Self-consistent Generalized Langevin Equation Research Articles

Non-equilibrium theory of arrested spinodal decomposition.

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.

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

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.

Microscopic Theory of the Long-Time Diffusivity and Intermediate-Time Anomalous Transport of a Nanoparticle in Polymer Melts

The transport of a single nanoparticle in unentangled and entangled polymer melts is studied based on a microscopic, force-level, self-consistent generalized Langevin equation approach. Nanoparticle self-motion and length-scale-dependent polymer relaxation enter as two parallel and competing channels of force relaxation. For entangled melts, particle long-time mobility exhibits size-dependent non-Stokes–Einstein behaviors controlled by (i) fast diffusion and length-scale-dependent collective relaxation of the unentangled matrix for particles smaller than the tube diameter and (ii) polymer relaxation via reptation for nanoparticles larger than the tube diameter. The two regimes are connected by a relatively sharp but continuous crossover. Recovery of hydrodynamic behavior requires the nanoparticle to be roughly 10 times larger than the tube diameter. Theoretical predictions for the diffusion constant are in good agreement with recent simulations of unentangled and lightly entangled melts, including the crossover regime where the nanoparticle and tube diameters are roughly equal. The calculations are also in reasonable accord with several experiments in entangled synthetic polymer melts and DNA solutions, although the activated hopping motion not included in the present theory may be important in a narrow intermediate crossover regime for sufficiently entangled polymers. The predictions of the theory for intermediate-time anomalous diffusion are of mixed quality based on comparison to simulations. The significant limitations identified are associated with our minimalist treatment of collective density fluctuation relaxation on intermediate time and length scales, including the possible importance of non-reptation relaxation processes in entangled melts.

Non-equilibrium dynamics of glass-forming liquid mixtures

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 n̄α(r, t) and for the covariance σ(αβ)(r, r'; t) ≡ δn(α)(r, t)δn(β)(r', t) of the fluctuations δn(α)(r, t) = n(α)(r, t) - n̄α(r, t). 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 C(αβ)(r, r'; t, t') ≡ δn(α)(r, t)δn(β)(r', t'). 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 n̄α(r, t), 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 F(αβ)(S)(k, τ; t). 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.

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

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 φexhibits 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)).

Aging of a homogeneously quenched colloidal glass-forming liquid

The nonequilibrium self-consistent generalized Langevin equation theory of colloid dynamics is used to describe the nonstationary 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.

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

In this work we propose a theory to describe the irreversible diffusive relaxation of the localconcentration of a colloidal dispersion that proceeds toward its stable thermodynamicequilibrium state, but which may in the process be trapped in metastable or dynamicallyarrested states. The central assumption of this theory is that the irreversiblerelaxation of the macroscopically observed mean value of the local concentration of colloidal particles is described by a diffusion equation involving a localmobility b*(r,t) that depends not only on the mean value but also on the covariance of the fluctuations . This diffusion equation must hence be solved simultaneously with the relaxation equation for thecovariance σ(r,r′;t), and here we also derive the corresponding relaxation equation. The dependence of the local mobilityb*(r,t) on the mean value and the covariance is determined by a self-consistent set of equationsinvolving now the spatially and temporally non-local time-dependent correlation functions,which in a uniform system in equilibrium reduces to the self-consistent generalizedLangevin equation (SCGLE) theory of colloid dynamics. The resulting general theoryconsiders the possibility that these relaxation processes occur under the influence ofexternal fields, such as gravitational forces acting in the process of sedimentation. In thispaper, however, we describe a simpler application, in which the system remainsspatially uniform during the irreversible relaxation process, and discuss the generalfeatures of the glass transition scenario predicted by this non-equilibrium theory.

Ionic and Wigner Glasses, Superionic Conductors, and Spinodal Electrostatic Gels: Dynamically Arrested Phases of the Primitive Model

Based on the recently proposed self-consistent generalized Langevin equation theory of dynamic arrest, in this letter we show that the ergodic-nonergodic phase diagram of a classical mixture of charged hard spheres (the so-called "primitive model" of ionic solutions and molten salts) includes arrested phases corresponding to nonconducting ionic glasses, partially arrested states that represent solid electrolytes (or "superionic" conductors), low-density colloidal Wigner glasses, and low-density electrostatic gels associated with arrested spinodal decomposition.

Glass transition in soft-sphere dispersions

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

Alternative View of Dynamic Arrest in Colloid-Polymer Mixtures

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.

Theory of dynamic arrest in colloidal mixtures

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.

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

The self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics isemployed to describe the ergodic–non-ergodic transition in model mono-disperse colloidaldispersions whose particles interact through hard-sphere plus short-ranged attractive forces.The ergodic–non-ergodic phase diagram in the temperature–concentration state space isdetermined for the hard-sphere plus attractive Yukawa model within the meanspherical approximation for the static structure factor by solving a remarkably simpleequation for the localization length of the colloidal particles. Finite real values ofthis property signals non-ergodicity and determines the non-ergodic parametersf(k) and fs(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 ofmode coupling theory. Although both theories coincide in the general features of this phasediagram, there are also clear qualitative differences. One of the most relevant is the SCGLEprediction that the ergodic–attractive glass transition does not preempt the gas–liquidphase transition, but always intersects the corresponding spinodal curve on itshigh-concentration side. We also calculate the ergodic–non-ergodic phase diagram for thesticky hard-sphere model to illustrate the dependence of the predicted SCGLE dynamicphase diagram on the choice of one important constituent element of the SCGLE theory.

Diffusion of colloidal fluids in random porous media

The diffusive relaxation of a colloidal fluid adsorbed in a porous medium depends on many factors, including the concentration and composition of the adsorbed colloidal fluid, the average structure of the porous matrix, and the nature of the colloid-colloid and colloid-substrate interactions. A simple manner to describe these effects is to model the porous medium as a set of spherical particles fixed in space at random positions with prescribed statistical structural properties. Within this model one may describe the relaxation of concentration fluctuations of the adsorbed fluid by simply setting to zero the short-time mobility of one species (the porous matrix) in a theory of the dynamics of equilibrium colloidal mixtures, or by extending such dynamic theory to explicitly consider the porous matrix as a random external field, as recently done in the framework of mode coupling theory [V. Krakoviack, Phys. Rev. Lett. 94, 065703 (2005)]. Here we consider the first approach and employ the self-consistent generalized Langevin equation (SCGLE) theory of the dynamics of equilibrium colloidal mixtures, to describe the dynamics of the mobile component. We focus on the short- and intermediate-time regimes, which we compare with Brownian dynamics simulations involving a binary mixture with screened Coulomb interactions for two models of the average static structure of the matrix: a porous matrix constructed by quenching configurations of an equilibrium mixture in which both species were first equilibrated together, and a preexisting matrix with prescribed average structure, in which we later add the mobile species. We conclude that in both cases, if the correct static structure factors are provided as input, the SCGLE theory correctly predicts the main features of the dynamics of the permeating fluid.

Simplified self-consistent theory of colloid dynamics

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.

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

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.

Erratum: Self-consistent generalized Langevin equation for colloidal mixtures [Phys. Rev. E72, 031107 (2005)

Erratum: Self-consistent generalized Langevin equation for colloidal mixtures [Phys. Rev. E<b>72</b>, 031107 (2005)

Self-consistent generalized Langevin equation for colloidal mixtures

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.

Self-consistent generalized Langevin equation for colloid dynamics.

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.