Lower Deposition Rates. The formation of stable glasses by vapor deposition at moderately low temperatures is due to enhanced mobility of molecules within a few nanometers of the free surface.5,8 Fast surface layers are then buried by additional deposition, locking in the low enthalpy structure. Lower deposition rates allow more time for configurational sampling in the mobile surface layer and thus more stable materials. For TNB and IMC vapor-deposited between Tg and 0.85Tg, calorimetric measurements show increased kinetic stabilization and lower enthalpies as the deposition rate is lowered to the smallest rate utilized (0.2 nm/s).8,25 Figure 3 shows the evolution of the concentration profile of TNB glasses prepared at 295 K and 0.02 nm/s (a rate 10 times lower than that of the sample shown in Figure 2). The qualitative features of the transformation of the stable glass are the same as for samples deposited at 0.2 nm/s. The growth front velocity is also very similar, as discussed below. Based on our experience at higher deposition rates,8 we expect that samples deposited at
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Figure 3. Evolution of d-TNB/h-TNB concentration profiles for a stable glass vapor-deposited at 295 K at a rate of 0.02 nm/s. The lower three samples were annealed at 345 K for 12 000, 27 300, and 55 000 s.
Figure 4. Depth of growth front from the free surface in stable TNB glasses as a function of annealing time at 345 K. Different symbols represent samples from independent depositions. The solid line is a linear fit with a zero intercept. Inset is a schematic of a low-energy conformation of TNB.
0.02 nm/s have higher density and lower enthalpy than those deposited at 0.2 nm/s, but we have not verified this. Growth Front Velocity. For the stable TNB glasses shown in Figures 2 and 3, transformation into the liquid occurs exclusively by propagation of a growth front through the solid. As seen in the figures, the front position at each annealing time can be readily determined by comparison of the annealed sample data with the overlaid as-deposited data. Figure 4 plots the position of the growth front near the free surface as a function of time for 7 stable glass samples annealed at 345 K. Each of these samples was deposited at 295 K with a deposition rate of 0.2 nm/s. The line shown is the best fit to the data with the constraint that the line passes through the origin. The excellent fit implies that the growth front starts at the free surface promptly upon annealing and propagates through the sample at a constant velocity Vgr during annealing at 345 K. The data in Figures 2 and 3 indicate that there is a measurable difference in front propagation from the free surface and glass/ substrate interface. For TNB, the substrate-initiated front propagates up to twice as far into the stable glass at a given annealing time. While the growth front behavior at the free surface is quite reproducible (Figure 4), considerable variability is observed at the glass/substrate interface. We do not understand the origin of this variability or even why a growth front is
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Figure 5. Growth front velocities measured from the free surface of stable IMC and TNB films at various annealing temperatures (relative to Tg for each material). Blue squares and open red diamond are for TNB deposited at 295 K at 0.2 nm/s determined from SIMS and neutron reflectivity, respectively. Other SIMS measurements shown are as follows: TNB deposited at 295 K at 0.02 nm/s (black star), TNB deposited at 310 K at 0.2 nm/s (green circle), and IMC deposited at 265 K at 0.2 nm/s (orange triangle). The solid line is the Fickian diffusion coefficient for the supercooled liquid of TNB.22 Inset is a schematic of IMC.
initiated at the glass/substrate interface. We speculate that the interfacial energy between TNB and the substrate surface is important. The free surface-initiated growth front velocities Vgr are shown in Figure 5 for TNB stable glasses and also an IMC sample, as a function of annealing temperature. This plot shows SIMS data for the two TNB deposition conditions discussed thus far (0.2 and 0.02 nm/s at 295 K). Data from a third TNB deposition condition (0.2 nm/s at 310 K) is also shown; the results of this deposition are discussed below. When Vgr for these three deposition conditions are compared at the annealing temperature of 345 K (Tg - 2 K), the values are within a factor of 3, with the most stable TNB glass (0.02 nm/s at 295 K) showing the slowest growth front. The open red diamond at 345 K was determined from neutron reflectivity experiments21 on stable TNB glasses prepared under the same conditions discussed here. In analyzing the neutron reflectivity results, we have assumed the presence of two growth fronts propagating into the film at the same rate. Good agreement is observed between the SIMS and neutron reflectivity results. The temperature dependence of Vgr is similar to the temperature dependence of self-diffusion in the supercooled liquid of TNB22 (solid curve in Figure 5). As previously discussed,10,18 this suggests a mechanism for the solid-liquid transformation in which the rate at which individual molecules or small clusters leave the glass is controlled by molecular mobility in the liquid phase. Across this temperature range, the time required to diffuse one molecular diameter in the supercooled liquid is similar to the time required for the transforming growth front to propagate one molecular layer. Mobility of TNB Liquid Behind Growth Front. After passage of the growth front through the stable glass, a liquid is formed. Molecular mobility in this liquid is clearly much faster than mobility in the stable glass. We initially expected that the mobile material would be the ordinary supercooled liquid, with dynamics matching those observed in glasses deposited near Tg, such as shown in Figure 1. However, the data indicates that the transformed liquid has diffusion that is 2-3 times faster than the ordinary supercooled liquid, as we describe in this section. Figure 6 shows two concentrations profiles observed during the annealing of a TNB stable glass (deposited at 295 K at 0.2
Figure 6. Concentration profile of d-TNB for a stable glass when annealed at 345 K. This sample was vapor-deposited at 295 K and 0.2 nm/s and annealed for 28 000 or 62 000 s (thick black lines). The asdeposited profile is overlaid (thin black line). Calculations assume Fickian diffusion behind the growth front: D ) 3.5 × 10-17 cm2/s (blue squares) and D ) 1.0 × 10-16 cm2/s (red ×). Residuals are shown below each data set. The known supercooled liquid diffusion coefficient (blue squares) does not describe the data.
nm/s). The thin gray line is the as-deposited profile, and the thick black line is the data after annealing at T ) 345 K. As seen in previous figures of stable TNB glasses, the material in the center of the film shows no measurable molecular motion on the length scales probed by SIMS, even after 62000 s at Tg - 2 K. The red and blue symbols shown in Figure 6 are calculated concentration profiles (method described in section II). These calculations assume a sharp boundary between the stable glass and the liquid that propagates at the measured growth velocity; diffusion is Fickian behind the growth front and no diffusion occurs ahead of the growth front. The blue squares are the profiles predicted with a diffusion coefficient for the transformed liquid equal to that measured for the ordinary supercooled liquid at the same temperature (D ) 3.5 × 10-17 cm2/s). These calculations clearly underpredict the composition changes that have occurred behind the growth front. The red crosses show the best fit to the experimental data, which yields D ) 1.0 × 10-16 cm2/s. The residuals below each data set emphasize the improved fit represented by the larger D value. The red concentration profile for the longer annealing time (t ) 62 000 s) was found using this larger value of D and two different starting times: the as-deposited (t ) 0 s) and the previous data set (t ) 28 500 s). The two resulting profiles are nearly indistinguishable, and both provide an excellent fit to the measured data. Thus, the liquid formed by transforming the stable glass is not only more mobile than the ordinary supercooled liquid, but it maintains this high mobility for very long times. Based on dielectric relaxation measurements, the structural relaxation time of the ordinary supercooled liquid at 345 K is about 100 s. Figure 6 demonstrates that the liquid generated by the growth front can remain more mobile than the ordinary SCL for at least several hundred τR. (The data
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Figure 7. Self-diffusion coefficients for the supercooled liquid of TNB (squares, from ref 22) and the liquid prepared by transforming TNB stable glasses (open circles). Lines are guides to the eye.
shown in Figure 6 were previously shown in Figure 2 of ref 10. In that figure, the h-TNB concentration profile was mistakenly identified as the d-TNB concentration.) The model used to determine the diffusion coefficient behind the growth front assumes a sharp boundary that moves linearly in time. It is possible that the front is not completely planar, and we have tested the effect a more diffuse boundary would have on the fitting. The data in Figure 6A (top) can be reasonably fit with a front as wide as 10 nm. This was determined by averaging a series of simulations with a small range of growth front velocities. The data in Figure 6B (bottom) at longer times is consistent with a total front width of 20 nm. It should be noted that these values represent upper bounds on the possible diffuseness of the front and that the data is consistent with a sharp boundary under all experimental conditions. These upper limits depend strongly on concentration profile and annealing time. In future work, we will attempt to place stronger bounds on the sharpness of the interface by using optimal concentration profiles. The color scheme used in Figure 6 will be used consistently for all following figures: blue squares are concentration profiles calculated using the known supercooled liquid diffusion coefficient, while red crosses show the concentration profiles calculated with a D value 2-5 times larger. The temperature dependence of diffusion in the liquid produced by the transformation of the stable TNB glass is shown in Figure 7 (open points). For reference, filled points show diffusion coefficients reported previously22 for the supercooled liquid (as shown in Figure 1). Across the temperature range from Tg - 2 K to Tg + 11 K, the newly created liquid consistently showed greater mobility than the ordinary supercooled liquid by a factor of 2-3. The enhanced mobility of the liquid created by the transformed stable glass has been observed in all TNB samples prepared at 295 K, including more than 10 separate depositions. In none of those experiments were we able to see the evolution of the enhanced-mobility liquid into the conventional supercooled liquid. The accessible time range is limited by the diffusion of the isotopically labeled materials; further information on diffusion is inaccessible once the concentration profiles become flat. Furthermore, the SIMS experimental technique loses resolution and sensitivity with films exceeding roughly 1000 nm. This thickness limit effectively places a time limit on such diffusion experiments. In order to circumvent these limitations, we have investigated the transformation kinetics of somewhat less stable TNB glasses. Figure 8 shows the concentration evolution for a TNB glass
Figure 8. Evolution of d-TNB concentration profiles (thick black lines) for a sample vapor deposited at 310 K and 0.2 nm/s. Calculated composition profiles are shown using D ) 7.0 × 10-17 cm2/s (red ×) or 3.5 × 10-17 cm2/s (blue square) behind the growth fronts. Note that the known supercooled liquid D (blue squares) adequately describes the bottom panel but not the t ) 15 000 s data. Each calculation is based upon the experimental concentration profile for the previous time step. For the third panel, D ) 6.0 × 10-18 cm2/s in front of the growth front; in all other cases, D ) 0 in this region.
prepared at 310 K (0.89 Tg) and 0.2 nm/s. Deposition at 310 K yields a glass that is less dense than does deposition at 295 K;21 based upon experiments on IMC, samples deposited at 0.89 Tg will have a higher enthalpy than those deposited at 0.85Tg. As shown in Figure 8, TNB films prepared at 310 K initially give rise to the unexpectedly mobile liquid but at later times the liquid mobility matches that of the known supercooled liquid. In the first 15 000 s of annealing at 345 K, this sample shows front propagation. Similar to Figure 6, the mobility behind the front was found to be roughly 2 times greater than the ordinary supercooled liquid; i.e., the red symbols provide a better fit than the blue symbols. At intermediate times, the surface-initiated growth front and a bulk process begin to compete as mechanisms for transformation to the liquid, as shown in the third panel. Here the red crosses associated with the larger D are only slightly better than the blue squares; both calculated curves use the previous (t ) 15 000 s) data set as the starting profile. For these calculations, we used a value of D ) 6.0 × 10-18 cm2/s in the region in front of the growth front (for all other calculation D ) 0 in this region). This nonzero value likely results because bulk diffusion begins at some late point during this annealing period; in the calculations a single D is used for the entire annealing period. Finally, at long times molecular motion becomes homogeneously Fickian, with a diffusion coefficient equal to that of the ordinary supercooled liquid. This is demonstrated in the bottom panel (t ) 55 000 s). Here the D value for the supercooled liquid (blue) describes the data better than the larger D value. Thus, while a moderately stable TNB glass initially transforms into the extra-high mobility liquid, it eventually evolves to the
Growth Fronts in Stable Glasses
Figure 9. d-IMC concentration profile (thick black lines) for stable indomethacin glass when annealed at 319 K. Samples were vapordeposited at 265 K at 0.2 nm/s. The previous data set is overlaid (thin black lines) and used as the starting profile for calculated curves. Calculated concentration curves with D ) 3.0 × 10-15 cm2/s (red ×) and D ) 6.0 × 10-16 cm2/s (blue squares) behind the growth fronts are shown. Note that the known supercooled liquid D (blue squares) does not describe the data.
ordinary supercooled liquid. This suggests that the more stable glasses deposited at 0.85Tg will do likewise given sufficient time. Stable IMC Glasses. To date, TNB is the only material for which the transformation mechanism of the stable glass has been carefully investigated. To test how general the growth front mechanism may be, we have prepared isotopically labeled films of a second organic glass former, indomethacin (IMC). Other aspects of stable glass formation of IMC have been previously reported.4,6,8,24,25 Figures 9 and 10 show the evolution of the concentration profiles of stable IMC glasses during annealing at 319 K () Tg + 4 K). These samples were vapor-deposited at 265 K () 0.85Tg) at 0.2 nm/s. Similar to the case of TNB, a growth front is observed to propagate from the free surface under conditions where the concentration profile remains unchanged in much of the sample. The value of Vgr is reasonably welldefined and is plotted as a triangle in Figure 5 along with data for TNB. On the T - Tg scale, the free surface growth fronts for stable glasses of TNB and IMC propagate at very similar rates. In contrast to TNB, no growth front is observed at the IMC glass/substrate interface. In addition, in one stable IMC sample (Figure 10) counterpropagating growth fronts were initiated in the interior of the film (at a position about 275 nm from the free surface, indicated by arrow). This internal front appears to arise from the position at which the deposition was paused for several hours. Occasional similar pauses in TNB depositions between isotopic layers have not shown such effects, and neither have shorter pauses of up to tens of minutes during IMC depositions. Growth front velocities for IMC stable glasses can
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Figure 10. d-IMC concentration profile for stable indomethacin glass when annealed at 319 K. Samples were vapor-deposited at 265 K at 0.2 nm/s; for this sample, the deposition was paused for several hours at the position marked by the arrow. The as-deposited profile is overlaid in the second and third panels (thin lines). Growth fronts initiated from the free surface and from the position where the deposition was paused (arrow). Calculated curves on 21 600 s data are Fickian predictions using the t ) 12 700 s data as the initial profile. Diffusion coefficients are D ) 6.0 × 10-16 cm2/s (blue squares) and D ) 1.3 × 10-15 cm2/s (red ×). Residuals are shown in the bottom panel. The known supercooled liquid diffusion coefficient (blue squares) does not describe the data.
be estimated for the three fronts shown in Figure 10 and agree within a factor of 2 with the value obtained from Figure 9. The data in Figures 9 and 10 also allow us to determine the mobility of the liquid produced after the growth front moves though the IMC stable glass. In this regard, we find that IMC is again very similar to TNB, with mobility behind the front faster than the ordinary supercooled liquid by a factor of about 2-5. The supercooled liquid D values for IMC were determined in a manner analogous to the experiment shown in Figure 1, using vapor deposition onto substrates very near Tg; these values will be reported elsewhere. In the second and third panels of Figure 9, we compare the experimental data to calculations of the concentration profiles. As illustrated in the third panel, the D value for supercooled IMC (blue squares) is clearly too small in comparison with the data. The red symbols illustrate that a D value 5 times larger describes the data very well. The fifth panel of Figure 10 shows a similar comparison. At very long times, diffusion in this sample is still more than twice as fast as would be expected for the supercooled liquid. The residuals at the bottom of Figure 10 demonstrate the discrepancy between the data and the calculated profiles. IV. Discussion Why Do Growth Fronts Occur When Stable Glasses Are Heated? We have proposed10 that growth fronts arise as a result of kinetic facilitation29-31 and the particular initial conditions for our experiments. Kinetic facilitation expresses the idea that mobility is locally required to create mobility in an otherwise jammed system. Vapor-deposited stable glasses of TNB and
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IMC are very efficiently packed. For thin films, surfaces and interfaces can initiate the transformation to the liquid with little or no induction period. For the free surface, we suppose that this occurs because of the high mobility of the surface. Once the temperature is raised above Tg, highly mobile molecules within a few nanometers of the surface liberate adjacent molecules from the stable glass. This gives rise to a planar growth front, which propagates uniformly into the film at a velocity determined by the rate at which individual molecules can free themselves from the solid and join the neighboring liquid. Our experiments establish that this rate has roughly the same temperature dependence as the diffusion coefficient of the supercooled liquid. A recent theoretical treatment of the glass-to-liquid transformation predicts such a growth front and at least qualitatively agrees with the observations reported here. An adapted version of the random first-order transition theory (RFOT) of Wolynes32 predicts an inhomogeneous transformation when an aged glass is heated with the front velocity dependent upon the mobility of the two phases.18 Reference 18 predicts values for Vgr that scale with the difference in molecular mobility on either side of the growth front and estimate front thicknesses of a few molecular diameters. For very low mobility glasses, their work predicts a growth front velocity that is dependent upon the relaxation time of the generated liquid rather than the relaxation time of the initial solid phase. This ran counter to our expectations before conducting these experiments, but is consistent with the results shown in Figure 5. For three TNB glasses of varying stability, we observe only a factor of 3 variation in Vgr. The absolute growth front velocity predicted by eq 18 of ref 18 is about an order of magnitude larger than the experimentally observed value at 345 K (Figure 5). For this calculation, we took the “particle” spacing (r0) as 0.52 nm, approximated the width of the interface ξ ≈ ξ0 ≈ 5r0, and used the inverse dielectric relaxation time28 as the liquid mobility. Our observation that Vgr has the same temperature dependence as diffusion in the supercooled liquid closely parallels the analysis of ref 18. If we modify eq 18 of ref 18 to account for this, then the discrepancy between the predicted growth velocity and the experimental value becomes larger. In a recent publication,24 we used wide-angle X-ray scattering to show that stable glasses of IMC (same deposition temperature and rate as used for the sample in Figure 9) have a different local packing arrangement than the supercooled liquid or the ordinary glass. The data presented there are consistent with the view that a first-order phase transition (polyamorphism)33-38 would occur near Tg - 20 K if supercooled IMC could be cooled infinitely slowly. We do not know what role this structural change might play in the growth fronts shown in Figure 9. Our current thinking is that the kinetic facilitation view of the growth front is appropriate whether or not the stable glass turns out to be associated with a different liquid phase. What Can Initiate a Growth Front? As discussed above, the initiation of a growth front at the free surface is natural given the high mobility in the top few nanometers of the glass.39-42 This high mobility allows a uniform plane of transformation sites to generate a liquid growth front. In most TNB samples, a substrate-initiated front was also observed, but this front propagation was less reproducible between samples as compared to the free surface. We expect that the growth velocities measured from both external film interfaces will become equal at steady state, and some preliminary results support this view. While we do not understand in detail why the glass/substrate interface would generate a growth
Swallen et al. front, it seems plausible that interactions with the substrate might give rise to packing arrangements that allow higher mobility. For one sample of IMC, a growth front was also observed at an internal interface when deposition of the sample was halted for several hours before additional glass layers were added (see Figure 10). Perhaps this pause provided the then-external surface additional time to equilibrate into a state that provided suitable sites for initiating a front. We cannot rule out the possibility of monolayer thickness adsorption of background material in the vacuum chamber, such as water. In either case, whatever induces the internal growth front must be present essentially everywhere on the internal interface in order to generate the observed changes in the concentration profile. While the transformation mechanism for the stable glass into the liquid for thin films (
