Structural relaxation of polydisperse hard spheres: comparison of the mode-coupling theory to a Langevin dynamics simulation

Decoupling of rotation and translation at the colloidal glass transition

In dense particle systems, the coupling of rotation and translation motion becomes intricate. Here, we report the results of confocal fluorescence microscopy where simultaneous recording of translational and rotational particle trajectories from a bidisperse colloidal dispersion is achieved by spiking the samples with rotational probe particles. The latter consist of colloidal particles containing two fluorescently labeled cores suited for tracking the particle's orientation. A comparison of the experimental data with event driven Brownian simulations gives insights into the system's structure and dynamics close to the glass transition and sheds new light onto the translation-rotation coupling. The data show that with increasing volume fractions, translational dynamics slows down drastically, whereas rotational dynamics changes very little. We find convincing agreement between simulation and experiments, even though the simulations neglect far-field hydrodynamic interactions. An additional analysis of the glass transition following mode coupling theory works well for the structural dynamics but indicates a decoupling of the diffusion of the smaller particle species. Shear stress correlations do not decorrelate in the simulated glass states and are not affected by rotational motion.

Rotational inertia-induced glassy transition in chiral particle systems

The dense active matter exhibits characteristics reminiscent of traditional glassy phenomena, yet the role of rotational inertia in glass dynamics remains elusive. In this study, we investigate the glass dynamics of chiral active particles influenced by rotational inertia. Rotational inertia endows exponential memory to particle orientation, restricting its alteration and amplifying the effective persistence time. At lower spinning frequencies, the diffusion coefficient exhibits a peak function relative to rotational inertia for shorter persistence times, while it steadily increases with rotational inertia for longer persistence times. In the realm of high-frequency spinning, the impact of rotational inertia on diffusion behavior becomes more pronounced, resulting in a nonmonotonic and intricate relationship between the diffusion coefficient and rotational inertia. Consequently, the introduction of rotational inertia significantly alters the glassy dynamics of chiral active particles, allowing for the control over transitions between fluid and glassy states by modulating rotational inertia. Moreover, our findings indicate that at a specific spinning temperature, there exists an optimal spinning frequency at which the diffusion coefficient attains its maximum value.

Rescaled mode-coupling scheme for the quantitative description of experimentally observed colloid dynamics

We describe experimentally observed collective dynamics in colloidal suspensions of model hard-sphere particles using a modified mode coupling theory (MCT). This rescaled MCT is capable of describing quantitatively the wave-vector and time-dependent diffusion in these systems. Intermediate scattering functions of liquidlike structured dispersions are determined by means of static and dynamic light-scattering experiments. The structure and short-time dynamics of the systems can be described quantitatively employing a multicomponent Percus-Yevick ansatz for the partial structure factors and an effective, one-component description of hydrodynamic interactions based on the semianalytical δγ expansion. Combined with a recently proposed empirical modification of MCT in which memory functions are calculated using effective structure factors at rescaled number densities, the scheme is able to model the collective dynamics over the entire accessible time and wave-vector range and predicts the volume-fraction-dependence of long-time self-diffusion coefficients and the zero-shear viscosity quantitatively. This highlights the potential of MCT as a practical tool for the quantitative analysis and prediction of experimental observations.

Microscopic theory for nonequilibrium correlation functions in dense active fluids

One of the key hallmarks of dense active matter in the liquid, supercooled, and solid phases is the so-called equal-time velocity correlations. Crucially, these correlations can emerge spontaneously, i.e., they require no explicit alignment interactions, and therefore represent a generic feature of dense active matter. This indicates that for a meaningful comparison or possible mapping between active and passive liquids one not only needs to understand their structural properties, but also the impact of these velocity correlations. This has already prompted several simulation and theoretical studies, though they are mostly focused on athermal systems and thus overlook the effect of translational diffusion. Here, we present a fully microscopic method to calculate nonequilibrium correlations in two-dimensional systems of thermal active Brownian particles (ABPs). We use the integration through transients formalism together with (active) mode-coupling theory and analytically calculate qualitatively consistent static structure factors and active velocity correlations. We complement our theoretical results with simulations of both thermal and athermal ABPs which exemplify the disruptive role that thermal noise has on velocity correlations.

Cooperative activated hopping dynamics in binary glass-forming liquids: effects of the size ratio, composition, and interparticle interactions

Slow dynamics in supercooled and glassy liquids is an important research topic in soft matter physics. Compared to the traditionally focused one-component systems, glassy dynamics in mixture systems adds in a rich set of new complexities, which are fundamentally interesting and also relevant for many technological applications. In this paper, we apply the recently developed self-consistent cooperative hopping theory (SCCHT) to systematically investigate the effects of the size ratio, composition and interparticle interactions on the cooperative activated hopping dynamics of matrix (in larger size) and penetrant (in smaller size) particles in varied binary sphere mixture model systems, with a specific focus on ultrahigh mixture packing fractions that mimic the deeply supercooled glass transition conditions for molecular/polymeric mixture materials. Analysis shows that in these high activation barrier cases, the long-range elastic distortion associated with a matrix particle hopping over its cage confinement always generates an elastic barrier of a nonnegligible magnitude, although the ratio between the elastic barrier and local barrier contribution is sensitively dependent on all three mixture-specific system factors considered in this work. SCCHT predicts two general scenarios of penetrant-matrix cooperative activated hopping dynamics: matrix/penetrant co-hopping (regime 1) or the penetrant mean barrier hopping time shorter than that of the matrix (regime 2). Increasing the penetrant-to-matrix size ratio or the penetrant-matrix cross-attraction strength is found to universally enlarge the composition window of regime 1. Diverse dynamical properties characterising different aspects of the cooperative activated hopping process, including the penetrant and matrix transient localization lengths, penetrant and matrix hopping jump distances, different types of local and elastic activated barriers, and matrix long-time diffusivity, relaxation time and dynamic fragility are quantitatively studied against a wide range of variations over the three system factors. Of particular interest is the universal "anti-plasticization" phenomenon achievable for sufficiently strong cross-attractive interactions. The prospects this work opens for the exploration of a wide variety of polymer-based mixture materials are briefly discussed at the end.

Emergent structural correlations in dense liquids

The complete quantitative description of the structure of dense and supercooled liquids remains a notoriously difficult problem in statistical physics. Most studies to date focus solely on two-body structural correlations, and only a handful of papers have sought to consider additional three-body correlations. Here, we go beyond the state of the art by extracting many-body static structure factors from molecular dynamics simulations and by deriving accurate approximations up to the six-body structure factor via density functional theory. We find that supercooling manifestly increases four-body correlations, akin to the two- and three-body case. However, at small wave numbers, we observe that the four-point structure of a liquid drastically changes upon supercooling, both qualitatively and quantitatively, which is not the case in two-point structural correlations. This indicates that theories of the structure or dynamics of dense liquids should incorporate many-body correlations beyond the two-particle level to fully capture their intricate behavior.

Computer simulations and mode-coupling theory of glass-forming confined hard-sphere fluids

We present mode-coupling theory (MCT) results for densely packed hard-sphere fluids confined between two parallel walls and compare them quantitatively to computer simulations. The numerical solution of MCT is calculated using the full system of matrix-valued integro-differential equations. We investigate several dynamical properties of supercooled liquids including scattering functions, frequency-dependent susceptibilities, and mean-square displacements. Close to the glass transition, we find quantitative agreement between the coherent scattering function predicted from theory and that evaluated from simulations, which enables us to make quantitative statements on caging and relaxation dynamics of the confined hard-sphere fluid.

Glassy Dynamics in Chiral Fluids

Chiral active matter is enjoying a rapid increase of interest, spurred by the rich variety of asymmetries that can be attained in, e.g., the shape or self-propulsion mechanism of active particles. Though this has already led to the observance of so-called chiral crystals, active chiral glasses remain largely unexplored. A possible reason for this could be the naive expectation that interactions dominate the glassy dynamics and the details of the active motion become increasingly less relevant. Here, we show that quite the opposite is true by studying the glassy dynamics of interacting chiral active Brownian particles. We demonstrate that when our chiral fluid is pushed to glassy conditions, it exhibits highly nontrivial dynamics, especially compared to a standard linear active fluid such as common active Brownian particles. Despite the added complexity, we are still able to present a full rationalization for all identified dynamical regimes. Most notably, we introduce a new "hammering" mechanism, unique to rapidly spinning particles in high-density conditions, that can fluidize a chiral active solid.

Active glassy dynamics is unaffected by the microscopic details of self-propulsion

Recent years have seen a rapid increase of interest in dense active materials, which, in the disordered state, share striking similarities with the conventional passive glass-forming matter. For such passive glassy materials, it is well established (at least in three dimensions) that the details of the microscopic dynamics, e.g., Newtonian or Brownian, do not influence the long-time glassy behavior. Here, we investigate whether this still holds true in the non-equilibrium active case by considering two simple and widely used active particle models, i.e., active Ornstein-Uhlenbeck particles (AOUPs) and active Brownian particles (ABPs). In particular, we seek to gain more insight into the role of the self-propulsion mechanism on the glassy dynamics by deriving a mode-coupling theory (MCT) for thermal AOUPs, which can be directly compared to a recently developed MCT for ABPs. Both theories explicitly take into account the active degrees of freedom. We solve the AOUP- and ABP-MCT equations in two dimensions and demonstrate that both models give almost identical results for the intermediate scattering function over a large variety of control parameters (packing fractions, active speeds, and persistence times). We also confirm this theoretical equivalence between the different self-propulsion mechanisms numerically via simulations of a polydisperse mixture of active quasi-hard spheres, thereby establishing that, at least for these model systems, the microscopic details of self-propulsion do not alter the active glassy behavior.

Choice of diameters in a polydisperse model glassformer: Deterministic or stochastic?

In particle-based computer simulations of polydisperse glassforming systems, the particle diameters σ=σ_{1},⋯,σ_{N} of a system with N particles are chosen with the intention to approximate a desired distribution density f with the corresponding histogram. One method to accomplish this is to draw each diameter randomly and independently from the density f. We refer to this stochastic scheme as model S. Alternatively, one can apply a deterministic method, assigning an appropriate set of N values to the diameters. We refer to this method as model D. We show that, for sample-to-sample fluctuations, especially for the glassy dynamics at low temperatures, it matters whether one chooses model S or model D. Using molecular dynamics computer simulations, we investigate a three-dimensional polydisperse nonadditive soft-sphere system with f(s)∼s^{-3}. The swap Monte Carlo method is employed to obtain equilibrated samples at very low temperatures. We show that for model S the sample-to-sample fluctuations due to the quenched disorder imposed by the diameters σ can be explained by an effective packing fraction. Dynamic susceptibilities in model S can be split into two terms: one that is of thermal nature and can be identified with the susceptibility of model D, and another one originating from the disorder in σ. At low temperatures the latter contribution is the dominating term in the dynamic susceptibility. Our study clarifies the pros and cons of the use of models S and D in practice.

Mild stratification in drying films of colloidal mixtures

Size stratification of bidisperse colloidal mixtures during vertical drying was investigated using the implicit solvent Langevin dynamics (LD) simulation and the explicit solvent lattice Boltzmann (LB) method. Simulations were performed for the Péclet number (Pe) over a wide range of 1-1000. In the case of a low size ratio of 2, mild stratification was observed in both simulation methods, in contrast to distinct stratification with thick "small-on-top" or "large-on-top" layers. The LD simulations exhibited a "small-on-top" stratification or mixed state. In contrast, the LB simulations exhibited a "large-on-top" or mixed state, according to the variation in Pe. The results demonstrated that the explicit solvent reduced the collective diffusion under moderate Pe conditions. This suppressed the steep concentration gradient of small particles in the packed region of particles near the air-solvent interface. Thus, distinguishable stratification patterns were obtained for the implicit and explicit solvent models.

Cage Length Controls the Nonmonotonic Dynamics of Active Glassy Matter

Dense active matter is gaining widespread interest due to its remarkable similarity with conventional glass-forming materials. However, active matter is inherently out of equilibrium and even simple models such as active Brownian particles (ABPs) and active Ornstein-Uhlenbeck particles (AOUPs) behave markedly differently from their passive counterparts. Controversially, this difference has been shown to manifest itself via either a speedup, slowdown, or nonmonotonic change of the glassy relaxation dynamics. Here we rationalize these seemingly contrasting views on the departure from equilibrium by identifying the ratio of the short-time length scale to the cage length, i.e., the length scale of local particle caging, as a vital and unifying control parameter for active glassy matter. In particular, we explore the glassy dynamics of both thermal and athermal ABPs and AOUPs upon increasing the persistence time. We find that for all studied systems there is an optimum of the dynamics; this optimum occurs when the cage length coincides with the corresponding short-time length scale of the system, which is either the persistence length for athermal systems or a combination of the persistence length and a diffusive length scale for thermal systems. This new insight, for which we also provide a simple physical argument, allows us to reconcile and explain the manifestly disparate departures from equilibrium reported in many previous studies of dense active materials.

Tracer dynamics in crowded active-particle suspensions

We discuss the dynamics of active Brownian particles (ABPs) in crowded environments through the mean-squared displacement (MSD) of active and passive tracer particles in both active and passive host systems. Exact equations for the MSD are derived using a projection operator technique, extending to dense systems the known solution for a single ABP. The interaction of the tracer particle with the host particles gives rise to strong memory effects. Evaluating these approximately in the framework of a recently developed mode-coupling theory for active Brownian particles (ABP-MCT), we discuss the various dynamical regimes that emerge: While self-propelled motion gives rise to super-diffusive MSD, at high densities, this competes with an interaction-induced sub-diffusive regime. The predictions of the theory are shown to be in good agreement with results obtained from an event-driven Brownian dynamics (ED-BD) simulation scheme for the dynamics of two-dimensional active Brownian hard disks.

Generalized mode-coupling theory for mixtures of Brownian particles

Generalized mode-coupling theory (GMCT) has recently emerged as a promising first-principles theory to study the poorly understood dynamics of glass-forming materials. Formulated as a hierarchical extension of standard mode-coupling theory (MCT), it is able to systematically improve its predictions by including the exact dynamics of higher-order correlation functions into its hierarchy. However, in contrast to Newtonian dynamics, a fully generalized version of the theory based on Brownian dynamics is still lacking. To close this gap, we provide a detailed derivation of GMCT for colloidal mixtures obeying a many-body Smoluchowski equation. We demonstrate that a hierarchy of coupled equations can again be established and show that these, consistent with standard MCT, are identical to the ones obtained from Newtonian GMCT when taking the overdamped limit. Consequently, the nontrivial similarity between Brownian and Newtonian MCT is maintained for our multicomponent GMCT. As a proof of principle, we also solve the generalized mode-coupling equations for the binary Kob-Andersen Lennard-Jones mixture undergoing Brownian dynamics and confirm the improved predictive power of the theory upon using more levels of the GMCT hierarchy of equations.

Mode-coupling theory for tagged-particle motion of active Brownian particles

We derive a mode-coupling theory (MCT) to describe the dynamics of a tracer particle that is embedded in a dense system of active Brownian particles (ABPs) in two spatial dimensions. The ABP undergo translational and rotational Brownian motion and are equipped with a fixed self-propulsion speed along their orientational vector that describes their active motility. The resulting equations of motion for the tagged-particle density-correlation functions describe the various cases of tracer dynamics close to the glass transition: that of a single active particle in a glass-forming passive host suspensions, that of a passive colloidal particle in a suspension of ABP, and that of active tracers in a bath of active particles. Numerical results are presented for these cases assuming hard-sphere interactions among the particles. The qualitative and quantitative accuracy of the theory is tested against event-driven Brownian dynamics (ED-BD) simulations of active and passive hard disks. Simulation and theory are found in quantitative agreement, provided one adjusts the overall density (as known from the passive description of glassy dynamics), and allows for a rescaling of self-propulsion velocities in the active host system. These adjustments account for the fact that ABP-MCT generally overestimates the tendency for kinetic arrest. We confirm in the simulations a peculiar feature of the transient and stationary dynamical density-correlation functions regarding their lack of symmetry under time reversal, demonstrating the nonequilibrium nature of the system and how it manifests itself in the theory.

Tagged-particle motion of Percus-Yevick hard spheres from first principles

We develop a first-principles-based generalized mode-coupling theory (GMCT) for the tagged-particle motion of glassy systems. This theory establishes a hierarchy of coupled integro-differential equations for self-multi-point density correlation functions, which can formally be extended up to infinite order. We use our GMCT framework to calculate the self-nonergodicity parameters and the self-intermediate scattering function for the Percus-Yevick hard-sphere system based on the first few levels of the GMCT hierarchy. We also test the scaling laws in the α- and β-relaxation regimes near the glass-transition singularity. Furthermore, we study the mean-square displacement and the Stokes-Einstein relation in the supercooled regime. We find that qualitatively our GMCT results share many similarities with the well-established predictions from standard mode-coupling theory, but the quantitative results change, and typically improve, by increasing the GMCT closure level. However, we also demonstrate on general theoretical grounds that the current GMCT framework is unable to account for violation of the Stokes-Einstein relation, underlining the need for further improvements in the first-principles description of glassy dynamics.

Multi-component generalized mode-coupling theory: predicting dynamics from structure in glassy mixtures

The emergence of glassy dynamics and the glass transition in dense disordered systems is still not fully understood theoretically. Mode-coupling theory (MCT) has shown to be effective in describing some of the non-trivial features of glass formation, but it cannot explain the full glassy phenomenology due to the strong approximations on which it is based. Generalized mode-coupling theory (GMCT) is a hierarchical extension of the theory, which is able to outclass MCT by carefully describing the dynamics of higher-order correlations in its generalized framework. Unfortunately, the theory has so far only been developed for single-component systems and as a result works poorly for highly polydisperse materials. In this paper, we solve this problem by developing GMCT for multi-component systems. We use it to predict the glassy dynamics of the binary Kob-Andersen Lennard-Jones mixture, as well as its purely repulsive Weeks-Chandler-Andersen analogue. Our results show that each additional level of the GMCT hierarchy gradually improves the predictive power of GMCT beyond its previous limit. This implies that our theory is able to harvest more information from the static correlations, thus being able to better understand the role of attraction in supercooled liquids from a first-principles perspective.

Glassy dynamics of a binary Voronoi fluid: a mode-coupling analysis

The binary Voronoi mixture is a fluid model whose interactions are derived from the Voronoi-Laguerre tessellation of the configurations of the system. The resulting interactions are local and many-body. Here we perform molecular-dynamics (MD) simulations of an equimolar mixture that is weakly polydisperse and additive. For the first time we study the structural relaxation of this mixture in the supercooled-liquid regime. From the simulations we determine the time- and temperature-dependent coherent and incoherent scattering functions for a large range of wave vectors, as well as the mean-square displacements of both particle species. We perform a detailed analysis of the dynamics by comparing the MD results with the first-principles-based idealized mode-coupling theory (MCT). To this end, we employ two approaches: fits to the asymptotic predictions of the theory, and fit-parameter-free binary MCT calculations based on static-structure-factor input from the simulations. We find that many-body interactions of the Voronoi mixture do not lead to strong qualitative differences relative to similar analyses carried out for simple liquids with pair-wise interactions. For instance, the fits give an exponent parameter λ ≈ 0.746 comparable to typical values found for simple liquids, the wavevector dependence of the Kohlrausch relaxation time is in good qualitative agreement with literature results for polydisperse hard spheres, and the MCT calculations based on static input overestimate the critical temperature, albeit only by a factor of about 1.2. This overestimation appears to be weak relative to other well-studied supercooled-liquid models such as the binary Kob-Andersen Lennard-Jones mixture. Overall, the agreement between MCT and simulation suggests that it is possible to predict several microscopic dynamic properties with qualitative, and in some cases near-quantitative, accuracy based solely on static two-point structural correlations, even though the system itself is inherently governed by many-body interactions.

Universality in Driven and Equilibrium Hard Sphere Liquid Dynamics

We demonstrate that the time evolution of the van Hove dynamical pair correlation function is governed by adiabatic forces that arise from the free energy and by superadiabatic forces that are induced by the flow of the van Hove function. The superadiabatic forces consist of drag, viscous, and structural contributions, as occur in active Brownian particles, in liquids under shear and in lane forming mixtures. For hard sphere liquids, we present a power functional theory that predicts these universal force fields in quantitative agreement with our Brownian dynamics simulation results.

Effective structure of a system with continuous polydispersity

In a system of N particles, with continuous size polydispersity, there exists an N(N - 1) number of partial structure factors, making it analytically less tractable. A common practice is to treat the system as an effective one component system, which is known to exhibit an artificial softening of the structure. The aim of this study is to describe the system in terms of M pseudospecies such that we can avoid this artificial softening but, at the same time, have a value of M ≪ N. We use potential energy and pair excess entropy to estimate an optimum number of species, M₀. We then define the maximum width of polydispersity, Δσ₀, that can be treated as a monodisperse system. We show that M₀ depends on the degree and type of polydispersity and also on the nature of the interaction potential, whereas Δσ₀ weakly depends on the type of polydispersity but shows a stronger dependence on the type of interaction potential. Systems with a softer interaction potential have a higher tolerance with respect to polydispersity. Interestingly, M₀ is independent of system size, making this study more relevant for bigger systems. Our study reveals that even 1% polydispersity cannot be treated as an effective monodisperse system. Thus, while studying the role of polydispersity by using the structure of an effective one component system, care must be taken in decoupling the role of polydispersity from that of the artificial softening of the structure.

Generalized mode-coupling theory of the glass transition. I. Numerical results for Percus-Yevick hard spheres

Mode-coupling theory (MCT) constitutes one of the few first-principles-based approaches to describe the physics of the glass transition, but the theory's inherent approximations compromise its accuracy in the activated glassy regime. Here, we show that microscopic generalized mode-coupling theory (GMCT), a recently proposed hierarchical framework to systematically improve upon MCT, provides a promising pathway toward a more accurate first-principles description of glassy dynamics. We present a comprehensive numerical analysis for Percus-Yevick hard spheres by performing explicitly wavenumber- and time-dependent GMCT calculations up to sixth order. Specifically, we calculate the location of the critical point, the associated non-ergodicity parameters, and the time-dependent dynamics of the density correlators at both absolute and reduced packing fractions, and we test several universal scaling relations in the α- and β-relaxation regimes. It is found that higher-order GMCT can successfully remedy some of MCT's pathologies, including an underestimation of the critical glass transition density and an overestimation of the hard-sphere fragility. Furthermore, we numerically demonstrate that the celebrated scaling laws of MCT are preserved in GMCT and that the predicted critical exponents manifestly improve as more levels are incorporated in the GMCT hierarchy. Although formally the GMCT equations should be solved up to infinite order to reach full convergence, our finite-order GMCT calculations unambiguously reveal a uniform convergence pattern for the dynamics. We thus argue that GMCT can provide a feasible and controlled means to bypass MCT's main uncontrolled approximation, offering hope for the future development of a quantitative first-principles theory of the glass transition.

Generalized mode-coupling theory of the glass transition. II. Analytical scaling laws

Generalized mode-coupling theory (GMCT) constitutes a systematically correctable, first-principles theory to study the dynamics of supercooled liquids and the glass transition. It is a hierarchical framework that, through the incorporation of increasingly many particle density correlations, can remedy some of the inherent limitations of the ideal mode-coupling theory (MCT). However, despite MCT's limitations, the ideal theory also enjoys several remarkable successes, notably including the analytical scaling laws for the α- and β-relaxation dynamics. Here, we mathematically derive similar scaling laws for arbitrary-order multi-point density correlation functions obtained from GMCT under arbitrary mean-field closure levels. More specifically, we analytically derive the asymptotic and preasymptotic solutions for the long-time limits of multi-point density correlators, the critical dynamics with two power-law decays, the factorization scaling laws in the β-relaxation regime, and the time-density superposition principle in the α-relaxation regime. The two characteristic power-law-divergent relaxation times for the two-step decay and the non-trivial relation between their exponents are also obtained. The validity ranges of the leading-order scaling laws are also provided by considering the leading preasymptotic corrections. Furthermore, we test these solutions for the Percus-Yevick hard-sphere system. We demonstrate that GMCT preserves all the celebrated scaling laws of MCT while quantitatively improving the exponents, rendering the theory a promising candidate for an ultimately quantitative first-principles theory of glassy dynamics.

Composition fluctuations in polydisperse liquids: Glasslike effects well above the glass transition

We study a two-dimensional glass-forming system of slightly polydisperse (LJ) particles using molecular dynamics simulations and demonstrate that in the liquid regime (well above the vitrification temperature) this model shows a number of features typical of the glass transition: (i) the relation between compressibility and structure factor S(q) is strongly violated; (ii) the dynamical structure factor S(q,t) at low q shows a two-step relaxation; (iii) the time-dependent heat capacity c_{v}(t) shows a long-time power-law tail. We show that these phenomena can be rationalized with the idea of composition fluctuations and provide a quantitative theory for the effects (i) and (ii). It implies that such effects must be inherent in all polydisperse colloidal models, including binary LJ mixtures.

Wave-Vector Dependence of the Dynamics in Supercooled Metallic Liquids

We present a detailed investigation of the wave-vector dependence of collective atomic motion in Au_{49}Cu_{26.9}Si_{16.3}Ag_{5.5}Pd_{2.3} and Pd_{42.5}Cu_{27}Ni_{9.5}P_{21} supercooled liquids close to the glass transition temperature. Using x-ray photon correlation spectroscopy in a previously uncovered spatial range of only a few interatomic distances, we show that the microscopic structural relaxation process mimics the structure and presents a marked slowing down at the main average interparticle distance. This behavior is accompanied by dramatic changes in the shape of the intermediate scattering functions, which suggest the presence of large dynamical heterogeneities at length scales corresponding to a few particle diameters. A ballisticlike mechanism of particle motion seems to govern the structural relaxation of the two systems in the highly viscous phase, likely associated with hopping of caged particles in agreement with theoretical studies.

Solvable Models of Supercooled Liquids in Three Dimensions

We introduce a supercooled liquid model and obtain parameter-free quantitative predictions that are in excellent agreement with numerical simulations, notably in the hard low-temperature region characterized by strong deviations from mode-coupling-theory behavior. The model is the Fredrickson-Andersen kinetically constrained model on the three-dimensional M-layer lattice. The agreement has implications beyond the specific model considered because the theory is potentially valid for many more systems, including realistic models and actual supercooled liquids.

Critical force in active microrheology

Soft solids like colloidal glasses exhibit a yield stress, above which the system starts to flow. The microscopic analogon in microrheology is the untrapping or depinning of a tracer particle subject to an external force exceeding a threshold value in a glassy host. We characterize this delocalization transition based on a bifurcation analysis of the corresponding mode-coupling theory equations. A schematic model that allows analytical progress is presented first, and the full physical model is studied numerically next. This analysis yields a continuous dynamic transition with a critical power-law decay of the probe correlation functions with exponent -1/2. To compare with simulations with a limited duration, a finite-time analysis is performed, which yields reasonable results for not-too-small wave vectors. The theoretically predicted findings are verified by Langevin dynamics simulations. For small wave vectors we find anomalous behavior for the probe position correlation function, which can be traced back to a wave-vector divergence of the critical amplitude. In addition, we propose and test three methods to extract the critical force from experimental data, which provide the same value of the critical force when applied to the finite-time theory or simulations.

Refractive index matched, nearly hard polymer colloids

Refractive index matched particles serve as essential model systems for colloid scientists, providing nearly hard spheres to explore structure and dynamics. The poly(methyl methacrylate) latexes typically used are often refractive index matched by dispersing them in binary solvent mixtures, but this can lead to undesirable changes, such as particle charging or swelling. To avoid these shortcomings, we have synthesized refractive index matched colloids using polymerization-induced self-assembly (PISA) rather than as polymer latexes. The crucial difference is that these diblock copolymer nanoparticles consist of a single core-forming polymer in a single non-ionizable solvent. The diblock copolymer chosen was poly(stearyl methacrylate)-poly(2,2,2-trifluoroethyl methacrylate) (PSMA-PTFEMA), which self-assembles to form PTFEMA core spheres in n-alkanes. By monitoring scattered light intensity, n-tetradecane was found to be the optimal solvent for matching the refractive index of such nanoparticles. As expected for PISA syntheses, the diameter of the colloids can be controlled by varying the PTFEMA degree of polymerization. Concentrated dispersions were prepared, and the diffusion of the PSMA-PTFEMA nanoparticles as a function of volume fraction was measured. These diblock copolymer nanoparticles are a promising new system of transparent spheres for future colloidal studies.

q-Independent Slow Dynamics in Atomic and Molecular Systems

Investigating million-atom systems for very long simulation times, we demonstrate that the collective density-density correlation time (τ_{α}) in simulated supercooled water and silica becomes wave-vector independent (q^{0}) when the probing wavelength is several times larger than the interparticle distance. The q independence of the collective density-density correlation functions, a feature clearly observed in light-scattering studies of some soft-matter systems, is thus a genuine feature of many (but not all) slow-dynamics systems, either atomic, molecular, or colloidal. Indeed, we show that when the dynamics of the density fluctuations includes particle-type diffusion, as in the case of the Lennard-Jones binary-mixture model, the q^{0} regime does not set in and the relaxation time continues to scale as τ_{α}∼q^{-2} even at small q.

Binary colloidal glasses: linear viscoelasticity and its link to the microscopic structure and dynamics

We study the relation between the microscopic structure and dynamics and the macroscopic rheological response of glass-forming colloidal suspensions, namely binary colloidal hard-sphere mixtures with large size asymmetry (1 : 5) that span a large range of mixture compositions close to the glass transition. The dynamical shear moduli are measured by oscillatory rheology and the structure and dynamics on the single-particle level by confocal microscopy. The data are compared with Brownian Dynamics simulations and predictions from mode-coupling theory based on the Percus-Yevick approximation. Experiments, simulations and theory consistently observe a strong decrease of the intermediate-frequency mechanical moduli combined with faster dynamics at intermediate mixing ratios and hence a non-monotonic dependence of these parameters but a localization of the large particles which decreases monotonically as the fraction of small particles is increased. We find that the Generalized-Stokes Einstein relation applied to the mean square displacements of the two components leads to a reasonable estimate of the shear moduli of the mixtures and hence links the rheological response to the particle dynamics which in turn reflects the microscopic structure.

Bypassing sluggishness: SWAP algorithm and glassiness in high dimensions

The recent implementation of a swap Monte Carlo algorithm (SWAP) for polydisperse glass forming mixtures bypasses computational sluggishness and closes the gap between experimental and simulation timescales in physical dimensions d=2 and 3. Here, we consider suitably optimized systems in d=2,3,⋯,8 to obtain insights into the performance and underlying physics of SWAP. We show that the speedup obtained decays rapidly with increasing the dimension. SWAP nonetheless delays systematically the onset of the activated dynamics by an amount that remains seemingly finite in the limit d→∞. This shows that the glassy dynamics in high dimensions d>3 is now computationally accessible using SWAP, thus opening the door for the systematic consideration of finite-dimensional deviations from the mean-field description.

Continuous-time random-walk approach to supercooled liquids: Self-part of the van Hove function and related quantities

From equilibrium molecular dynamics (MD) simulations of a bead-spring model for short-chain glass-forming polymer melts we calculate several quantities characterizing the single-monomer dynamics near the (extrapolated) critical temperature [Formula: see text] of mode-coupling theory: the mean-square displacement g₀(t), the non-Gaussian parameter [Formula: see text] and the self-part of the van Hove function [Formula: see text] which measures the distribution of monomer displacements r in time t. We also determine these quantities from a continuous-time random walk (CTRW) approach. The CTRW is defined in terms of various probability distributions which we know from previous analysis. Utilizing these distributions the CTRW can be solved numerically and compared to the MD data with no adjustable parameter. The MD results reveal the heterogeneous and non-Gaussian single-particle dynamics of the supercooled melt near [Formula: see text]. In the time window of the early [Formula: see text] relaxation [Formula: see text] is large and [Formula: see text] is broad, reflecting the coexistence of monomer displacements that are much smaller ("slow particles") and much larger ("fast particles") than the average at time t, i.e. than [Formula: see text]. For large r the tail of [Formula: see text] is compatible with an exponential decay, as found for many glassy systems. The CTRW can reproduce the spatiotemporal dependence of [Formula: see text] at a qualitative to semiquantitative level. However, it is not quantitatively accurate in the studied temperature regime, although the agreement with the MD data improves upon cooling. In the early [Formula: see text] regime we also analyze the MD results for [Formula: see text] via the space-time factorization theorem predicted by ideal mode-coupling theory. While we find the factorization to be well satisfied for small r, both above and below [Formula: see text] , deviations occur for larger r comprising the tail of [Formula: see text]. The CTRW analysis suggests that single-particle "hops" are a contributing factor for these deviations.

Hard-sphere-like dynamics in highly concentrated alpha-crystallin suspensions

The dynamics of concentrated suspensions of the eye-lens protein alpha crystallin have been measured using x-ray photon correlation spectroscopy. Measurements were made at wave vectors corresponding to the first peak in the hard-sphere structure factor and volume fractions close to the critical volume fraction for the glass transition. Langevin dynamics simulations were also performed in parallel to the experiments. The intermediate scattering function f(q,τ) could be fit using a stretched exponential decay for both experiments and numerical simulations. The measured relaxation times show good agreement with simulations for polydisperse hard-sphere colloids.

Dramatic slowing of compositional relaxations in the approach to the glass transition for a bimodal colloidal suspension

Molecular dynamics simulation was used to study a model colloidal suspension with two species of slightly different sized colloidal particles in an explicit solvent. In this work we calculated the four interdiffusion coefficients for the ternary system, which were then used to calculate the decay coefficients D_{±} of the two independent diffusive modes. We found that the slower D_{-} decay mode, which is associated with the system's ability to undergo compositional changes, was responsible for the long-time decay in the intermediate scattering function. We also found that a decrease in D_{-} to negligible values at a packing fraction of Φ_{g}=0.592 resulted in an extreme slow-down in the long-time decay of the intermediate scattering function often associated with the glass transition. Above Φ_{g}, the system formed a long-lived metastable state that did not relax to its equilibrium crystal state within the simulation time window. We concluded that the inhibition of crystallization was caused by the inability of the quenched fluid to undergo the compositional changes needed for the formation of the equilibrium crystal.

Nonergodicity parameters of confined hard-sphere glasses

Within a recently developed mode-coupling theory for fluids confined to a slit we elaborate numerical results for the long-time limits of suitably generalized intermediate scattering functions. The theory requires as input the density profile perpendicular to the plates, which we obtain from density functional theory within the fundamental-measure framework, as well as symmetry-adapted static structure factors, which can be calculated relying on the inhomogeneous Percus-Yevick closure. Our calculations for the nonergodicity parameters for both the collective as well as for the self motion are in qualitative agreement with our extensive event-driven molecular dynamics simulations for the intermediate scattering functions for slightly polydisperse hard-sphere systems at high packing fraction. We show that the variation of the nonergodicity parameters as a function of the wavenumber correlates with the in-plane static structure factors, while subtle effects become apparent in the structure factors and relaxation times of higher mode indices. A criterion to predict the multiple reentrant from the variation of the in-plane static structure is presented.

Describing magnetorheology under a colloidal glass approach

The equilibrium structure and dynamics of magnetorheological (MR) fluids are studied in this work by simulations, where particles are modeled as dipoles with a quasihard spherical core. Upon increasing the interaction strength, controlled experimentally by the magnetic field, elongated clusters grow and, for intense fields, thick columns form, aligned with the field. The dynamics of the system is monitored by the mean-squared displacement and density correlation functions, which show an increasing slowing down with the attraction strength. The correlation function shows a two-step decay, with a separation between microscopic and long time dynamics, a typical hallmark of undercooled fluids. We have therefore analyzed the dynamics of this MR fluid using the typical concepts for undercooled fluids. Thus, the second decay of the density correlation function is fitted with a stretched exponential, and the wave-vector dependence of the fitting parameters studied. Both the amplitude and the time scale oscillate in phase with the structure factor. Our results support the idea that the magnetorheological effect is in fact the manifestation of a colloidal system approaching an attractive glass transition (or gel transition).

Active microrheology in a colloidal glass

We study the dynamics of a probe particle driven by a constant force through a colloidal glass of hard spheres. This nonequilibrium and anisotropic problem is investigated using a new implementation of the mode-coupling approximation with multiple relaxation channels and Langevin dynamics simulations. A force threshold is found, below which the probe remains localized, while above it the probe acquires a finite velocity. We focus on the localized regime, comparing theory and simulations concerning the dynamics in the length scale of the cage and the properties of the transition to the delocalized regime, such as the critical power-law decay of the probe correlation function. Probe van Hove functions predicted by the theory show exponential tails reminiscent of an intermittent dynamics of the probe. This scenario is microscopically supported by simulations.

Non-monotonic effect of confinement on the glass transition

The relaxation dynamics of glass forming liquids and their structure are influenced in the vicinity of confining walls. This effect has mostly been observed to be a monotonic function of the slit width. Recently, a qualitatively new behaviour has been uncovered by Mittal and coworkers, who reported that the single particle dynamics in a hard-sphere fluid confined in a planar slit varies in a non-monotonic way as the slit width is decreased from five to roughly two particle diametres (Mittal et al 2008 Phys. Rev. Lett. 100 145901). In view of the great potential of this effect for applications in those fields of science and industry, where liquids occur under strong confinement (e.g. nano-technology), the number of researchers studying various aspects and consequences of this non-monotonic behaviour has been rapidly growing. This review aims at providing an overview of the research activity in this newly emerging field. We first briefly discuss how competing mechanisms such as packing effects and short-range attraction may lead to a non-monotonic glass transition scenario in the bulk. We then analyse confinement effects on the dynamics of fluids using a thermodynamic route which relates the single particle dynamics to the excess entropy. Moreover, relating the diffusive dynamics to the Widom's insertion probability, the oscillations of the local dynamics with density at moderate densities are fairly well described. At high densities belonging to the supercooled regime, however, this approach breaks down signaling the onset of strongly collective effects. Indeed, confinement introduces a new length scale which in the limit of high densities and small pore sizes competes with the short-range local order of the fluid. This gives rise to a non-monotonic dependence of the packing structure on confinement, with a corresponding effect on the dynamics of structural relaxation. This non-monotonic effect occurs also in the case of a cone-plate type channel, where the degree of confinement varies with distance from the apex. This is a very promising issue for future research with the possibility of uncovering the existence of alternating glassy and liquid-like domains.

Quantitative approximation schemes for glasses

By means of a systematic expansion around the infinite-dimensional solution, we obtain an approximation scheme to compute properties of glasses in low dimensions. The resulting equations take as input the thermodynamic and structural properties of the equilibrium liquid, and from this they allow one to compute properties of the glass. They are therefore similar in spirit to the Mode Coupling approximation scheme. Our scheme becomes exact, by construction, in dimension d→∞, and it can be improved systematically by adding more terms in the expansion.

Microscopic Dynamics of Supercooled Liquids from First Principles

The transition from a liquid to a glass remains one of the most poorly understood phenomena in condensed matter physics, and still no fully microscopic theory exists that can describe the dynamics of supercooled liquids in a quantitative manner over all relevant time scales. Here, we present a theoretical framework that yields near-quantitative accuracy for the time-dependent correlation functions of a glass-forming system over a broad density range. Our approach requires only simple static structural information as input and is based entirely on first principles. Owing to its ab initio nature, the framework offers a unique platform to study the relation between structure and dynamics in glass-forming matter, and paves the way towards a systematically correctable and ultimately fully quantitative theory of microscopic glassy dynamics.

Numerical study of long-time dynamics and ergodic-nonergodic transitions in dense simple fluids

Since the mid-1980s, mode-coupling theory (MCT) has been the de facto theoretic description of dense fluids and the transition from the fluid state to the glassy state. MCT, however, is limited by the approximations used in its construction and lacks an unambiguous mechanism to institute corrections. We use recent results from a new theoretical framework--developed from first principles via a self-consistent perturbation expansion in terms of an effective two-body potential--to numerically explore the kinetics of systems of classical particles, specifically hard spheres governed by Smoluchowski dynamics. We present here a full solution for such a system to the kinetic equation governing the density-density time correlation function and show that the function exhibits the characteristic two-step decay of supercooled fluids and an ergodic-nonergodic transition to a dynamically arrested state. Unlike many previous numerical studies--and in stark contrast to experiment--we have access to the full time and wave-number range of the correlation function with great precision and are able to track the solution unprecedentedly close to the transition, covering nearly 15 decades in scaled time. Using asymptotic approximation techniques analogous to those developed for MCT, we fit the solution to predicted forms and extract critical parameters. We find complete qualitative agreement with known glassy behavior (e.g. power-law divergence of the α-relaxation time scale in the ergodic phase and square-root growth of the glass form factors in the nonergodic phase), as well as some limited quantitative agreement [e.g. the transition at packing fraction η*=0.60149761(10)], consistent with previous static solutions under this theory and with comparable colloidal suspension experiments. However, most importantly, we establish that this new theory is able to reproduce the salient features seen in other theories, experiments, and simulations but has the advantages of being derived from first principles and possessing a clear mechanism for making systematic corrections.

Simulated glass-forming polymer melts: dynamic scattering functions, chain length effects, and mode-coupling theory analysis

We present molecular-dynamics simulations for a fully flexible model of polymer melts with different chain length N ranging from short oligomers (N = 4) to values near the entanglement length (N = 64). For these systems we explore the structural relaxation of the supercooled melt near the critical temperature T c of mode-coupling theory (MCT). Coherent and incoherent scattering functions are analyzed in terms of the idealized MCT. For temperatures T > T c we provide evidence for the space-time factorization property of the β relaxation and for the time-temperature superposition principle (TTSP) of the α relaxation, and we also discuss deviations from these predictions for T ≈ T c. For T larger than the smallest temperature where the TTSP holds we perform a quantitative analysis of the dynamics with the asymptotic MCT predictions for the late β regime. Within MCT a key quantity, in addition to T c, is the exponent parameter λ. For the fully flexible polymer models studied we find that λ is independent of N and has a value (λ = 0.735 ) typical of simple glass-forming liquids. On the other hand, the critical temperature increases with chain length toward an asymptotic value T c (∞) . This increase can be described by T c (∞) - T c(N) ∼ 1/N and may be interpreted in terms of the N dependence of the monomer density ρ, if we assume that the MCT glass transition is ruled by a soft-sphere-like constant coupling parameter Γ c = ρ c T c (-1/4), where ρ c is the monomer density at T c. In addition, we also estimate T c from a Hansen-Verlet-like criterion and MCT calculations based on structural input from the simulation. For our polymer model both the Hansen-Verlet criterion and the MCT calculations suggest T c to decrease with increasing chain length, in contrast to the direct analysis of the simulation data.

Hard sphere-like glass transition in eye lens α-crystallin solutions

We study the equilibrium liquid structure and dynamics of dilute and concentrated bovine eye lens α-crystallin solutions, using small-angle X-ray scattering, static and dynamic light scattering, viscometry, molecular dynamics simulations, and mode-coupling theory. We find that a polydisperse Percus-Yevick hard-sphere liquid-structure model accurately reproduces both static light scattering data and small-angle X-ray scattering liquid structure data from α-crystallin solutions over an extended range of protein concentrations up to 290 mg/mL or 49% vol fraction and up to ca. 330 mg/mL for static light scattering. The measured dynamic light scattering and viscosity properties are also consistent with those of hard-sphere colloids and show power laws characteristic of an approach toward a glass transition at α-crystallin volume fractions near 58%. Dynamic light scattering at a volume fraction beyond the glass transition indicates formation of an arrested state. We further perform event-driven molecular dynamics simulations of polydisperse hard-sphere systems and use mode-coupling theory to compare the measured dynamic power laws with those of hard-sphere models. The static and dynamic data, simulations, and analysis show that aqueous eye lens α-crystallin solutions exhibit a glass transition at high concentrations that is similar to those found in hard-sphere colloidal systems. The α-crystallin glass transition could have implications for the molecular basis of presbyopia and the kinetics of molecular change during cataractogenesis.

Continuous-time random-walk approach to supercooled liquids. II. Mean-square displacements in polymer melts

The continuous-time random walk (CTRW) describes the single-particle dynamics as a series of jumps separated by random waiting times. This description is applied to analyze trajectories from molecular dynamics (MD) simulations of a supercooled polymer melt. Based on the algorithm presented by Helfferich et al. [Phys. Rev. E 89, 042603 (2014)], we detect jump events of the monomers. As a function of temperature and chain length, we examine key distributions of the CTRW: the jump-length distribution (JLD), the waiting-time distribution (WTD), and the persistence-time distribution (PTD), i.e., the distribution of waiting times for the first jump. For the equilibrium (polymer) liquid under consideration, we verify that the PTD is determined by the WTD. For the mean-square displacement (MSD) of a monomer, the results for the CTRW model are compared with the underlying MD data. The MD data exhibit two regimes of subdiffusive behavior, one for the early α process and another at later times due to chain connectivity. By contrast, the analytical solution of the CTRW yields diffusive behavior for the MSD at all times. Empirically, we can account for the effect of chain connectivity in Monte Carlo simulations of the CTRW. The results of these simulations are then in good agreement with the MD data in the connectivity-dominated regime, but not in the early α regime where they systematically underestimate the MSD from the MD.

Continuous-time random-walk approach to supercooled liquids. I. Different definitions of particle jumps and their consequences

Single-particle trajectories in supercooled liquids display long periods of localization interrupted by "fast moves." This observation suggests a modeling by a continuous-time random walk (CTRW). We perform molecular dynamics simulations of equilibrated short-chain polymer melts near the critical temperature of mode-coupling theory Tc and extract "moves" from the monomer trajectories. We show that not all moves comply with the conditions of a CTRW. Strong forward-backward correlations are found in the supercooled state. A refinement procedure is suggested to exclude these moves from the analysis. We discuss the repercussions of the refinement on the jump-length and waiting-time distributions as well as on characteristic time scales, such as the average waiting time ("exchange time") and the average time for the first move ("persistence time"). The refinement modifies the temperature (T) dependence of these time scales. For instance, the average waiting time changes from an Arrhenius-type to a Vogel-Fulcher-type T dependence. We discuss this observation in the context of the bifurcation of the α process and (Johari) β process found in many glass-forming materials to occur near Tc. Our analysis lays the foundation for a study of the jump-length and waiting-time distributions, their temperature and chain-length dependencies, and the modeling of the monomer dynamics by a CTRW approach in the companion paper [J. Helfferich et al., Phys. Rev. E 89, 042604 (2014)].