Glass-transition Singularities Research Articles

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.

Mode-coupling theory of the glass transition for colloidal liquids in slit geometry

ABSTRACT We provide a detailed derivation of the mode-coupling equations for a colloidal liquid confined by two parallel smooth walls. We introduce irreducible memory kernels for the different relaxation channels thereby extending the projection operator technique to colloidal liquids in slit geometry. Investigating both the collective dynamics as well as the tagged-particle motion, we prove that the mode-coupling functional assumes the same form as in the Newtonian case corroborating the universality of the glass-transition singularity with respect to the microscopic dynamics.

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

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

Higher-order glass-transition singularities in nano-confined states

The self-intermediate scattering functions of confined BPM at 360 K.

Qualitative features at the glass crossover

We discuss generic features of the dynamics of glass-forming liquids close to the glass transition singularity of the idealized mode-coupling theory (MCT). The analysis is based on a recent theory by one of the authors for the intermediate-time dynamics (β relaxation), derived by applying dynamical field-theory techniques to the idealized MCT. Taking the long-time asymptote of the β-relaxation regime as a proxy for universal aspects of the slow structural relaxation, we explain three prominent features of the dynamical crossover: the change from a power-law to exponential increase in the structural relaxation time, the replacement of the Stokes-Einstein relation between diffusion and viscosity by a fractional law, and two distinct growth regimes of the thermal susceptibility that has been associated to dynamical heterogeneities.

Glass transitions and scaling laws within an alternative mode-coupling theory.

Idealized glass transitions are discussed within an alternative mode-coupling theory (TMCT) proposed by Tokuyama [Physica A 395, 31 (2014)]. This is done in order to identify common ground with and differences from the conventional mode-coupling theory (MCT). It is proven that both theories imply the same scaling laws for the transition dynamics, which are characterized by two power-law decay functions and two diverging power-law time scales. However, the values for the corresponding anomalous exponents calculated within both theories differ from each other. It is proven that the TMCT, contrary to the MCT, does not describe transitions with continuously vanishing arrested parts of the correlation functions. It is also demonstrated for a schematic model that the TMCT does not lead to the MCT scenarios either for transition-line crossings or for the appearance of higher-order glass-transition singularities.

Higher-Order Singularities without Glass-Glass Transitions

Within the framework of mode-coupling theory, the glass-transition scenario is investigated for a system of particles interacting with a hard-core repulsion and an additional square-shoulder soft core at larger distances. The static structure is calculated from the potential in Percus-Yevick approximation. For certain widths of the shoulder, the exponent parameter λ along the glass-transition line shows a double peak. At both peaks, λ can reach unity indicating the existence of higher-order glass-transition singularities. It is shown that these higher-order singularities originate from a line of avoided glass-glass transitions.

Free energy landscape approach to glass transition

As a Landau-type theory for the glass transition, we present a free energy landscape (FEL) picture which provides a unified understanding of glass transition singularities and show that the FEL can really be calculated by a theoretical and a computational approach. We first give a clear definition of the FEL and argue that there are two kinds of cooperative rearranging region. One is the region defined by the difference between two adjacent basins which could be called simultaneously rearranged region and the other is the atoms involved in the excited state between two adjacent basins. Exploiting the density functional theory, we obtain the FEL for a relaxation process which is characterized by a string motion and determine the size of the cooperatively rearranging region. We also show that the FEL can be determined by the principal component analysis for time dependent configuration obtained by the MD simulation.

Critical decay at higher-order glass-transition singularities

Within the mode-coupling theory for the evolution of structural relaxation in glass-formingsystems, it is shown that the correlation functions for density fluctuations for states atA3- andA4-glass-transition singularities can be presented as an asymptoticseries in increasing inverse powers of the logarithm of the timet: , where gn(x) = pn(lnx)/xn withpn denoting somepolynomial and x = ln(t/t0). The results are demonstrated for schematic models describing the system by solely one ortwo correlators and also for a colloid model with a square-well-interaction potential.

Dynamics in colloidal liquids near a crossing of glass- and gel-transition lines.

Within the mode-coupling theory for ideal glass transitions, the mean-squared displacement and the correlation function for density fluctuations are evaluated for a colloidal liquid of particles interacting with a square-well potential for states near the crossing of the line for transitions to a gel with the line for transitions to a glass. It is demonstrated how the dynamics is ruled by the interplay of the mechanisms of arrest due to hard-core repulsion and due to attraction-induced bond formation as well as by a nearby higher-order glass-transition singularity. Application of the universal relaxation laws for the slow dynamics near glass-transition singularities explains the qualitative features of the calculated time dependence of the mean-squared displacement, which are in accord with the findings obtained in molecular-dynamics simulation studies by Zaccarelli et al. [Phys. Rev. E 66, 041402 (2002)]. Correlation functions found by photon-correlation spectroscopy in a micellar system by Mallamace et al. [Phys. Rev. Lett. 84, 5431 (2000)] can be interpreted qualitatively as a crossover from gel to glass dynamics.

Evidence of a higher-order singularity in dense short-ranged attractive colloids.

We study a model in which particles interact through a hard-core repulsion complemented by a short-ranged attractive potential of the kind found in colloidal suspensions. Combining theoretical and numerical work we locate the line of higher-order glass-transition singularities and its end point-named A4-on the fluid-glass line. Close to the A4 point, we detect logarithmic decay of density correlations and a sublinear power-law increase of the mean square displacement, for time intervals up to 4 orders of magnitude. We establish the presence of the A4 singularity by studying how the range of the potential affects the time window where anomalous dynamics is observed.

Logarithmic relaxation in a colloidal system.

The slow dynamics for a colloidal suspension of particles interacting with a hard-core repulsion complemented by a short-ranged attraction is discussed within the frame of mode-coupling theory for ideal glass transitions for parameter points near a higher-order glass-transition singularity. The solutions of the equations of motion for the density correlation functions are solved for the square-well system in quantitative detail by asymptotic expansion using the distance of the three control parameters-packing fraction, attraction strength and attraction range-from their critical values as small parameters. For given wave vectors, distinguished surfaces in parameter space are identified where the next-to-leading-order contributions for the expansion vanish so that the decay functions exhibit a logarithmic decay over large time intervals. For both coherent and tagged particle dynamics the leading-order logarithmic decay is accessible in the liquid regime for wave vectors of several times the principal peak in the structure factor. The logarithmic decay in the correlation function is manifested in the mean-squared displacement as a subdiffusive power law with an exponent varying sensitively with the control parameters. Shifting parameters through the distinguished surfaces, the correlation functions and the logarithm of the mean-squared displacement considered as functions of the logarithm of the time exhibit a crossover from concave to convex behavior, and a similar scenario is obtained when varying the wave vector.

Higher-order glass-transition singularities in systems with short-ranged attractivepotentials

Within the mode-coupling theory for the evolution of structural relaxation, theA4-glass-transitionsingularities are identified for systems of particles interacting with a hard-sphererepulsion complemented by different short-ranged potentials: Baxter’ssingular potential regularized by a large-wavevector cut-off, a model for theAsakura–Oosawa depletion attraction, a triangular potential, a Yukawaattraction, and a square-well potential. The regular potentials yield criticalpacking fractions, critical Debye–Waller factors, and critical amplitudes very closeto each other. The elastic moduli and the particle localization lengths forcorresponding states of the Yukawa system and the square-well system may differby up to 20 and 10%, respectively.

Logarithmic relaxation in glass-forming systems.

Within the mode-coupling theory for ideal glass transitions, an analysis of the correlation functions of glass-forming systems for states near higher-order glass-transition singularities is presented. It is shown that the solutions of the equations of motion can be asymptotically expanded in polynomials of the logarithm of time t. In leading order, a ln(t) law is obtained, and the leading corrections are given by a fourth-order polynomial. The correlators interpolate between three scenarios. First, there are surfaces in parameter space where the dominant corrections to the ln(t) law vanish, so that the logarithmic decay governs the structural relaxation process. Second, the dynamics due to the higher-order singularity can describe the initial and intermediate part of the alpha process thereby reducing the range of validity of von Schweidler's law and leading to strong alpha relaxation stretching. Third, the ln(t) law can replace the critical decay law of the beta process, leading to a particularly large crossover interval between the end of the transient and the beginning of the alpha process. This may lead to susceptibility spectra below the band of microscopic excitations exhibiting two peaks. Typical results of the theory are demonstrated for models dealing with one and two correlation functions.

Mode-coupling theory of colloids with short-range attractions

Within the framework of the mode-coupling theory of super-cooledliquids, we investigate new phenomena in colloidal systems on approachto their glass transitions. When the inter-particle potentialcontains an attractive part, besides the usual repulsive hard core,two intersecting liquid-glass transition lines appear, one of whichextends to low densities, while the other one, at high densities,shows a re-entrant behaviour. In the glassy region a newtype of transition appears between two different types ofglasses. The complex phenomenology can be described in terms of higherorder glass transition singularities. The various glass phases arecharacterized by means of their viscoelastic properties.The glass driven by attractions has been associatedwith particle gels, andthe other glass is the well known repulsive colloidal glass. Thesecorrespondences, in association with the new predictions of glassybehaviour, mean that such phenomena may be expected in colloidal systemswith, for example, strong depletion or other short-ranged attractivepotentials.

Higher-order glass-transition singularities in colloidal systems with attractive interactions.

The transition from a liquid to a glass in colloidal suspensions of particles interacting through a hard core plus an attractive square-well potential is studied within the mode-coupling-theory framework. When the width of the attractive potential is much shorter than the hard-core diameter, a reentrant behavior of the liquid-glass line and a glass-glass-transition line are found in the temperature-density plane of the model. For small well-width values, the glass-glass-transition line terminates in a third-order bifurcation point, i.e., in a A3 (cusp) singularity. On increasing the square-well width, the glass-glass line disappears, giving rise to a fourth-order A4 (swallow-tail) singularity at a critical well width. Close to the A3 and A4 singularities the decay of the density correlators shows stretching of huge dynamical windows, in particular logarithmic time dependence.

Microscopicderivation of jump rate distribution and the glass transition

A general framework is given for deriving the jump rate distributionfunction and the long-time behaviour of the waiting timedistribution for the jump is shown to be a power-law function whoseexponent is related to the contribution of fast modes relative tothat of slow modes in the density fluctuation. A unified explanation forthe vitrification process and glass transition singularities ispresented.

Kinetic glass transition in a micellar system with short-range attractive interaction

We show that percolation and structural arrest transitions coexist in different regions of the phase diagram of a copolymer-micellar system and relate them to short-range intermicellar attraction. The intermediate scattering function shows a nonergodic transition along a temperature and concentration dependent line. Analyses show a logarithmic time dependence, attributed to a higher-order glass transition singularity predicted by mode-coupling theory, followed by a power law.

Recent tests of the mode-coupling theory for glassy dynamics

An overview is given of recent tests of the mode-coupling theory for the evolution of structural relaxation in glass-forming liquids. Emphasis is put on comparisons between the leading-order asymptotic formulae derived for the dynamics near glass transition singularities and the results of neutron scattering, depolarized light scattering, impulsive stimulated light scattering and dielectric-loss spectroscopy for conventional liquids. The tests based on photon-correlation spectroscopy results for the glassy dynamics of colloids and the findings of molecular dynamics simulations for model systems are also considered.

Equations of structural relaxation

In the mode coupling theory of the liquid to glass transition the long time structural relaxation follows from equations solely determined by equilibrium structural parameters. The present extension of these structural relaxation equations to arbitrarily short times on the one hand allows calculations unaffected by model assumptions about the microscopic dynamics and on the other hand supplies new starting points for analytical studies. As a first application, power-law like structural relaxation at a glass-transition singularity is explicitly proven for a special schematic MCT model.