Crossover from equilibration to aging: Nonequilibrium theory versus simulations

Non-equilibrium view of the amorphous solidification of liquids with competing interactions

The interplay between short-range attractions and long-range repulsions (SALR) characterizes the so-called liquids with competing interactions, which are known to exhibit a variety of equilibrium and non-equilibrium phases. The theoretical description of the phenomenology associated with glassy or gel states in these systems has to take into account both the presence of thermodynamic instabilities (such as those defining the spinodal line and the so called λ line) and the limited capability to describe genuine non-equilibrium processes from first principles. Here, we report the first application of the non-equilibrium self-consistent generalized Langevin equation theory to the description of the dynamical arrest processes that occur in SALR systems after being instantaneously quenched into a state point in the regions of thermodynamic instability. The physical scenario predicted by this theory reveals an amazing interplay between the thermodynamically driven instabilities, favoring equilibrium macro- and micro-phase separation, and the kinetic arrest mechanisms, favoring non-equilibrium amorphous solidification of the liquid into an unexpected variety of glass and gel states.

Phase separation and dynamical arrest of protein solutions dominated by short-range attractions

The interplay of liquid-liquid phase separation (LLPS) and dynamical arrest can lead to the formation of gels and glasses, which is relevant for such diverse fields as condensed matter physics, materials science, food engineering, and the pharmaceutical industry. In this context, protein solutions exhibit remarkable equilibrium and non-equilibrium behaviors. In the regime where attractive and repulsive forces compete, it has been demonstrated, for example, that the location of the dynamical arrest line seems to be independent of ionic strength, so that the arrest lines at different ionic screening lengths overlap, in contrast to the LLPS coexistence curves, which strongly depend on the salt concentration. In this work, we show that the same phenomenology can also be observed when the electrostatic repulsions are largely screened, and the range and strength of the attractions are varied. In particular, using lysozyme in brine as a model system, the metastable gas-liquid binodal and the dynamical arrest line as well as the second virial coefficient have been determined for various solution conditions by cloud-point measurements, optical microscopy, centrifugation experiments, and light scattering. With the aim of understanding this new experimental phenomenology, we apply the non-equilibrium self-consistent generalized Langevin equation theory to a simple model system with only excluded volume plus short-range attractions, to study the dependence of the predicted arrest lines on the range of the attractive interaction. The theoretical predictions find a good qualitative agreement with experiments when the range of the attraction is not too small compared with the size of the protein.

Structural relaxation, dynamical arrest, and aging in soft-sphere liquids

We investigate the structural relaxation of a soft-sphere liquid quenched isochorically (ϕ = 0.7) and instantaneously to different temperatures T_f above and below the glass transition. For this, we combine extensive Brownian dynamics simulations and theoretical calculations based on the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory. The response of the liquid to a quench generally consists of a sub-linear increase of the α-relaxation time with system's age. Approaching the ideal glass-transition temperature from above (T_f > T^a), sub-aging appears as a transient process describing a broad equilibration crossover for quenches to nearly arrested states. This allows us to empirically determine an equilibration timescale t^eq(T_f) that becomes increasingly longer as T_f approaches T^a. For quenches inside the glass (T_f ≤ T^a), the growth rate of the structural relaxation time becomes progressively larger as T_f decreases and, unlike the equilibration scenario, τ_α remains evolving within the whole observation time-window. These features are consistently found in theory and simulations with remarkable semi-quantitative agreement and coincide with those revealed in a previous and complementary study [P. Mendoza-Méndez et al., Phys. Rev. 96, 022608 (2017)] that considered a sequence of quenches with fixed final temperature T_f = 0 but increasing ϕ toward the hard-sphere dynamical arrest volume fraction ϕ_HS ^a=0.582. The NE-SCGLE analysis, however, unveils various fundamental aspects of the glass transition, involving the abrupt passage from the ordinary equilibration scenario to the persistent aging effects that are characteristic of glass-forming liquids. The theory also explains that, within the time window of any experimental observation, this can only be observed as a continuous crossover.

From Subaging to Hyperaging in Structural Glasses

We demonstrate nonequilibrium scaling laws for the aging and equilibration dynamics in glass formers that emerge from combining a relaxation equation for the static structure with the equilibrium scaling laws of glassy dynamics. Different scaling regimes are predicted for the evolution of the structural relaxation time τ with age (waiting time t_{w}), depending on the depth of the quench from the liquid into the glass: "simple" aging (τ∼t_{w}) applies for quenches close to the critical point of mode-coupling theory (MCT) and implies "subaging" (τ≈t_{w}^{δ} with δ<1) as a broad equilibration crossover for quenches to nearly arrested equilibrium states; "hyperaging" (or superaging, τ∼t_{w}^{δ^{'}} with δ^{'}>1) emerges for quenches deep into the glass. The latter is cut off by non-mean-field fluctuations that we account for within a recent extension of MCT, the stochastic β-relaxation theory (SBR). We exemplify the scaling laws with a schematic model that quantitatively fits simulation data.

Distance-as-time in physical aging

Although it has been known for half a century that the physical aging of glasses in experiments is described well by a linear thermal-history convolution integral over the so-called material time, the microscopic definition and interpretation of the material time remains a mystery. We propose that the material-time increase over a given time interval reflects the distance traveled by the system's particles. Different possible distance measures are discussed, starting from the standard mean-square displacement and its inherent-state version that excludes the vibrational contribution. The viewpoint adopted, which is inspired by and closely related to pioneering works of Cugliandolo and Kurchan from the 1990s, implies a "geometric reversibility" and a "unique-triangle property" characterizing the system's path in configuration space during aging. Both of these properties are inherited from equilibrium, and they are here confirmed by computer simulations of an aging binary Lennard-Jones system. Our simulations moreover show that the slow particles control the material time. This motivates a "dynamic-rigidity-percolation" picture of physical aging. The numerical data show that the material time is dominated by the slowest particles' inherent mean-square displacement, which is conveniently quantified by the inherent harmonic mean-square displacement. This distance measure collapses data for potential-energy aging well in the sense that the normalized relaxation functions following different temperature jumps are almost the same function of the material time. Finally, the standard Tool-Narayanaswamy linear material-time convolution-integral description of physical aging is derived from the assumption that when time is replaced by distance in the above sense, an aging system is described by the same expression as that of linear-response theory.

"Inner clocks" of glass-forming liquids

Providing a physically sound explanation of aging phenomena in non-equilibrium amorphous materialsis a challenging problem in modern statistical thermodynamics. The slow evolution of physical propertiesafter quenches of control parameters is empirically well interpreted via the concept of material time (orinternal clock), based on the Tool-Narayanaswamy-Moynihan (TNM) model. Yet, the fundamental reasonsof its striking success remain unclear. We propose a microscopic rationale behind the material time onthe basis of the linear laws of irreversible thermodynamics and its extension that treats the correspondingkinetic coefficients as state functions of a slowly evolving material state. Our interpretation is based onthe recognition that the same mathematical structure governs both the Tool model and the recently devel-oped non-equilibrium extension of the self-consistent generalized Langevin equation theory (NE-SCGLE),guided by the universal principles of Onsager's theory of irreversible processes. This identification opensthe way for a generalization of the material-time concept to aging systems where several relaxation modeswith very different equilibration processes must be considered, and partially frozen glasses manifest theappearance of partial ergodicity breaking, and hence materials with multiple very distinct inner clocks.

Non-equilibrium relaxation and aging in the dynamics of a dipolar fluid quenched towards the glass transition

The recently developed non-equilibrium self-consistent generalized Langevin equation theory of the dynamics of liquids of non-spherically interacting particles [2016J. Phys. Chem. B1207975] is applied to the description of the irreversible relaxation of a thermally and mechanically quenched dipolar fluid. Specifically, we consider a dipolar hard-sphere liquid quenched (att_w= 0) from full equilibrium conditions towards different ergodic-non-ergodic transitions. Qualitatively different scenarios are predicted by the theory for the time evolution of the system after the quench (t_w> 0), that depend on both the kind of transition approached and the specific features of the protocol of preparation. Each of these scenarios is characterized by the kinetics displayed by a set of structural correlations, and also by the development of two characteristic times describing the relaxation of the translational and rotational dynamics, allowing us to highlight the crossover from equilibration to aging in the system and leading to the prediction of different underlying mechanisms and relaxation laws for the dynamics at each of the glass transitions explored.

On a fundamental description of the Kovacs' kinetic signatures in glass-forming systems

The time-evolution equation for the time-dependent static structure factor of the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory was used to investigate the kinetics of glass-forming systems under isochoric conditions. The kinetics are studied within the framework of the fictive temperature (T_F) of the glassy structure. We solve for the kinetics of T_F(t) and the time-dependent structure factor and find that they are different but closely related by a function that depends only on temperature. Furthermore, we are able to solve for the evolution of T_F(t) in a set of temperature-jump histories referred to as the Kovacs' signatures. We demonstrate that the NE-SCGLE theory reproduces all the Kovacs' signatures, namely, intrinsic isotherm, asymmetry of approach, and memory effect. In addition, we extend the theory into largely unexplored, deep glassy state, regions that are below the notionally "ideal" glass temperature.

Vitrification is a spontaneous non-equilibrium transition driven by osmotic pressure

Persistent dynamics in colloidal glasses suggest the existence of a non-equilibrium driving force for structural relaxation during glassy aging. But the implicit assumption in the literature that colloidal glasses form within the metastable state bypasses the search for a driving force for vitrification and glassy aging and its connection with a metastable state. The natural relation of osmotic pressure to number-density gradients motivates us to investigate the osmotic pressure as this driving force. We use dynamic simulation to quench a polydisperse hard-sphere colloidal liquid into the putative glass region while monitoring structural relaxation and osmotic pressure. Following quenches to various depths in volume fractionϕ(whereϕ_RCP≈ 0.678 for 7% polydispersity), the osmotic pressure overshoots its metastable value, then decreases with age toward the metastable pressure, driving redistribution of coordination number and interparticle voids that smooths structural heterogeneity with age. For quenches to 0.56 ⩽ϕ⩽ 0.58, accessible post-quench volume redistributes with age, allowing the glass to relax into a strong supercooled liquid and easily reach a metastable state. At higher volume fractions, 0.59 ⩽ϕ< 0.64, this redistribution encounters a barrier that is subsequently overcome by osmotic pressure, allowing the system to relax toward the metastable state. But forϕ⩾ 0.64, the overshoot is small compared to the high metastable pressure; redistribution of volume stops as particles acquire contacts and get stuck, freezing the system far from the metastable state. Overall, the osmotic pressure drives structural rearrangements responsible for both vitrification and glassy age-relaxation. The connection of energy, pressure, and structure identifies the glass transition, 0.63 <ϕ_g⩽ 0.64. We leverage the connection of osmotic pressure to energy density to put forth the mechanistic view that relaxation of structural heterogeneity in colloidal glasses occurs via individual particle motion driven by osmotic pressure, and is a spontaneous energy minimization process that drives the glass off and back to the metastable state.

"Dense diffusion" in colloidal glasses: short-ranged long-time self-diffusion as a mechanistic model for relaxation dynamics

Despite decades of exploration of the colloidal glass transition, mechanistic explanation of glassy relaxation processes has remained murky. State-of-the-art theoretical models of the colloidal glass transition such as random first order transition theory, active barrier hopping theory, and non-equilibrium self-consistent generalized Langevin theory assert that relaxation reported at volume fractions above the ideal mode coupling theory prediction φg,MCT requires some sort of activated process, and that cooperative motion plays a central role. However, discrepancies between predicted and measured values of φg and ambiguity in the role of cooperative dynamics persist. Underlying both issues is the challenge of conducting deep concentration quenches without flow and the difficulty in accessing particle-scale dynamics. These two challenges have led to widespread use of fitting methods to identify divergence, but most a priori assume divergent behavior; and without access to detailed particle dynamics, it is challenging to produce evidence of collective dynamics. We address these limitations by conducting dynamic simulations accompanied by experiments to quench a colloidal liquid into the putative glass by triggering an increase in particle size, and thus volume fraction, at constant particle number density. Quenches are performed from the liquid to final volume fractions 0.56 ≤ φ ≤ 0.63. The glass is allowed to age for long times, and relaxation dynamics are monitored throughout the simulation. Overall, correlated motion acts to release dynamics from the glassy plateau - but only over length scales much smaller than a particle size - allowing self-diffusion to re-emerge; self-diffusion then relaxes the glass into an intransient diffusive state, which persists for φ < 0.60. We observe similar relaxation dynamics up to φ = 0.63 before achieving the intransient state. We find that this long-time self-diffusion is short-ranged: analysis of mean-square displacement reveals a glassy cage size a fraction of a particle size that shrinks with quench depth, i.e. increasing volume fraction. Thus the equivalence between cage size and particle size found in the liquid breaks down in the glass, which we confirm by examining the self-intermediate scattering function over a range of wave numbers. The colloidal glass transition can hence be viewed mechanistically as a shift in the long-time self-diffusion from long-ranged to short-ranged exploration of configurations. This shift takes place without diverging dynamics: there is a smooth transition as particle mobility decreases dramatically with concomitant emergence of a dense local configuration space that permits sampling of many configurations via local particle motion.

Glass-transition asymptotics in two theories of glassy dynamics: Self-consistent generalized Langevin equation and mode-coupling theory

We contrast the generic features of structural relaxation close to the idealized glass transition that are predicted by the self-consistent generalized Langevin equation theory (SCGLE) against those that are predicted by the mode-coupling theory of the glass transition (MCT). We present an asymptotic solution close to conditions of kinetic arrest that is valid for both theories, despite the different starting points that are adopted in deriving them. This in particular provides the same level of understanding of the asymptotic dynamics in the SCGLE as was previously done only for MCT. We discuss similarities and different predictions of the two theories for kinetic arrest in standard glass-forming models, as exemplified through the hard-sphere system. Qualitative differences are found for models where a decoupling of relaxation modes is predicted, such as the generalized Gaussian core model, or binary hard-sphere mixtures of particles with very disparate sizes. These differences, which arise in the distinct treatment of the memory kernels associated to self- and collective motion of particles, lead to distinct scenarios that are predicted by each theory for partially arrested states and in the vicinity of higher-order glass-transition singularities.

Interference between the glass, gel, and gas-liquid transitions

Recent experiments and computer simulations have revealed intriguing phenomenological fingerprints of the interference between the ordinary equilibrium gas-liquid phase transition and the non-equilibrium glass and gel transitions. We thus now know, for example, that the liquid-gas spinodal line and the glass transition loci intersect at a finite temperature and density, that when the gel and the glass transitions meet, mechanisms for multistep relaxation emerge, and that the formation of gels exhibits puzzling latency effects. In this work we demonstrate that the kinetic perspective of the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory of irreversible processes in liquids provides a unifying first-principles microscopic theoretical framework to describe these and other phenomena associated with spinodal decomposition, gelation, glass transition, and their combinations. The resulting scenario is in reality the competition between two kinetically limiting behaviors, associated with the two distinct dynamic arrest transitions in which the liquid-glass line is predicted to bifurcate at low densities, below its intersection with the spinodal line.

Glassy dynamics in asymmetric binary mixtures of hard spheres

We perform a systematic and detailed study of the glass transition in highly asymmetric binary mixtures of colloidal hard spheres, combining differential dynamic microscopy experiments, event-driven molecular dynamics simulations, and theoretical calculations, exploring the whole state diagram and determining the self-dynamics and collective dynamics of both species. Two distinct glassy states involving different dynamical arrest transitions are consistently described, namely, a double glass with the simultaneous arrest of the self-dynamics and collective dynamics of both species, and a single glass of large particles in which the self-dynamics of the small species remains ergodic. In the single-glass scenario, spatial modulations in the collective dynamics of both species occur due to the structure of the large spheres, a feature not observed in the double-glass domain. The theoretical results, obtained within the self-consistent generalized Langevin equation formalism, are in agreement with both simulations and experimental data, thus providing a stringent validation of this theoretical framework in the description of dynamical arrest in highly asymmetric mixtures. Our findings are summarized in a state diagram that classifies the various amorphous states of highly asymmetric mixtures by their dynamical arrest mechanisms.

Glassy relaxation slows down by increasing mobility

We find a striking trend reversal in the relaxation dynamics of mixtures with strong dynamical asymmetry. Simulations by both Brownian and Newtonian dynamics reveal that in mixtures of fast and slow hard spheres, above a critical density, the dynamics becomes slower upon increasing the mobility of the fast particles. Below that density, the same increase in mobility speeds up the dynamics. The critical density itself can be identified with the glass transition of the mode-coupling theory that does not depend on the dynamical asymmetry. The asymptotic dynamics close to the critical density is universal, but strong pre-asymptotic effects prevail in particular when the dynamical asymmetry also involves size asymmetry. Our observations reconcile earlier findings, where a strong dependence on kinetic parameters was found for the glassy dynamics, with the paradigm that the glass transition is determined by the properties of configuration space alone.

Cage dynamics leads to double relaxation of the intermediate scattering function in a binary colloidal system

A system of binary colloids where one fraction of particles is aggregating by forming irreversible bonds and the other fraction of particles only interacts as hard spheres, is simulated using Brownian cluster dynamics. These aggregating species always formed percolating clusters for the case of diffusing hard spheres while for the static case, formation of percolating clusters depended on the fraction of static hard spheres in the system. The dynamics of the hard spheres inside the percolating clusters was studied by restarting the simulation after the kinetics of aggregation was arrested. Two cases were studied, one where the percolated particles moved within the bonds or cage dynamics was allowed and another where the movement within the bonds was not allowed or the cages were static. The hard spheres showed anomalous diffusion in both cases. The mean square displacement showed that for the case of dynamic cages we always had diffusive hard spheres irrespective of the fraction of hard spheres for volume fractions below 0.49. Static cages, depending on the fraction of hard spheres, showed either diffusive or arrested behavior of hard spheres. The intermediate scattering function of only the hard sphere particles showed double relaxation similar to the colloidal glass system for low volume fraction, where the fraction of hard sphere particles was small. For higher fractions we observed only a single stretched exponential. We could differentiate between slow and fast particles for both static and dynamic cages. For the case of static cages the hard spheres were permanently stuck inside the cages while for the case of dynamic cages almost all the hard spheres were moving in and out of the cages.