Experimental observation of the marginal glass phase in a colloidal glass

Andrew P. Hammond; Eric I Corwin

REVIEW 4 major objections 4 minor 1 cited by

Experimental observation of the marginal glass phase in a colloidal glass

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that a densifying colloidal glass passes through a predicted 'marginal' phase, visible as logarithmic growth in a single tracer particle's motion.

desk verdict A clever short-time MSD assay reports the first 3D thermal colloidal marginal-glass sequence, but the central phase claim rests on one tracer's log-shaped MSD taken while the sample is still densifying. read the letter →

arxiv 1908.08152 v2 pith:6O3UEJX4 submitted 2019-08-22 cond-mat.soft cond-mat.stat-mech

Andrew P. Hammond , Eric I Corwin This is my paper · ORCID

classification cond-mat.softcond-mat.stat-mech

keywords marginalglassGardnerphasecolloidalmeansquareddisplacementreplicatheoryofglassesstablereentranttransitionsingle-particletracking

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a single tracer colloid in a slowly densifying three-dimensional colloidal glass acts as a probe of the glass phase diagram, and that the probe's mean squared displacement reveals four successive regimes: a sub-diffusive liquid, a stable glass with a flat plateau, a marginal glass whose motion grows logarithmically in time, and a return to stable glass at the highest densities. The measurements are taken in the short-time ballistic-to-cage regime, which sidesteps the slow aging that makes long-time glass measurements difficult. The authors interpret the logarithmic regime as the signature of the marginal, or Gardner, glass phase predicted by replica mean-field theory, and they take the overall sequence as direct experimental evidence that this phase exists in three-dimensional thermal colloids. A sympathetic reader would care because the result would carry the Gardner transition out of infinite-dimensional theory and simulation and into laboratory matter.

What carries the argument

The central object is the drift-subtracted mean squared displacement of a single tracer particle, computed from 64,000-frames-per-second video and corrected for localization noise. Its shape is classified against a crossover time $\tau^*$ where the tracer first feels its neighbors: beyond $\tau^*$, a flat plateau means a smooth energy basin (stable glass), while a logarithmic law $\mathrm{MSD}(\tau) = a \log(\tau/\tau^*) + x^*$ means a basin hierarchically split into sub-basins (marginal glass). Slow gravity-driven densification sweeps the packing fraction upward by about four percent over hundreds of minutes at fixed temperature, so one continuous trajectory of MSD curves maps out successive phases. The logarithmic fit parameter $a$ is the quantitative handle on how fast the system explores the sub-basin hierarchy.

What would settle it

Track several tracer colloids in the same sedimenting sample and measure the local packing fraction around each one. If the marginal-glass claim is right, the logarithmic regime should begin for every tracer when its local density crosses the same value, and the logarithmic slope should be reproducible across tracers at matched density; if the logarithmic shape depends on the chosen tracer or on global sedimentation time rather than local density, the central claim fails. A second check is to compare a sedimenting sample with its drift subtracted using a fixed reference and with an intentionally immobilized tracer; if the logarithmic regime disappears under those controls, it was an artifact.

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Extended reading notes

Core claim

The paper's central claim is that the short-time mean squared displacement (MSD) of one tracer colloid in a sedimenting colloidal glass is a phase assay, and by that assay a fixed-temperature system traverses stable glass, marginal glass, and stable glass again as density rises. At the lowest densities the MSD is ballistic at short times and sub-diffusive at long times, the liquid signal. After roughly 180 minutes of sedimentation a flat plateau appears and shrinks with increasing density; this is the smooth-basin signature of a stable glass. The plateau does not shrink to zero. Instead, from roughly 340 to 420 minutes the MSD is well fit by a logarithm growing toward the earlier plateau value, which the paper identifies with exploration of a hierarchy of sub-basins within sub-basins, the marginal phase. At the highest densities the logarithmic regime vanishes and a plateau reappears at almost the same height, the re-entrant stable glass. The paper compares this sequence to the mean-field phase diagram for soft-sphere glasses and finds agreement.

Load-bearing premise

The load-bearing premise is that a logarithmically growing mean squared displacement in one tracer particle is an unambiguous fingerprint of the marginal glass phase, rather than a product of ordinary aging, polydispersity, residual drift, or the tracer's particular local cage.

Editorial extensions

If this is right

If the marginal phase is real in three-dimensional colloids, then the replica-theory phase diagram, including the re-entrant stable-glass branch, applies to laboratory materials rather than only to infinite-dimensional models.
The short-time MSD protocol gives a practical assay for locating the Gardner transition: repeating the sedimentation experiment at other temperatures would map the boundaries of the marginal phase.
The logarithmic growth observed here implies that the tracer's cage is not a smooth bowl but a nested hierarchy of barriers, so caged dynamics in dense colloids should retain long-time memory of the sub-basin structure.
Because the re-entrant plateau appears within the same sample without a change in temperature, the stable-to-marginal-to-stable sequence is a density-driven transition, and its sharpness can be tested by measuring the logarithmic slope as a function of elapsed time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One extension the authors leave implicit is that the same short-time MSD protocol could be applied to many tracers simultaneously; if the Gardner interpretation is right, each tracer should enter the logarithmic regime at the same local density, and the spread of those onsets would measure the sharpness of the transition.
A natural further test is to examine higher-order statistics, such as the non-Gaussian parameter or displacement kurtosis, over the same trajectories; a hierarchical sub-basin landscape would predict intermittent rearrangements whose statistics differ from a smooth cage.
The protocol could be transferred to other glass formers, including emulsions or granular systems with thermalized grains, to see whether the same logarithmic regime appears whenever a glass is densified at fixed temperature.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here is the short version: this paper is worth reading and worth sending to referees, but the main claim is not yet supported by the evidence as presented. The genuinely new thing is the assay and the reported sequence: stable glass, marginal glass, reentrant stable glass in a sedimenting 3D thermal colloidal glass, identified via the short-time single-tracer MSD. Prior experimental work was a 2D granular system (Seguin-Dauchot) and indirect dielectric measurements (Geirhos et al.); nobody has claimed this reentrant sequence in a 3D thermal colloidal system. The idea of probing the ballistic-to-caged crossover is clever and avoids the long-time-equilibration problem.

What the paper does well: it is honest about the contested status of the Gardner transition, citing the simulations that argue it may be avoided in finite dimensions (refs 12 and 27). It explicitly states that the system is still densifying throughout, and the density never stops changing. The references are appropriate and include competing work; there is no self-citation problem beyond citing their own earlier method paper, which is relevant. No circularity: the phase diagram is taken from theory and used as a benchmark; the log-fit parameters are descriptive.

Soft spots, in proportion. The decisive classification is of the blue MSD curves as logarithmic via Eq. (2). The system is still compacting at all measured times; logarithmic growth is a generic symptom of slow aging or cage compaction in dense suspensions. The paper offers no comparison against aging or power-law models, no error bars, no replicate tracers, no local packing fraction near the tracer, and no quantitative criterion for what counts as a 'good logarithmic fit.' There are also internal timing inconsistencies: the Fig. 4 caption says logarithmic from 320 to 400 min, while the text says 'after about 340' and later 'after about 380.' These could be minor editing slips, but they matter because the phase is assigned post-hoc to a window of elapsed time. Also, 'these are the only curves to demonstrate a good logarithmic fit' is asserted rather than shown. The data availability is stated, but 'upon request' is weak; there is no repository.

My read: the central sequence might well be real, but the current evidence underdetermines it. This is a plausibility argument, not a demonstration. For peer review: send it out, but expect heavy revision. The paper deserves referee time because it is the first experimental report of this reentrant sequence and it could sharpen the Gardner debate. I would ask for quantitative fits with alternative models, an ensemble of tracers, and at least one fixed-density control or a local density measurement near the tracer.

Referee Report

4 major / 4 minor

Summary. The paper reports a single-particle tracking experiment in a sedimenting colloidal glass. The authors measure the short-time mean squared displacement (MSD) of one tracer colloid every 20 minutes and classify the MSD shape into four regimes as the sample densifies: subdiffusive liquid, stable glass (plateau), marginal glass (logarithmic growth), and reentrant stable glass (plateau). They interpret this sequence as direct experimental evidence for the marginal/Gardner glass phase predicted by replica mean-field theory and compare it qualitatively with the reentrant phase diagram for soft spheres.

Significance. If substantiated, the observation of a logarithmic MSD segment between two plateau regimes in a three-dimensional thermal colloidal glass would be a notable experimental confirmation of replica-theory predictions and would validate the short-time MSD as a probe of deep-glass phases. The experimental design has clear strengths: high-speed imaging at 64,000 fps, sub-pixel localization with a stated 1.5 nm localization error, explicit drift subtraction, and a defined free-colloid comparison for identifying the cage-onset lag time. The claim is timely and the technique is potentially powerful. However, as presented the phase assignment is underdetermined by the data: it rests on one tracer, on an unquantified goodness-of-fit criterion, and on measurements taken while the sample is still densifying. The central conclusion therefore needs substantially stronger quantitative and statistical support.

major comments (4)

[§2, Fig. 4, Eq. (2)] The assignment of the blue MSD curves to the marginal phase rests entirely on the curves being "well fit" by the logarithmic form in Eq. (2), but the classification is not quantitative: no goodness-of-fit measure, no comparison against plateau, power-law, stretched-exponential, or aging-based functional forms, and no uncertainty in the fitted parameters a and x* is given. The problem is compounded by the fact that the sample is never at a fixed state point: Fig. 2 states that the density increases monotonically even at the longest times, so every 20-minute MSD is recorded while the glass is aging and compacting. Logarithmic MSD growth is a generic signature of slow caging, compaction, or aging in dense suspensions. To make the Gardner-phase claim load-bearing, the authors need either fixed-density control measurements or a quantitative model comparison that distinguishes hierarchical sub-basin exploration from ordinary aging.
[§2, Figs. 4 and 5] The entire phase sequence is inferred from a single tracer particle. The paper reports no error bars on the MSD curves, no ensemble average over multiple tracer particles, no replicate MSDs at the same elapsed time, and no local volume-fraction measurement in the tracer's neighborhood. The text refers to behavior "across all samples" and to supplemental runs, but the supporting data are not shown or quantified in the main text. Because every subsequent claim about the reentrant marginal glass depends on the shape classification of these particular curves, the analysis needs to be repeated over several tracers and reported with statistical uncertainty and an explicit, quantified criterion for what constitutes a "good logarithmic fit."
[Fig. 4 caption and §2] The time intervals defining the marginal phase are internally inconsistent: the Fig. 4 caption states that the MSD is logarithmic from 320 to 400 minutes, whereas the text says the plateau-to-logarithm transition occurs "after about 340 minute" and later "after about 380 minutes"; the reported duration of the marginal phase is "about 80 minutes" in one sentence and "an average of 165 minutes" in the next. Since the phase boundaries are defined by these intervals, a table listing elapsed time, fitted parameters, and fit-quality measures for each run is needed to remove the ambiguity.
[§2 and Fig. 1] The comparison to the theoretical phase diagram is only qualitative and the density axis is not calibrated. Fig. 1 is a sketch adapted from Refs. [6,14], the experiment measures only a global change in density from PIV (Fig. 2), and the local packing fraction around the tracked tracer is not measured. The claim of agreement with the mean-field reentrant phase diagram would require converting the global densification curve into a local volume fraction at the tracer's position and locating the phase boundaries with uncertainties; as it stands, the comparison is an interpretation rather than a quantitative test.

minor comments (4)

[Eq. (1)] Equation (1) has a typographical imbalance in parentheses: the expression contains an unmatched closing parenthesis after the first term. Please correct the typography.
[Fig. 4] The color coding of the four regimes is difficult to parse in printed or grayscale versions; please add distinct line styles or markers in addition to color.
[Introduction] The statement that "the transition from ballistic to logarithmic behavior in the MSD has never been observed" should be qualified with respect to Ref. [28], which reports Gardner-phase evidence in a quasi-two-dimensional granular system; the distinction between that prior work and the present experiment should be made explicit.
[Data Availability] Because the entire inference rests on a single tracked trajectory, please consider depositing the raw tracked positions, the analysis scripts, and the fitted parameters in a permanent public repository rather than providing them only upon request.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper uses an external theoretical prediction (logarithmic MSD in the marginal phase) as an operational diagnostic and then reports agreement with that prediction; no fitted parameter is renamed as a prediction and no load-bearing argument reduces to the paper's own inputs.

full rationale

The paper's central claim is that a densifying colloidal glass exhibits MSD signatures consistent with the replica-theory phase sequence: sub-diffusive liquid, stable-glass plateau, marginal-glass logarithmic growth, and reentrant stable-glass plateau. The derivation chain is not circular: the replica-theory prediction of logarithmic MSD is imported from external theory (refs. 1-8, 14), not derived from the present data. The experimental analysis fits Eq. (2) (MSD = a log(tau/tau*) + x*) to certain curves, but this fit is used only as a diagnostic to classify observed behavior, not as an input that constructs the phase diagram or forces the qualitative comparison to Fig. 1. The classification criterion is clearly stated: curves that are 'well fit by a logarithm' are assigned to the marginal phase, while plateau curves are assigned to the stable phase. This is an operational test of an external prediction, not a self-derivation. The only self-citation is ref. 32, which validates the drift-subtraction and localization-error subtraction technique used to measure the MSD; this is a methodological support, not the load-bearing evidence for the marginal phase. No uniqueness theorem is imported from the authors' prior work, no ansatz is smuggled in via citation, and no known empirical pattern is merely renamed as a new result. Concerns about whether a single tracer particle or ongoing sedimentation can uniquely identify the Gardner phase are empirical underdetermination or correctness risks, not circularity under the definitions used here. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entity is postulated. The analysis imports the marginal/Gardner phase concept from replica theory and relies on several unverified experimental assumptions: the uniqueness of the logarithmic MSD signature, the sedimentation-time-to-density mapping, single-tracer representativeness, and the localization-error subtraction procedure.

free parameters (4)

a (logarithmic slope in Eq. 2) = not reported
Fit to each MSD curve in the alleged marginal phase; no reported uncertainty. Descriptive, but central to identifying logarithmic growth.
x* (logarithmic offset) = not reported
Together with a, sets the logarithmic fit; no uncertainties reported.
tau* (crossover lag time) = not reported
Defined as the time when the MSD deviates by more than 10% from the Clerx-Schram free-colloid expression; the threshold is chosen by hand.
10% deviation threshold = 10%
Ad hoc criterion for defining tau*; no sensitivity analysis is provided.

assumptions (5)

domain assumption Replica mean-field theory's mapping: stable glass gives a flat MSD plateau, while marginal/Gardner glass gives logarithmic MSD growth.
Central interpretive key; taken from refs 5-8 and prior simulations, not independently tested here.
domain assumption Sedimentation time maps monotonically to local packing fraction around the tracer, following the same path as the equilibrium phase diagram at fixed temperature.
The phase sequence is read against elapsed time; no absolute or local density measurement is reported.
domain assumption A single tracer particle's drift-subtracted MSD is representative of the bulk glass phase.
No ensemble averaging over many tracers or independent samples is shown in the main text; supplemental runs are only mentioned.
domain assumption Localization errors are independent and identically distributed and can be subtracted to resolve MSD below the nominal noise floor.
Needed because reported MSD values are a few nanometers while the per-frame localization error is about 1.5 nm; relies on prior work (ref 32).
domain assumption Soft-sphere thermal colloids can access densities above jamming, producing a reentrant stable glass.
From theory (refs 6 and 14), used to explain the final plateau.

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Cite this review

Pith. "Pith review of Experimental observation of the marginal glass phase in a colloidal glass." pith.science (2026). https://pith.science/paper/6O3UEJX4

@misc{pith2026190808152,
  author       = {Pith},
  title        = {Pith review of: Experimental observation of the marginal glass phase in a colloidal glass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6O3UEJX4}},
  note         = {Machine review of arXiv:1908.08152}
}

read the original abstract

The replica theory of glasses predicts that in the infinite dimensional mean field limit there exist two distinct glassy phases of matter: stable glass and marginal glass. We have developed a technique to experimentally probe these phases of matter using a colloidal glass. We avoid the difficulties inherent in measuring the long time behavior of glasses by instead focusing on the very short time dynamics of the ballistic to caged transition. We track a single tracer particle within a slowly densifying glass and measure the resulting mean squared displacement (MSD). By analyzing the MSD we find that upon densification our colloidal system moves through several states of matter. At lowest densities it is a sub-diffusive liquid. Next it behaves as a stable glass, marked by the appearance of a plateau in the MSD whose magnitude shrinks with increasing density. However, this shrinking plateau does not shrink to zero, instead at higher densities the system behaves as a marginal glass, marked by logarithmic growth in the MSD towards that previous plateau value. Finally, at the highest experimental densities the system returns to the stable glass phase. This provides direct experimental evidence for the existence of a marginal glass in 3d.

Figures

Figures reproduced from arXiv: 1908.08152 by the authors.

**Figure 2.** Global change in density relative to the initially shaken sample versus time for a representative colloidal system. Even at the longest times studied the system is still slowly densifying. Inset: Image of the colloidal glass during sedimentation. The interstitial fluid has been index matched to the PMMA beads so they are nearly invisible. The black tracer particles are thus revealed. signature similar to that employ… view at source ↗

**Figure 4.** MSDs from a sedimenting tracer-colloid plotted on a log-log scale. The MSDs were taken every 20 minutes. As a guide to the eye, we show ballistic and diffusive asymptotes demonstrating that at short times the motion is near ballistic and at long times the curves are all sub-diffusive. There are four important MSD regimes with a representative highlighted: from shaken to 160 minutes the MSD is sub-diffusive (green, 1… view at source ↗

**Figure 5.** Representative MSD curves from the four time steps. Superimposed on each MSD is the best fit logarithm. A: MSD in the sub-diffusive regime, taken at 100 minutes with a logarithmic fit. B: MSD in plateau regime, 260 minutes. C: MSD in logarithmic regime, 400 minutes. D: MSD in reentrant plateau regime, 580 minutes. then beyond this lag time it should be well fit by MSD(τ ) = a × log τ τ ∗ + x ∗ [2] where a charac… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Jamming Energy Landscape is Hierarchical and Ultrametric
cond-mat.soft 2019-08 conditional novelty 6.0 of 10

Nearby energy minima of 3D jammed soft-sphere packings are hierarchically clustered and approach an ultrametric metric in the thermodynamic limit, supporting Gardner physics in athermal jamming.

Reference graph

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Methods Our experimental system consists of nominally 50µm diame- ter PMMA colloids (Cospheric PMPMS-1.2 45-53um> 95%) with 95% falling within a range of 45-53µm, sphericity greater than 99%, and density 1.20 g/cm3. The degree of polydisper- sity is chosen to frustrate crystallization. These colloids are suspended in a mixture of tetrahydronapthalene, dec...
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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

The degree of polydisper- sity is chosen to frustrate crystallization

Methods Our experimental system consists of nominally 50µm diame- ter PMMA colloids (Cospheric PMPMS-1.2 45-53um> 95%) with 95% falling within a range of 45-53µm, sphericity greater than 99%, and density 1.20 g/cm3. The degree of polydisper- sity is chosen to frustrate crystallization. These colloids are suspended in a mixture of tetrahydronapthalene, dec...

[2] [2]

Results and Discussion As shown in ﬁgure 4, we ﬁnd a substantial change of behav- ior with increasing time and thus increasing density. The MSD of the tracer-colloid just after it has been shaken (green curves) shows ballistic motion at short times turning over to a subdiﬀusive motion at long times, consistent with that of a colloid in a dense liquid susp...

work page

[3] [3]

We track an indi- vidual colloid deep within the glass at short times and ﬁnd the associated MSD

Conclusions We have created a colloidal glass at ﬁxed temperature and tracked it through the liquid, stable glass, and marginal glass phases as a function of increasing density. We track an indi- vidual colloid deep within the glass at short times and ﬁnd the associated MSD. This experiment lays the groundwork for a complete experimental determination of ...

[4] [4]

Nature Communications 5:3725

Charbonneau P , Kurchan J, Parisi G, Urbani P , Zamponi F (2014) Fractal free energy land- scapes in structural glasses. Nature Communications 5:3725

work page 2014

[5] [5]

It is striking that not only do we see individual signatures of each phase but that they combine to begin to map out a phase diagram which agrees with the theoretical predictions

work page

[6] [6]

Nuclear Physics B 257:747–765

Gardner E (1985) Spin glasses with p-spin interactions. Nuclear Physics B 257:747–765

work page 1985

[7] [7]

The Journal of Chemical Physics 131(2):024504

Brito C, Wyart M (2009) Geometric interpretation of previtriﬁcation in hard sphere liquids. The Journal of Chemical Physics 131(2):024504

work page 2009

[8] [8]

Parisi G, Zamponi F (2010) Mean-ﬁeld theory of hard sphere glasses and jamming. Rev. Mod. Phys. 82(1):789–845

work page 2010