Mode-coupling Theory Equations Research Articles

An Improved Integration Scheme for Mode-Coupling-Theory Equations

Within the mode-coupling theory (MCT) of the glass transition, we reconsider the numerical schemes to evaluate the MCT functional. Here we propose nonuniform discretizations of the wave number, in contrast to the standard equidistant grid, in order to decrease the number of grid points without losing accuracy. We discuss in detail how the integration scheme on the new grids has to be modified from standard Riemann integration. We benchmark our approach by solving the MCT equations numerically for mono-disperse hard disks and hard spheres and by computing the critical packing fraction and the nonergodicity parameters. Our results show that significant improvements in performance can be obtained by employing a nonuniform grid.

Dynamical properties of densely packed confined hard-sphere fluids.

Numerical solutions of the mode-coupling theory (MCT) equations for a hard-sphere fluid confined between two parallel hard walls are elaborated. The governing equations feature multiple parallel relaxation channels which significantly complicate their numerical integration. We investigate the intermediate scattering functions and the susceptibility spectra close to structural arrest and compare to an asymptotic analysis of the MCT equations. We corroborate that the data converge in the β-scaling regime to two asymptotic power laws, viz. the critical decay and the von Schweidler law. The numerical results reveal a nonmonotonic dependence of the power-law exponents on the slab width and a nontrivial kink in the low-frequency susceptibility spectra. We also find qualitative agreement of these theoretical results to event-driven molecular dynamics simulations of polydisperse hard-sphere systems. In particular, the nontrivial dependence of the dynamical properties on the slab width is well reproduced.

Description of glass transition kinetics in 3D [formula omitted] model in terms of gauge field theory

A gauge theory of the glass transition in the frustrated XY model being simplest model containing topologically nontrivial excitations is considered. Investigation of the transition kinetics of this system finds that the three-dimensional system exhibits the Vogel–Fulcher–Tammann criticality going with its freezing into a spin glass. It is analytically shown that the system demonstrates all glass transition properties, like the logarithmic relaxation, and corresponding behaviour of linear and non-linear susceptibility. The mode-coupling theory equation in the Zwanziger–Mori representation also is derived in framework of our approach. The findings provide insights into the topological origin of glass formation, that allows to make progress in understanding glass-transition processes in more intricate systems.

Glass transition in hard-core fluids and beyond, using an effective static structure in the mode coupling theory.

The dynamical arrest in classical fluids is studied using a simple modification of the mode coupling theory (MCT) aimed at correcting its overestimation of the tendency to glass formation while preserving its overall structure. As in previous attempts, the modification is based on the idea of tempering the static pair correlations used as input. It is implemented in this work by computing the static structure at a different state point than the one used to solve the MCT equation for the intermediate scattering function, using the pure hard-sphere glass for calibration. The location of the glass transition predicted from this modification is found to agree with simulations data for a variety of systems --pure fluids and mixtures with either purely repulsive interaction potentials or ones with attractive contributions. Besides improving the predictions in the long-time limit, and so reducing the non-ergodicity domain, the same modification works as well for the time-dependent correlators.

Diffusion of a Rouse chain in porous media: A mode-coupling-theory study.

We use a kinetic mode-coupling theory (MCT) combining with generalized Langevin equation (GLE) to study the diffusion and conformational dynamics of a bead-spring Rouse chain (RC) dissolved in porous media. The media contains fluid particles and immobile matrix ones wherein the latter leads to the lack of translational invariance. The friction kernel ζ(t) used in the GLE can be obtained directly by adopting a simple density-functional approach in which the density correlators calculated by MCT equations of porous media serve as inputs. Due to cage effects generated by surrounding particles, ζ(t) shows a very long tail memory in the high volume fraction of fluid and matrix. It is found that the long-time center-of-mass diffusion constant D_{CM} of the RC decreases with the increment of volume fraction, influencing more strongly by the matrix particles than by the fluid ones. The auto-correlation function (ACF) of the end-to-end distance fluctuation can also be calculated theoretically based on GLE. Of particular interest is that the power-law region of ACF has a nearly fixed length in logarithmic scale when it shifts to longer time range, with increasing the volume fraction of media particles. Moreover, the effect of lack of translational invariance has been investigated by comparing the results between fluid-matrix and pure fluid cases under identical total volume fraction.

Understanding the approximations of mode-coupling theory for sheared steady states of colloids.

The lack of clarity of various mode-coupling theory (MCT) approximations, even in equilibrium, makes it hard to understand the relation between various MCT approaches for sheared steady states as well as their regime of validity. Here we try to understand these approximations indirectly by deriving the MCT equations through two different approaches for a colloidal system under shear, first through a microscopic approach, as suggested by Zaccarelli etal., and second through fluctuating hydrodynamics, where the approximations used in the derivation are quite clear. The qualitative similarity of our theory with a number of existing theories show that linear response theory might play a role in various approximations employed in deriving those theories and one needs to be careful while applying them for systems arbitrarily far away from equilibrium, such as a granular system or when shear is very strong. As a by-product of our calculation, we obtain the extension of the Yvon-Born-Green (YBG) equation for a sheared system and under the assumption of random-phase approximation, the YBG equation yields the distorted structure factor that was earlier obtained through different approaches.

Mode-coupling approach for the slow dynamics of a liquid on a spherical substrate.

We study the dynamics of a one-component liquid constrained on a spherical substrate, a 2-sphere, and investigate how the mode-coupling theory (MCT) can describe the new features brought by the presence of curvature. To this end we have derived the MCT equations in a spherical geometry. We find that, as seen from the MCT, the slow dynamics of liquids in curved space at low temperature does not qualitatively differ from that of glass-forming liquids in Euclidean space. The MCT predicts the right trend for the evolution of the relaxation slowdown with curvature but is dramatically off at a quantitative level.

Quasi-equilibrium in glassy dynamics: a liquid theory approach

We introduce a quasi-equilibrium formalism in the theory of liquids in order to obtain a set of coarse grained dynamical equations for the description of long time glassy relaxation. Our scheme allows to use typical approximations devised for equilibrium to study glassy dynamics. After introducing dynamical Ornstein–Zernike relations, we focus on the hypernetted chain (HNC) approximation and a recent closure scheme developed by Szamel. In both cases we get dynamical equations that have the structure of the mode-coupling theory (MCT) equations in the long time regime. The HNC approach, that was so far used to get equilibrium quantities is thus generalized to a fully consistent scheme where long-time dynamic quantities can also be computed. In the context of this approximation we get an asymptotic description of both equilibrium glassy dynamics at high temperature and of aging dynamics at low temperature. The Szamel approximation on the other hand is shown to lead to the canonical MCT equations obtained by Götze for equilibrium dynamics. We clarify the way phase space is sampled according to MCT during dynamical relaxation.

Long-wavelength fluctuations lead to a model of the glass crossover

The effect of long-wavelength fluctuations on the Mode-Coupling-Theory (MCT) dynamical singularity at Tc in the β-regime is studied by means of the standard field-theoretical procedure for a genuine second-order phase transition. The resulting perturbative loop expansion can be resummed leading to an extension of the MCT equation for the critical correlator with local random fluctuations of the separation parameter. The corresponding model explains both qualitatively and quantitatively why the MCT dynamical singularity is transformed into a crossover from relaxational to activated dynamics. Dynamical Heterogeneities emerge naturally as the ergodicity-restoring mechanism instead of ad hoc hopping processes.

Determining Critical TemperatureTcin a Molecular Dynamics-Simulated Glass Forming Ni0.8Zr0.2-System

In this paper molecular dynamics computer simulations is used to investigate a critical temperature Tc for a dynamical glass transition as proposed by the mode-coupling theory (MCT) of dense liquids in a glass forming Ni0.8Zr0.2-system. The critical temperature Tc are analyzed from different quantities and checked the consistency of the estimated values, i.e. from (i) the non-vanishing nonergodicity parameters as asymptotic solutions of the MCT equations in the arrested state, (ii) the gm-parameters describing the approach of the melt towards the arrested state on the ergodic side, (iii) the diffusion coefficients in the melt. The resulting Tc values are found to agree within about 10-15 %.

Statistical-mechanical theory of nonlinear density fluctuations near the glass transition

The Tokuyama–Mori type projection-operator method is employed to study the dynamics of nonlinear density fluctuations near the glass transition. A linear non-Markov time-convolutionless equation for the scattering function Fα(q,t) is first derived from the Newton equation with the memory function ψα(q,t), where α=c for the coherent–intermediate scattering function and s for the self–intermediate scattering function. In order to calculate ψα(q,t), the Mori type projection-operator method is then used and a linear non-Markov time-convolution equation for ψα(q,t) is derived with the memory function φα(q,t). In order to calculate φα(q,t), the same binary approximation as that used in the mode-coupling theory (MCT) is also employed and hence φα(q,t) is shown to be identical with that obtained by MCT. Thus, the coupled equations are finally derived to calculate the scattering functions, which are quite different from the so-called ideal MCT equation. The most important difference between the present theory and MCT appears in the Debye–Waller factor fα(q). In MCT it is given by fα(q)=Γα(q)/(Γα(q)+1), where Γα(q) is the long-time limit of the memory function φα(q,t). On the other hand, in the present theory it is given by fα(q)=exp[−1/Γα(q)]. Thus, it is expected that the critical temperature Tc of the present theory would be much lower than that of MCT. The other differences are also discussed.

Nonequilibrium mode-coupling theory for uniformly sheared underdamped systems

Nonequilibrium mode-coupling theory (MCT) for uniformly sheared underdamped systems is developed, starting from the microscopic thermostated Sllod equation and the corresponding Liouville equation. Special attention is paid to the translational invariance in the sheared frame, which requires an appropriate definition of the transient time correlators. The derived MCT equation satisfies the alignment of the wave vectors and is manifestly translationally invariant. Isothermal condition is implemented by the introduction of current fluctuation in the dissipative coupling to the thermostat. This current fluctuation grows in the α relaxation regime, which generates a pronounced relaxation of the yield stress compared to the overdamped case. This result fills the gap between the molecular dynamics simulation and the overdamped MCT reported previously. The response to a perturbation of the shear rate demonstrates an inertia effect which is not observed in the overdamped case. Our theory turns out to be a nontrivial extension of the theory by Fuchs and Cates [J. Rheol. 53, 957 (2009)] to underdamped systems. Since our starting point is identical to that of Chong and Kim [Phys. Rev. E 79, 021203 (2009)], the contradictions between Fuchs-Cates and Chong-Kim are resolved.

Critical slowing down exponents in structural glasses: Random orthogonal and related models

An important prediction of Mode-Coupling-Theory (MCT) is the relationship between the power- law decay exponents in the {\beta} regime. In the original structural glass context this relationship follows from the MCT equations that are obtained making rather uncontrolled approximations and {\lambda} has to be treated like a tunable parameter. It is known that a certain class of mean-field spin-glass models is exactly described by MCT equations. In this context, the physical meaning of the so called parameter exponent {\lambda} has recently been unveiled, giving a method to compute it exactly in a static framework. In this paper we exploit this new technique to compute the critical slowing down exponents in a class of mean-field Ising spin-glass models including, as special cases, the Sherrington-Kirkpatrick model, the p-spin model and the Random Orthogonal model.

From caging to Rouse dynamics in polymer melts with intramolecular barriers: A critical test of the mode coupling theory

By means of computer simulations and solution of the equations of the mode coupling theory (MCT), we investigate the role of the intramolecular barriers on several dynamic aspects of nonentangled polymers. The investigated dynamic range extends from the caging regime characteristic of glass-formers to the relaxation of the chain Rouse modes. We review our recent work on this question, provide new results, and critically discuss the limitations of the theory. Solutions of the MCT for the structural relaxation reproduce qualitative trends of simulations for weak and moderate barriers. However, a progressive discrepancy is revealed as the limit of stiff chains is approached. This disagreement does not seem related with dynamic heterogeneities, which indeed are not enhanced by increasing barrier strength. It is not connected either with the breakdown of the convolution approximation for three-point static correlations, which retains its validity for stiff chains. These findings suggest the need of an improvement of the MCT equations for polymer melts. Concerning the relaxation of the chain degrees of freedom, MCT provides a microscopic basis for time scales from chain reorientation down to the caging regime. It rationalizes, from first principles, the observed deviations from the Rouse model on increasing the barrier strength. These include anomalous scaling of relaxation times, long-time plateaux, and nonmonotonous wavelength dependence of the mode correlators.

Chain dynamics in nonentangled polymer melts: A first-principle approach for the role of intramolecular barriers

By means of simulations and numerical solutions of the Mode Coupling Theory (MCT), we investigate the role of intramolecular barriers on the chain dynamics of nonentangled polymer melts. We present a global picture by studying the relaxation of the Rouse modes for a wide range of barrier strength, from fully-flexible to stiff chains. Simulations reveal, on increasing the barrier strength, strong deviations from the Rouse model, as anomalous scaling of relaxation times, long-time plateaux, and nonmonotonic wavelength dependence of the mode correlators. These highly non-trivial dynamics are accounted for by the solutions of the MCT equations. We conclude that MCT constitutes a general, first-principle, approach for chain dynamics in nonentangled polymer melts.

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

We analyze the slow glassy structural relaxation as measured through collective and tagged-particle density correlation functions obtained from Brownian dynamics simulations for a polydisperse system of quasi-hard spheres in the framework of the mode-coupling theory (MCT) of the glass transition. Asymptotic analyses show good agreement for the collective dynamics when polydispersity effects are taken into account in a multicomponent calculation, but qualitative disagreement at small q when the system is treated as effectively monodisperse. The origin of the different small-q behavior is attributed to the interplay between interdiffusion processes and structural relaxation. Numerical solutions of the MCT equations are obtained taking properly binned partial static structure factors from the simulations as input. Accounting for a shift in the critical density, the collective density correlation functions are well described by the theory at all densities investigated in the simulations, with quantitative agreement best around the maxima of the static structure factor and worst around its minima. A parameter-free comparison of the tagged-particle dynamics however reveals large quantitative errors for small wave numbers that are connected to the well-known decoupling of self-diffusion from structural relaxation and to dynamical heterogeneities. While deviations from MCT behavior are clearly seen in the tagged-particle quantities for densities close to and on the liquid side of the MCT glass transition, no such deviations are seen in the collective dynamics.

The role of intramolecular barriers on the glass transition of polymers: Computer simulations versus mode coupling theory

We present computer simulations of a simple bead-spring model for polymer melts with intramolecular barriers. By systematically tuning the strength of the barriers, we investigate their role on the glass transition. Dynamic observables are analyzed within the framework of the mode coupling theory (MCT). Critical nonergodicity parameters, critical temperatures, and dynamic exponents are obtained from consistent fits of simulation data to MCT asymptotic laws. The so-obtained MCT lambda-exponent increases from standard values for fully flexible chains to values close to the upper limit for stiff chains. In analogy with systems exhibiting higher-order MCT transitions, we suggest that the observed large lambda-values arise form the interplay between two distinct mechanisms for dynamic arrest: general packing effects and polymer-specific intramolecular barriers. We compare simulation results with numerical solutions of the MCT equations for polymer systems, within the polymer reference interaction site model (PRISM) for static correlations. We verify that the approximations introduced by the PRISM are fulfilled by simulations, with the same quality for all the range of investigated barrier strength. The numerical solutions reproduce the qualitative trends of simulations for the dependence of the nonergodicity parameters and critical temperatures on the barrier strength. In particular, the increase in the barrier strength at fixed density increases the localization length and the critical temperature. However the qualitative agreement between theory and simulation breaks in the limit of stiff chains. We discuss the possible origin of this feature.

Atomic structure and diffusion in Cu60Zr40 metallic liquid and glass: molecular dynamics simulations

Temperature effects on the structural evolution and diffusivity of Cu60Zr40 in the liquid and glassy states were studied by molecular dynamics simulations using the Finnis–Sinclair potential. The pair distribution functions and common-neighbor analysis were used to investigate the structural variations. It is found that the amount of pentagonal bipyramids increases sharply in a short temperature range of about 200 K above the glass transition temperature Tg, leading to the increasing of the icosahedral cluster centered by Cu atom and larger Kasper polyhedral cluster centered by Zr atom. The mean square displacement and the self part of the van Hove function were calculated to evaluate the relaxation and transport properties. The cage effect is found to appear at 1100 K. The mode-coupling theory equation is more suitable than the Vogel–Fulcher–Tammann equation to describe the change in diffusivity in the liquid state. The critical temperature Tc, a predicted glass transition temperature in mode-coupling theory, for Cu60Zr40 glass former is 1008.2 K. At temperatures near Tc, there is a dynamic crossover. In supercooled liquid state, the dynamic heterogeneity is detected. The comparison of the activation enthalpies of the two components shows that the small atoms transport faster obeying the size effect.

Mode-Coupling Theory of Sheared Dense Granular Liquids

Mode-coupling theory (MCT) of sheared dense granular liquids %in the vicinity of jamming transition is formulated. Starting from the Liouville equation of granular particles, the generalized Langevin equation is derived with the aid of the projection operator technique. The MCT equation for the density correlation function obtained from the generalized Langevin equation is almost equivalent to MCT equation for elastic particles under the shear. It is found that there should be the plateau in the density correlation function.

Dynamic glass transition in two dimensions

The question of the existence of a structural glass transition in two dimensions is studied using mode coupling theory (MCT). We determine the explicit d dependence of the memory functional of mode coupling for one-component systems. Applied to two dimensions we solve the MCT equations numerically for monodisperse hard disks. A dynamic glass transition is found at a critical packing fraction phi(c)d=2 approximately equal 0.697 which is above phi(c)d=3 approximately equal 0.516 by about 35%. Phi(c)d scales approximately with phi(rcp)d, the value for random close packing, at least for d=2, 3. Quantities characterizing the local, cooperative "cage motion" do not differ much for d=2 and d=3, and we, e.g., find the Lindemann criterion for the localization length at the glass transition. The final relaxation obeys the superposition principle, collapsing remarkably well onto a Kohlrausch law. The d=2 MCT results are in qualitative agreement with existing results from Monte Carlo and molecular dynamics simulations. The mean-squared displacements measured experimentally for a quasi-two-dimensional binary system of dipolar hard spheres can be described satisfactorily by MCT for monodisperse hard disks over four decades in time provided the experimental control parameter Gamma (which measures the strength of dipolar interactions) and the packing fraction phi are properly related to each other.