“Inner clocks” of glass-forming liquids

Amorphous states of matter that derive from the non-equilibrium solidification of supercooled liquids, such as most glasses and gels, are ubiquitous in nature and of significant technological and scientific relevance.^1–4 One of the most essential fingerprints experimentally observed in these non-equilibrium amorphous states is the slow evolution of physical properties (e.g., volume and transport coefficients) with the “waiting” time t after preparation, a phenomenon commonly referred to as physical aging.^5–16 The wealth of experimental data on the equilibration and aging processes of glass- and gel-forming liquids, thus, becomes a key to test the predictive power of available theories of glasses.^17–20 Many such data are successfully fitted by phenomenological models, and it remains as a challenge for statistical-mechanics theories to reconcile the phenomenology with fundamental principles.^21–24 

Many of these phenomenological models were built out of the need to include essential effects such as the existence of several (not only one) relaxation modes or the presence of dynamical heterogeneities.^14,16 For example, the need to describe measured thermal and viscoelastic properties during vitrification and devitrification under a wide variety of experimental conditions and thermal histories led to the development of powerful phenomenological tools, such as the Tool–Narayanaswamy–Moynihan (TNM^5–7) and the Kovacs–Aklonis–Hutchinson–Ramos (KAHR¹²) models. These models are commonly used in industry to predict aging effects,²⁵ and their development involved a rich discussion of many relevant issues.¹⁴ For our present purpose, however, we shall focus on the TNM model, and to simplify the discussion, we only highlight the essential feature of Tool’s original proposal,⁵ namely, to model the highly nonlinear aging processes observed in a given observable X(t) in terms of a simple mathematical representation: The model posits the generic nonlinear equation

where τ(X(t)) is a characteristic relaxation time that itself exhibits aging and that is postulated to depend on waiting time only through its dependence on X(t). A change of variable from the time t to the “material” (or “inner”) time ζ, defined as⁶ 

allows us to write the solution of Eq. (1) as X(t) = X*(ζ(t)) with the function X*(ζ) that obeys the ordinary linear differential equation

The solution of Eq. (1) with the arbitrary initial condition X(t = 0) = X₀ is, thus, given by

Despite its general success, the applicability of this simple version of the TNM model is known to be limited, in practice, and this prompted the proposal of alternative independent empirical models.¹⁴ In fact, already in the original proposal, Narayanaswamy considered extended versions.⁶ The resulting description turns out to capture the essence of the behavior experimentally observed in many glass forming materials, although the fundamental reason for this is not fully understood.^16,26 Determining the boundaries of validity of the TNM model and the origin of its corresponding limitations is still active matter of current research.²⁷ Only recently, careful long-time experiments could establish the very existence of an internal “clock” as a manifestation of more fundamental physical principles in a real material.²⁸ 

A general statistical mechanics theory of irreversible processes in liquids was proposed two decades ago.^29,30 This approach, referred to as the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory, predicts many relevant universal signatures of the glass and gel transitions—including aging phenomena—and more specific features reflecting the role of the particular molecular interactions acting in the specific system considered. For example, for simple liquids with purely repulsive interparticle interactions, this theoretical framework accurately describes the non-stationary and non-equilibrium process of the formation of high-density hard-sphere-like glasses,^30–32 whereas for liquids with repulsive plus attractive interactions at low densities and temperatures, it predicts the formation of sponge-like gels and porous glasses by arrested spinodal decomposition.^33–36 

The NE-SCGLE was formulated in Ref. 29 and can be summarized in terms of a set of equations that describe the non-equilibrium evolution of the structural and dynamical properties of a simple glass-forming liquid [we refer here for simplicity to a generic system of N identical spherical particles in a volume V that interact through a radially symmetric pair potential U(r)]. The most central of these equations describes the time evolution of the non-equilibrium structure factor (SF) S(k; t) ≡⟨δn(k) δn(−k)⟩_t, where the brackets indicate the average over the time-dependent non-equilibrium statistical ensemble and δn(k) is the Fourier-transformed (FT) fluctuating particle-number density $n(r)=∑1Nδ(r−ri)/N$⁠. For a homogeneous system, instantaneously quenched at t = 0 from an initial equilibrated state corresponding to density and temperature $(n̄(i),T(i))$⁠, toward new values $(n̄(f),T(f))$⁠, the NE-SCGLE for the SF reads

The evolution of the non-equilibrium SF is driven by a thermodynamic force given by $E(k;n,T)$⁠, the FT of the second functional derivative $E[r;n,T]$ of a postulated Helmholtz free energy density-functional $F[n;T]$⁠, i.e., $E[∣r−r′∣;n;T]≡δ2βF[n;T]/δn(r)δn(r′)$⁠,³⁷ evaluated at the uniform (bulk) density and temperature fields³⁰ $n(r,t)=n̄≡N/V$ and T(r, t) = T, where N, V, and T are, respectively, the total particle number, volume, and temperature of the final state of the quench, and β⁻¹ = k_BT with the Boltzmann constant k_B. The factor D⁰ in the NE-SCGLE relaxation [Eq. (5)] arises from the diffusive relaxation of density fluctuations and is related to the short-time friction coefficient ζ⁰ = k_BT/D⁰ in colloidal suspensions (or a suitable kinetic or “Doppler” friction in the case of molecular liquids^38,39).

More crucially, the factor b(t) in Eq. (5) is a non-equilibrium time-dependent mobility function²⁹ that can be identified, in the NE-SCGLE framework, with the normalized instantaneous long-time self-diffusion coefficient b(t) ≡ D_L(t)/D⁰. It is the crucial ingredient in the theory of aging, as it describes the kinetically slowed down evolution of structural quantities. It needs to be calculated by a suitable theory of (near-stationary) structural relaxation and kinetic arrest; in this context, b(t) is typically evaluated within the SCGLE framework, although a suitable extension of mode-coupling theory (MCT) would give qualitatively similar results.

Equations (1) and (5) bear an obvious resemblance: identifying S(k; t) as a length-scale dependent (k-dependent) observable X(t) within Tool’s model and $1/n̄(f)E(k;n̄(f),T(f))$ as the asymptotic value X_∞, we observe that NE-SCGLE encodes the mathematical structure of the generic relaxation postulated by Tool. In particular, a change of variable from time t to

allows us to write the solution of Eq. (5) as S(k; t) = S*(k; u(t)) in terms of the function S*(k; u) solving the linear differential equation

with a k-dependent relaxation factor $α(k)≡[2k2D0n̄(f)E(k;n̄(f),T(f))]$ and initial condition S*(k; u = 0) = S(k; t = 0) = S⁽ⁱ⁾(k); thus,

Hence, the NE-SCGLE proposes an unambiguous microscopic identification of the relaxation rate [1/τ(X(t))] corresponding to the variable X(t) = S(k; t). Remarkably, this factorizes, 1/τ(k; t) = α(k)b(t), and thus, the corresponding “material” time equally factorizes,

The quantity u(t) can be regarded as an “intrinsic” material time that governs the relaxation of density fluctuations on all length scales.

In this contribution, we aim to link the two different approaches to aging just highlighted—representative of a more practical-oriented materials-science and engineering methodology and of the statistical-physics method to predict macroscopic behavior from microscopic intermolecular forces—by exploiting the mathematical similitude between the phenomenological Tool model described by Eqs. (1)–(4) and the NE-SCGLE framework of Eqs. (5)–(9). We argue that this bears a deep physical significance as it reflects a remarkable connection in their fundamental roots, namely, the universal principles of Onsager’s theory of irreversible processes.

Relating classical and irreversible thermodynamics to glass science has, indeed, a long tradition, as illustrated since the 1950s by Davies and Jones,^40,41 in the 1970s by Moynihan et al.^7,8 and as discussed more recently by Bailey et al.⁴² In fact, the elegance and simplicity of the TNM model rest on its fundamental roots in the linear laws of irreversible thermodynamics, but it is notably augmented by Tool’s assumption that the relaxation times vary with time through their dependence on the state variables themselves. In irreversible-thermodynamic terms, this means that the kinetic coefficients of the transport equations (such as viscosity, conductivity, and diffusion coefficients) are treated as state functions. The empirical success of the TNM model clearly demands the theoretical investigation and prediction of the corresponding “kinetic equations of state,” which the NE-SCGLE theory incorporates in the very heart of its formulation.

As we will discuss below, the NE-SCGLE supports the definition of an intrinsic material time u(t) that is generic to a wide range of observables and, at the same time, suggests an extension of the TNM concept toward materials characterized by multiple intrinsic times, namely, those where fundamentally different modes of relaxation exist. We also hope to trigger by our discussion the reconsideration from a microscopic starting point of various phenomenological concepts of enormous practical and fundamental impact,¹⁴ whose discussion has been the subject of both historical arguments and enlightening reviews. Illustrative examples include the concepts of time–temperature superposition (TTS) and thermo-rheological simplicity (reviewed in Refs. 43 and 44) and the notions of fictive temperature and effective time, introduced decades ago by Struik.⁹ In close relationship with the fundamental description of Kovacs’ kinetic signatures in glass-forming systems and of the concept of fictive temperature^4,10,11 (see Ref. 45 for a recent discussion), the present work is meant to advance along this program by focusing on the fundamental relationship between the empirically successful TNM model and the microscopic foundation of the NE-SCGLE formalism.

We start in Sec. II by discussing the Tool model as a manifestation of the linear laws of irreversible thermodynamics, modified by TNM’s recognition that the kinetic coefficients may be considered state functions and, hence, depend on the current value of the state variables. These linear laws, applied to several coupled state variables, naturally extend the original Tool’s one-relaxation time model⁵ to the multi-variable model proposed by Narayanaswamy.⁶ Section III addresses the problem of defining and predicting the kinetic equations of state, i.e., the dependence of the kinetic coefficients on the state variables. The route proposed by NE-SCLGE theory rests on the universal laws of irreversible thermodynamics, although with several important additions.²⁹ The most important among them is the central role played by the kinetic equation of state L = L[X], which the resulting non-equilibrium microscopic theory predicts in an approximate, but systematic manner. For clarity, in Subsection III B, we provide an illustrative example on the use of the NE-SCGLE to calculate the intrinsic material time of a glass-forming model system. In Sec. IV, we show how the NE-SCGLE framework allows us to extend TNM’s concept of material time for the more sophisticated case of glass forming systems involving non-spherically interacting particles, leading to multiple relaxation modes originated in the dynamical coupling between the translational and orientational degrees of freedom. In Sec. IV, we also illustrate the use of this extended model for the description of aging phenomena in a model dipolar fluid. Finally, in Sec. V, we summarize our conclusions.

Let us start by reviewing the TNM framework from the perspective of the linear laws of irreversible thermodynamics, modified upon the recognition that the corresponding kinetic coefficients are state functions. We, then, explain how such linear laws applied to several state variables naturally extend Tool’s original model⁵ to the multi-variable model of Narayanaswamy.⁶ 

Tool’s Eq. (1) can be understood in a simple manner as a direct extension of the universal laws of irreversible thermodynamics.⁴⁶ For this, let the variable X(t) represent an extensive thermodynamic property. The most relevant of these laws states that the relaxation of this extensive variable toward its equilibrium value, X^eq, is governed close to equilibrium by a linear relationship,

between the “flux” [i.e., the time rate of change of X(t)] and the corresponding “thermodynamic force” ΔF ≡ F[X] − F[X^eq], where the function F[X] is the respective conjugate variable, F[X] = dS[X]/dX, with S[X] being the entropy⁴⁷ and L being the corresponding Onsager kinetic coefficient. This fundamental law is one of the pillars of the TNM model.

The fact that “flux” and “force” are linearly related, however, does not imply that Eq. (10) is a linear equation. One obvious reason is that the thermodynamic equation of state F = F[X] does not involve a linear relationship between ΔF and ΔX ≡ X − X^eq. Such “thermodynamic” non-linearity is expected and well understood, and Boltzmann’s fundamental postulate S[X] = k_B ln W[X]⁴⁸ allows us, in principle, to determine the precise function F[X] = dS[X]/dX. A less obvious non-linearity of Eq. (10), however, arises if one allows for the proportionality “constant” L to be a state function L = L[X]. In contrast to the thermodynamic equation of state F = F[X], no general fundamental principle analogous to Boltzmann’s postulate is available to assure the existence of or to determine the “kinetic equation of state” L = L[X] from first principles.

The very postulate of existence of a kinetic equation of state L = L[X] already offers a glimpse of one of the most fascinating predictions of Eq. (10). Written as

this equation assures us that any equilibrium state X^eq is a stationary solution since in the limit t → ∞, the factor $F[X(t)]−F[Xeq]$ on the right-hand side vanishes whenever X(t → ∞) = X^eq. The inverse statement, however, is not true. In other words, not every stationary solution of this equation corresponds to an equilibrium state. The reason for this is the possibility that the kinetic coefficient L[X(t)] asymptotically vanishes at long times (i.e., L[X(t → ∞)] = 0), thus establishing conditions for stationarity without the equilibrium condition ΔF = 0 being required. Thermodynamic equilibrium states have long been understood as a consequence of the fundamental and general laws of classical⁴⁷ and statistical⁴⁸ thermodynamics. In contrast, the second class of stationary solutions, denoted by X^a, in which the transport mechanisms represented by L[X] become arrested (L[X^a] = 0), find their vivid physical realization in the formation of rather common non-equilibrium states of matter, such as glasses and gels. Demonstrating the existence of these states is one of the central aspects of the NE-SCGLE theory.

The notorious asymmetry in the understanding of equilibrium vs non-equilibrium stationary states of matter derives from the fact that equilibrium states can be understood solely in terms of a maximum entropy principle combined with Boltzmann’s postulate, without having to appeal to the fact that they are stationary solutions of any non-equilibrium process. This is in stark contrast with non-equilibrium stationary states, which can only be understood as stationary solutions of non-linear transport equations of the kind in Eq. (11).

The full significance of Onsager’s principle and its non-equilibrium extension comes to light in the multi-variable generalization of Eq. (10). Consider ν thermodynamic extensive variables X₁(t), X₂(t), …, X_ν(t) grouped as the components of a vector X(t) ≡ [X₁(t), X₂(t), …, X_ν(t)], and let S = S[X] be the fundamental thermodynamic relation.⁴⁷ Then, F_i[X] ≡ (∂S[X]/∂X_i) are the corresponding intensive conjugate variables, and to first order,⁴⁶ 

where ΔX(t) ≡ X(t) − X^eq, ΔF[X(t)] ≡ F[X(t)] − F[X^eq], and L is the matrix of Onsager kinetic coefficients.

For small enough derivation from equilibrium, the linearization of F[X] in [X(t) − X^eq] yields

where the elements $Eij[X]$ of the matrix $E[X]$ are the thermodynamic susceptibilities $Eij[X]≡−kB−1(∂Fi(X)/∂Xj)=−kB−1(∂2S(X)/∂Xj∂Xi)$⁠.

This linearization suggests a natural extension toward a multi-variable Tool model, based on the ad-hoc assumption at L and $E$ can be considered as kinetic state functions to describe the evolution of an observable X(t) toward is asymptotic value X_∞,

with the relaxation-rate matrix $τ−1(t)≡L[X(t)]⋅E[X(t)]$⁠.

Equation (14) is formally solved as

where the time-ordered exponential $E(t,0)=exp+[−∫0tdt′τ−1(t′)]$ is the unique solution of the differential equation ∂_tE(t, 0) = −τ⁻¹(t)E(t, 0) to the initial condition E(0, 0) = 1. In principle, the matrix τ⁻¹ can be diagonalized, and it will have non-negative eigenvalues $τi−1≥0$ because it is the product of a symmetric matrix L and a matrix $E$ that for reasons of thermodynamic stability is assumed to be positive (semi-)definite. The individual modes of relaxation in the set of observables X_i that correspond to the eigenbasis of the relaxation-rate matrix, then, follow

Note the subtle difference to Eq. (1): each of the corresponding relaxation rates now is a state function that, in general, depends on all the variables grouped in the vector X(t). In addition, the typical experimental setup prescribes a set of X_i that do not correspond to an eigenbasis of τ⁻¹ so that, in general, the individual X_i(t) given by the matrix exponential in Eq. (15) do not decay as simple exponentials, but rather as superpositions of exponentials. This addresses Narayanaswamy’s concern⁶ that Tool’s original model⁵ in Eq. (1) “postulates a simple exponential relaxation mechanism governed by a single (time-dependent) relaxation time. In fact, most glasses exhibit a more complex nonexponential relaxation behavior that can, however, be characterized by a distribution or spectrum of relaxation times.” In addition, there need not exist a single scalar material time because different modes of τ⁻¹ can have very different temperature dependence.

The formal solution, Eq. (15), suggests to define a “material-time matrix,”

As we exemplify below, for the discussion of the long-time fate of aging systems, in the limit t → ∞, we essentially distinguish between the following cases: if τ⁻¹ approaches a matrix that is similar to a strictly positive definite matrix, asymptotically, ζ(t) ∼ ζ_∞t, i.e., the material time becomes asymptotically linearly related to physical time, but with a non-trivial prefactor matrix ζ_∞ that reflects the slow dynamics of the equilibrating system. In the second case, when the relaxation of all relevant modes of the system becomes kinetically arrested, τ⁻¹ approaches the zero matrix, and in this case, the material is characterized by an asymptotically constant set of material times, which can be expressed as a set of eigenvalues ζ_i (or equivalently, a set of finite intrinsic material-time values u_i). In this case, the physical aging of the system forever “gets stuck” at a finite internal clock-time.

The multi-variable generalization of the TNM model displays a third possibility: in systems with kinetic arrest but partial equilibration of some of the dynamical modes, the relaxation-rate matrix τ⁻¹(t) may have blocks with different temperature dependence and, in particular, may develop zero-blocks for large t while remaining non-zero, in general. In this case, the matrix ζ(t) will have sub-blocks that grow asymptotically with t and ones that approach constants, i.e., a mixture between the two cases described above that are already present in the single-variable TNM model. As discussed below, this situation arises, for example, in the kinetic arrest of a dipolar hard-core fluid, where translational degrees of freedom display kinetic arrest, while the rotational degrees of freedom can remain ergodic and, thus, equilibrate. Other systems where such partial equilibration is known are mixtures of very disparate particles. The partial equilibration, then, manifests as a subset of material-time eigenvalues that remain stuck at finite values and a separate subset that grows indefinitely with physical time.

Section II dealt with the TNM model of aging, viewed as the phenomenological manifestation of the universal laws of irreversible thermodynamics. In this section, we now shift our perspective to that of the NE-SCGLE theory. We will focus, first of all, on the identification of the functions b(t) and u(t) appearing in Eqs. (5) and (6) as, respectively, the kinetic coefficient and intrinsic material time of the variable S(k; t), as determined from the solution of the NE-SCGLE system of equations.

The main conceptual link between Tool’s Eq. (1) for a generic property X(t) and the relaxation equation [Eq. (5)] for the structure factor S(k; t) is the mobility function b(t) whose time integral $u(t)=∫0tb(t′)dt′$ is an intrinsic material time. Recall that, in essence, the mobility b(t) is the long-time self-diffusion coefficient and, hence, plays the role of the kinetic coefficient of this particular problem. Thus, in the context of the NE-SCGLE theory, it is a state function, which means that it depends on t through its functional dependence on S(k; t), a dependence that shall be denoted by b(t) = b[S(t)].

It has widely been recognized that the static structure factor is a “master property” of structural relaxation in the sense that—at least within the usual mean-field description of the glass transition as proposed by MCT or SCGLE—S(k) governs the long-time dynamics. Extending this notion to the non-equilibrium aging dynamics, we consider the kinetic coefficients in the TNM-like models to be functionals of S(k; t). In particular, within the NE-SCGLE, relaxation equations to determine b[S(t)] have been derived as a straightforward extension of the equilibrium SCGLE theory; a similar program could be carried out on the basis of MCT. In essence, the qualitative description given by both theories concerning the asymptotic structural-relaxation dynamics for long times will be identical.⁴⁹ 

Within the NE-SCGLE, the mobility b(t) is expressed through a generalized friction kernel,

For the t-evolving and τ-dependent friction kernel Δζ*(τ; t), the following approximate expression is derived:²⁹ 

where F(k, τ; t) ≡ N⁻¹⟨δn(k, t + τ)δn(−k, t)⟩ is the non-equilibrium collective intermediate scattering function (ISF), whose self-part, F_S(k, τ; t), is defined as $FS(k,τ;t)≡⟨expik⋅ΔrT(t,τ)⟩$⁠, with $ΔrT(t,τ)≡rT(t+τ)−rT(t)$ being the displacement of a tracer particle.

These equations are complemented with the corresponding memory-function equations for the ISFs F(k, τ; t) and F_S(k, τ; t), written approximately in terms of their Laplace transforms (LTs), F(k, z; t) and F_S(k, z; t), as

and

where $λ(k)≡1/[1+(k/kc)2]$ and k_c is an empirically determined parameter.³¹ 

Equations (18)–(21) constitute a closed system of equations that summarizes the NE-SCGLE theory, together with Eq. (5). They determine the kinetic coefficient b(t) as a (rather involved) functional b(t) = b[S(t)] of the non-equilibrium static structure factor S(k; t) and, thus, constitute a kinetic equation of state associated with the variable S(k; t). Once the thermodynamic input $E(k;n̄(f);T(f))$ has been determined, then the solution of these equations simultaneously yields predictions for S(k; t) and for the dynamic properties b(t), Δζ*(τ; t), F(k, τ; t), and F_S(k, τ; t).

The previous concepts are better illustrated by means of a concrete example involving the solution of the NE-SCGLEs. Despite their seemingly complex appearance and non-linear character, the mathematical features of Eq. (5)—and particularly Eq. (8)—render their numerical solution relatively manageable. The reader is referred to Ref. 30 for a thorough and detailed explanation of the numerical methods involved and to Ref. 50 for access to a code with the corresponding computational program. Here, we only illustrate the results for the glass-forming model liquid also studied in Ref. 30, which consists of N identical spherical particles interacting through short-ranged and purely repulsive interactions. We refer to the Weeks–Chandler–Andersen (WCA) model,⁵¹ whose pair potential U(r) vanishes for r larger than the distance σ of soft contact, but which for r ≤ σ is given by

The state space of this system is spanned by the reduced number density $n*≡n̄σ3$ and the reduced temperature T* ≡ k_BT/ϵ, here denoted simply as n and T, so that the volume fraction is ϕ = πn/6. For the determination of the thermodynamic property $E(k;n̄(f),T(f))$ entering in Eq. (5), we employ the same approximation as in Ref. 30 based on the Percus–Yevick/Verlet–Weis (PY/VW) approximation^52–54 for the hard-sphere liquid with an effective hard-sphere diameter σ_HS(T) determined by the blip function method.⁵⁵ 

The solution of the NE-SCGLEs describes the evolution of the properties S(k; t), b(t), Δζ*(τ; t), F(k, τ; t), and F_S(k, τ; t) for times t > 0 after the system, initially at equilibrium at temperature T⁽ⁱ⁾ and volume fraction ϕ, is instantaneously and isochorically quenched (at t = 0) to a final temperature T^(f). Here, however, we shall focus only on the behavior of the mobility function b(t) and on its integral u(t), the intrinsic material time of the variable S(k; t). As discussed in Ref. 30, the mobility function actually depends on both the initial and final temperatures, T⁽ⁱ⁾ and T^(f), and on the volume fraction ϕ of the quench, b(t) = b(t; ϕ, T⁽ⁱ⁾, T^(f)). Its long-time asymptotic limit b_a(ϕ, T^(f)) ≡ b(t → ∞; ϕ, T⁽ⁱ⁾, T^(f)), however, is independent of T⁽ⁱ⁾ and serves to identify two qualitatively different scenarios predicted by the NE-SCGLE theory for the relaxation of the system after quenching.

The two scenarios are (i) equilibration, for which b_a(ϕ, T^(f)) is identical to its finite and positive equilibrium value, b_a(ϕ, T^(f)) = b^eq(ϕ, T^(f)) > 0, thus defining a condition to predict whether the system would be able to reach a new equilibrium state, and (ii) aging, where b_a(ϕ, T^(f)) vanishes, and hence, the system is predicted to age endlessly. As illustrated in Fig. 1(a), using this criterion, the state space (ϕ, T) of the WCA system can be divided in two regions delimited by the dynamic arrest (solid) line T = T_c(ϕ), above which the system will equilibrate (ergodic region) and below which it will become arrested (non-ergodic or glass region). Let us mention in passing that full kinetic arrest at T_c is a mean-field simplification, as is well understood in the context of both MCT and SCGLE. For the present qualitative discussion, we do not consider additional relaxation modes that might be active in the idealized glass.

In the inset of Fig. 1(a), we plot the NE-SCGLE results for the asymptotic value b_a(ϕ, T^(f)) as a function of T^(f) along the isochore ϕ = 0.66. These results determine that, at this isochore, the dynamic arrest temperature is T_c(ϕ = 0.66) ≈ 0.0915. At T_c(ϕ), the kinetic parameter b_a(ϕ, T) passes continuously from its vanishing value below to its equilibrium non-zero value b^eq(ϕ, T) above. As also illustrated in the inset of Fig. 1(a), when T^(f) approaches T_c from above, b^eq(ϕ, T) vanishes as $beq(ϕ,T)∼(T−Tc(ϕ))μ$ with exponent μ = 2.2.

Figure 1(a) also shows the illustrative isochore ϕ = 0.66 and indicates with two downward arrows two representative isochoric quenches along this isochore. Both quenches start at the same initial temperature, T⁽ⁱ⁾ = 0.04, but end at different final temperatures T^(f). The shallower quench (T^(f) = 0.2 > T_c) represents a typical equilibration process, whereas the deeper quench (T = 0.07 < T_c) is meant to illustrate a process of dynamic arrest. In Figs. 1(b) and 1(c), we plot the full non-equilibrium evolution of, respectively, the kinetic parameter b(t; ϕ, T) (in terms of its inverse) and the intrinsic material time u(t; ϕ, T^(f)) as a function of the waiting time t, indicating with the thick solid and thick dashed lines these two illustrative quenches of Fig. 1(a). Figures 1(b) and 1(c) also include other similar quenches (thinner lines) to emphasize the crossover from the equilibration regime [solid lines, T^(f) > T_c(ϕ)] to the aging regime [dashed lines, T^(f) < T_c(ϕ)]. The thick dotted line corresponds to a quench to the critical temperature T_c(ϕ) ≈ 0.0915.

These results exhibit some relevant predictions of the NE-SCGLE theory whose detailed analysis, however, lies outside the scope of this article. Instead, we refer the reader to Ref. 30 for a detailed analysis of results analogous to those summarized in Fig. 1(b), but in which the various lines correspond to quenches with the same initial and final temperatures (T_i = 0.1 and T = 0), along different isochores. Here, we only highlight the predicted existence of the two regimes: equilibration for quenches with T^(f) > T_c(ϕ) [solid curves in Figs. 1(b) and 1(c)] and aging for T^(f) < T_c(ϕ) (dashed curves). In the equilibration regime, we also note that b⁻¹(t; ϕ, T^(f)) reaches its equilibrium value 1/b^eq(ϕ, T^(f)) after an equilibration time-scale t^eq(T^(f)), highlighted by the black circles in Fig. 1(b). The equilibration time t^eq grows asymptotically linearly with the relaxation time, t^eq ∼ 1/b^eq; both quantities, furthermore, increase strongly and eventually diverge as T^(f) approaches the critical value T_c from above.

Figure 1(c) displays the corresponding evolution of the intrinsic material time $u(t;ϕ,T(f))≡∫0tb(t′;ϕ,T(f))dt′$⁠. One observes that u(t) exhibits a short-time behavior given by u(t) ≈ b_i × t, which is determined by the initial condition b_i ≡ b^eq(ϕ, T⁽ⁱ⁾). At longer times, u(t) shows a crossover that can be easily understood in terms of the long-time behavior of b(t). For T^(f) > T_c, for instance, the long-time limit of u(t) would be approximately given by u(t) ≈ A + b^eq(ϕ, T^(f))t, where $A≈∫0teq[b(t′)−ba]dt′$ and t^eq(T^(f)) is the corresponding equilibration time-scale. Hence, the material time u(t) continues to grow as a linear function of time when the system has reached equilibrium. For T^(f) < T_c, instead, b(t) → 0 as t → ∞, so u(t) reaches an arrested plateau value $ua(ϕ,T(f))≈∫0tab(t′)dt′$⁠, with t^a being the time scale required by b(t) to reach its asymptotic arrested value b^a = 0, which becomes shorter with decreasing T^(f). The value u^a depends on the specific quench and, hence, serves as a nonequilibrium state parameter, augmenting the set of thermodynamic control variables that are needed to uniquely specify a material.

Let us mention in passing that the NE-SCGLE, in the spirit of a mean-field theory of the glass transition, expresses a number of physically relevant properties as functionals of S(k; t) so that these can, in principle, be easily evaluated on the basis of the results discussed above. Examples include the internal energy E(t) and the pressure p(t) of a simple liquid with only pair interactions whose configurational contributions ΔE(t) and Δp(t) are given, respectively, by the so-called “energy” and “pressure” equations,

and

In these equations, the non-equilibrium radial distribution function g(r; t) can be related with S(k; t) just as it is typically done in equilibrium conditions, i.e., [S(k; t) − 1]/n is the Fourier transform of [g(r; t) − 1]. Remarkably, these relationships, which are well known from equilibrium statistical physics, can be shown to also hold under non-equilibrium conditions by following their standard derivation,^48,55 starting from the definition of g(r; t) as a statistical average, but avoiding the use of the canonical (or any other equilibrium) ensemble. The discussion of these aspects is beyond the scope of the present contribution and will be reported elsewhere.

Similarly, many other relevant dynamical, transport, and rheological properties may be written at least approximately in terms of S(k; t) or of the dynamical scattering functions, which through Eqs. (18)–(21) again depend on S(k; t). A simple example is provided by the mean squared displacement W(τ; t) ≡ ⟨(r(t + τ) −r(t))²⟩/6, which may be written within the NE-SCGLE in terms of the friction function Δζ*(τ; t) through the following exact equation:

An equally important example is provided by the non-equilibrium rheological or viscoelastic properties of the system, which may be represented by the complex-valued dynamic shear modulus G(ω; t) or by the dynamic shear viscosity η(ω; t). Either of these properties determines the macroscopic stress induced in a viscoelastic liquid upon the application of a low-amplitude oscillatory shear strain of frequency ω.⁵⁶ These two properties are related by G(ω; t) = iωη(ω; t). An approximate determination of these properties may be provided by the non-equilibrium Stokes–Einstein relation,⁵⁷ ζ(ω; t) ≈ 3πσ × η(ω; t), where ζ(ω; t) is the Fourier–Laplace transform of ζ(τ; t) ≡ ζ₀[2δ(t) + Δζ*(τ; t)], with ζ₀ being the short-time friction coefficient ζ₀ ≡ k_BT/D₀. Thus, η(ω; t) may be approximated by

where η₀ ≡ ζ₀/3πσ is the short-time (or infinite frequency) viscosity. In its zero-frequency limit, this equation leads, using the definition of b(t) in Eq. (18), to the approximate identification of the non-equilibrium static viscosity η(t) ≡ η(ω = 0; t) with the inverse of the mobility function b(t), η(t) ≈ η₀/b(t). This means that the non-equilibrium linear viscosity η(t) is also expected to exhibit the same qualitative behavior as the results for b⁻¹(t; ϕ, T) in Fig. 1(b), which can also, then, be read as the predicted scaled viscosity η(t; ϕ, T)/η₀.

The simple glass-forming model liquid discussed in Sec. III B, formed by identical spherical particles, is, of course, an idealization. It allows us, however, to illustrate important features of the NE-SCGLE description of aging when a single relaxation mechanism is governed by one time-evolving relaxation time. A more interesting situation arises in systems where multiple relaxation mechanisms are active. This is the case, for instance, in dense and highly asymmetric colloidal mixtures of hard spheres⁵⁸ and also in suspensions where the pair interactions between colloids depend on particle’s orientations.^59–61 An interesting question that arises in this context, thus, is if the application of the NE-SCGLE theory to a simple model that incorporates, for example, rotational–translational dynamical coupling (or, alternatively, coupling in the dynamics of multiple species) could also serve to extend TNM’s model of aging to such more complex conditions.

With this idea in mind, in this section, we shall focus on the application of the NE-SCGLE framework to the description of aging in a model system whose interparticle interaction is not radially symmetric. We refer to a dipolar hard-sphere (DHS) fluid,⁵⁵ formed by N axially symmetric identical particles in a volume V, interacting through a hard sphere repulsion of diameter σ, which shall be denoted as U_HS(r₁₂), plus a dipole–dipole interaction so that the total pair potential reads

where r₁ and r₂ are vectors describing the positions of the particles, r₁₂ ≡ (r₂ − r₁), r₁₂ = |r₁₂|, and $r̂12=r12/r12$ and $μ̂1$ and $μ̂2$ are unit vectors indicating the particles’ orientations so that the dipolar tensor $D(r̂12,μ̂1,μ̂2)≡μ̂1⋅μ̂2−3(μ̂1⋅r̂12)(μ̂2⋅r̂12)$ contains all the orientational features of the interaction. A detailed characterization of this system has been provided recently in the context of the NE-SCGLE theory.⁶² Here, we only bring together some of the most important elements in the context of our present discussion.

The non-spherical extension of the NE-SCGLE theory was formulated in Refs. 63 and 64 in terms of a set of equations describing the non-equilibrium evolution of the structural and dynamical properties of a system represented by Eq. (27) and has been systematically employed⁶² to understand the dynamically arrested states already detected in extensive molecular dynamics (MD) simulations of this model.^65,66 We summarize the relevant ingredients of the non-spherical version of the NE-SCGLE theory in the  Appendix.

Similar to the spherical case discussed previously, the most central equation of the extended NE-SCGLE describes the time-evolution of the non-equilibrium structure factor, which for liquids involving non-spherical interactions can be defined as $S(k,μ̂;k′,μ̂′;t)≡⟨δn(k,μ̂;t)δn*(k′,μ̂′;t)⟩$⁠. Here, $δn(k,μ̂;t)$ is the Fourier transform of the fluctuation in the local number density $n(r,μ̂;t)≡N−1/2∑a=1Nδ(r−ra(t))δ(μ̂−μ̂a(t))$ of particles at the position r and with orientation $μ̂$ at time t.⁶⁷ In practice, however, one focuses on the coefficients of the spherical harmonic expansion of $n(k,μ̂;t)$⁠,^63,65–68 commonly referred to as tensorial density modes and denoted as $n̂lm(k,t)$ [see Eq. (A1)]. These coefficients, in turn, allow us to define the corresponding projections of $S(k,μ̂;k′,μ̂′;t)$⁠, namely,^65,66 $Slm:l′m′(k,k′,t)≡⟨n̂lm(k;t)n̂l′m′*(k′;t)⟩$⁠. From symmetry considerations and within well-defined approximations,⁶² only a finite subset of its isotropic and homogeneous diagonal components, denoted as S_lm(k, t) ≡ S_lm:lm(k, −k, t), are involved in the central NE-SCGLE, which reads⁶⁴ 

Just as its spherically symmetric counterpart, Eq. (5), this equation describes the non-equilibrium isochoric response of the DHS system to an instantaneous quench at t = 0 from an arbitrary initial temperature T⁽ⁱ⁾ to a final value T^(f), while the system is constrained to remain spatially homogeneous and isotropic, with fixed bulk number density n = N/V. Alternatively, it also describes the isothermal response after an instantaneous compression from an initial volume fraction ϕ⁽ⁱ⁾ to a final value ϕ^(f).

In Eq. (28), $D0T$ and $D0R$ are, respectively, the translational and rotational free-diffusion coefficients of the constituent dipolar particles. The functions $Elm(f)(k)$⁠, on the other hand, are spherical harmonic projections of the second functional derivative of the free energy with respect to the local number density $n(r,μ̂)$⁠, evaluated at the final state point of a quench, i.e., $Elm(f)(k)≡Elm(k;n(f),T(f))$ (see Subsection 2 of the  Appendix). As mentioned before, this intrinsically thermodynamic property is an externally provided input for the theory, which, for the DHS model, can be determined following the steps outlined in Refs. 62 and 66. The other relevant elements appearing in Eq. (28) are the time-dependent translational and rotational mobility functions b_T(t) and b_R(t). They are calculated with the non-spherical extension of the NE-SCGLE theory summarized in Eqs. (A4)–(A9). We highlight in the following the physical notions underlying the extended Tool–Narayanaswamy–Moynihan model suggested by the non-equilibrium evolution of the tensorial components S_lm(k; t) in Eq. (28).

One simple manner to identify an extended TNM model, inspired in the NE-SCGLE theory for fluids with coupled rotational and translational dynamics, is again based on the recognition of the mathematical structure of Eq. (28), which has exactly the same structure of the generic relaxation equation [Eq. (1)] postulated by Tool, namely,

with the relaxation time τ_lm(t) given by

which can also be written as

with $αlm(k)≡2k2D0Tn̄(f)Elm(k;Tf)$ and $βlm(k)≡2l(l+1)D0Rn̄(f)Elm(k;Tf)$⁠. In fact, Eq. (29) corresponds to the multi-variable extension of Tool’s model, Eq. (14), with a relaxation-rate matrix that is diagonal in (lm), (l′m′).

This immediately allows us to identify the corresponding TNM material time ζ_lm(k; t) for the relaxation of each tensorial component S_lm(k; t) as

i.e., a linear combination of the translational intrinsic material time and a corresponding rotational intrinsic material time,

It is also straightforward to show that the solution of Eq. (29) can be written as $Slm(k;t)=Slm*(k;uT(t),uR(t))$ with the function $Slm*(k;uT,uR)$ defined as

where $Slm(i)(k)≡Slm(k;t=0)$⁠. Note that, in the absence of non-spherical interactions, the only relevant projections reduce to (l = 0, m = 0)-projections, so β_lm(k) = 0 and Eqs. (33)–(35) reduce to Eqs. (6)–(8) for a mono-component system of spherical particles.

Before considering specific results, let us highlight that Eq. (28) already reveals remarkable features regarding the possible non-equilibrium states of the DHS model system defined in Eq. (27) (and of any other that can be generically represented by this equation). The functional dependence of both b_T(t) and b_R(t) on the spherical harmonic projections S_lm(k; t) [see Eqs. (A4)–(A7)] introduces strong non-linearities and, more crucially, implies the existence of two kinds of stationary solutions where ∂S_lm(k, t)/∂t = 0, which are both fundamentally different from equilibrium states.

As mentioned before, we can distinguish ordinary thermodynamic equilibrium conditions, where stationarity is attained because the factor $Slm(k;t)−1/nElm(f)(k)$ on the right-hand side of Eq. (28) vanishes, i.e., because all the spherical harmonic components S_lm(k; t) are able to reach their corresponding thermodynamic equilibrium values, $Slm(eq)(k)=[nElm(f)(k)]−1$⁠, at a finite t. This corresponds to quenches to states where both mobilities b^T(t) and b^R(t) attain finite long-time values $limt→∞bT(t)≡bTeq>0$ and $limt→∞bR(t)≡bReq>0$⁠.

Other categories of stationary solutions of Eq. (28) are those for which the asymptotic value of the kinetic factor $τlm−1(k,t)≡αlm(k)bT(t)+βlm(k)bR(t)$ vanishes for t → ∞. Due to the structure of Eq. (28), this condition can be fulfilled if either both b_T(t) and b_R(t) vanish asymptotically or, for l = 0, only b_T(t) vanishes since β_lm(k) ∝ l vanishes in this case. These two possibilities, in fact, correspond to totally or partially arrested states of matter as predicted by the NE-SCGLE and observed in extensive molecular dynamics simulations:⁶⁵ in the fully arrested state, both mobilities vanish, while in mixed-glass states, rotational degrees of freedom equilibrate, and thus, b_R(t) never vanishes.

Thus, this simple analysis of the possible classes of stationary solutions of Eq. (28) already announces the possibility of only two qualitatively distinct non-equilibrium amorphous states for the DHS model. As just mentioned, the existence of these states has been confirmed recently with the assistance of extensive MD simulations.⁶⁵ In fact, the use of the asymptotic long time solutions of the non-spherical NE-SCGLEs (see Subsection 4 of the  Appendix) allows for the development of the corresponding arrested state diagram of the DHS fluid shown in Fig. 2(a). This diagram identifies the region in the (ϕ, T) plane (with T ≡ k_BTσ³/μ²) corresponding to ergodic states, as well as two regions corresponding to the two aforementioned dynamically arrested states, and the boundaries between these regions. The different arrows in Fig. 1(a) illustrate two qualitatively distinct processes that could be used to bring the DHS system to each of the two arrested states, namely, isochoric quenches toward a fully arrested state (vertical arrows) and isothermal compressions toward partially arrested states (horizontal arrows). As in Fig. 1, solid arrows correspond to equilibration processes in which the final states also lie in the ergodic region (Q1 and C1), whereas the dotted arrows illustrate processes toward dynamically arrest states (Q2 and C2).

The full solution of the NE-SCGLE (outlined in Subsection 3 of the  Appendix) provides the isochoric (or isothermal) evolution for t > 0 of the physical properties involved, for example, the projections S_lm(k; t), the corresponding dynamic correlations F_lm(k, τ; t), and $FlmS(k,τ;t)$⁠, and also the friction memory kernels $ΔζT*(τ;t)$ and $ΔζR*(τ;t)$⁠. The detailed analysis of such of properties, however, is beyond the scope of this contribution and will be addressed elsewhere. Here, we only focus on the non-equilibrium behavior of the mobility functions b_T(t) and b_R(t), which, according to our previous discussion, define the kinetic coefficients $τlm−1(t)$⁠, and also on the behavior of the corresponding intrinsic material times u_T(t) and u_R(t), which allow us to identify the corresponding TNM material time ζ_lm(k, t) of each projection S_lm(k; t).

Figure 2(b) displays the evolution of the two relaxation times $bT−1(t)$ (upper panel) and $bR−1(t)$ (lower panel) for the isochoric quenches Q1 (thick solid lines) and Q2 (thick dashed lines) shown in Fig. 2(a), both starting from the same initial equilibrium state (ϕ = 0.4, T⁽ⁱ⁾ = 1) and ending at distinct final state points (ϕ = 0.4, T^(f) = 0.3) and (ϕ = 0.4, T⁽ⁱ⁾ = 0.05), respectively. Thus, these processes consider state points that lie above and below the transition temperature T_c(ϕ = 0.4) ≈ 0.075. In Fig. 2(b), we also include results for other similar quenches in order to highlight the crossover from equilibration (solid lines, T^(f) > T_c) to aging (dashed lines, T^(f) ≤ T_c). One notes first that, qualitatively, both $bT−1(t)$ and $bR−1(t)$ display the same behavior, i.e., both become increasingly larger with the decreasing final temperature and both reach their corresponding equilibrium values, $1/bTeq(t)$ and $1/bReq(t)$⁠, at a very similar time-scale, which increases with T^(f). As emphasized by the black circles in the upper and lower panels, the equilibration times $tTeq(T(f))$ and $tReq(T(f))$ follow a rather similar trend, increasing asymptotically linearly with the equilibration time, $1/bT,Req(ϕ,T(f))∼teq$⁠.

For quenches at-or-below T_c, instead, one observes an unbounded increase in both $bT−1(t)$ and $bR−1(t)$⁠, thus suggesting their simultaneous divergence as t → ∞. This indicates a gradual and simultaneous decrease in the translational and rotational diffusion (i.e., an increase in the viscosity of the system) and describes, thus, the formation of a fully arrested state in which both the orientational and translational dynamics slow down and undergo aging during the endless process of reaching a glassy state. Let us also mention that an analogous behavior is found for the two α-relaxation times, $τT(α)(k,t)$ and $τR(α)(k,t)$⁠, associated with the functions $F00S(k,t)$ and $F10S(k,t)$⁠, which describe the relaxation of, respectively, the translational and orientational dynamics.

These features are to be contrasted against the behavior that is found for the mobility functions upon isothermal compressions (such as C1 and C2) toward the region of partially arrested states. As shown in Fig. 2(c), for these processes, one observes a clear decoupling in both mobilities: $bT−1(t)$ again shows a crossover from equilibration (for compressions with final volume fraction ϕ^(f) < ϕ_c = 0.582) to an aging regime (ϕ^(f) ≥ ϕ_c = 0.582), whereas $bR−1(t)$ does not change appreciably in the whole volume fraction regime explored and, in fact, reaches an equilibrium value at a remarkably small time-scale. This scenario describes, thus, the formation of a partially arrested state, in which translational diffusion slows down and ages as in the process of formation of a hard-sphere glass, while the orientational dynamics of the dipolar particles remains ergodic, but in a disordered and only slowly evolving configuration of positions.

It is also illustrative to consider the “inverse” functions of the intrinsic material times in Eqs. (33) and (34), namely,

and

where $bT*(uT,uR)≡bT(t(uT,uR))$ and $bR*(uT,uR)≡bR(t(uT,uR))$⁠. One requires, of course, that these two expressions yield the same value of the actual (i.e., real) waiting time, t, so that the differential form

must be satisfied along a (real) physical trajectory [u_T(t), u_R(t)] in the (u_T, u_R)-plane.

Thus, for each process considered (quench or compression), the system traces a corresponding physical trajectory of points [u_T(t), u_R(t)] on the material-time plane (u_T, u_R) as the waiting time evolves from t = 0 toward t → ∞. Some general features of these trajectories can be implied from the inspection of the possible long-time limits, $bTa≡limt→∞bT(t)$ and $bRa≡limt→∞bR(t)$⁠.

For instance, if a given protocol drives the system to an arbitrary equilibrium state (e.g., Q1 and C1), then both b_T(t) and b_R(t) eventually reach finite equilibrium values $bTeq>0$ and $bReq>0$ within a finite waiting time window. Then, Eqs. (33) and (34) imply that both material times will evolve without bound as t runs from 0 to infinity so that the physical trajectory will start at the origin, (u_T, u_R) = (0, 0), and will tend to (u_T, u_R) = (∞, ∞). At asymptotically long times, the physical trajectory in the (u_T, u_R) plane will be a straight line with a prefactor that is determined by the specific quench. This is illustrated by the blue and purple solid lines in Figs. 3(c) and 3(f).

If, instead, the protocol drives the system to a fully arrested state (e.g., Q2), both asymptotic values $bTa$ and $bRa$ will be zero. In this scenario, Eqs. (33) and (34) imply that the real trajectory starts again at (u_T, u_R) = (0, 0) but now ends at a point $(uTa,uRa)$⁠, with $uTa≈∫0tabTt′dt′$ and $uRa≈∫0tabRt′dt′$⁠. This case is illustrated by the dashed line in Fig. 3(c).

Finally, a third possibility is a partially arrested state where $bTa$ vanishes, but $bRa$ attains an equilibrium value. In this case, u_T(t) becomes finite, while u_R(t) increases without bound; see the dashed line in Fig. 3(f).

We have discussed the concept of a material time in the context of aging in glasses, i.e., of the slow evolution of the physical properties of a system after a quench in control parameters. In particular, we have shown how the empirical Tool–Narayanaswamy–Moynihan model of aging can be seen as arising from the mathematical structure of equations that is suggested by the non-equilibrium extension of the self-consistent Langevin equation theory, NE-SCGLE. A key concept in this identification is that of a kinetic state function. Specifically, the theory suggests a structure where the relaxation rate of evolving observables during aging is expressed as a functional of the waiting-time dependent static structure function.

The generic mathematical structure of the TNM relaxation equation allows for two distinct stationary states: those corresponding to equilibration, where upon a quench within the ergodic-liquid state of a system, all relaxation rates remain non-vanishing, and those corresponding to quenches into the (ideal) glassy state, where the relaxation rates vanish. In the two cases, the material time evolves distinctly: upon equilibration, the inner clock of the material evolves asymptotically proportional to the physical time after the quench. For arrested states, the material time attains a finite asymptotic value.

More intriguingly, the extension of the NE-SCGLE to include multiple relaxation modes—exemplified in the present contribution through a model with distinct translational and rotational degrees of freedom—highlights the possibility of a third kind of stationary state, that of partial equilibration. Here, the aging process is characterized by two material times: one of which attains a finite long-time limit, while the other grows asymptotically proportionally with the real waiting time.

In principle, this highlights that a “material time” can only be defined with respect to a given observable, and not all observables need to follow the same internal clock; considering a set of observables that undergo aging, a corresponding set of material times would, in principle, be obtained. However, the NE-SCGLE theory encodes an interesting factorization property: for example, when considering the length-scale dependent relaxation of density fluctuations, the information on the length scale can be absorbed into a wave-number dependent prefactor. We expect, thus, that the aging process of a glass former, in general, is controlled by a (small) set of intrinsic material times u_i corresponding to structurally distinct relaxation channels—such as those connected to translational and rotational degrees of freedom in a system with anisotropic interactions or those related to different particle species in mixtures of very disparately sized particles. The final material properties of a quenched glass are, thus, characterized by a vector u^a of the asymptotic values of these intrinsic material times (some components possibly infinite in the case of partial equilibration), and the aging process is represented as a one-dimensional curve in the space spanned by the intrinsic material time components, parameterized by the physical waiting time.

We expect that the fundamental and universal laws of irreversible thermodynamics and the microscopic perspective provided by statistical mechanical theories of irreversible processes, such as the NE-SCGLE theory, may be instrumental in extending the descriptive power of the TNM model beyond its current limitations. Similarly, the use of the NE-SCGLE formalism should also enrich the discussion of other phenomenological concepts of great practical and fundamental relevance, such as the concepts of time–temperature superposition (TTS) and thermo-rheological simplicity,^43,44 and the notions of fictive temperature and effective time.⁹ In the absence of fundamental principles that allow for the identification of their microscopic origin, it has been difficult to assess their validity and to establish a clear relationship between them, with the concept of material clock. In this regard, it is important to mention that the TTS scaling for equilibrium liquids is a distinct prediction of mode-coupling theory^3,70–72 (and hence, also of the original equilibrium SCGLE theory⁴⁹). These first-principles equilibrium theories break down in the very high viscosity regime. Future work will address the combination of the NE-SCGLE formalism with extensions of MCT-like theories that account for the avoidance of the ideal glass transition.⁷³ This combination is expected to open a route to extend equilibrium glass-transition theories to the non-equilibrium regime and, hence, to provide a fundamental basis of a non-equilibrium extension of the notion of TTS and of its possible connection with the concept of a material clock.