Termination of Solid-State Crystal Growth in Molecular Glasses by

Apr 25, 2014 - growth according to D/u < 7 pm. This condition holds for all liquids exhibiting GC growth and suggests that the phenomenon is a solid-s...
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Termination of Solid-State Crystal Growth in Molecular Glasses by Fluidity Daniele Musumeci, C. Travis Powell, M. D. Ediger, and Lian Yu* School of Pharmacy and Department of Chemistry, University of WisconsinMadison, Madison, Wisconsin 53705, United States S Supporting Information *

ABSTRACT: Fast crystal growth can abruptly emerge as molecular liquids are cooled to become glasses, a phenomenon presently unknown for other materials. This glass-tocrystal (GC) mode can cause crystallization rates orders of magnitude faster than those predicted by standard models. While GC growth is known for 12 systems, its features vary greatly with growth rates spanning a factor of 104. We report that the general condition for GC growth to exist is that liquid diffusion be slow relative to crystal growth according to D/u < 7 pm. This condition holds for all liquids exhibiting GC growth and suggests that the phenomenon is a solid-state process terminated by fluidity. GC growth must solidify several molecular layers before rearrangement by diffusion. We propose that GC growth propagates by a nonequilibrium crystal/liquid interface 3 nm wide that solidifies by local mobility. These results explain the prevalence of GC growth among organic liquids and guide its discovery in other materials. SECTION: Glasses, Colloids, Polymers, and Soft Matter termination as glasses are heated to gain fluidity. To understand this transition, we studied the mobility of liquids at the termination of GC growth and its dependence on the velocity of crystal growth. Our study examined all known systems of GC growth, which have widely different growth velocities, including indomethacin (IMC), whose GC growth is reported here for the first time and is the slowest on record. To maximize the coverage of our study, we considered three related measures of fluidity: diffusivity, structural relaxation time, and Tg-scaled temperature. We first describe the condition for GC growth in IMC and oterphenyl (OTP),2,11 for which diffusion data are available.12,13 Figure 1 shows the rates of crystal growth and diffusion in the two systems. Crystal growth in IMC was measured in this work in glass films 40−130 μm thick prepared between microscope coverslips.4 Upon cooling toward Tg, both IMC and OTP enter a regime in which crystal growth is slaved to diffusion, u ∝ D. For IMC, the relation u ∝ D holds for three polymorphs (α, γ, and δ) crystallizing from the same liquid.14 This relation is anticipated by standard models5−7 and justifies the name “diffusion-controlled crystal growth”. With further cooling, however, crystal growth abruptly becomes faster, outpacing diffusion-limited growth by 104 times. The termination temperature Tt for GC growth (arrows in Figure 1) is the temperature below which compact crystalline domains expand at a constant velocity and above which the growth mode is disrupted. Figure 2 shows this transition for γ

G

lasses are amorphous solids that combine liquid-like spatial uniformity and solid-like strength, making them ideal for many applications. Amorphous materials must resist crystallization, and in this context, a remarkable phenomenon is the abrupt emergence of fast crystal growth as some liquids are cooled to form glasses.1−4 This glass-to-crystal (GC) mode causes crystal growth orders of magnitude faster than that predicted by standard models.5−7 This phenomenon was first observed in 1967 by Greet and Turnbull1 and, after a gap of 3 decades, was systematically studied by Oguni and co-workers.2,8 GC growth has received five different explanations: homogeneous-nucleation-based crystallization,2 tension-induced mobility,3 solid-state transformation by local mobility,4 percolative nanocrystallization,9 and shear-stress-induced mobility.10 At present, the phenomenon is known only for molecular glass formers. Because crystallization consists of nucleation and growth, it is worth noting that GC growth refers to the expansion of an existing polycrystalline domain, not the nucleation of a new crystalline phase. We report that while individual systems differ greatly, the general condition for the existence of GC growth is that liquid mobility be low relative to crystal growth according to D/u < 7 pm, where D is diffusivity and u is the velocity of crystal growth. This condition suggests that GC growth is a solid-state process terminated by fluidity and is consistent with its propagation by a nonequilibrium interface 3 nm wide. Our finding explains the prevalence of GC growth in molecular liquids, guides its discovery in other systems, and is relevant for understanding crystal growth in amorphous materials. A central puzzle about GC growth is its emergence as liquids are cooled near the glass transition temperature Tg and its © 2014 American Chemical Society

Received: March 28, 2014 Accepted: April 25, 2014 Published: April 25, 2014 1705

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Figure 1. Crystal growth velocities u and diffusivities D in the liquids of OTP and IMC. The Greek symbols refer to the polymorphs of IMC.

IMC, whose Tt is 306 K. As observed in other systems,4 heating above Tt need not fully eliminate GC growth; remnants of it can still be observed in the form of loose fibers. Although this fiber growth is imperfectly understood, it is experimentally straightforward to determine the termination temperature of GC growth. Among the systems of GC growth, IMC is noteworthy because it has the lowest velocity and lowest Tt relative to Tg (Tg − 9 K). IMC also exhibits GC growth in the α polymorph but at a rate 10 times slower (see the Supporting Information), making its Tt difficult to determine. A significant point revealed by the data in Figure 1 is that the faster GC growth of OTP terminates at a higher temperature relative to Tg, in a liquid of higher diffusivity, than the slower process of IMC. Despite this difference, the two systems share a common condition for the existence of GC growth: D/u < 7 pm. To see this condition, we plot Figure 1 data in the format u versus D (Figure 3a). For both systems, GC growth is active only above the line D/u = 7 pm. Active GC growth features low D/u values; upon heating, D increases faster than u, and D/u increases, reaching 7 pm at Tt (open circles). This relation indicates that GC growth exists only if diffusion is slow relative to crystal growth. Above Tt, GC growth terminates, and crystal growth velocity tracks diffusivity according to D/u ≈ 50 nm. The diffusion-controlled growth slows down near D = 10−11 m2/s as the systems approach the crystal melting points and lose the driving force for crystallization. What is the significance of the empirical condition for GC growth, D/u < 7 pm, and in particular, the very short distance of 7 pm? We show that the condition is equivalent to the

Figure 2. GC growth of γ IMC and its termination upon heating. (a) Growth at 303 K yielded compact, steadily advancing domains. (b) Growth at 308 K produced compact domains and loose fibers. (c) Growth at 313 K yielded only fibers whose growth slowed over time. Scale bar = 100 μm. (d) SEM image of the fibers. The SEM sample first grew at 303 K in compact domains (1) and then at 308 K as loose fibers (2).

requirement that the crystal/liquid interface must advance at least several molecular layers before they are rearranged by diffusion. The time for an average molecule to diffuse its diameter a is given by τD = a2/(6D), where a ≈ 0.8 nm for OTP and IMC. During τD, the distance of crystal growth is uτD = ua2/(6D). This distance is determined mainly by the ratio D/ u for a is approximately constant. This distance has the following significance: if uτD < a, diffusion can rearrange molecules faster than crystallization solidifies them; if uτD > a, molecules are solidified by crystallization before they are significantly rearranged by diffusion. Figure 3b shows uτD as a function of Tg-shifted temperature for OTP and IMC. For diffusion-controlled growth, we obtain uτD ≈ 1 pm ≪ a, as expected for this growth mode. In GC growth, however, uτD can exceed a by orders of magnitude, which signifies a process much faster than diffusion. With heating, uτD decreases (because D increases faster than u), reaching 15 nm at the termination of GC growth (marked by open circles in Figure 1706

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We now show that the condition for GC growth obtained for OTP and IMC is general. At present, direct testing of the condition D/u < 7 pm is possible only with OTP and IMC, owing to the limited diffusion data, but we shall take advantage of the more abundant structural relaxation times τα from dielectric spectroscopy (Table 1). τα is known to describe the Table 1. Conditions under Which GC Growth Terminatesa system toluene IPB DMP salol salol I salol II DPCP DPCH OTP TPE ROYYT04 ROY-Y ROY-OP ROY-R TP NIF β IMC γ

Tg cal, K

Tt, K

log u at Tt, m/s

log τα at Tt, s

117.5 129 195 222

222 230 246

116 127 190 226 223 222 224.5 233 249

−8.28 −9.48 −9.30 −7.14 −7.5 −8.5 −7.77 −8.05 −7.76

1.2 3.0 3.5 −0.4 0.4 0.9 0.5 0.4 0.0

260

257 269

−8.02 −7.2

0.2 −0.5

−7.4 −7.5 −8.2 −6.1 −8.5 −10.37

0.4 0.4 1.4

272 315 315

265 265 261 280 316 306

1.4 4.6

log D at Tt, m2/s

reference 8, 8, 8, 8, 3,

−19.0

−21.5

17, 19, 20, 20, 20,

18 20 21 22 22

23, 24 23, 24 2, 11, 12, 25 8, 26 4, 27

28 29, 30 this work, 13, 31

a

Key: IPB: isopropylbenzene; DMP: dimethyl phthalate; DPCP: 1,2diphenylcyclopentene; DPCH: 1,2-diphenylcyclohexene; OTP: oterphenyl; TPE: triphenylethylene; ROY: 5-methyl-2-[(2nitrophenyl)amino]-3-thiophenecarbonitrile; TP: testosterone propionate; NIF: nifedipine; and IMC: indomethacin. τα is the mostprobable structural relaxation time. Tg cal: calorimetric Tg measured at 10 K/min.

time scale for viscous flow and molecular rotation. While one might expect D and τα to be inversely proportional to each other, the actual relation that describes the available data on molecular liquids (OTP,12 IMC,13 and tris-naphthylbenzene16) is

Figure 3. (a) Crystal growth rates of OTP and IMC plotted against liquid diffusivity. Diffusivity values below 10−20 m2/s for OTP and below 10−21 m2/s for IMC were obtained by extrapolating the experimental data (Figure 1) to lower temperatures. Despite their different growth rates, OTP and IMC develop GC growth under the same condition D/u < 7 pm. (b) Distance of crystal growth during one diffusion time uτD versus temperature. This distance greatly exceeds the molecular dimension a (0.8 nm) in active GC growth, approaches a at its termination, and is much smaller than a in diffusion-controlled growth.

Dτα0.68 = 10−19

(1)

where D and τα are in m /s and s, respectively. Equation 1 holds for τα = 10−1−105 s (the range in which GC growth has been observed to terminate) and reproduces the existing data to within 0.25 decade. To the extent that eq 1 holds in general, the condition for GC growth D/u < 7 pm is alternatively stated as uτα0.68 > 1.5 × 10−8, where u is in m/s and τα in s. We shall see next that this condition is observed experimentally. Figure 4a plots the velocity of GC growth at Tt versus the corresponding τα for all systems for which there are data (Table 1). The liquids of ROY and salol crystallize in several polymorphs, and all of the data are plotted. This plot shows τα in reverse order so that fluidity increases left-to-right, as in Figure 3a. The points in Figure 4a are equivalent to the circled points in Figure 3a, which mark the termination of active GC growth in OTP and IMC with increasing fluidity. A correlation is evident between the two properties plotted, signifying that faster GC growth can persist at higher fluidity (shorter τα). The quantitative condition for the existence of GC growth is 2

3b). Note that this distance is calculated using the diffusivity of the bulk liquid not the liquid near the crystal. Liquid mobility is known to increase near a growing crystal by 4 times for OTP.15 This effect would decrease the terminal value of uτD to ∼3 nm, or several molecular layers. We indicate this minimal distance of GC growth with the horizontal bar in Figure 3b, whose width spans 3−15 nm. For GC growth to exist, the crystal must solidify the nearest few layers of molecules before rearrangement by diffusion. In this sense, GC growth is an inherently solid-state process that is disrupted by the onset of fluidity.4 1707

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characterized by a nearly constant “fragility index”, defined as m = d log τα/d(Tg/T). For the liquids of this study (Table 1), τα near Tg is well reproduced by the equation log τα = log τα(Tg) + m(Tg/T − 1), where m = 80, τα(Tg) = 20 s, and Tg is the calorimetric Tg measured at 10 K/min. (We use Tg cal, not the “100 s Tg” from dielectric spectroscopy to make this analysis independent from the previous and to include testosterone proprionate,28 whose Tg cal is known but not τα, owing to fast crystallization.) This relation allows the previous condition for GC growth, Inequality 2, to be restated as (T − Tg) Tg

(3)

To test the above condition, Figure 4b plots the velocity of GC growth at Tt against Tt/Tg for all available systems (Table 1). This plot shows a positive correlation between the two properties, indicating that faster GC growth persists up to a higher temperature relative to Tg. Furthermore, despite the scatter, the experimental points are in full agreement with the predicted relation. Thus, assuming typical dynamics of organic liquids, the condition for GC growth expressed with T/Tg as the measure of fluidity (Figure 4b) is equivalent to that with τα as the measure (Figure 4a). We conclude that all three conditions above for the existence of GC growth are equivalent and reducible to the simple statement D/u < 7 pm. Our finding is consistent with the notion that GC growth propagates through a crystal/glass interface structured such that molecules join the crystal by local oscillations and rearrangement of the molecules by fluidity disrupts the process.4 Studies of crystal/liquid interfaces have found their thickness to be a few particle diameters and their structure to differ from that of the crystal or the liquid.32,33 The known interfacial thickness agrees with the minimal distance of GC growth in one diffusion time (3 nm). Such interfacial control of GC growth is supported by the fact that active growth can be halted by a brief heating above Tt;4 the brief heating would rearrange the requisite growth interface, which must be redeveloped upon returning to the initial temperature of active growth. Because it is altered by fluidity, the GC growth interface must differ from the equilibrated crystal/liquid interface. The ease with which a given system develops and sustains such an out-of-equilibrium interface may determine its ability to exhibit GC growth. Can GC growth be observed in other liquids? At present, the emergence of GC growth is known for 12 molecular liquids whose molar masses range from 92 to 358 g/mol. These liquids exhibit faster-than-diffusion crystal growth (GC mode) at low fluidity and diffusion-limited crystal growth at high fluidity. In contrast to these systems, some liquids are known to have only diffusion-limited crystal growth; for example, crystal growth velocities in silicates are well coupled to viscosity, down to the Tg’s, with no evidence of GC growth.34 For these liquids, it would be valuable to investigate the existence of GC growth with the condition D/u < 7 pm as a guide to experiments. This condition implies that slower GC growth exists at lower temperatures relative to Tg, a requirement that increases the observation time and could have prevented previous detection of the phenomenon. In this context, the fragility of organic liquids (rapid change of dynamics with temperature near Tg) sharpens the crossing in temperature of the ratio D/u through the 7 pm threshold, making the onset of GC growth easily observable; the process would be less spectacular in stronger liquids. In atomic liquids (e.g., metals6 and argon35), crystal

Figure 4. (a) u versus τα at the termination temperature Tt for GC growth in 11 liquids. (b) u at Tt versus Tt/Tg for the same liquids, minus TPE (no Tg cal data) and plus testosterone propionate (TP). Panels (a) and (b) share the same legend.

uτα0.64 > 3 × 10−8

< 0.16 + 0.020 log u

(2)

which is nearly identical with the condition derived from D/u < 7 pm and the known relation between D and τα (eq 1). The small discrepancy is within experimental error. Thus, the simpler condition D/u < 7 pm accurately describes the existence of GC growth in all liquids available for this test (n = 11) on the assumption that eq 1 is obeyed in all systems. These systems differ substantially in intermolecular forces, Tg, and crystal growth rates, and yet their GC growth terminates if a common condition is met. It is noteworthy that a liquid’s Tg has no apparent influence on the condition under which GC growth terminates. We now consider the condition for GC growth using a third measure of liquid mobility, the Tg-scaled temperature. At Tg, different liquids have similar relaxation times; for organic liquids, the change of τα upon departure from Tg is 1708

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growth can achieve velocities fast relative to diffusion, even at high fluidity. Crystal growth in atomic liquids thus has the character of GC growth in molecular liquids but is not disrupted by fluidity. This difference between molecular and atomic liquids likely stems from their different structures. In atomic liquids, fluidity exchanges only the positions of spherical particles, and the resulting perturbation on the structure of the crystal/liquid interface may be insufficient to halt fast crystal growth. In molecular liquids, however, fluidity alters both the positions and orientations of molecules and may more effectively damage the ordered crystal/liquid interface that sustains GC growth. In more complex liquids (more components, larger molecules), crystal growth might require structural reorganizations too large for local oscillations to deliver. In this sense, molecular liquids exhibiting GC growth have the “optimal” degree of complexity to possess fast GC growth at low fluidity and diffusion-limited growth at high fluidity. In summary, we report the condition under which the fast glass−crystal growth mode emerges in supercooled liquids: D/ u < 7 pm. While individual systems vary greatly with respect to growth rates and termination temperatures, this condition holds for all systems known to exhibit the phenomenon. This finding supports the view that GC growth is an inherently solidstate process disrupted by fluidity: GC growth propagates by a nonequilibrium crystal/glass interface ∼3 nm wide, which is continuously solidified by local mobility and whose reorganization by fluidity halts the process. This phenomenon will not be observed in the manner of organic liquids if fluidity cannot efficiently restructure the crystal/liquid interface (possibly the case for simple atomic liquids) or if crystal growth requires structural reorganization too large to occur by local oscillations (the case for complex, multicomponent liquids). GC growth is expected in systems of intermediate complexity, and its existence can be searched for under the condition reported here. Our finding is relevant for developing stable amorphous materials, especially organic glasses for electronic and biomedical applications, and for advancing the models of crystal growth. In future studies, it would be valuable to evaluate and refine the models of GC growth against the existence condition identified here. Up to this point, it has been standard to use bulk fluidity to represent the kinetic barriers for crystal growth. The emergence of fast GC growth in deeply supercooled liquids motivates the search for additional measures of molecular mobility relevant for crystal growth.

AUTHOR INFORMATION

Corresponding Author

*(L.Y.) E-mail: [email protected]. Phone: (608) 263 2263. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the National Science Foundation (DMR-1234320) for supporting this work and a Graduate Research Fellowship to C.T.P.

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EXPERIMENTAL METHODS IMC (99% purity, γ polymorph) was obtained from SigmaAldrich and used as received. Crystal growth was measured with a light microscope in glass films 40−130 μm thick prepared between microscope coverslips.4 For SEM analysis, the top coverslip was removed, and the sample was gold-coated and scanned on a field-emission LEO 1530. Polymorphs were identified by Raman microscopy (Thermo DXR) and X-ray diffraction (Bruker D8 Advance).

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ASSOCIATED CONTENT

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GC growth rates and SEM images of GC crystals for α IMC. This material is available free of charge via the Internet at http://pubs.acs.org. 1709

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