Dynamic entropy as a measure of caging and persistent particle motion in supercooled liquids

A dynamical system approach to relaxation in glass-forming liquids

The "classical" thermodynamic and statistical mechanical theories of Gibbs and Boltzmann are both predicated on axiomatic assumptions whose applicability is hard to ascertain. Theoretical objections and an increasing number of observed deviations from these theories have led to sustained efforts to develop an improved mathematical and physical foundation for them, and the search for appropriate extensions that are generally applicable to condensed materials at low temperatures (T) and high material densities where the assumptions of these theories start to become particularly questionable. These theoretical efforts have largely focused on minimal models of condensed material systems, such as the Fermi-Ulam-Pasta-Tsingou model, and other simplified models of condensed materials that are amenable to numerical and analytic treatments and that can serve to illuminate essential features of relaxation processes in condensed materials under conditions approaching integrable dynamics where clear departures from classical thermodynamics and dynamics can be generally expected. These studies indicate an apparently general multi-step relaxation process, corresponding to an initial "fast" relaxation process (termed the fast β-relaxation in the context of cooled liquids), followed by a longer "equipartition time", namely, the α-relaxation time τα in the context of cooled liquids. This relaxation timescale can be enormously longer than the fast β-relaxation time τβ so that τα is the primary parameter governing the rate at which the material comes into equilibrium, and thus is a natural focus of theoretical attention. Since the dynamics of these simplified dynamical systems, originally intended as simplified models of real crystalline materials exhibiting anharmonic interactions, greatly resemble the observed relaxation dynamics of both heated crystals and cooled liquids, we adapt this dynamical system approach to the practical matter of estimating relaxation times in both cooled liquids and crystals at elevated temperatures, which we identify as weakly non-integrable dynamical systems.

Dynamic Entropy of Two-Dimensional Active Brownian Systems in Colloidal Plasmas

We analyze the experimental data on the motion of active Brownian micrograins in RF discharge plasmas. In the experiments, two types of microparticles were used: first—plastic grains fully covered with metal, and second—Janus particles with a thin metal cap. We have tracked the trajectories of the separate grains and plotted the pair correlation functions of the observed structures. To examine the motion of the grains, we studied the dependencies of the MFPT dynamic entropy on the coarsening parameter, the fractal dimension of the system on its mean kinetic temperature, and the mean localization area of the grain on its mean kinetic temperature. Based on the obtained results, we conclude that the character of motion of our active Brownian systems changes as the power of an illuminating laser (and, therefore, the mean kinetic temperature of the grains) increases. Janus particles change their trajectories from more chaotic to spiral-like ones; in the case of fully covered particles, we observe the dynamical phase transition from the more ordered structure to the less ordered one.

Breakdown of the Stokes-Einstein relation in supercooled liquids: A cage-jump perspective

The breakdown of the Stokes-Einstein relation in supercooled liquids, which is the increase in the ratio τ_ατ_D between the two macroscopic times for structural relaxation and diffusion on decreasing the temperature, is commonly ascribed to dynamic heterogeneities, but a clear-cut microscopic interpretation is still lacking. Here, we tackle this issue exploiting the single-particle cage-jump framework to analyze molecular dynamics simulations of soft disk assemblies and supercooled water. We find that τ_ατ_D∝⟨t_p⟩⟨t_c⟩, where ⟨t_p⟩ and ⟨t_c⟩ are the cage-jump times characterizing slow and fast particles, respectively. We further clarify that this scaling does not arise from a simple term-by-term proportionality; rather, the relations τ_α∝⟨t_p⟩⟨Δr_J ²⟩ and τ_D∝⟨t_c⟩⟨Δr_J ²⟩ effectively connect the macroscopic and microscopic timescales, with the mean square jump length ⟨Δr_J ²⟩ shrinking on cooling. Our work provides a microscopic perspective on the Stokes-Einstein breakdown and generalizes previous results on lattice models to the case of more realistic glass-formers.

Physical aging and compressed exponential behaviors in a model soft colloidal system

Diffusing wave spectroscopy (DWS)-based micro-rheology has been used in different optical geometries (backscattering and transmission) as well as different sample thicknesses in order to probe system dynamics at different length scales [D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer. J. Phys., 1990, 51(18), 2101-2127]. Previous study from this lab [Q. Li, X. Peng, G. B. McKenna. Soft Matter, 2017, 13(7), 1396-1404] indicates the DWS-based micro-rheology observes the system non-equilibrium behaviors differently from macro-rheology. The object of the present work was to further explore the non-equilibrium dynamics and to address the range of utility of DWS as a micro-rheological method. A thermo-sensitive core-shell colloidal system was investigated both during aging and subsequent to aging into a metastable equilibrium state using temperature-jump induced volume fraction-jump experiments. We find that in the non-equilibrium state, significant differences in the measured dynamics are observed for the different geometries and length scales. Compressed exponential relaxations for the autocorrelation function g2(t) were observed for large length scales. However, upon converting the g2(t) data to the mean square displacement (MSD), such differences with length scale diminished and the long-time MSD behavior was consistent with diffusive behavior. These observations in the non-equilibrium behaviors for different length scales leads to questioning of some interpretations in the current field of light scattering-based micro-rheology and provides a possibility to interrogate the aging mechanisms in colloidal glasses from a broader perspective than normally considered in measurements of g2(t) using DWS-based micro-rheology.

Perspective: Excess-entropy scaling

This article gives an overview of excess-entropy scaling, the 1977 discovery by Rosenfeld that entropy determines properties of liquids like viscosity, diffusion constant, and heat conductivity. We give examples from computer simulations confirming this intriguing connection between dynamics and thermodynamics, counterexamples, and experimental validations. Recent uses in application-related contexts are reviewed, and theories proposed for the origin of excess-entropy scaling are briefly summarized. It is shown that if two thermodynamic state points of a liquid have the same microscopic dynamics, they must have the same excess entropy. In this case, the potential-energy function exhibits a symmetry termed hidden scale invariance, stating that the ordering of the potential energies of configurations is maintained if these are scaled uniformly to a different density. This property leads to the isomorph theory, which provides a general framework for excess-entropy scaling and illuminates, in particular, why this does not apply rigorously and universally. It remains an open question whether all aspects of excess-entropy scaling and related regularities reflect hidden scale invariance in one form or other.

Particle localization and hyperuniformity of polymer-grafted nanoparticle materials

The properties of materials largely reflect the degree and character of the localization of the molecules comprising them so that the study and characterization of particle localization has central significance in both fundamental science and material design. Soft materials are often comprised of deformable molecules and many of their unique properties derive from the distinct nature of particle localization. We study localization in a model material composed of soft particles, hard nanoparticles with grafted layers of polymers, where the molecular characteristics of the grafted layers allow us to "tune" the softness of their interactions. Soft particles are particular interesting because spatial localization can occur such that density fluctuations on large length scales are suppressed, while the material is disordered at intermediate length scales; such materials are called "disordered hyperuniform". We use molecular dynamics simulation to study a liquid composed of polymer-grafted nanoparticles (GNP), which exhibit a reversible self-assembly into dynamic polymeric GNP structures below a temperature threshold, suggesting a liquid-gel transition. We calculate a number of spatial and temporal correlations and we find a significant suppression of density fluctuations upon cooling at large length scales, making these materials promising for the practical fabrication of "hyperuniform" materials.

Heterogeneous diffusion, viscosity, and the Stokes-Einstein relation in binary liquids

We investigate the origin of the breakdown of the Stokes-Einstein relation (SER) between diffusivity and viscosity in undercooled melts. A binary Lennard-Jones system, as a model for a metallic melt, is studied by molecular dynamics. A weak breakdown at high temperatures can be understood from the collectivization of motion, seen in the isotope effect. The strong breakdown at lower temperatures is connected to an increase in dynamic heterogeneity. On relevant time scales some particles diffuse much faster than the average or than predicted by the SER. The van Hove self-correlation function allows one to unambiguously identify slow particles. Their diffusivity is even less than predicted by the SER. The time span of these particles being slow particles, before their first conversion to be a fast one, is larger than the decay time of the stress correlation. The contribution of the slow particles to the viscosity rises rapidly upon cooling. Not only the diffusion but also the viscosity shows a dynamically heterogeneous scenario. We can define a "slow" viscosity. The SER is recovered as the relation between slow diffusivity and slow viscosity.

Consequences of anomalous diffusion in disordered systems under cyclic forcing

We study the particle scale response of a 2D frictionless disk system to bulk forcing via cyclic shear with reversal amplitude γ_{r}. We find a subdiffusive γ_{r}-dependent regime, which is consistent with models of anomalous diffusion with scale-invariant cage dynamics, and a crossover to diffusive grain motion at high γ_{r}. Analysis of local displacements of a particle relative to its cage of neighbors reveals a key distinction from thermal systems. Particles are moved by fluctuations of their cage of neighbors rather than rattling in their cage, indicating a distinct cage-breaking mechanism.

Glassy Interfacial Dynamics of Ni Nanoparticles: Part I Colored Noise, Dynamic Heterogeneity and Collective Atomic Motion

Most condensed materials exhibit a significant fraction of atoms, molecules or particles that are strongly interacting with each other, while being configured geometrically at any instant of time in an 'amorphous' state having a relatively uniform density. Recently, both simulations and experiments have revealed that the dynamics of diverse condensed amorphous materials is generally characterized by significant heterogeneity in the local mobility and by progressively increasing collective motion upon cooling that takes the form of string-like collective particle rearrangements. The direct experimental observation of this type of collective motion, which has been directly linked to the growing relaxation times of glass-forming materials, and its quantification under different thermodynamic conditions, has so far been restricted to colloidal and driven granular fluids. The present work addresses the fundamental problem of how to determine the scale of this type of collective motion in materials composed of molecules or atoms. The basic premise of our work is that large scale dynamic particle clustering in amorphous materials must give rise to large fluctuations in particle mobility so that transport properties, especially those related to particle mobility, should naturally exhibit noise related to the cooperative motion scale. In our initial exploratory study seeking a relationship of this kind, we find 1/fα or 'colored noise', in both potential energy and particle displacements fluctuations of the atoms within the glassy interfacial layer of Ni nanoparticles (NPs). A direct relation between the particle displacement (mobility) noise exponent α and the average polymerization index of the string-like collective motion L is observed for a range of NP sizes, temperatures and for surface doping of the NPs with other metal atoms (Ag, Au, Pt) to change of fragility of the glassy interfacial layer at the surface of the Ni NPs. We also introduce a successful analytic model to understand this relationship between α and L.

Manifestations of dynamical facilitation in glassy materials

By characterizing the dynamics of idealized lattice models with a tunable kinetic constraint, we explore the different ways in which dynamical facilitation manifests itself within the local dynamics of glassy materials. Dynamical facilitation is characterized both by a mobility transfer function, the propensity for highly mobile regions to arise near regions that were previously mobile, and by a facilitation volume, the effect of an initial dynamical event on subsequent dynamics within a region surrounding it. Sustained bursts of dynamical activity-avalanches-are shown to occur in kinetically constrained models, but, contrary to recent claims, we find that the decreasing spatiotemporal extent of avalanches with increased supercooling previously observed in granular experiments does not imply diminishing facilitation. Viewed within the context of existing simulation and experimental evidence, our findings show that dynamical facilitation plays a significant role in the dynamics of systems investigated over the range of state points accessible to molecular simulations and granular experiments.

Cooperativity at different space and time scales in multiscale protein dynamics

A method proposed by Matsunaga [Phys. Rev. Lett. 99, 238103 (2007)] is applied to simple stochastic models and two model proteins composed of 46 amino beads with three different kinds of residues. The method, which is based on the combination of the principal component analysis and the finite size Lyapunov exponent, characterize the coarse-grained dynamics in different spatiotemporal hierarchies in protein dynamics. The application of the method to model proteins reveals that the low-indexed (large-variance) principal components carry less-divergent, regularized dynamics at the coarse-grained scales on a less-frustrated energy landscape, whereas this less-divergent nature is less pronounced for a protein model with a more frustrated energy landscape. It is also revealed that our technique can differentiate the collective motions on the projected principal component space inherent to the system and the apparent collective behavior which can appear even in high-dimensional stochastic systems.

Decoupling of exchange and persistence times in atomistic models of glass formers

With molecular dynamics simulations of a fluid mixture of classical particles interacting with pairwise additive Weeks-Chandler-Andersen potentials, we consider the time series of particle displacements and thereby determine the distributions for local persistence times and local exchange times. These basic characterizations of glassy dynamics are studied over a range of supercooled conditions and were shown to have behaviors, most notably decoupling, similar to those found in kinetically constrained lattice models of structural glasses. Implications are noted.

Dynamical first-order phase transition in kinetically constrained models of glasses

We show that the dynamics of kinetically constrained models of glass formers takes place at a first-order coexistence line between active and inactive dynamical phases. We prove this by computing the large-deviation functions of suitable space-time observables, such as the number of configuration changes in a trajectory. We present analytic results for dynamic facilitated models in a mean-field approximation, and numerical results for the Fredrickson-Andersen model, the East model, and constrained lattice gases, in various dimensions. This dynamical first-order transition is generic in kinetically constrained models, and we expect it to be present in systems with fully jammed states.

Local polymer dynamics under strong connectivity constraints: The dendrimer case

The characteristics of local motion are explored by molecular dynamics simulations in a series of AB(2)-type dendrimer melts. Systems of generations 3-5 were simulated in a wide temperature range, allowing the assessment of effects associated with molecular size, proximity to the detected glasslike transitions, and the strong connectivity constraints imposed by the dendritic topology. Investigation of the mechanisms involved in local motion at short temporal and spatial scales revealed the connection between the non-Gaussian nature of monomer displacements to alpha-relaxation and the caging/decaging process under different degrees of confinement. In the latter mechanism, two characteristic localization lengths were identified: at the low temperature limit spatial localization was realized within approximately 10% of the nearest neighbor distance while at temperatures higher than the glass transition, the existence of an analogous length scale is ascribed to the geometric constraints due to the dense connectivity pattern. As the results from this study are discussed in comparison to the behavior observed in linear polymers and supercooled liquids, new insight is provided on the universal/specific mechanisms involved in local dynamics of different glass-forming systems.

On the Breakdown of the Stokes−Einstein Law in Supercooled Liquids

The wavevector-dependent shear viscosity, eta(k), is evaluated for a range of temperatures in a supercooled binary Lennard-Jones liquid. The mode coupling theory of Keyes and Oppenheim (Phys. Rev. A 1973, 8, 937) expresses the self-diffusion constant, D, in terms of eta(k). Replacing eta(k) with the usual viscosity, eta identical with eta(k = 0), yields the Stokes-Einstein law. It is found that the breakdown of the SE law in this system is well described by keeping the simulated k-dependence. Simply put, bath processes on all length scales (wavevectors) contribute to D, the system is much less viscous at finite k, and thus D exceeds the SE estimate based upon eta. The functional form of eta(k) allows for the estimation of a correlation length that grows with decreasing T.

Subdiffusion and cage effect in a sheared granular material

We investigate experimentally the diffusion properties of a bidimensional bidisperse dry granular material under quasistatic cyclic shear. The comparison of these properties with results obtained both in computer simulations of hard spheres systems and Lennard-Jones liquids and experiments on colloidal systems near the glass transition demonstrates a strong analogy between the statistical behavior of granular matter and these systems, despite their intrinsic microscopic differences (thermal vs athermal). More specifically, we study in detail the cage dynamics responsible for the subdiffusion in the slow relaxation regime, and obtain the values of relevant time and length scales.

Single particle jumps in a binary Lennard-Jones system below the glass transition

We study a binary Lennard-Jones system below the glass transition with molecular dynamics simulations. To investigate the dynamics we focus on events (jumps) where a particle escapes the cage formed by its neighbors. Using single particle trajectories we define a jump by comparing for each particle its fluctuations with its changes in average position. We find two kinds of jumps: "reversible jumps," where a particle jumps back and forth between two or more average positions, and "irreversible jumps," where a particle does not return to any of its former average positions, i.e., successfully escapes its cage. For all investigated temperatures both kinds of particles jump and both irreversible and reversible jumps occur. With increasing temperature, relaxation is enhanced by an increasing number of jumps and growing jump lengths in position and potential energy. However, the waiting time between two successive jumps is independent of temperature. This temperature independence might be due to aging, which is present in our system. We therefore also present a comparison of simulation data with three different histories. The ratio of irreversible to reversible jumps is also increasing with increasing temperature, which we interpret as a consequence of the increased likelihood of changes in the cages, i.e., a blocking of the "entrance" back into the previous cage. In accordance with this interpretation, the fluctuations both in position and energy are increasing with increasing temperature. A comparison of the fluctuations of jumping particles and nonjumping particles indicates that jumping particles are more mobile even when not jumping. The jumps in energy normalized by their fluctuations are decreasing with increasing temperature, which is consistent with relaxation being increasingly driven by thermal fluctuations. In accordance with subdiffusive behavior are the distributions of waiting times and jump lengths in position.

Relationship between dynamical heterogeneities and stretched exponential relaxation

We identify dynamical heterogeneities as an essential prerequisite for stretched exponential relaxation in dynamically frustrated systems. This heterogeneity takes the form of ordered domains of finite but diverging lifetime for particles in atomic or molecular systems, or spin states in magnetic materials. At the onset of the dynamical heterogeneity, the distribution of time intervals spent in such domains or traps becomes stretched exponential at long times. We rigorously show that once this is the case the autocorrelation function of the renewal process formed by these time intervals is also a stretched exponential at long times.

Hopping in a supercooled Lennard-Jones liquid: metabasins, waiting time distribution, and diffusion

We investigate the jump motion among potential energy minima of a Lennard-Jones model glass former by extensive computer simulation. From the time series of minima energies, it becomes clear that the energy landscape is organized in superstructures called metabasins. We show that diffusion can be pictured as a random walk among metabasins, and that the whole temperature dependence resides in the distribution of waiting times. The waiting time distribution exhibits algebraic decays: tau(-1/2) for very short times and tau(-alpha) for longer times, where alpha approximately 2 near T(c). We demonstrate that solely the waiting times in the very stable basins account for the temperature dependence of the diffusion constant.

Trap models and slow dynamics in supercooled liquids

The predictions of a class of phenomenological trap models of supercooled liquids are tested via computer simulation of a model glass-forming liquid. It is found that a model with a Gaussian distribution of trap energies provides a good description of the landscape dynamics, even at temperatures above T(c), the critical temperature of mode-coupling theory. A scenario is discussed whereby deep traps are composed of collections of inherent structures above T(c) and single inherent structures below T(c). Deviations from the simple Gaussian trap picture are quantified and discussed.

Properties of cage rearrangements observed near the colloidal glass transition

We use confocal microscopy to study particle motion in colloidal systems. Near the glass transition, motion is inhibited, as particles spend time trapped in transient "cages" formed by neighboring particles. We measure the cage sizes and lifetimes, which, respectively, shrink and grow as the glass transition approaches. Cage rearrangements are more prevalent in regions with lower concentrations and higher disorder. Neighboring rearranging particles typically move in parallel directions, although a nontrivial fraction moves in antiparallel directions, usually from particle pairs with initial separations corresponding to local maxima and minima of the pair correlation function g(r), respectively.

Ising spin glass that closely resembles the physical glass transition

We consider a modification of the one-dimensional Ising model in an external field in which the higher-energy spin state is assumed to be P-fold degenerate. The model shows a transition that becomes first order in the limit of infinite coupling constant. Here we report a study of the dynamical properties of the model by computer simulation in the vicinity of this transition, under the assumption that the model evolves by single spin flips with Metropolis bias, but with certain forbidden flips. The result is a model that exhibits many well-known empirical properties of the physical glass transition, including the "Kauzmann paradox," the Vogel law, stretched-exponential relaxation, and dynamic heterogeneity.

Correlation of positron annihilation and other dynamic properties in small molecule glass-forming substances

Positron annihilation spectroscopy is used to characterize the ortho-positronium lifetime, tau(3), in a broad range of small molecule organic glasses over a wide temperature range. The magnitude and thermal variations of tau(3) reflect changes in the dielectric alpha-relaxation time tau(alpha) and its non-Arrhenius or "fragile" characteristics. tau(3) also displays striking similarities with the fast relaxations reflected in the hydrogen-weighted mean square atomic displacements <u(2)>. The parallel temperature variations of <u(2)(T/T(g))>, tau(3)(T/T(g)), and tau(alpha)(T/T(g)), and their similar patterns of change between the different glasses, are discussed in terms of anharmonicity in the intermolecular caging potentials and the subnanometer density heterogeneities.

Spatially correlated dynamics in a simulated glass-forming polymer melt: analysis of clustering phenomena

In recent years, experimental and computational studies have demonstrated that the dynamics of glass-forming liquids are spatially heterogeneous, exhibiting regions of temporarily enhanced or diminished mobility. Here we present a detailed analysis of dynamical heterogeneity in a simulated "bead-spring" model of a low-molecular-weight polymer melt. We investigate the transient nature and size distribution of clusters of "mobile" chain segments (monomers) as the polymer melt is cooled toward its glass transition. We also explore the dependence of this clustering on the way in which the mobile subset is defined. We show that the mean cluster size is time dependent with a peak at intermediate time, and that the mean cluster size at the peak time grows with decreasing temperature T. We show that for each T a particular fraction of particles maximizes the mean cluster size at some characteristic time, and this fraction depends on T. The growing size of the clusters demonstrates the growing range of correlated motion, previously reported for this same system [C. Beneman et al. Nature (London) 399, 246 (1999)]. The distribution of cluster sizes approaches a power law near the mode-coupling temperature, similar to behavior reported for a simulated binary mixture and a dense colloidal suspension, but with a different exponent. We calculate the correlation length of the clusters, and show that it exhibits similar temperature- and time-dependent behavior as the mean cluster size, with a maximum at intermediate time. We show that the characteristic time of the maximum cluster size follows the scaling predicted by mode-coupling theory (MCT) for the beta time scale, revealing a possible connection between spatially heterogeneous dynamics and MCT.

Viscous flow and jump dynamics in molecular supercooled liquids. I. Translations

The transport and relaxation properties of a molecular supercooled liquid on an isobar are studied by molecular dynamics. The molecule is a rigid heteronuclear biatomic system. The diffusivity is fitted over four orders of magnitude by the power law D proportional to (T-T(c))(gamma(D)), with gamma(D)=1.93+/-0.02 and T(c)=0.458+/-0.002. The self-part of the intermediate scattering function F(s)(k(max),t) exhibits a steplike behavior at the lowest temperatures. On cooling, the increase of the related relaxation time tau(alpha) tracks the diffusivity, i.e., tau(alpha) proportional to (k(2)(max)D)(-1). At the lowest temperatures, fractions of highly mobile and trapped molecules are also evidenced. Translational jumps are also evidenced. The duration of the jumps exhibits a distribution. The distribution of the waiting times before a jump takes place, psi(t), is exponential at higher temperatures. At lower temperatures a power-law divergence is evidenced at short times, psi(t) proportional to t(xi-1) with 0<xi<or=1, which is ascribed to intermittency. The shear viscosity is fitted by the power law eta proportional to (T-T(c))(gamma(eta)), with gamma(eta)=-2.20+/-0.03 at the lowest temperatures. At higher temperatures the Stokes-Einstein relation fits the data if stick boundary conditions are assumed. The product D eta/T increases at lower temperatures, and the Stokes-Einstein relation breaks down at a temperature which is close to the one where the intermittency is evidenced by psi(t). A precursor effect of the breakdown is observed, which manifests itself as an apparent stick-slip transition.

Viscous flow and jump dynamics in molecular supercooled liquids. II. Rotations

The rotational dynamics of a supercooled model liquid of rigid A-B dumbbells interacting via a Lennard-Jones potential is investigated along one single isobar. The time-temperature superposition principle, one key prediction of mode-coupling theory (MCT), was studied for the orientational correlation functions C(l). In agreement with previous studies we found that the scaling of C(l) in a narrow region at long times is better at high-l values. However, on a wider time interval the scaling works fairly better at low-l values. Consistently, we observed the remarkable temperature dependence of the rotational correlation time tau(1) as a power law in T-T(c) over more than three orders of magnitude and the increasing deviations from that law on increasing l (T(c) is the MCT critical temperature). For 0.7<T<2, good agreement with the diffusion model is found. For lower temperatures the agreement becomes poorer, and the results are also only partially accounted for by the jump-rotation model. The angular Van Hove function shows that in this region a meaningful fraction of the sample reorientates by jumps of about 180 degrees. The distribution of the waiting times in the angular sites cuts exponentially at long times. At lower temperatures it decays at short times as t(xi-1), with xi=0.34+/-0.04 at T=0.5, in analogy with the translational case. The breakdown of the Debye-Stokes-Einstein relation is observed at lower temperatures, where the rotational correlation times diverge more weakly than the viscosity.