Non-equilibrium Onsager–Machlup theory

Ricardo Peredo-Ortiz, Luis Fernando Elizondo-Aguilera, Pedro Ezequiel Ramírez-Gonzalez, Edilio Lázaro, Patricia Mendoza-Méndez and M. Medina-Noyola

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or ‘piecewise’) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein–Uhlenbeck stochastic process (which is globally stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager–Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager–Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the non-equilibrium self-consistent generalised Langevin equation (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

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Non-Equilibrium View of the Amorphous Solidification of Liquids with Competing Interactions

Ana Gabriela Carretas-Talamante, Jesús Benigno Zepeda-López, Edilio Lázaro-Lázaro,Luis Fernando Elizondo-Aguilera, and Magdaleno Medina-Noyola

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 to 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.

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

Jan Hansen, Carolyn J. Moll, Leticia López-Flores, Ramon Castaneda-Priego, M. Medina-Noyola, Stefan U. Egelhaaf and Florian Platten

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 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 (NE-SCGLE) 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.

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

Patricia Mendoza-Méndez, Ricardo Peredo-Ortiz, Edilio Lázaro, Martín Chavez-Paez, Honorina Ruiz-Estrada, Felipe Pacheco-Vázquez, M. Medina-Noyola, Luis F. Elizondo-Aguilera

We investigate the structural relaxation of a soft-sphere liquid quenched isochorically (φ = 0.7) and instantaneously to different temperatures Tf 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 (Tf > Ta ) 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(Tf) that becomes increasingly longer as Tf approaches Ta . For quenches inside the glass (Tf less than or equal to Ta) the growth rate of the structural relaxation time becomes progressively larger as Tf decreases and, unlike the equilibration scenario, τα 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 [Phys. Rev. 96, 022608 (2017)] that considered a sequence of quenches with fixed final temperature Tf = 0 but increasing φ towards the hard-sphere dynamical arrest volume fraction φ a HS = 0.582. The NESCGLE 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.

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From Subaging to Hyperaging in Structural Glasses

Luis F. Elizondo-Aguilera, Tommaso Rizzo, and Thomas Voigtmann

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 tw), depending on the depth of the quench from the liquid into the glass: “simple” aging (τ∼tw) applies for quenches close to the critical point of mode-coupling theory (MCT) and implies “subaging” (τ≈twδ with δ<1) as a broad equilibration crossover for quenches to nearly arrested equilibrium states; “hyperaging” (or superaging, τ∼twδ′ 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.

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"Inner clocks" of glass-forming liquids

Ricardo Peredo-Ortiz, Magdaleno Medina-Noyola, Thomas Voigtmann, et al.

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.

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Spatially heterogeneous dynamics and locally arrested density fluctuations from first principles

J. Lira-Escobedo, J. R. Vélez-Cordero and Pedro Ezequiel Ramirez-Gonzalez

We present a first-principles formalism for studying dynamical heterogeneities in glass-forming liquids. Based on the non-equilibrium self-consistent generalized Langevin equation theory, we were able to describe the time-dependent local density profile during the particle interchange among small regions of the fluid. The final form of the diffusion equation contains both the contribution of the chemical potential gradient written in terms of a coarse-grained density and a collective diffusion coefficient as well as the effect of a history-dependent mobility factor. With this diffusion equation, we captured interesting phenomena in glass-forming liquids such as the cases when a strong density gradient is accompanied by a very low mobility factor attributable to the denser part: in such circumstances, the density profile falls into an arrested state even in the presence of a density gradient. On the other hand, we also show that above a certain critical temperature, which depends on the volume fraction, any density heterogeneity relaxes to a uniform state in a finite time, known as equilibration time. We further show that such equilibration time varies little with the temperature in diluted systems but can change drastically with temperature in concentrated systems.

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

Ricardo Peredo-Ortiz, M. Medina-Noyola, Gabriel Perez-Ángel, Thomas Voigtmann and Luis Fernando Elizondo-Aguilera

The recently developed non-equilibrium self-consistent generalized Langevin equation theory of the dynamics of liquids of non-spherically interacting particles [2016 J. Phys. Chem. B 120 7975] 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 (at tw = 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 (tw > 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.

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

J. Lira-Escobedo, Patricia Mendoza-Méndez, M. Medina-Noyola, Gregory B. Mckenna and Pedro Ezequiel Ramírez-Gonzalez

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 (TF) of the glassy structure. We solve for the kinetics of TF(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 TF(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.

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Effect of attractions on the hard-sphere dynamic universality class

Leticia López-Flores, José Manuel Olais-Govea, Martín Chávez-Páez, M. Medina-Noyola

The fundamental understanding of the dynamic and transport properties of liquids is crucial for the better processing of most materials. The usefulness of this understanding increases when it involves general scaling rules, such as the concept of the hard-sphere dynamic universality class, which provides a unifying scaling of the dynamics of soft-sphere repulsive systems. A relevant question is how far this concept extends to systems that also involve attractive interactions. To answer this question, in this work we performed systematic molecular and Brownian dynamics simulations with the Lennard-Jones system in a wide range of temperatures and densities and verify the extent to which its static and dynamic properties map onto those of the hard-sphere system. We determine that in most of the fluid regime, the Lennard-Jones liquid exhibits the same dynamic equivalence with the hard-sphere system as most purely repulsive fluids, thus establishing the degree of its inclusion in the hard-sphere dynamic universality class.

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Waiting-time dependent non-equilibrium phase diagram of simple glass- and gel-forming liquids

Jesús Benigno Zepeda-López, and Magdaleno Medina-Noyola

Under numerous circumstances, many soft and hard materials are present in a puzzling wealth of non-equilibrium amorphous states, whose properties are not stationary and depend on preparation. They are often summarized in unconventional “phase diagrams” that exhibit new “phases” and/or “transitions” in which time, however, is an essential variable. This work proposes a solution to the problem of theoretically defining and predicting these non-equilibrium phases and their time-evolving phase diagrams, given the underlying molecular interactions. We demonstrate that these non-equilibrium phases and the corresponding non-stationary (i.e., aging) phase diagrams can indeed be defined and predicted using the kinetic perspective of a novel non-equilibrium statistical mechanical theory of irreversible processes. This is illustrated with the theoretical description of the transient process of dynamic arrest into non-equilibrium amorphous solid phases of an instantaneously quenched simple model fluid involving repulsive hard-sphere plus attractive square well pair interactions.

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Spherical harmonic projections of the static structure factor of the dipolar hard sphere model: Theory vs simulations

Luis F. Elizondo-Aguilera , Ernesto C. Cortés-Morales, Pablo F. Zubieta-Rico, Magdaleno Medina-Noyola, Ramón Castañeda-Priego , Thomas Voigtmann , and Gabriel Pérez-Ángel

We investigate the static correlations of a dipolar fluid in terms of the irreducible coefficients of the spherical harmonic expansion of the static structure factor. To this end, we develop a theoretical framework based on a soft-core version of Wertheim’s solution of the mean spherical approximation (MSA), which renders the analytical determination of such coefficients possible. The accuracy of this approximation is tested by a comparison against the results obtained with the assistance of extensive molecular dynamics simulations at different regimes of concentration and temperature. Crucial aspects for the comparison of the results provided by the two methods are carefully discussed, concerning the different reference frames used in theory and simulations to describe rotations and orientations, and leading to important differences in the behavior of correlation functions with the same combination of spherical harmonic indices. We find a remarkable agreement between the two approaches in the fluid regime, thus providing a first stringent comparison of the irreducible coefficients of the spherical harmonic expansion of the dipolar fluid’s static structure factor, provided by the MSA theory and molecular dynamics simulations.

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Arrested dynamics of the dipolar hard sphere model

Luis Fernando Elizondo-Aguilera, Ernesto Carlos Cortes-Morales, Pablo Fernando Zubieta-Rico, M. Medina-Noyola, Ramón Castaneda-Priego, Thomas Voigtmann, Gabril Perez

We report the combined results of molecular dynamics simulations and theoretical calculations concerning various dynamical arrest transitions in a model system representing a dipolar fluid, namely, N (soft core) rigid spheres interacting through a truncated dipole-dipole potential. By exploring different regimes of concentration and temperature, we find three distinct scenarios for the slowing down of the dynamics of the translational and orientational degrees of freedom: At low (η=0.2) and intermediate (η=0.4) volume fractions, both dynamics are strongly coupled and become simultaneously arrested upon cooling. At high concentrations (η≤0.6), the translational dynamics shows the features of an ordinary glass transition, either by compressing or cooling down the system, but with the orientations remaining ergodic, thus indicating the existence of partially arrested states. In this density regime, but at lower temperatures, the relaxation of the orientational dynamics also freezes. The physical scenario provided by the simulations is discussed and compared against results obtained with the self-consistent generalized Langevin equation theory, and both provide a consistent description of the dynamical arrest transitions in the system. Our results are summarized in an arrested states diagram which qualitatively organizes the simulation data and provides a generic picture of the glass transitions of a dipolar fluid.

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

José Manuel Olais-Govea, Leticia López-Flores, Jesús Benigno Zepeda-López and Magdaleno Medina-Noyola

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.

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

Luis Fernando Elizondo-Aguilera, and Th. Voigtmann

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.

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Glassy dynamics in asymmetric binary mixtures of hard spheres

Edilio Lázaro-Lázaro, Jorge Adrián Perera-Burgos, Patrick Laermann, Tatjana Sentjabrskaja, Gabriel Pérez-Ángel, Marco Laurati, Stefan U. Egelhaaf, Magdaleno Medina-Noyola, Thomas Voigtmann, Ramón Castañeda-Priego, and Luis Fernando Elizondo-Aguilera

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.

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The Subtle Kinetics of Arrested Spinodal Decomposition: Colloidal Gels and Porous Glasses

José Manuel Olais-Govea, Leticia Lopez-Flores, and M. Medina-Noyola

The non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory of irreversible processes in liquids has been proposed as a theoretical framework capable of predicting the age- and preparation-dependent properties of highly ubiquitous non-equilibrium amorphous solids, such as like glasses and gels. By this formalism, we discuss the main kinetic features of the irreversible relaxation of simple liquids involved in the arrested spinodal decomposition of suddenly and deeply quenched. At some lower temperature we identify, by means of a latency time within which particles retain a finite apparently stationary mobility, the crossover from full phase separation to arrested spinodal decomposition which leads to recognize the onset of gelation.

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Non-equilibrium kinetics of the transformation of liquids into physical gels

José Manuel Olais-Govea, Leticia López-Flores, Martín Chávez-Páez and M. Medina-Noyola

A major stumbling block for statistical physics and materials science has been the lack of a universal principle that allows us to understand and predict elementary structural, morphological, and dynamical properties of nonequilibrium amorphous states of matter. The recently developed nonequilibrium self-consistent generalized Langevin equation theory, however, has been shown to provide a fundamental tool for the understanding of the most essential features of the transformation of liquids into amorphous solids, such as their aging kinetics or their dependence on the protocol of fabrication. In this work we focus on the predicted kinetics of one of the main fingerprints of the formation of gels by arrested spinodal decomposition of suddenly and deeply quenched simple liquids, namely, the arrest of structural parameters associated with the morphological evolution from the initially uniform fluid, to the dynamically arrested spongelike amorphous material. The comparison of the theoretical predictions (based on a simple specific model system), with simulation and experimental data measured on similar but more complex materials, suggests the universality of the predicted scenario.

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Anomalous dynamic arrest of non-interacting spheres (“polymer”) diluted in a hard-sphere (“colloid”) liquid

E. Lázaro-Lázaro, J. A. Moreno-Razo, and M. Medina-Noyola

Upon compression, the equilibrium hard-sphere liquid [pair potential uHS(r)] freezes at a packing fraction ϕf = 0.494 or, if crystallization is prevented, becomes metastable up to its glass transition at ϕg ≈ 0.58. Throughout the fluid regime (ϕ < ϕg), we are, thus, certain that this model liquid does not exhibit any form of kinetic arrest. If, however, a small portion of these spheres (packing fraction ϕ2 ≪ ϕ) happen to ignore each other [u22(r) = 0] but do not ignore the remaining “normal” hard spheres [u12(r) = u21(r) = u11(r) = uHS(r)], whose packing fraction is thus ϕ1 = ϕ − ϕ2, they run the risk of becoming dynamically arrested before they demix from the “normal” particles. This unexpected and counterintuitive scenario was first theoretically predicted and then confirmed by simulations.

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Crossover from equilibration to aging: Nonequilibrium theory versus simulations

Patricia Mendoza-Méndez, Edilio Lázaro, Luis E. Sánchez-Diaz, Pedro Ezequiel Ramirez-Gonzalez, Gabriel Perez and M. Medina Noyola

Understanding glasses and the glass transition requires comprehending the nature of the crossover from the ergodic (or equilibrium) regime, in which the stationary properties of the system have no history dependence, to the mysterious glass transition region, where the measured properties are nonstationary and depend on the protocol of preparation. In this work we use nonequilibrium molecular dynamics simulations to test the main features of the crossover predicted by the molecular version of the recently developed multicomponent nonequilibrium self-consistent generalized Langevin equation theory. According to this theory, the glass transition involves the abrupt passage from the ordinary pattern of full equilibration to the aging scenario characteristic of glass-forming liquids. The same theory explains that this abrupt transition will always be observed as a blurred crossover due to the unavoidable finiteness of the time window of any experimental observation. We find that within their finite waiting-time window, the simulations confirm the general trends predicted by the theory.

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Self-consistent generalized Langevin equation theory of the dynamics of multicomponent atomic liquids

Edilio Lázaro, Luis Fernando Elizondo-Aguilera, Patricia Mendoza-Méndez, Jorge Adrián Perera Burgos, Gabriel Perez-Angel, Ramon Castaneda-Priego and M. Medina Noyola

A fundamental challenge of the theory of liquids is to understand the similarities and differences in the macroscopic dynamics of both colloidal and atomic liquids, which originate in the (Newtonian or Brownian) nature of the microscopic motion of their constituents. Starting from the recently-discovered long-time dynamic equivalence between a colloidal and an atomic liquid that share the same interparticle pair potential, in this work we develop a self-consistent generalized Langevin equation (SCGLE) theory for the dynamics of equilibrium multicomponent atomic liquids, applicable as an approximate but quantitative theory describing the long-time diffusive dynamical properties of simple equilibrium atomic liquids. When complemented with a Gaussian-like approximation, this theory is also able to provide a reasonable representation of the passage from ballistic to diffusive behavior. We illustrate the applicability of the resulting theory with three particular examples, namely, a monodisperse and a polydisperse monocomponent hard-sphere liquid, and a highly size-asymmetric binary hard-sphere mixture. To assess the quantitative accuracy of our results, we perform event-driven molecular dynamics simulations, which corroborate the general features of the theoretical predictions.

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Equilibration and Aging of Liquids of Non-Spherically Interacting Particles

Ernesto Carlos Cortes-Morales, Luis Fernando Elizondo-Aguilera and M. Medina-Noyola

The non-equilibrium self-consistent generalized Langevin equation theory of irreversible processes in liquids is extended to describe the positional and orientational thermal fluctuations of the instantaneous local concentration profile $n(\mathbf{r},\Omega,t)$ of a suddenly-quenched colloidal liquid of particles interacting through non spherically-symmetric pairwise interactions, whose mean value $\overline{n}(\mathbf{r},\Omega,t)$ is constrained to remain uniform and isotropic, $\overline{n}(\mathbf{r},\Omega,t)=\overline{n}(t)$. Such self-consistent theory is cast in terms of the time-evolution equation of the covariance $\sigma(t)=\overline{\delta n_{lm}(\mathbf{k};t) \delta n^{\dagger}_{lm}(\mathbf{k};t)}$ of the fluctuations $\delta n_{lm}(\mathbf{k};t)=n_{lm}(\mathbf{k};t) -\overline{n_{lm}}(\mathbf{k};t)$ of the spherical harmonics projections $n_{lm}(\mathbf{k};t)$ of the Fourier transform of $n(\mathbf{r},\Omega,t)$. The resulting theory describes the non-equilibrium evolution after a sudden temperature quench of both, the static structure factor projections $S_{lm}(k,t)$ and the two-time correlation function $F_{lm}(k,\tau;t)\equiv\overline{\delta n_{lm}(\mathbf{k},t)\delta n_{lm}(\mathbf{k},t+\tau)}$, where $\tau$ is the correlation \emph{delay} time and $t$ is the \emph{evolution} or \emph{waiting} time after the quench. As a concrete and illustrative application we use the resulting self-consistent equations to describe the irreversible processes of equilibration or aging of the orientational degrees of freedom of a system of strongly interacting classical dipoles with quenched positional disorder.

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Localization and dynamical arrest of colloidal fluids in a disordered matrix of polydisperse obstacles

Luis Fernando Elizondo-Aguilera and M. Medina-Noyola

The mobility of a colloidal particle in a crowded and confined environment may be severely reduced by its interactions with other mobile colloidal particles and the fixed obstacles through which it diffuses. The latter may be modelled as an array of obstacles with random fixed positions. In this contribution, we report on the effects of the size-polydispersity of such fixed obstacles on the immobilization and dynamical arrest of the diffusing colloidal particles. This complex system is modelled as a monodisperse Brownian hard-sphere fluid diffusing through a polydisperse matrix of fixed hard spheres with a given size distribution. In the Lorentz gas limit (absence of interactions between the mobile particles), we first develop a simple excluded-volume theory to describe the localization transition of the tracer mobile particles. To take into account the interactions among the mobile particles, we adapt the multi-component self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics, which also allows us to calculate the dynamical arrest transition line, and in general, all the dynamical properties of the mobile particles (mean-squared displacement, self-diffusion coefficient, etc.). The scenarios described by both approaches in the Lorentz gas limit are qualitatively consistent, but the SCGLE formalism describes the dependence of the dynamics of the adsorbed fluid on the polydispersity of the porous matrix at arbitrary concentrations of the mobile spheres and arbitrary volume fractions of the obstacles. Two mechanisms for dynamical arrest (glass transition and localization) are analyzed and we also discuss the crossover between them using the SCGLEs.

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Non-equilibrium Theory of Arrested Spinodal Decomposition

José Manuel Olais-Govea, Leticia López-Flores, and M. Medina-Noyola

The non-equilibrium self-consistent generalized Langevin equation theory of irreversible relaxation [P. E. Ramŕez-González and M. Medina-Noyola, Phys. Rev. E 82, 061503 (2010); 82, 061504 (2010)] is applied to the description of the non-equilibrium processes involved in the spinodal decomposition of suddenly and deeply quenched simple liquids. For model liquids with hard-sphere plus attractive (Yukawa or square well) pair potential, the theory predicts that the spinodal curve, besides being the threshold of the thermodynamic stability of homogeneous states, is also the borderline between the regions of ergodic and non-ergodic homogeneous states. It also predicts that the high-density liquid-glass transition line, whose high-temperature limit corresponds to the well-known hard-sphere glass transition, at lower temperature intersects the spinodal curve and continues inside the spinodal region as a glass-glass transition line. Within the region bounded from below by this low-temperature glass-glass transition and from above by the spinodal dynamic arrest line, we can recognize two distinct domains with qualitatively different temperature dependence of various physical properties. We interpret these two domains as corresponding to full gas-liquid phase separation conditions and to the formation of physical gels by arrested spinodal decomposition. The resulting theoretical scenario is consistent with the corresponding experimental observations in a specific colloidal model system.

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Dynamics of a suspension of interacting yolk-shell particles

L. E. Sánchez Díaz†, E. C. Cortes-Morales‡, X. Li†, Wei-Ren Chen†, and M. Medina-Noyola

In this work we study the self-diffusion properties of a liquid of hollow spherical particles (shells)bearing a smaller solid sphere in their interior (yolks). We model this system using purely repulsive hard-body interactions between all (shell and yolk) particles, but assume the presence of a background ideal solvent such that all the particles execute free Brownian motion between collisions,characterized by short-time self-diffusion coefficients D0s for the shells and D0y for the yolks. Using a softened version of these interparticle potentials we perform Brownian dynamics simulations to determine the mean squared displacement and intermediate scattering function of the yolk-shell complex. These results can be understood in terms of a set of effective Langevin equations for the N interacting shell particles, pre-averaged over the yolks' degrees of freedom, from which an approximate self-consistent description of the simulated self-diffusion properties can be derived. Here we compare the theoretical and simulated results between them, and with the results for the same system in the absence of yolks. We find that the yolks, which have no effect on the shell-shell static structure, influence the dynamic properties in a predictable manner, fully captured by the theory.

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Self-consistent generalized Langevin-equation theory for liquids of nonspherically interacting particles

Luis Fernando Elizondo-Aguilera, P.F. Zubieta-Rico, Honorina Ruiz, and O. Alarcón-Waess

A self-consistent generalized Langevin-equation theory is proposed to describe the self- and collective dynamics of a liquid of linear Brownian particles. The equations of motion for the spherical harmonics projections of the collective and self-intermediate-scattering functions, F_{lm,lm}(k,t) and F_{lm,lm}^{S}(k,t), are derived as a contraction of the description involving the stochastic equations of the corresponding tensorial one-particle density n_{lm}(k,t) and the translational (α=T) and rotational (α=R) current densities j_{lm}^{α}(k,t). Similar to the spherical case, these dynamic equations require as an external input the equilibrium structural properties of the system contained in the projections of the static structure factor, denoted by S_{lm,lm}(k). Complementing these exact equations with simple (Vineyard-like) approximate relations for the collective and the self-memory functions we propose a closed self-consistent set of equations for the dynamic properties involved. In the long-time asymptotic limit, these equations become the so-called bifurcation equations, whose solutions (the nonergodicity parameters) can be written, extending the spherical case, in terms of one translational and one orientational scalar dynamic order parameter, γ_{T} and γ_{R}, which characterize the possible dynamical arrest transitions of the system. As a concrete illustrative application of this theory we determine the dynamic arrest diagram of the dipolar hard-sphere fluid. In qualitative agreement with mode coupling theory, the present self-consistent equations also predict three different regions in the state space spanned by the macroscopic control parameters η (volume fraction) and T* (scaled temperature): a region of fully ergodic states, a region of mixed states, in which the translational degrees of freedom become arrested while the orientational degrees of freedom remain ergodic, and a region of fully nonergodic states.

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Non-equilibrium dynamics of glass-forming liquid mixtures

Luis E. Sánchez-Diáz, Edilio Lázaro, José Manuel Olais-Govea, and M. Medina-Noyola

The non-equilibrium self-consistent generalized Langevin equation theory of irreversible processes in glass-forming liquids [P. Ramírez-González and M. Medina-Noyola, Phys. Rev. E 82, 061503 (2010)] is extended here to multi-component systems. The resulting theory describes the statistical properties of the instantaneous local particle concentration profiles nα(r, t) of species α in terms of the coupled time-evolution equations for the mean value [Formula: see text] and for the covariance [Formula: see text] of the fluctuations [Formula: see text]. As in the monocomponent case, these two coarse-grained equations involve a local mobility function bα(r, t) for each species, written in terms of the memory function of the two-time correlation function [Formula: see text]. If the system is constrained to remain spatially uniform and subjected to a non-equilibrium preparation protocol described by a given temperature and composition change program T(t) and [Formula: see text], these equations predict the irreversible structural relaxation of the partial static structure factors Sαβ(k; t) and of the (collective and self) intermediate scattering functions Fαβ(k, τ; t) and [Formula: see text]. We illustrate the applicability of the resulting theory with two examples involving simple model mixtures subjected to an instantaneous temperature quench: an electroneutral binary mixture of equally sized and oppositely charged hard-spheres, and a binary mixture of soft-spheres of moderate size-asymmetry.

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The discontinuous ideal glass transition, a soft crossover at finite waiting times

P. Mendoza-Méndez, E. Lázaro-Lázaro1, L. E. Sánchez-Díaz, P.E. Ramírez-González, G. Pérez-Ángel, and M. Medina-Noyola

The \emph{non-equilibrium} self-consistent generalized Langevin equation (NE-SCGLE) theory of irreversible processes in liquids is shown to provide a coherent and conceptually simple picture of the crossover from ergodic equilibration to non-equilibrium aging in structural glass-forming liquids. According to this picture, the glass transition is in essence a discontinuous, ``mode-coupling''--like transition, characterized by the abrupt passage from ergodic to dynamically arrested states and by the divergence of the \emph{equilibrium} $\alpha$-relaxation time at the transition. The same picture, however, also predicts that such discontinuous and singular scenario will be blurred in real life by the unavoidable finiteness of the time window of any experimental observation. The comparison of these predictions with pertinent simulation experiments involving hard-sphere glass-forming liquids reconciles the lack of observable diverging relaxation times in (real or simulated) experiments, with the predicted critical condition at which the \emph{equilibrium} $\alpha$-relaxation time is expected to diverge.

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Dynamic equivalences in the hard-sphere dynamic universality class

Leticia Lopez-Flores, Honorina Ruiz, Martín Chávez-Páez, M. Medina-Noyola

We perform systematic simulation experiments on model systems with soft-sphere repulsive interactions to test the predicted dynamic equivalence between soft-sphere liquids with similar static structure. For this we compare the simulated dynamics (mean squared displacement, intermediate scattering function, α-relaxation time, etc.) of different soft-sphere systems, between them and with the hard-sphere liquid. We then show that the referred dynamic equivalence does not depend on the (Newtonian or Brownian) nature of the microscopic laws of motion of the constituent particles, and hence, applies independently to colloidal and to atomic simple liquids. Finally, we verify another more recently proposed dynamic equivalence, this time between the long-time dynamics of an atomic liquid and its corresponding Brownian fluid (i.e., the Brownian system with the same interaction potential).

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Equilibration and aging of dense soft-sphere glass-forming liquids

Luis E. Sánchez-Diaz, Pedro Ezequiel Ramírez-Gonzalez and M. Medina-Noyola

The recently developed nonequilibrium extension of the self-consistent generalized Langevin equation theory of irreversible relaxation [Ramírez-González and Medina-Noyola, Phys. Rev. E 82, 061503 (2010); Ramírez-González and Medina-Noyola, Phys. Rev. E 82, 061504 (2010)] is applied to the description of the irreversible process of equilibration and aging of a glass-forming soft-sphere liquid that follows a sudden temperature quench, within the constraint that the local mean particle density remains uniform and constant. For these particular conditions, this theory describes the nonequilibrium evolution of the static structure factor S(k;t) and of the dynamic properties, such as the self-intermediate scattering function F_{S}(k,τ;t), where τ is the correlation delay time and t is the evolution or waiting time after the quench. Specific predictions are presented for the deepest quench (to zero temperature). The predicted evolution of the α-relaxation time τ_{α}(t) as a function of t allows us to define the equilibration time t^{eq}(ϕ), as the time after which τ_{α}(t) has attained its equilibrium value τ_{α}^{eq}(ϕ). It is predicted that both, t^{eq}(ϕ) and τ_{α}^{eq}(ϕ), diverge as ϕ→ϕ^{(a)}, where ϕ^{(a)} is the hard-sphere dynamic-arrest volume fraction ϕ^{(a)}(≈0.582), thus suggesting that the measurement of equilibrium properties at and above ϕ^{(a)} is experimentally impossible. The theory also predicts that for fixed finite waiting times t, the plot of τ_{α}(t;ϕ) as a function of ϕ exhibits two regimes, corresponding to samples that have fully equilibrated within this waiting time (ϕ≤ϕ^{(c)}(t)), and to samples for which equilibration is not yet complete (ϕ≥ϕ^{(c)}(t)). The crossover volume fraction ϕ^{(c)}(t) increases with t but saturates to the value ϕ^{(a)}.

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Molecular-Brownian Correspondence in the Hard-sphere Dynamic Universality Class

Leticia Lopez-Flores, Honorina Ruiz, Martín Chávez-Páez and M. Medina Noyola

We perform systematic simulation experiments on model systems with soft-sphere repulsive interactions to test the predicted dynamic equivalence between soft-sphere liquids with similar static structure. For this we compare the simulated dynamics (mean squared displacement, intermediate scattering function, {\alpha}-relaxation time, etc.) of different soft-sphere systems, between them and with the hard-sphere liquid. We then show that the referred dynamic equivalence does not depend on the (Newtonian or Brownian) nature of the microscopic laws of motion of the constituent particles, and hence, applies independently to colloidal and to atomic simple liquids. In addition, we verify another more recently-proposed dynamic equivalence, this time between the long-time dynamics of a Brownian fluid and its corresponding atomic liquid (i.e., the atomic system with the same interaction potential).

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Irreversible Equilibration and Aging in Glass-forming Liquids

Luis E. Sánchez-Diaz, Pedro Ezequiel Ramirez-Gonzalez and M. Medina-Noyola

We review the recently-proposed non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory of irreversible processes in liquids, and describe the scenario that emerges from its application to the equilibration (or absence of equilibration!) of quenched glass-forming liquids. This theory extends to non-equilibrium conditions the SCGLE theory of dynamic arrest, which (just like the well-known mode coupling theory) determines the boundary of the ergodic domain of the system. In this first systematic application of the non-equilibrium theory we consider a model soft-sphere glass-forming liquid, initially at an ergodic equilibrium state, suddenly quenched to a lower final temperature that lies either (a) also in the ergodic domain, or (b) in the region of dynamically arrested states. In the first case the liquid will equilibrate within a finite equilibration time teq , while in the second the theory predicts that the liquid will age forever, (i.e., teq = ∞). The dynamic arrest boundary is thus predicted to determine the crossover from equilibration to aging, and to be characterized by the divergence of the equilibration time. In either case the theory predicts the irreversible t-evolution of the measured static structure factor S(k;t) and of the dynamic properties such as the self-intermediate scattering function FS (k, τ;t).

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Overdamped van Hove function of atomic liquids

Leticia Lopez-Flores, Laura Yeomans, Matín Chávez-Páez and M. Medina Noyola

Using the generalized Langevin equation formalism and the process of contraction of the description we derive a general memory function equation for the thermal fluctuations of the local density of a simple atomic liquid. From the analysis of the long-time limit of this equation, a striking equivalence is suggested between the long-time dynamics of the atomic liquid and the dynamics of the corresponding \emph{Brownian} liquid. This dynamic equivalence is confirmed here by comparing molecular and Brownian dynamics simulations of the self-intermediate scattering function and the long-time self-diffusion coefficient for the hard-sphere liquid.

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Generalized Langevin Equation for Tracer Diffusion in Atomic Liquids

Patricia Mendoza-Méndez, Leticia López-Flores, Alejandro Vizcarra-Rendón, Luis E. Sánchez-Díaz, and Magdaleno Medina-Noyola

We derive the time-evolution equation that describes the Brownian motion of labeled individual tracer particles in a simple model atomic liquid (i.e., a system of $N$ particles whose motion is governed by Newton's second law, and interacting through spherically symmetric pairwise potentials). We base our derivation on the generalized Langevin equation formalism, and find that the resulting time evolution equation is formally identical to the generalized Langevin equation that describes the Brownian motion of individual tracer particles in a colloidal suspension in the absence of hydrodynamic interactions. This formal dynamic equivalence implies the long-time indistinguishability of some dynamic properties of both systems, such as their mean squared displacement, upon a well-defined time scaling. This prediction is tested here by comparing the results of molecular and Brownian dynamics simulations performed on the hard sphere system.

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Density-Temperature-Softness Scaling of the Dynamics of Glass-Forming Soft-Sphere Liquids

Pedro E. Ramírez-González, Heriberto Acuña-Campa, and Magdaleno Medina-Noyola

We employ the principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E 68, 011405 (2003)] to describe the interplay of the effects of varying the density n, the temperature T, and the softness (characterized by a softness parameter ν(-1)) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is the existence of a dynamic universality class associated with the hard-sphere fluid, constituted by the soft-sphere systems whose dynamic parameters depend on n, T, and ν only through the reduced density n*≡nσ(HS)(T*,ν). A number of scaling properties observed in recent experiments and simulations involving glass-forming fluids with repulsive short-range interactions are found to be a direct manifestation of this general dynamic equivalence principle.

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Equilibrium structure of the multi-component screened charged hard-sphere fluid

Luis E. Sánchez-Díaz, Arlette Méndez-MAldonado, Minerva González-Melchor and Honorina Ruiz

The generalized mean spherical approximation of the structural properties of the binary charge-symmetric fluid of screened charged hard-spheres of the same diameter, i.e., the screened restricted primitive model, is extended to include binary charge-asymmetric and multi-component fluids. Molecular dynamics simulation data are generated to assess the accuracy of the corresponding theoretical predictions.

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Dynamic equivalence between atomic and colloidal liquids

Leticia López-Flores, Patricia Mendoza-Méndez, Luis E. Sánchez-Díaz, Laura L. Yeomans-Reyna, Alejandro Vizcarra-Rendón4, Gabriel Pérez-Ángel5, Martín Chávez-Páez and Magdaleno Medina-Noyola

We show that the kinetic-theoretical self-diffusion coefficient of an atomic fluid plays the same role as the short-time self-diffusion coefficient D_S in a colloidal liquid, in the sense that the dynamic properties of the former, at times much longer than the mean free time, and properly scaled with D_S, will indistinguishable from those of a colloidal liquid with the same interaction potential. One important consequence of such dynamic equivalence is that the ratio D_L/ D_S of the long-time to the short-time self-diffusion coefficients must then be the same for both, an atomic and a colloidal system characterized by the same inter-particle interactions. This naturally extends to atomic fluids a well-known dynamic criterion for freezing of colloidal liquids[Phys. Rev. Lett. 70, 1557 (1993)]. We corroborate these predictions by comparing molecular and Brownian dynamics simulations on (soft- and hard-sphere) model systems, representative of what we may refer to as the "hard-sphere" dynamic universality class.

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Equilibration of Concentrated Hard Sphere Fluids

Gabriel Pérez-Ángel, Luis Enrique Sánchez-Díaz, Pedro E. Ramírez-González, Rigoberto Juárez-Maldonado, Alejandro Vizcarra-Rendón, and Magdaleno Medina-Noyola

We report a systematic molecular dynamics study of the isochoric equilibration of hard-sphere fluids in their metastable regime close to the glass transition. The thermalization process starts with the system prepared in a nonequilibrium state with the desired final volume fraction ϕ for which we can obtain a well-defined nonequilibrium static structure factor S(0)(k;ϕ). The evolution of the α-relaxation time τ(α)(k) and long-time self-diffusion coefficient D(L) as a function of the evolution time t(w) is then monitored for an array of volume fractions. For a given waiting time the plot of τ(α)(k;ϕ,t(w)) as a function of ϕ exhibits two regimes corresponding to samples that have fully equilibrated within this waiting time [ϕ≤ϕ(c)(t(w))] and to samples for which equilibration is not yet complete [ϕ≥ϕ(c)(t(w))]. The crossover volume fraction ϕ(c)(t(w)) increases with t(w) but seems to saturate to a value ϕ(a)≡ϕ(c)(t(w)→∞)≈0.582. We also find that the waiting time t(w)(eq)(ϕ) required to equilibrate a system grows faster than the corresponding equilibrium relaxation time, t(w)(eq)(ϕ)≈0.27[τ(α)(eq)(k;ϕ)](1.43), and that both characteristic times increase strongly as ϕ approaches ϕ(a), thus suggesting that the measurement of equilibrium properties at and above ϕ(a) is experimentally impossible.

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Aging of a Homogeneously Quenched Colloidal Glass-forming Liquid

P. E. Ramírez-Gonzalez and M. Medina-Noyola

The non-equilibrium self-consistent generalized Langevin equation theory of colloid dynamics is used to describe the non-stationary aging processes occurring in a suddenly quenched model colloidal liquid with hard-sphere plus short-ranged attractive interactions, whose static structure factor and van Hove function evolve irreversibly from the initial conditions before the quench to a final, dynamically arrested state. The comparison of our numerical results with available simulation data are highly encouraging.

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General Non-equilibrium Theory of Colloid Dynamics

P. E. Ramírez-Gonzalez and M. Medina-Noyola

A nonequilibrium extension of Onsager's canonical theory of thermal fluctuations is employed to derive a self-consistent theory for the description of the statistical properties of the instantaneous local concentration profile n(r,t) of a colloidal liquid in terms of the coupled time-evolution equations of its mean value n(r,t) and of the covariance [Formula in text] of its fluctuations δn(r,t)=n(r,t)-n(r,t). These two coarse-grained equations involve a local mobility function b(r,t) which, in its turn, is written in terms of the memory function of the two-time correlation function [Formula in text]. For given effective interactions between colloidal particles and applied external fields, the resulting self-consistent theory is aimed at describing the evolution of a strongly correlated colloidal liquid from an initial state with arbitrary mean and covariance n(0)(r) and σ(0)(r,r') toward its equilibrium state characterized by the equilibrium local concentration profile n(eq)(r) and equilibrium covariance σ(eq)(r,r'). This theory also provides a general theoretical framework to describe irreversible processes associated with dynamic arrest transitions, such as aging, and the effects of spatial heterogeneities.

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Generalized mean spherical approximation for the multicomponent restricted primitive model

L. E. Sánchez-Díaz, A. Vizcarra-Rendón, and M. Medina-Noyola

The generalized mean spherical approximation of Stell and Sun [J. Chem. Phys. 63, 5333 (1975)] for the binary charge-symmetric restricted primitive model (electroneutral mixture of equally sized hard spheres) is extended to charge-asymmetric binary electrolytes and to the generally multicomponent, but still restricted (i.e., equally sized) primitive model.

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Non-equilibrium relaxation and near-arrest dynamics in colloidal suspensions

P. E. Ramírez-Gonzalez and M. Medina-Noyola

In this work we propose a theory to describe the irreversible diffusive relaxation of the local concentration of a colloidal dispersion that proceeds toward its stable thermodynamic equilibrium state, but which may in the process be trapped in metastable or dynamically arrested states. The central assumption of this theory is that the irreversible relaxation of the macroscopically observed mean value [Formula: see text] of the local concentration of colloidal particles is described by a diffusion equation involving a local mobility b(*)(r,t) that depends not only on the mean value [Formula: see text] but also on the covariance [Formula: see text] of the fluctuations [Formula: see text]. This diffusion equation must hence be solved simultaneously with the relaxation equation for the covariance σ(r,r';t), and here we also derive the corresponding relaxation equation. The dependence of the local mobility b(*)(r,t) on the mean value and the covariance is determined by a self-consistent set of equations involving now the spatially and temporally non-local time-dependent correlation functions, which in a uniform system in equilibrium reduces to the self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics. The resulting general theory considers the possibility that these relaxation processes occur under the influence of external fields, such as gravitational forces acting in the process of sedimentation. In this paper, however, we describe a simpler application, in which the system remains spatially uniform during the irreversible relaxation process, and discuss the general features of the glass transition scenario predicted by this non-equilibrium theory.

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Generalized Onsager-Machlup's theory of thermal fluctuations for non-equilibrium systems

M. Medina-Noyola

In this work a generalization of Onsager-Machlup's theory of time-dependent thermal fluctuations of equilibrium systems is proposed, to the case in which the system relaxes irreversibly along a non-equilibrium trajectory that can be approximated as a sequence of stationary states. This generalization is summarized by a canonical description of the dependence of the two-time correlation function C(t+\tau,t), and of the equal-time correlation function \sigma(t)= C(t,t) (the covariance of the fluctuations), on the non-equilibrium relaxation time t.

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Glass transition in soft-sphere dispersions

P. E. Ramírez-Gonzalez and M. Medina-Noyola

The concept of dynamic equivalence among mono-disperse soft-sphere fluids is employed in the framework of the self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics to calculate the ideal glass transition phase diagram of model soft-sphere colloidal dispersions in the softness–concentration state space. The slow dynamics predicted by this theory near the glass transition is compared with available experimental data for the decay of the intermediate scattering function of colloidal dispersions of soft-microgel particles. Increasing deviations from this simple scheme occur for increasingly softer potentials, and this is studied here using the Rogers–Young static structure factor of the soft-sphere systems as the input of the SCGLE theory, without assuming a priori the validity of the equivalence principle above.

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Alternative View of Dynamic Arrest in Colloid-Polymer Mixtures

R. Juárez-Maldonado and M. Medina-Noyola

Colloid-polymer mixtures are frequently viewed as an effective one-component fluid (the colloid) with polymer-mediated depletion interactions. This view, together with conventional mode coupling theory, constitutes the current description of the reentrant glass transition experimentally observed in these systems. A more fundamental view is to consider these systems as what they actually are, namely, genuine highly size-asymmetric binary colloidal mixtures. In this Letter we demonstrate that the recently developed multicomponent self-consistent generalized Langevin equation theory of dynamic arrest correctly predicts the observed reentrance in excellent quantitative agreement with the experimental glass transition line of a colloid-polymer mixture. In this scenario the polymer plays a much more active dynamic role than in the conventional one-component description.

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Glass-liquid-glass reentrance in mono-component colloidal dispersions

P. E. Ramirez-Gonzalez, A. Vizcarra-Rendón, F. de J. Guevara-Rodríguez and M. Medina-Noyola

The self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics is employed to describe the ergodic-non-ergodic transition in model mono-disperse colloidal dispersions whose particles interact through hard-sphere plus short-ranged attractive forces. The ergodic-non-ergodic phase diagram in the temperature-concentration state space is determined for the hard-sphere plus attractive Yukawa model within the mean spherical approximation for the static structure factor by solving a remarkably simple equation for the localization length of the colloidal particles. Finite real values of this property signals non-ergodicity and determines the non-ergodic parameters f(k) and f(s)(k). The resulting phase diagram for this system, which involves the existence of reentrant (repulsive and attractive) glass states, is compared with the corresponding prediction of mode coupling theory. Although both theories coincide in the general features of this phase diagram, there are also clear qualitative differences. One of the most relevant is the SCGLE prediction that the ergodic-attractive glass transition does not preempt the gas-liquid phase transition, but always intersects the corresponding spinodal curve on its high-concentration side. We also calculate the ergodic-non-ergodic phase diagram for the sticky hard-sphere model to illustrate the dependence of the predicted SCGLE dynamic phase diagram on the choice of one important constituent element of the SCGLE theory.

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Theory of dynamic arrest in colloidal mixtures

R. Juárez-Maldonado and M. Medina-Noyola

We present a first-principles theory of dynamic arrest in colloidal mixtures based on the multicomponent self-consistent generalized Langevin equation theory of colloid dynamics [M. A. Chávez-Rojo and M. Medina-Noyola, Phys. Rev. E 72, 031107 (2005); M. A. Chávez-Rojo and M. Medina-Noyola, Phys. Rev. E76, 039902 (2007)]. We illustrate its application with a description of dynamic arrest in two simple model colloidal mixtures: namely, hard-sphere and repulsive Yukawa binary mixtures. Our results include observation of the two patterns of dynamic arrest, one in which both species become simultaneously arrested and the other involving the sequential arrest of the two species. The latter case gives rise to mixed states in which one species is arrested while the other species remains mobile. We also derive the ("bifurcation" or fixed-point") equations for the nonergodic parameters of the system, which takes the surprisingly simple form of a system of coupled equations for the localization length of the particles of each species. The solution of this system of equations indicates unambiguously which species is arrested (finite localization length) and which species remains ergodic (infinite localization length). As a result, we are able to draw the entire ergodic-nonergodic phase diagram of the binary hard-sphere mixture.

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Simplified Self-Consistent Theory of Colloid Dynamics

R. Juárez-Maldonado, M. A. Chávez-Rojo, P. E.Ramírez-González, L. Yeomans-Reyna, and M. Medina-Noyola

One of the main elements of the self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics [Phys. Rev. E 62, 3382 (2000); 72, 031107 (2005)] is the introduction of exact short-time moment conditions in its formulation. The need to previously calculate these exact short-time properties constitutes a practical barrier for its application. In this Brief Report, we report that a simplified version of this theory, in which this short-time information is eliminated, leads to the same results in the intermediate and long-time regimes. Deviations are only observed at short times, and are not qualitatively or quantitatively important. This is illustrated by comparing the two versions of the theory for representative model systems.

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Dynamic arrest within the self-consistent generalized Langevin equation of colloid dynamics

L. Yeomans-Reyna, M. A. Chávez-Rojo, P. E. Ramírez-González, R. Juárez-Maldonado,M. Chávez-Páez, and M. Medina-Noyola

This paper presents a recently developed theory of colloid dynamics as an alternative approach to the description of phenomena of dynamic arrest in monodisperse colloidal systems. Such theory, referred to as the self-consistent generalized Langevin equation (SCGLE) theory, was devised to describe the tracer and collective diffusion properties of colloidal dispersions in the short- and intermediate-time regimes. Its self-consistent character, however, introduces a nonlinear dynamic feedback, leading to the prediction of dynamic arrest in these systems, similar to that exhibited by the well-established mode coupling theory of the ideal glass transition. The full numerical solution of this self-consistent theory provides in principle a route to the location of the fluid-glass transition in the space of macroscopic parameters of the system, given the interparticle forces (i.e., a nonequilibrium analog of the statistical-thermodynamic prediction of an equilibrium phase diagram). In this paper we focus on the derivation from the same self-consistent theory of the more straightforward route to the location of the fluid-glass transition boundary, consisting of the equation for the nonergodic parameters, whose nonzero values are the signature of the glass state. This allows us to decide if a system, at given macroscopic conditions, is in an ergodic or in a dynamically arrested state, given the microscopic interactions, which enter only through the static structure factor. We present a selection of results that illustrate the concrete application of our theory to model colloidal systems. This involves the comparison of the predictions of our theory with available experimental data for the nonergodic parameters of model dispersions with hard-sphere and with screened Coulomb interactions.

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Self-consistent generalized Langevin equation for colloidal mixtures

Marco Antonio Chávez-Rojo and Magdaleno Medina-Noyola

A self-consistent theory of collective and tracer diffusion in colloidal mixtures is presented. This theory is based on exact results for the partial intermediate scattering functions derived within the framework of the generalized Langevin equation formalism, plus a number of conceptually simple and sensible approximations. The first of these consists of a Vineyard-like approximation between collective and tracer diffusion, which writes the collective dynamics in terms of the memory function related to tracer diffusion. The second consists of interpolating this only unknown memory function between its two exact limits at small and large wave vectors; for this, a phenomenologically determined, but not arbitrary, interpolating function is introduced: a Lorentzian with its inflection point located at the first minimum of the partial static structure factor. The small wave-vector exact limit involves a time-dependent friction function, for which we take a general approximate result, previously derived within the generalized Langevin equation formalism. This general result expresses the time-dependent friction function in terms of the partial intermediate scattering functions, thus closing the system of equations into a fully self-consistent scheme. This extends to mixtures a recently proposed self-consistent theory developed for monodisperse suspensions [Yeomans-Reyna and Medina-Noyola, Phys. Rev. E 64, 066114 (2001)]. As an illustration of its quantitative accuracy, its application to a simple model of a binary dispersion in the absence of hydrodynamic interactions is reported.

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Self-consistent theory of collective Brownian dynamics: Theory versus simulation

Laura Yeomans-Reyna, Heriberto Acuña-Campa, Felipe de Jesús Guevara-Rodríguez, and Magdaleno Medina-Noyola

A recently developed theory of collective diffusion in colloidal suspensions is tested regarding the quantitative accuracy of its description of the dynamics of monodisperse model colloidal systems without hydrodynamic interactions. The idea is to exhibit the isolated effects of the direct interactions, which constitute the main microscopic relaxation mechanism, in the absence of other effects, such as hydrodynamic interactions. Here we compare the numerical solution of the fully self-consistent theory with the results of Brownian dynamics simulation of the van Hove function G(r,t) and/or the intermediate scattering function F(k,t) of four simple model systems. Two of them are representative of short-ranged soft-core repulsive interactions [(sigma/r)(mu), with mu>1], in two and in three dimensions. The other two involve long-ranged repulsive forces in two (dipolar, r(-3) potential) and in three (screened Coulomb, or repulsive Yukawa interactions) dimensions. We find that the theory, without any sort of adjustable parameters or rescaling prescriptions, provides an excellent approximate description of the collective dynamics of these model systems, particularly in the short- and intermediate-time regimes. We also compare our results with those of the single exponential approximation and with the competing mode-mode coupling theory.

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Self-consistent generalized Langevin equation for colloid dynamics

Laura Yeomans-Reyna and Magdaleno Medina-Noyola

We present a general self-consistent theory of colloid dynamics which, for a system without hydrodynamic interactions, allows us to calculate F(k,t), and its self-diffusion counterpart F(S)(k,t), given the effective interaction pair potential u(r) between colloidal particles, and the corresponding equilibrium static structural properties. This theory is build upon the exact results for F(k,t) and F(S)(k,t) in terms of a hierarchy of memory functions, derived from the application of the generalized Langevin equation formalism, plus the proposal of Vineyard-like connections between F(k,t) and F(S)(k,t) through their respective memory functions, and a closure relation between these memory functions and the time-dependent friction function Delta zeta(t). As an illustrative application, we present and analyze a selection of numerical results of this theory in the short- and intermediate-time regimes, as applied to a two-dimensional repulsive Yukawa Brownian fluid. For this system, we find that our theory accurately describes the dynamic properties contained in F(k,t) in a wide range of conditions, including strongly correlated systems, at the longest times available from our computer simulations.

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Vineyard-like approximations for colloid dynamics

Laura Yeomans-Reyna, Heriberto Acuña-Campa, and Magdaleno Medina-Noyola

In this paper we propose a hierarchy of higher-order Vineyard-like approximations for colloidal systems. These consist of approximate expressions for the intermediate scattering function F(k,t) in terms of the self-intermediate scattering function F(s)(k,t) (or some memory function associated with it), and of other static structural properties of the suspension. In order to assess the accuracy of the proposed approximations, we perform Brownian dynamics simulations in a simple model system (a two-dimensional Yukawa Brownian fluid), in which we determine F(k,t), F(s)(k,t), and the required static structural properties. We study proposals for "second-order" and "third-order" Vineyard-like approximations. We find that the detailed structure of the relationship between the corresponding collective and self-memory functions turns out to be most important, as quantified by our simulation results.

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Overdamped van Hove function of colloidal suspensions

Laura Yeomans-Reyna and Magdaleno Medina-Noyola

The generalized-hydrodynamic theory for collective diffusion of a monodisperse colloidal suspension is developed in the framework of the Onsager-Machlup theory of time-dependent fluctuations. The time evolution of the intermediate scattering function F(k,t) is derived as a contraction of the description involving the instantaneous particle number concentration, the particle current, and the stress tensor of the Brownian fluid as state variables. We show that the proper overdamped limit of this equation requires the explicit separation of the stress tensor in its mutually orthogonal kinetic and configurational contributions. Analogous results also follow for the self-intermediate scattering function F(s)(k,t). We show that neglecting the non-Markovian part of the configurational stress tensor memory, one recovers the single exponential memory approximation (based on sum rules derived from the Smoluchowski equation) for both F(s)(k,t) and F(k,t). We suggest simple approximate manners to relate the collective and the self-memory functions, leading to Vineyard-like approximate relations between F(s)(k,t) and F(k,t).

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Long-Time Self-Diffusion in Concentrated Colloidal Dispersions

Magdaleno Medina-Noyola

A theory of self-diffusion in concentrated dispersions is presented, in which the effects of hydrodynamic and direct interactions between macroparticles are virtually decoupled. The long-time self-diffusion coefficient DLs is written in terms of the radial distribution function of the suspended particles and of their short-time self-diffusion coefficient DSs. The predictions for the volume fraction dependence of DLs in hard-sphere suspensions are in excellent agreement with available experimental results.

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The fluctuation-dissipation theorem for non-Markov processes and their contractions: The role of the stationarity condition

Magdaleno Medina-Noyola and José Luis Del-Río-Correa

A demonstration is given of the equivalence between the stationarity condition of an N-dimensional stochastic process a(t), defined as the solution of a generalized Langevin equation with random initial values, with the (“second”) fluctuation-dissipation theorem. As a result, it is shown that a similar relation also holds for any stochastic process obtained as a projection of a(t) into a subspace of the original space.

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The generalized Langevin equation as a contraction of the description. An approach to tracer diffusion

Magdaleno Medina-Noyola

An effective Langevin equation for a tracer Brownian particle immersed in a macrofluid of other diffusing particles is derived as a contraction of the description involving the stochastic equations for the local concentration and the local current of the macrofluid particles. The resulting Langevin equation contains the effects of the interactions with the other diffusing particles in a temporally non-local friction term plus a fluctuating force representing the random, diffusion-driven departures from spherical symmetry of the distribution of macrofluid particles around the tracer. This fluctuating force satisfies a fluctuation–dissipation relation with the effective time-dependent friction. This program is fully developed here only in the absence of hydrodynamic interactions, although the formal aspects of its extension are also suggested. The results derived here, however, are found to provide a unifying framework to describe, for example, self-friction and electrolyte friction in suspensions of charged colloidal particles within the same theoretical scheme.

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