Growth law, temperature dependence, and geometry of the dynamical correlation length in aging structural glasses

When a glass-forming liquid is quenched below its glass transition, its relaxation slows down progressively with the time elapsed since the quench (aging), and the regions over which particles move in a correlated way grow. Parisi's simulations of an aging binary soft-sphere glass [J. Phys. Chem. B 103, 4128 (1999)] showed that the dynamical correlation length grows with waiting time, consistent with $R(t_w) \sim t_w^{1/6}$, while the number of correlated particles grows as $I(t_w) \sim t_w^{1/2}$, and asked what happens at lower temperatures and deeper in the energy-barrier-dominated (activated) regime, and whether the correlated (fluidized) domains are compact or fractal. Later work has confirmed the $t_w^{1/6}$ growth near the glass transition in a second glass model, found domains close to compact (fractal dimension about 2.9, rising toward 3 at maximal correlation), and found a logarithmically growing static length coexisting with the power-law dynamical length; mode-coupling theory for aging glasses predicts temperature-independent exponents. The surviving open core is the behavior of the aging dynamical correlation length well inside the activated regime and at substantially lower temperatures — persistent power law versus crossover to slower growth or saturation — together with a conclusive determination of the compact-versus-fractal geometry of the correlated domains. As context, indirect, ensemble-averaged evidence of growing cooperative relaxation during aging exists in aging clay suspensions and superspin magnets, but no direct experimental observation of the growing aging length in a structural glass has been made.

Background

When a liquid is cooled rapidly enough that it can neither crystallize nor equilibrate, it falls out of equilibrium at the glass transition temperature $T_g$ and becomes a glass. Below $T_g$ the material ages: its properties keep evolving with the time elapsed since preparation, so that a measurement depends on the waiting time $t_w$ (the time between quenching the sample and probing it) in addition to temperature and density. Aging was first characterized numerically in a Lennard-Jones glass (a standard model pair potential) by Kob and Barrat (1997), who showed that the relaxation time grows with $t_w$ and that the fluctuation-dissipation theorem is violated during aging.

Relaxation in viscous liquids is spatially heterogeneous: particles do not relax independently but in transient groups. This dynamical heterogeneity is quantified by four-point density correlation functions, which measure how spatially correlated the local change of the density is between two times; their spatial integral, the four-point susceptibility $\chi_4(t)$, counts the number of particles that relax in a correlated way (Glotzer, Novikov and Schröder 2000; Lačević et al 2003). A growth of $\chi_4$ on cooling implies a growing dynamical correlation length, and multipoint-susceptibility measurements gave the first direct experimental evidence for such a growing length near $T_g$ in molecular and colloidal glass formers (Berthier et al 2005). Growing lengths, static and dynamic, are the central quantitative pillar of cooperative-relaxation pictures of the glass transition, such as the Adam-Gibbs scenario (rearranging regions grow as the configurational entropy per particle decreases) and the random first-order transition theory (reviewed by Berthier and Biroli 2011; Karmakar, Dasgupta and Sastry 2016).

Parisi (1999) measured the growing dynamical length during aging, before the four-point formalism was standard, using an overlap-based construction. The overlap between two configurations separated by a time interval $t$ after a waiting time $t_w$ is $q(t_w,t) = N^{-1}\sum_i w(|\vec r_i(t_w+t)-\vec r_i(t_w)|)$, where $w(x)$ is a smooth window function equal to 1 for $x \ll a$ and 0 for $x \gg a$, with $a$ a fraction of a particle diameter. Each particle carries a local overlap $q_i$; particles that have lost their cage and rearranged have low $q_i$ and define fluidized regions. Writing $\sigma_i = 2(q_i - q)$, the spatial correlation $f(r,t_w) = \langle\sigma_i\sigma_j\rangle_{|x_i-x_j|=r}$ and its envelope decay define the dynamical correlation length $R(t_w)$, while its integral $I(t_w) = \int d^3x\, f(x,t_w)$ is an integrated susceptibility, roughly the number of particles that move in a correlated way. If the fluidized regions are compact three-dimensional domains of radius $R$, then $I \propto R^3$; if they are fractal, with fractal dimension $d_f < 3$, then $I \propto R^{d_f}$ grows more slowly than $R^3$. A closely related earlier observation is that equilibrium supercooled liquids relax via string-like cooperative groups of particles (Donati et al 1998).

The simulated system was a 50:50 binary mixture of soft spheres (diameter ratio 1.2, repulsive $|x|^{-12}$ potential with a cutoff) of $N = 27000$ particles, studied by Monte Carlo dynamics after a quench from high temperature. Its thermodynamic control parameter is $\Gamma = \beta^{1/4}$ ($\beta$ the inverse temperature in reduced units); the mode-coupling-like crossover (the point below which the relaxation time appears to diverge as a power law in idealized mode-coupling theory) occurs at $\Gamma_c \approx 1.45$, and the simulations were performed at $\Gamma = 1.8$. Mean-field-inspired theories predict that below the crossover the approach to equilibrium has two regimes: an early regime dominated by entropic barriers, where the slowing down is controlled by the configurational entropy (the number of available rearrangement pathways) rather than by energetic cost, and a later regime dominated by energy barriers, whose crossing makes the dynamics activated and strongly temperature-dependent (Coluzzi and Parisi 1998). Parisi (1997) had earlier studied short-time aging in the same soft-sphere model.

In the entropy-dominated regime Parisi (1999) found that $R(t_w)$ is well approximated by a power of $t_w$ with exponent about 0.16, and $I(t_w)$ by a power with exponent about 0.52. The rearrangement time of a region of size $R$ diverges approximately as $R^6$, so with the aging identification of the current relaxation time with the waiting time, $\tau(t_w) \simeq t_w$, the measured exponents are consistent with the exact values speculated in the source, $R \propto t_w^{1/6}$ and, for compact domains, $I \propto t_w^{1/2}$. The measurement was restricted to a single state point $\Gamma = 1.8$, and at the longest simulated times the energy relaxation was entering the barrier-dominated regime.

What remains open is the behavior of this aging dynamical correlation length outside the validated window: at lower temperatures (higher $\Gamma$), deeper in the barrier-dominated regime; and the compact-versus-fractal geometry of the correlated domains. As context for possible direct tests, relevant experimental settings include multipoint susceptibilities of molecular glass formers measured during aging, dielectric-noise measurements on aging polymer glasses, and direct confocal microscopy of colloidal, microgel, and clay glasses, where particle-resolved dynamics are accessible.

Problem Statement

Determine the growth of the dynamical correlation length during the aging of a structural glass, outside the single state point and time window in which it was first measured. The source (Parisi 1999) explicitly asks for extending the studies to longer times and larger systems to see whether the behavior of $R(t_w)$ changes when the energy relaxation enters the barrier-dominated regime, and its own scaling construction — the expectation $I \propto R^3$ for compact fluidized domains, the measured $I$ exponent, and the explicitly conjectural exact exponents $1/6$ and $1/2$ — leaves open the geometry of the correlated domains; these two linked issues, temperature/regime dependence and domain geometry, are the parts of the target here. Later literature has partially settled the geometry question, confirmed the near-$T_g$ growth, and produced indirect experimental evidence of growing cooperative relaxation during aging in clay and superspin systems.

(1) Temperature and regime dependence. Does the power-law growth $R(t_w) \sim t_w^{1/6}$ (with $I(t_w) \sim t_w^{1/2}$) found in the entropy-dominated regime persist at lower temperatures (higher $\Gamma$), or do the exponents change? In particular, deep in the energy-barrier-dominated (activated) regime, where the relaxation time is expected to be controlled by barriers that grow with domain size, does $R(t_w)$ follow a different scaling — for example the logarithmic-in-$t_w$ growth found for static amorphous-order lengths — or does it saturate?

(2) Geometry. Are the fluidized (correlated-rearrangement) domains compact, $I \propto R^3$, or fractal, $I \propto R^{d_f}$ with $d_f < 3$? Does their geometry change with age or temperature? The fractal dimension and its waiting-time and temperature dependence have not been fixed; the already-reported value $d_f \approx 2.9$ is not conclusive.

The common scientific objective is a single characterization of the aging dynamical length — its growth law and its regime and temperature dependence, together with the geometry of the correlated domains; the two issues are the natural parts of that objective, not independent problems.

References

1. G. Parisi, "An Increasing Correlation Length in Off-Equilibrium Glasses", J. Phys. Chem. B 103, 4128-4131 (1999), DOI 10.1021/jp983967m, https://doi.org/10.1021/jp983967m
2. G. Parisi, "A divergent correlation length in off-equilibrium glasses", arXiv:cond-mat/9801034 (1998), https://arxiv.org/abs/cond-mat/9801034
3. G. Parisi, "Short-time aging in binary glasses", J. Phys. A: Math. Gen. 30, L765 (1997), DOI 10.1088/0305-4470/30/22/004, https://doi.org/10.1088/0305-4470/30/22/004
4. W. Kob and J.-L. Barrat, "Aging Effects in a Lennard-Jones Glass", Phys. Rev. Lett. 78, 4581-4584 (1997), DOI 10.1103/PhysRevLett.78.4581, https://doi.org/10.1103/PhysRevLett.78.4581
5. B. Coluzzi and G. Parisi, "On the approach to the equilibrium and the equilibrium properties of a glass-forming model", J. Phys. A: Math. Gen. 31 (1998), DOI 10.1088/0305-4470/31/19/004, https://doi.org/10.1088/0305-4470/31/19/004
6. C. Donati, J. F. Douglas, W. Kob, S. J. Plimpton, P. H. Poole, S. C. Glotzer, "Stringlike Cooperative Motion in a Supercooled Liquid", Phys. Rev. Lett. 80, 2338 (1998), DOI 10.1103/PhysRevLett.80.2338, https://doi.org/10.1103/PhysRevLett.80.2338
7. S. C. Glotzer, V. N. Novikov, T. B. Schrøder, "Time-dependent, four-point density correlation function description of dynamical heterogeneity and decoupling in supercooled liquids", J. Chem. Phys. 112, 509-512 (2000), DOI 10.1063/1.480541, https://doi.org/10.1063/1.480541
8. N. Lačević, F. W. Starr, T. B. Schrøder, S. C. Glotzer, "Spatially heterogeneous dynamics investigated via a time-dependent four-point density correlation function", J. Chem. Phys. 119, 7372 (2003), DOI 10.1063/1.1605094, https://doi.org/10.1063/1.1605094
9. E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, "Off-equilibrium dynamics at very low temperatures in three-dimensional spin glasses", J. Phys. A: Math. Gen. 33, 2373-2382 (2000), DOI 10.1088/0305-4470/33/12/305, https://doi.org/10.1088/0305-4470/33/12/305
10. E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, D. A. Weitz, "Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition", Science 287, 627 (2000), DOI 10.1126/science.287.5453.627, https://doi.org/10.1126/science.287.5453.627
11. R. E. Courtland and E. R. Weeks, "Direct visualization of ageing in colloidal glasses", J. Phys.: Condens. Matter 15, S359 (2003), DOI 10.1088/0953-8984/15/1/349, https://doi.org/10.1088/0953-8984/15/1/349
12. L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, D. El Masri, D. L'Hôte, F. Ladieu, M. Pierno, "Direct Experimental Evidence of a Growing Length Scale Accompanying the Glass Transition", Science 310, 1797 (2005), DOI 10.1126/science.1120714, https://doi.org/10.1126/science.1120714
13. G. C. Cianci, R. E. Courtland, E. R. Weeks, "Correlations of structure and dynamics in an aging colloidal glass", Solid State Commun. 139, 599-604 (2006), DOI 10.1016/j.ssc.2006.04.039, https://doi.org/10.1016/j.ssc.2006.04.039
14. L. Berthier, G. Biroli, J.-P. Bouchaud, W. Kob, K. Miyazaki, D. R. Reichman, "Spontaneous and induced dynamic fluctuations in glass formers. I. General results and dependence on ensemble and dynamics", J. Chem. Phys. 126, 184503 (2007), DOI 10.1063/1.2721554, https://doi.org/10.1063/1.2721554
15. H. E. Castillo and A. Parsaeian, "Local fluctuations in the ageing of a simple structural glass", Nat. Phys. 3, 26-28 (2007), DOI 10.1038/nphys482, https://doi.org/10.1038/nphys482
16. L. D. C. Jaubert, C. Chamon, L. F. Cugliandolo, M. Picco, "Growing dynamical length, scaling and heterogeneities in the 3d Edwards-Anderson model", J. Stat. Mech. P05001 (2007), DOI 10.1088/1742-5468/2007/05/P05001, arXiv:cond-mat/0701116, https://arxiv.org/abs/cond-mat/0701116
17. E. R. Weeks, J. C. Crocker, D. A. Weitz, "Short- and long-range correlated motion observed in colloidal glasses and liquids", J. Phys.: Condens. Matter 19, 205131 (2007), DOI 10.1088/0953-8984/19/20/205131, https://doi.org/10.1088/0953-8984/19/20/205131
18. A. Parsaeian and H. E. Castillo, "Growth of spatial correlations in the aging of a simple structural glass", Phys. Rev. E 78, 060105(R) (2008), DOI 10.1103/PhysRevE.78.060105, https://doi.org/10.1103/PhysRevE.78.060105
19. H. Oukris and N. E. Israeloff, "Nanoscale non-equilibrium dynamics and the fluctuation-dissipation relation in an ageing polymer glass", Nat. Phys. 6, 135-138 (2010), DOI 10.1038/nphys1482, https://doi.org/10.1038/nphys1482
20. E. Wandersman, V. Dupuis, E. Dubois, R. Perzynski, S. Nakamae, E. Vincent, "Growth of a dynamical correlation length in an aging superspin glass", Europhys. Lett. 84, 37011 (2008), DOI 10.1209/0295-5075/84/37011, https://doi.org/10.1209/0295-5075/84/37011
21. S. Jabbari-Farouji, R. Zargar, G. H. Wegdam, D. Bonn, "Dynamical heterogeneity in aging colloidal glasses of Laponite", Soft Matter 8, 5507 (2012), DOI 10.1039/c2sm25171j, https://doi.org/10.1039/c2sm25171j
22. S. Nakamae, C. Crauste-Thibierge, D. L'Hôte, E. Vincent, E. Dubois, V. Dupuis, R. Perzynski, "Dynamic correlation length growth in superspin glass: Bridging experiments and simulations", Appl. Phys. Lett. 101, 242409 (2012), DOI 10.1063/1.4769840, https://doi.org/10.1063/1.4769840
23. P. Gadige, D. Saha, S. K. Behera, R. Bandyopadhyay, "Study of dynamical heterogeneities in colloidal nanoclay suspensions approaching dynamical arrest", Sci. Rep. 7, 8017 (2017), DOI 10.1038/s41598-017-08495-9, https://doi.org/10.1038/s41598-017-08495-9
24. L. Berthier and G. Biroli, "Theoretical perspective on the glass transition and amorphous materials", Rev. Mod. Phys. 83, 587 (2011), DOI 10.1103/RevModPhys.83.587, https://doi.org/10.1103/RevModPhys.83.587
25. S. K. Nandi and S. Ramaswamy, "How Do Glassy Domains Grow?", Phys. Rev. Lett. 109, 115702 (2012), DOI 10.1103/PhysRevLett.109.115702, https://doi.org/10.1103/PhysRevLett.109.115702
26. A. Smessaert and J. Rottler, "Distribution of local relaxation events in an aging three-dimensional glass: Spatiotemporal correlation and dynamical heterogeneity", Phys. Rev. E 88, 022314 (2013), DOI 10.1103/PhysRevE.88.022314, https://doi.org/10.1103/PhysRevE.88.022314
27. S. Karmakar, C. Dasgupta, S. Sastry, "Length scales in glass-forming liquids and related systems: a review", Rep. Prog. Phys. 79, 016601 (2016), DOI 10.1088/0034-4885/79/1/016601, https://doi.org/10.1088/0034-4885/79/1/016601
28. K. Vollmayr-Lee, C. H. Gorman, H. E. Castillo, "Universal scaling in the aging of the strong glass former SiO2", J. Chem. Phys. 144, 234510 (2016), DOI 10.1063/1.4953911, https://doi.org/10.1063/1.4953911
29. S. Nath and S. Karmakar, "Growth of Dynamic and Static Correlations in the Aging Dynamics of a Glass-Forming Liquid", arXiv:2512.17168 (2025), https://arxiv.org/abs/2512.17168