Calorimetric glass transition explained by hierarchical dynamic facilitation

Abstract

The glass transition refers to the nonequilibrium process by which an equilibrium liquid is transformed to a nonequilibrium disordered solid, or vice versa. Associated response functions, such as heat capacities, are markedly different on cooling than on heating, and the response to melting a glass depends markedly on the cooling protocol by which the glass was formed. This paper shows how this irreversible behavior can be interpreted quantitatively in terms of an East-model picture of localized excitations (or soft spots) in which molecules can move with a specific direction, and from which excitations with the same directionality of motion can appear or disappear in adjacent regions. As a result of these facilitated dynamics, excitations become correlated in a hierarchical fashion. These correlations are manifested in the dynamic heterogeneity of the supercooled liquid phase. Although equilibrium thermodynamics is virtually featureless, a nonequilibrium glass phase emerges when the model is driven out of equilibrium with a finite cooling rate. The correlation length of this emergent phase is large and increases with decreasing cooling rate. A spatially and temporally resolved fictive temperature encodes memory of its preparation. Parameters characterizing the model can be determined from reversible transport data, and with these parameters, predictions of the model agree well with irreversible differential scanning calorimetry.

When a supercooled liquid is cooled significantly below its glass transition temperature, the system freezes into an amorphous vitrified state (1, 2). This process encodes a unique nonequilibrium structural signature within the glassy material, a signature that is not apparent in equilibrium correlation functions but is evident from thermodynamic properties and response functions, such as the enthalpy, specific volume, heat capacity, or thermal expansivity. The complex temperature variation of these properties for systems heated from the vitrified state reveals that glassy materials bear a distinct memory of the protocol by which they were created. Here, we show how this irreversible behavior can be understood in terms of the nonequilibrium behavior of a model we have used in the past to treat reversible behaviors of glass-forming liquids (3–6). In so doing, we uncover an underlying dynamical transition between equilibrium melts with no trivial spatial correlations and nonequilibrium glasses with correlation lengths that are both large and dependent on the rate at which the glass is prepared.

The class of experimental protocols we consider is illustrated in Fig. 1A. The temperature T is a function of time with a rate of temperature change, ν = ΔT/Δt, that is negative for cooling (ν = ν_c < 0) and positive for heating (ν = ν_h > 0). The structural relaxation time of the liquid, τ, increases as the temperature decreases. At a sufficiently low temperature, |dτ/dT| exceeds |1/ν_c|. At this stage, the liquid begins to fall out of equilibrium, forming a glass. The process is continuous but precipitous. A glass transition temperature, T_g, is defined in various ways but always such that it is a temperature in this precipitous region. Heat capacities measured by differential scanning calorimetry (DSC) (7, 8) and corresponding enthalpies are typical experimental indicators of the transition. Their behaviors are illustrated schematically in Fig. 1 B and C.

Fig. 1.

Schematics of a standard temperature cycle (A) and corresponding thermodynamic quantities (B and C). For theoretical analysis, a continuously changing temperature T is treated as a discrete sequence, k = 1, 2, …, with each step having a time duration Δt. The working temperature at the (k + 1)th step differs from that at the kth step by an amount ΔT. The rates of cooling and heating, ν ≡ ΔT/Δt, are denoted ν_c and ν_h, respectively. Enthalpy H (B) and the corresponding heat capacity C_p (C) for a system that is cooled and then heated through the glass transition (blue and red) are shown. H^eq and H^glass refer to the enthalpy of the liquid and glass, respectively. The irreversible glass transition occurs over the range of temperatures where the blue and red curves differ. A standard definition of fictive temperature, T_f, is given in terms of its temperature derivative in that range: , where . As the system tends to equilibrium, T_f → T.

A most important feature illustrated in the figure is time asymmetry. Cooling produces a monotonic decrease in heat capacity, whereas heating produces a nonmonotonic and anomalous response, the size of which (we discuss below) depends on the rate at which the glass was produced by cooling. Hysteresis between equilibrium phases lacks this asymmetry. We do not focus on the end points of those curves: the heat capacities of the glass and liquid phases, and , respectively. The difference, , one may argue (9), is decoupled from dynamics and understood in terms of time-independent elastic responses of the two phases (9, 10). In contrast, the irreversible transitional behaviors found on cooling and heating can only be understood in terms of dynamics. It is these irreversible behaviors we consider here.

To do so, we generalize the concept of fictive temperature (11, 12) to a spatially resolved field, . To define this quantity and describe its utility, note that a time-dependent working temperature T(t) (i.e., the temperature of a surrounding bath) equilibrates quickly throughout a glass or glass-forming material on a time scale that is effectively instantaneous in comparison to structural equilibration times. Further, structural relaxation occurs locally and intermittently. A short time after a local region surrounding position happens to reorganize, say, at time t′, the configurational contribution to a local thermodynamic property, such as an enthalpy density , will be locked at a value, h_eq(T(t′)), typical of the equilibrium distribution of values for temperature T(t′), and it will change only after the region reorganizes again. Therefore, we define to be the working temperature of the system during the most recent structural reorganization of the region surrounding . As such, extensive thermodynamic properties, such as configurational enthalpy H, can be written as

The net enthalpy is this configurational part plus a regular contribution from molecular vibrations. The latter is simply characterized by the working temperature.

Tool’s original formulation (11, 12) is similar, but without considering spatial variation. In reality, different domains within the system lose and regain equilibrium at different temperatures. Thus, the nonequilibrium distribution of particle arrangements is never representative of an equilibrium configuration at any temperature (12–14). Phenomenological improvements to Tool’s original approach (8), including the Tool–Narayanaswami–Moynihan (TNM) model (7, 15), introduce equations of motion for the fictive temperature, but still in mean field. These models require extensive parameterization to correct for neglected microscopic details. Improvements (e.g., refs. 16–19) offer phenomenological approaches that go beyond mean field, but still with many parameters and without a concrete microscopic model for dynamics.

In contrast, we consider a spatially resolved fictive temperature field that is coupled to underlying dynamics exhibiting fluctuations on all but the smallest trivial scale. Specifically, we use “East-like” models. The original East model (3) is one of many kinetically constrained models (KCMs), which are models that have been invented to study glassy dynamics in idealized contexts (20, 21). The East model, in particular, treats a 1D lattice with excitations distributed at random on the lattice stretching from from right (“east”) to left (“west”). The dynamics of the model are nontrivial because the birth and death of an excitation require the presence of another excitation immediately to its east. At low enough excitation concentrations, this directional facilitation leads to hierarchical dynamics (22), where relaxation over a domain of length ℓ requires an energy J_ℓ that grows logarithmically with ℓ (23). It thus encodes dynamical behavior as posited by Palmer et al. (22).

Generalizations of the East model to higher dimensions exhibit similar scaling. Most importantly, the behaviors of these East-like models (directional facilitation and concomitant logarithmic scaling) have been shown to emerge from molecular dynamics of atomistic models of glass-forming liquids (24), and this scaling has also been shown to collapse the seemingly disparate results of thousands of measurements of reversible structural relaxation in supercooled melts (25). The correspondence between behaviors of real materials and East-like behavior follows from the two essential features of glass-forming liquids. First, the possibility of local reorganization is coincident with a local soft spot, and small movements within such a region can lead to a softening of an adjacent region (24); this feature is the origin of facilitation. Second, the direction of a displacement or movement is preserved by the movements in the newly softened region (24); this feature is the origin of directionality or string-like motions (26). East-like models contain both features and essentially nothing else.

East Model Illustrates Fictive Temperature, Domain Size, and Glassy Memory

To illustrate how East-like models work, we consider first the original East model with N lattice sites, i = 1, 2, …, N, each of which can be in one of two states, n_i = 0 or 1. The latter represents an excited site. Site i can excite or de-excite, provided the adjacent neighbor to the east is excited. This dynamics are encoded in the transition rates for that site, k_i,0→1 = n_i−1c/(1 − c) and k_i,1→0 = n_i−1, where c ≡ 〈n_i〉 and c/(1 − c) = exp(−1/T). We use units where excitation energy over Boltzmann’s constant is unity and the time of a single integration step is unity. The spatial variable, , is the dimensionless lattice-site label i.

The transition rates obey detailed balance; thus, the model relaxes to its equilibrium state. In this state, excitations are distributed at random, where P(ℓ) = c exp(−cℓ) is the probability that a tagged excitation is separated from its nearest excited neighbor by ℓ. Integration of the stochastic dynamics for this model can be done in various ways (27). For the results shown in this section, it is carried out with a continuous time algorithm (28) starting from an equilibrated high-temperature state. Each cooling or heating step in a cycle depicted in Fig. 1A is propagated for many integration time steps up to the time Δt. The next cooling or heating step proceeds from the configuration obtained at the end of the previous step but with the temperature shifted by the amount νΔt.

Whereas the equilibrium behavior is unstructured, the dynamics of the model become notably heterogeneous below the onset temperature, T_o ≈ 1. In that low-temperature regime, the structural relaxation time is non-Arrhenius, obeying the parabolic law (25)

where J² = 1/2 ln 2 (23, 29) and τ_mf is the structural relaxation time in mean field, which has Arrhenius temperature dependence and is an accurate approximation for τ when T ≳ T_o. The universal super-Arrhenius behavior (Eq. 2) is found in real systems for T < T_o, but with different J and different τ_mf. [In most glass-forming liquids, the temperature variation of τ_mf is negligible (25).]

The universal behaviors of fluctuations or dynamic heterogeneities for T < T_o can be quantified with a persistence field (30), . Here, δ[X] stands for Kronecker’s delta, which is 1 when X = 0, and 0 otherwise, and κ_i(t) = |n_i(t) − n_i(t − 1)| indicates whether a change in state (or kink) has occurred at site i and time t. Thus, the persistence field at site i, p_i(t, t′), is 1 if and only if there has been no dynamical activity at site i over the time frame between t′ and t. This quantity provides a recursive relationship for the fictive temperature field, T_f,i(t). Specifically, with working temperature T(t),

When the East model is driven out of equilibrium with a temperature that changes in steps of time of duration Δt, the three fields together, n_i(t), p_i(t, t − Δt), and T_f,i(t), elucidate the resulting dynamics.

Fig. 2 depicts these fields for a system cooled and reheated through T_g. At the beginning and end of the trajectories, excitations initially assume a gas-like distribution (Fig. 3A). In that regime, trajectories show space-time bubbles characteristic of reversible dynamics of a glass former (4, 6). In the high-temperature regime, the system is at equilibrium and the distribution of interexcitation distances is exponential, with ℓ = 0 most probable. As the temperature is lowered, excitations annihilate to adjust to the lower equilibrium concentration and the average domain size grows, but ℓ = 0 is still the most probable. Because the activation energy to equilibrate a domain, J_ℓ, increases with ℓ, relaxation occurs more readily for excitations separated by smaller distances than by larger distances, a relationship we quantify in the following section.

Fig. 2.

Cooling and heating cycle for the East model and the emergence of a nonequilibrium striped phase. For this illustrated trajectory, the cooling and heating rates are |ν_c| = |ν_h| = 10⁻⁷, which is the value of |dτ/dT|⁻¹ for the East model at T ≈ 0.37. (A) Evolution of excitations as a function of temperature. Sites are colored according to the external temperature, T(t). On heating, the system retains memory of the cooling protocol, manifested by a striped configuration containing similarly sized domains. (B) Corresponding persistence field. Sites that persist during the kth step are colored white, whereas sites that relax are colored black. (C) Fictive temperature field derived from the persistence field. The color scheme indicates the value of the instantaneous fictive temperature at each lattice site.

Fig. 3.

Size distributions of glassy domains in the East model after cooling with frequency ν_c to the temperature T. Statistics are obtained by averaging over many trajectories, like the one illustrated in Fig.1. (A) Probability distribution of domain sizes P(ℓ) for excitations in the East model for a cooling rate of ν_c = −10⁻⁵. Above the kinetic glass transition temperature T_g ≈ 0.48, excitations obey ideal gas statistics with P(ℓ) ∝ exp(−cℓ). Below T_g, excitations develop pair correlations with the most probable domain size occurring near ℓ = 10. As the cooling rate slows, this characteristic domain size grows. (B) P(ℓ) for glasses formed by the East model at different cooling rates, all quenched to T = 0. The curves collapse when scaled by ℓ_eq(T_f), the equilibrium length corresponding to the spatially averaged fictive temperature of the glass. The black dashed line shows the variation with ℓ for the equilibrium distribution, P_eq(ℓ) ∝ exp(−ℓ/ℓ_eq).

At low temperatures, when the system begins to fall out of equilibrium, it does so over a series of stages during which ever smaller domains freeze into place. Due to the hierarchical nature of the dynamics, therefore, smaller domains acquire lower fictive temperatures. The result is seen in the complex spatial pattern for the fictive temperature field T_f,i illustrated in Fig. 2C. The frozen phase that eventually forms, corresponding to glass, is a striped phase in space-time. It contains an excess of large domains relative to that of the equilibrium phase, corresponding to melt or liquid. This result is quantified with the distributions shown in Fig. 3. After the formation of the glass (i.e., below T_g), the distribution peaks at a characteristic value of ℓ > 0. This length is the typical size of the smallest domains that cannot relax during the cooling process. It increases as the cooling rate decreases. Fig. 3B shows that the distributions collapse when scaled by the equilibrium domain length, (ℓ_eq(T) = exp(1/T), evaluated at the spatially averaged fictive temperature, .

Thus, there is a structural distinction between the glass and its melt. The former has a nontrivial static length that results from an irreversible dynamical process. The latter has no nontrivial static length. The frozen system and its equilibrium counterpart can have the same excitation concentration, c, but with different spatial distributions of excitations. An instantaneous quench to T = 0 will not produce this distinct structural behavior of the nonequilibrium system. A finite cooling rate or physical aging (31, 32) below T_g is required to produce this transition to a glass. An instantaneous quench of an equilibrium system will remove surging fluctuations surrounding anchored excitations but will otherwise leave the spatial distribution of excitations identical to that of the equilibrium system. The quenched system will thus melt as soon as temperature is returned to any finite value, equilibrating within a reversible relaxation time at that temperature. Precursors to the glass phase with finite correlation lengths emerge out of equilibrium after a quench to a low but finite temperature (31, 32).

Lowering the cooling rate produces larger glassy domain sizes; thus, lowering the cooling rate produces more stable glass. It is another manifestation of hierarchical dynamics (i.e., J_ℓ growing with increasing ℓ). An excitation can spawn new excitations, building lines of excitations typically as long as ℓ_eq(T). Longer lines will be exponentially rare. Thus, for T < T_g, excitations are isolated and cannot relax because ℓ_eq(T) ≪ ℓ_eq(T_f). As the system is heated from its frozen state, minor surges of activity can occur on short-length scales, which cause the fictive temperature to increase slightly. However, the glassy domains persist until, at warm enough temperatures, the system finally acquires sufficient energy to relax the domains of characteristic size. At this point, equilibration occurs rapidly. We will see shortly that this behavior is the essence of the asymmetrical responses characteristic of the calorimetric glass transition.

Hysteresis in response to a time-varying field is a general property of any system with dynamical bottlenecks or separations in time scales. Thus, it will be a property of all KCMs (e.g., refs. 33, 34). However, the time asymmetry or encoded memory of preparation discussed here is a property of hierarchical dynamics (35), as explained in the previous paragraphs. Domains in KCMs that have diffusive (i.e., nonhierarchical) dynamics, such as the one-spin facilitated Fredrickson–Andersen model (20) and the Backgammon model (21), tend to evolve toward equilibrium on heating, much like standard unconstrained systems [e.g., the Ising model (36)]. In contrast, hierarchical KCMs, such as the East model, must populate lower energy levels before relaxing larger domains. Further discussion and comparison between prior approaches and models are provided in SI Text.

Comparison with DSC Experiments

DSC determines a nonequilibrium heat capacity, or more precisely, the rate of change of enthalpy as a function of time. It is linked to the fictive temperature through Eq. 1. To a good approximation, the equilibrium density of configurational enthalpy is a linear function of temperature over the relevant range of temperatures. With that approximation, the nonequilibrium configurational heat capacity is , that is,

where ΔC_p is the difference between liquid and glass heat capacities outside the glass-transition region and T refers to the working temperature at the time when the nonequilibrium heat capacity is measured. The spatially averaged fictive temperature, T_f, is a functional of T(t) encoding preparation history. Its working-temperature derivative in Eq. 4 is evaluated at the measurement time. [Expressions more complicated than Eq. 4 apply when h_eq(T) is a nonlinear function of T and when the working temperature is a nonlinear function of t.]

The original East model requires generalization to account for dimensionality and differing energy scales of different experimental systems. The logarithmic growth of activation energy for domains of length ℓ is written as (24)

where σ stands for a principal structural length in the system (e.g., a molecular diameter) and J_σ is the activation energy for displacements or reorganization on that scale. For the original East model, σ = 1, J_σ = 1, and γ = 1/2 ln 2 set the effective energy barrier (29). More generally, σ, J_σ, and γ are material properties. We use the terminology East-like to refer to this general class of models. From Eq. 5, the time scale to relax a domain of length ℓ is

where τ_o is the mean field relaxation time at the onset temperature. (We neglect the temperature dependence in τ_mf and consider only T < T_o.) The parabolic law, Eq. 2, is a consequence because the equilibrium domain size grows exponentially with 1/T. Specifically,

where d_f is the fractal dimensionality of the dynamic heterogeneity. This gives . The original East model in dimension d has d_f = d (27).

Thus, when using an East-like model to interpret experimental results for , values for J, T_o, and τ_o are required. These are known from reversible transport data (25). Values for σ, γ, and d_f are also required. Numerical studies suggest that the last of these is universal, depending only on the dimensionality and symmetry of the system (24). As such, we take its value as that computed from 3D atomistic models of glass formers: d_f ≈ 0.8d = 2.4 (24). These numerical studies also find system-dependent values of γ ranging from 0.2 to 0.5. From a scaling argument (Materials and Methods), γ and σ can be collapsed to a reference value, γ₀ ≈ 0.275 and a reduced length σ/ℓ₀ that is of order 1. Therefore, each different system is treated with one fitting parameter, either σ/ℓ₀ or, equivalently, γ. The parameters used are collected in Table 1.

Table 1.

Material parameters for systems considered in Figs. 4 and 5

System		(J/To)*,†	ln (τo)*	γ‡	(σ/ℓo)§
Boron oxide (B2O3)¶	1,066	5.0	−19.0	0.293	0.8
Borosilicate crown glass (BSC)¶	2,002	3.5	−20.2	0.275	1.0
Dibutylphthalate (DBP)¶	340	6.2	−27.1	0.221	2.4
Glycerol (GLY)¶	338	6.2	−17.7	0.220	2.5
O-terphenyl (OTP)¶	341	12.9	−20.5	0.244	1.6
Propylene glycol (PG)¶	321	5.2	−17.7	0.237	1.8
Salol (SAL)¶	308	12.6	−19.6	0.275	1.0
Trisnaphthylbenzene (TNB)¶	510	10.8	−21.2	0.247	1.5

^*Reproduced from ref. 25.

^†Reported in log base e.

^‡Best fits to calorimetry data (Materials and Methods).

^§Reduced lengths implied by γ with γ₀ = 0.275 (Materials and Methods).

^¶Experimental heat capacity scans obtained from refs. 14 (BSC), 37 (B₂0₃), 38 (DBP and PG), 39 (GLY), 40 (OTP), 41 (SAL), and 42 (TNB).

For the relevant temperature regime, East-like dynamics can be treated by quadrature (31, 32). The procedure is an iterative algorithm derived from the fact that during a cooling or heating step of duration Δt, there is an upper limit to equilibrated domain size, Δℓ. From Eqs. 5 and 6, substituting Δt for τ_ℓ and Δℓ for ℓ,

This relationship prescribes a method for computing the nonequilibrium persistence and fictive temperature fields for an experimental system by relaxing domains smaller than Δℓ at each temperature step, using the Sollich–Evans superdomain construction (32) to account for domain regeneration properly during the relaxation process (Materials and Methods). The domain regeneration ensures that an equilibrium ensemble is maintained when the system is ergodic and T is not varying with time.

Fig. 4 A–H compares theory and experiment for eight typical glass formers in experiments. Where available, standard scans (40) are considered, with the cooling and heating rates equal to 20 K/min. Given J, T_o, and τ_o determined from equilibrium structural relaxation data (25), the value of γ is obtained iteratively by optimizing fits to the calorimetry data. The corresponding cooling and heating curves obtained from our model are depicted by blue and red lines, respectively, in Fig. 4. Optimizing γ proves important for quantitative agreement between theory and experiment. However, using a single typical value, γ ≈ 0.25, for all systems considered, yields qualitative agreement between theory and experiment. Fits of comparable quality to those shown in Fig. 4 can be obtained from TNM-like models, but requiring at least four different adjustable parameters (8) for each system considered.

Fig. 4.

Comparison with heat capacity data obtained through DSC. (A–H) Reduced heat capacity predicted by our model compared with experimental data. Experimental data are depicted by points, with triangles (▲) for cooling scans (where available) and circles (●) for heating scans. Red and blue lines represent heating and cooling curves obtained from our calculation. A summary of systems and parameters is given in Table 1. B2O3, boron oxide; BSC, borosilicate crown glass; DBP, dibutylphthalate; GLY, glycerol; OTP, o-terphenyl; PG, propylene glycol; SAL, salol; TNB, trisnaphthylbenzene.

Our East-like modeling also produces correct quantitative behavior for nonstandard cooling and heating rates. Fig. 5A shows DSC heating scans for glycerol for glasses created at several different cooling rates. Model curves are obtained using the parameters of Table 1 and depicted by lines. Slower cooling yields a more stable glass with larger domains, and thus a more prominent peak in the heat capacity on heating. Fig. 5B shows heating scans for a glass formed at a single cooling rate for B₂O₃. When the magnitude of the heating rate exceeds the cooling rate, the system is able to equilibrate further during heating and the heat capacity dips below the heat capacity of the glass, depicted by dashed black lines, before recovering. This behavior is particularly apparent for very large values of |ν_h/ν_c|, as illustrated by the predicted dataset for ν_c = −100 K/min.

Fig. 5.

Behavior of the heat capacity on heating from the glass. (A) Heating scans for a typical system (glycerol) for constant heating rate ν_h and variable cooling rate ν_c. Experimental data and simulated curves are depicted by points and lines, respectively. Slower cooling yields a more stable glass and a more prominent peak in the heat capacity on heating. (B) Heating cycles for a fixed cooling rate for a different system, B₂O₃. When the magnitude of the heating exceeds the cooling rate, the heat capacity dips below the glassy heat capacity before recovering. The extrapolated heat capacity of each glass is depicted by a dashed black line. The bottommost dataset with no corresponding experimental data points is a prediction of the model. Curves are offset for clarity.

A final qualitative behavior of interest regards the sharpness of the heat capacity to drop at T_g, which increases with decreasing cooling rates. Here, we define T_g as the temperature where is maximum on cooling. Empirically, the value of this nonequilibrium correlates with the fragility of the equilibrium relaxation time, (43). The East-like models exhibit this correlation too. Specifically, over the range of typical values for σ/ℓ₀, J, T_o, and τ_o, East-like models yield . The linear relationship for East-like models holds when relaxation is dominated by hierarchical dynamics. It diminishes in accuracy in the regime of relatively small fragility, m ≲ 30, where both mean field and hierarchical dynamics play significant roles. The proportionality constant in the linear relationship also varies with σ/ℓ₀ when this ratio differs by more than an order of magnitude from 1.

Other Experiments

Although we have focused here on configurational enthalpy and heat capacity, our model could be applied to a host of other nonequilibrium phenomena. For example, a specific volume and refractive index could be investigated with minor modifications. More complex behaviors, such as the calorimetry of remarkably stable glass (44) or the uptake of water vapor (45, 46), might potentially be considered as well.

The prediction of large nonequilibrium correlation lengths deserves both experimental and computational investigation. Keys et al. (24) describe a simulation scheme for identifying excitations (or soft spots) in the equilibrium melt based on irreversible particle displacements, which, conceivably, could allow for the direct measurement of a correlation length. However, the relative scarcity of particle displacements in the glass could render such methods intractable. An alternative method to identify excitations in the absence of their dynamics might involve localized soft modes (47, 48). Spatial correlations among soft regions will be reflected in the vibrational spectra of glass, which might then be used to infer a correlation length (49).

Experimental attempts to measure the nonequilibrium lengths might involve small-angle X-ray and neutron scattering. The distribution of these lengths reflects an anticorrelation between excitations that is present in the glass but absent in the liquid. It should be manifested in the rate at which the structure factor approaches the compressibility value as wave vector q → 0 (50). Detection will require sensitivity to variations of molecular density consistent with variations in the fictive temperature field and the coefficient of thermal expansion.

Materials and Methods

Our East-like model calculations are done on a d = 1 lattice with lattice spacing ℓ₀. At temperatures relevant to experimental glass transitions, straightforward numerical simulation of the East model is impractical, but for d = 1, we may exploit Sollich–Evans superdomain analysis (32). A superdomain analysis might be possible in higher dimensions, but such a method is not yet known. Rather, to compare with experiments, we map the d = 1 results obtained in this way to a d = 3 system with a characteristic length σ. Details on the calculations and the mapping are provided in this section.

Conversion to Lattice Units.

Structural relaxation occurs on a scale of characteristic length, typically a molecular diameter or the length 2π/q₀, where q₀ is the wave vector for the main peak of the structure factor. As a result, the parameter J obtained from structural relaxation data is directly related to excitation activation energy on a scale of this length, σ (24). In contrast, the principal length of an East-like model is a lattice spacing, ℓ₀. As a result of East-like scaling, J_ℓ = J₀[1 + γ₀ ln (ℓ/ℓ₀)] (Eq. 5) follows with

and

With these equations, the energy scales of the underlying East-like model are determined in terms of γ and . The value of γ accounts for entropy of relaxation pathways connecting excitations, which changes with length scale. Here, we are concerned with the large-length regime, which applies to structural relaxation. For the d = 1 East model, γ is 1/ln 2 at small-length scales, and it is 1/2 ln 2 for lengths larger than 1/c (29). In practice, γ (or, equivalently, σ/ℓ₀) can be determined by fitting calorimetry data. The equilibrium excitation concentration is given in reference lattice units by

The length of equilibration in the cooling time scale Δt is given by

Mapping Between 1D and 3D.

Regardless of dimensionality, structural relaxation is dictated by the distance between excitations, ℓ. Interexcitation distances (rather than excitation concentrations) are conserved when mapping between the d = 1 and d = 3 systems. For d > 1, relaxation between excitations occurs on irregular paths, such that the average length between excitations is given by . The equilibrated regions of the 1D proxy lattice are populated with excitations at the concentration 1/ℓ_eq.

The fictive temperature profile along a line connecting excitations in an East-like model should be roughly identical for the d = 3 and d = 1 systems. However, for d = 3, the domains are space-filling and the average fictive temperature must be reweighted with respect to the d = 1 case. A mapping of the spatially averaged fictive temperature from the d = 1 case to the d = 3 case that accounts for these facts is

Here, D is set of proxy lattice sites i ∪ {j ∈ N: i < j ≤ i + N(D) − 1} satisfying n_i = 1, n_j = 0, and n_i+N(D) = 1. The set of all such domains within the system is denoted {D}, and N(D) represents the number of lattice sites in domain D. We find that T_f is reasonably insensitive to details of the method chosen to mimic 3D domains with 1D domains treated in the calculation. For example, constructions where excitation fronts are assumed to move forward in a cone prove equally satisfactory (SI Text). The relative insensitivity to this detail is a result of the directional facilitation in East-like models.

Fictive Temperature Computation Algorithm.

The persistence field for a given system is propagated by running dynamics of the proxy d = 1 East-like model at a temperature corresponding to equilibrium concentration 1/ℓ_eq. This is done for a time Δt corresponding to the time scale for the equilibration length Δℓ computed from Eq. 12. At each step k, the persistence field is computed, the fictive temperature field is updated, and the average fictive temperature is computed from Eq. 13. The process is then repeated with the new working temperature for step k + 1. The computation is decomposed into small enough temperature steps to negate quadrature error. We find that using 1,000 steps is sufficiently large to approximate time dependence accurately for the nonequilibrium protocols considered here.

Standard Monte Carlo simulation of East-like models becomes prohibitively expensive when . The nonequilibrium systems we consider to compare with experiments are typically at . The “superdomain” construction of Sollich and Evans (31, 32) for computing persistence fields of East-like models is accurate and efficient in this region, and we therefore use this procedure. The superdomain construction derives from that fact that at low enough temperatures, relaxation in East-like models can be partitioned in terms of relaxation on different scales, and each separate scale is nearly independent of the others. Although the original Sollich–Evans method is designed to operate at a fixed temperature, we make a small modification to allow for multiple temperature steps.

At the end of each step k, the system is left with a set of nonequilibrated excitations called “superspins.” Each superspin possesses an equilibrated zone of length Δℓ, adjacent to the spin in the direction of facilitation. Excitations within these equilibrated zones are inconsequential in the Sollich–Evans procedure, and therefore not kept track of. An initial configuration for the k + 1 cooling or heating step is generated from the superspins of step k by populating excitations within equilibrated zones. The positions of these excitations are drawn at random from the Poisson distribution with an equilibrium concentration 1/ℓ_eq. Each calculation reported here is averaged over 10,000 trajectories for a system of size n = 32,768. The statistical error is less than or equal to the width of the lines used to represent the data. A user-friendly program for carrying out these calculations can be found at http://www.nottingham.ac.uk/~ppzjpg/GLASS/.