Experimental Observation of the Marginal Glass Phase in a Colloidal

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PHYSICS ever longer plateaus at ever greater length scales. In the thermo- dynamic limit, this series of discrete plateaus will merge into a continuous logarithmic growth of the MSD (7, 8). For soft spheres (i.e., particles with noninfinite resistance to deformation), the glass phase diagram (sketched in Fig. 1) has been proposed to exhibit a reentrant stable glass phase (6, 14). The athermal jamming density controls the location of the onset of marginality in phase space and thus controls the density at which the marginal phase is first encountered, forming a so- called Gardner dome about the athermal jamming point (6, 14, 26). That the system is able to explore densities above jamming is due to the soft sphere interaction and the presence of thermal excitatory energy. The existence of the marginal glass regime and its transition out of the stable glass regime has been extensively probed with numerical simulations. Examining hard and soft sphere systems, several of these simulations confirm the existence of the marginal glass (11), with some even including numerical MSDs with logarithmic-like scalings (9, 10, 13). However, other work using different algorithmic approaches contest this assessment, positing that the marginal phase is an avoided transition in low Fig. 2. Global change in density relative to the initially shaken sample ver- spatial dimensions (12, 27). Further, due to the high computa- sus time for a representative colloidal system. Even at the longest times tional cost in numerical simulations, the transition from ballistic studied, the system is still slowly densifying. (Inset) Image of the colloidal to logarithmic behavior in the MSD has never been observed. glass during sedimentation. The interstitial fluid has been index matched to Probing such a marginal phase in physical glasses is difficult, the PMMA beads, so they are nearly invisible. The black tracer particles are necessitating novel methods. The first experimental evidence thus revealed. compatible with the marginal glass phase was work done on an externally controlled pseudo-thermal two-dimensional disk sys- tion of elapsed time. We observe a clear signature of both the tem (28). That work was done by examining a direct replica stable and marginal phases. We then qualitatively compare these signature similar to that employed in numerical simulations. results to the reentrant glassy phase diagram for colloids (6, 14) Evidence for the Gardner phase was found by resetting the (Fig. 1) and find good agreement. system to identical starting points and observing random qua- sithermal excitations to evolve the system’s energy landscape into Methods a hierarchy of subbasins. A second work, studying the Johari– Our experimental system consists of nominally 50-µm diameter Goldstein relaxation in sorbitol and xylitol under cooling to low polymethyl methacrylate (PMMA) colloids (Cospheric PMPMS- temperature found an extreme broadening of the β relaxation 1.2; 45–53 µm; > 95%), with 95% falling within a range of distribution, consistent with the fractal roughening of the energy 45 to 53 µm and having sphericity of >99% and a density landscape predicted by the marginal phase (29). of 1.20 g/cm3. The degree of polydispersity is chosen to frus- In this work, we create a three-dimensional thermal colloidal trate crystallization. These colloids are suspended in a mixture glass and follow its behavior under densification. We use the of tetrahydronapthalene, decahydronapthalene, and cyclohexyl MSD as a signature to characterize the glass phase as a func- bromide. This three-component liquid is chosen to allow for pre- cise independent control of both the refractive index and density. To this are added a small number of dyed black polyethylene tracer-colloids with the same density and diameter as the PMMA colloids (Cospheric BKPMS-1.2; 45 to 53 µm). This colloidal suspension is placed into a sealed cuvette sample chamber with an Liquid imaging width of 3 mm, large enough to allow for imaging far Stable Glass from any boundaries. The sealed cuvette allows us to reset the system to a new configuration by shaking it. All imaging data were collected on a Nikon TE2000s microscope on a floating-stage optical table in a climate controlled room. Illumination is provided by a high-intensity red light- emitting diode (Thorlabs M660D2) with output of 940 mW. High-speed video was recorded on a Phantom Miro M310 high- speed camera running at 64,000 fps using an image size of Marginal 192 × 192 pixels with a total collection of 110,000 frames and Temperature Marginal Glass a time of 1.72 s. We used a 50× lens (Nikon LU plan ELWD Glass 50×/0.55 B, ∞/0, WD 10.1), giving us an image resolution of 0.4 µm per pixel. The cuvette being imaged is stabilized on the x microscope using a custom sample holder that ensured a flat imaging plane and a minimum of vibration. The imaging stage is then encased in a small cardboard box for thermal and acoustic isolation. Density We match the solutions’ index to the PMMA but introduce a Fig. 1. Sketch of a phase diagram for soft sphere glasses as a function of slight density mismatch between the fluid and the colloids caus- temperature and density (adapted from refs. 6 and 14). Our experiment ing the colloids to slowly sediment, resulting in a colloidal glass starts at the “X” and follows the arrows from the liquid to stable glass to whose packing fraction increases with increasing time. We mea- marginal glass and then further into the stable glass phase. sure the average sedimentation across many samples by imaging

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(dark end time to of (blue) progression start The from box. running red on color the overlayed by within is represented up video is blown 1.72-s and glass. entire colloid the the colloidal for of colloid top matched the index of motion the tracked The inside tracer-colloid diameter ∼50-µm 3. Fig. amn n Corwin and Hammond MSD(τ euearda etrtakn loih 3)t n the find to (31) algorithm tracking center radial a use We measure- these of sensitivity vibration extreme the to Due ∼ 1.5 ~ x ersnaiesiliaefo h atrdvdosoigthe showing video captured the from image still Representative (t = ) ) mfrec rm.Fo hs rce aa we data, tracked these From frame. each for nm stemaue oiino h atcea time at particle the of position measured the is hk ~ x rmtecvteinrwls(approximately walls inner cuvette the from µm (t + τ ) − ~ x (t )k 2 i kh − ~ x (t + τ ) − 10 µm ~ x (t This Inset. 10 )i 2 k, n t , m τ [1] is a ie tsol ewl tby fit this well beyond be then should logarithmic, it is time, motion lag more the If by col- (33). floating deviates We motion free for MSD loidal tracer. expression each Clerx–Schram the the when from of 10% time than motion the the as upon this characterize impinge to begins time particles lag the finding by begin floating we freely a the in on motion work diffusive (32). previous to colloid This in ballistic floor. MSD verified from noise this and crossover nominal from employed the was it below technique subtract measurements (i.e., can achieve time we to long distributed, extremely the identically an only and is affecting drift MSD, the the 0.1 Typically, of average. part time small a denote ets where snal h aea h rvospaeu(rne 8 min). 580 (orange; which plateau plateau, previous is the MSD is as the MSD same min, the the 520 min, min, to nearly 320 400 420 is to and from 320 180 min); between from 400 min); (blue; min); 200 logarithmic 100 (yellow; (green; plateau are subdiffusive is There is MSD the MSD subdiffusive. the all shaken min, are from 160 highlighted: curves to representative a the is with regimes times, motion MSD the long ballistic important four show times, at we short and eye, at the ballistic, to that near guide demonstrating a asymptotes As diffusive min. 20 and every taken were MSDs The 4. Fig. at MSD the of value hs Ssaeceryntlogarithmic. As not sample. clearly shaken are freshly MSDs these a 5A, in density, Fig. even in colloidal caging seen initial some high to the leads from of which results absence colloid The regime thermal (18). diffusive motion floating a subdiffusive free shows a instead and of (33) dense expectation around a subdiffusive the of in from a times colloid deviates a lag to of At that over suspension. with turning liquid consistent times, times, long the shows at short of curves) motion at (green MSD shaken motion The been has ballistic density. it with increasing after behavior just of thus tracer-colloid change and substantial a time find we increasing 4, Fig. in shown As Discussion and Results request. upon readers Availability. Data nodrt itnus ewe ifrn om fteMSD, the of forms different between distinguish to order In 10

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Fig. 5. Representative MSD curves from the four time steps. Superimposed on each MSD is the best ﬁt logarithm. (A) MSD in the subdiffusive regime, taken at 100 min with a logarithmic ﬁt. (B) MSD in the plateau regime (260 min). (C) MSD in the logarithmic regime (400 min). (D) MSD in the reentrant plateau regime (580 min).

As the colloidal density increases, the power law of the subdif- Taken as a whole, the behavior of the densifying colloidal glass fusive motion decreases toward zero. Starting at about 180 min, is consistent with the schematic phase diagram shown in Fig. 1. the MSD flattens to a plateau, indicating that the tracer-colloid The system begins in the liquid phase, at the point marked “X.” is trapped inside a cage of other colloids. This behavior is shown The experiment proceeds at a constant temperature but increas- in yellow in Fig. 4 on a log-log scale and on a semilog scale in ing density along the red line, passing through the stable glass Fig. 5B. This demonstrates that the system has entered into the phase, the marginal phase, and ending once again in a high- stable glass phase. The height of the plateaus shows a small but density stable glass, as shown in Fig. 5. It is striking that not only overall consistent trend toward smaller cages, driven by the slow do we see individual signatures of each phase but that they com- overall densification of the system. The transition between bal- bine to begin to map out a phase diagram that agrees with the listic motion and the plateau happens slowly, over nearly three theoretical predictions. decades, only reaching the flat plateau at τ ∼ 1 × 10−1 s. Across Conclusions all samples measured, the transition from subdiffusive to plateau We have created a colloidal glass at fixed temperature and happens after about 160 min. tracked it through the liquid, stable glass, and marginal glass After about 340 min, the constant plateau is replaced by phases as a function of increasing density. We track an indi- an even lower MSD approaching that earlier plateau value, as vidual colloid deep within the glass at short times and find the shown in blue in Fig. 4. This change in the MSD is indicative of associated MSD. This experiment lays the groundwork for a a change in phase of the underlying system. As seen in Fig. 5C, complete experimental determination of the glass phase diagram this motion is well fit by a logarithm. These are the only curves to by repeating such measurements over as broad a range of tem- demonstrate a good logarithmic fit. This behavior is indicative of peratures as possible. A full phase diagram will also allow for the system entering the marginal glass phase. In some systems, a careful exploration of the transition between phases to char- the logarithmic MSD as a function of starting time exhibits a acterize what, if any, phase transitions are present in colloidal progression toward smaller values of the slope-fitting parameter. glasses. Further, the techniques developed in this work explore This implies that small changes in packing fraction result in dra- a previously overlooked regime for glasses, that of the short- matic slowing down of dynamics. Across all samples measured, est time and length scales. Measuring behavior in this regime the transition from plateau to logarithmic happens after about is a potent tool with which to explore previously inaccessible 380 min. implications of the mean field theories of glasses and to probe The system remains in the marginal phase for about 80 min new material properties. The experimental observation of these (across all samples, an average of 165 min). After which, the distinct phases provides evidence for the validity of the replica MSDs consistently cease to be logarithmic and revert back to a theory of glasses in physical colloidal glasses and opens the door constant plateau, indicating a return to the stable glass regime, to an exploration of its validity in the wider world of glassy shown in orange curves in Figs. 4 and 5D. There is no further materials. evolution of the MSD during the experimental observation. In ACKNOWLEDGMENTS. We thank Dan Blair, Raghuveer Parthasarathy, and addition to the sedimenting tracer particle MSD shown in Figs. 4 Camille Scalliet for helpful discussions and the University of Oregon machine and 5, we have included similar data for other experimental runs and electrical shop staff. This work was supported by NSF Career Award in SI Appendix. DMR-1255370 and the Simons Foundation Grant 454939.

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PHYSICS