Tests of mode-coupling theory in two dimensions

We analyze the glassy dynamics of a binary mixtures of hard disks in two dimensions. Predictions of the Mode-Coupling theory(MCT) are tested with extensive Brownian dynamics simulations. Measuring the collective particle density correlation functions in the vicinity of the glass transition we verify four predicted mixing effects. For instance, for large size disparities, adding a small amount of small particles at fixed packing fraction leads to a speed up in the long time dynamics, while at small size disparity it leads to a slowing down. Qualitative features of the non-ergodicity parameters and the $\beta$-relaxation which both depend in a non-trivial way on the mixing ratio are found in the simulated correlators. Studying one system in detail we are able to determine its ideal MCT glass transition point as $\phi^c = 0.7948$ and test MCT predictions quantitatively.Comment: 12 pages, 18 figure

Comment on "Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition"

In the Letter [PRL 102, 085703 (2009)] Brambilla, et al. claimed to observe activated dynamics in colloidal hard spheres above the critical packing fraction of mode coupling theory (MCT). By performing microscopic MCT calculations, we show that polydispersity in their system shifts the critical packing fraction above the value determined by van Megen et al. for less polydisperse samples, and that the data agree with theory except for, possibly, the highest packing fraction.Comment: Comment in print in Phys. Rev. Lett.; for accompanying reply see arXiv Brambilla et al. (Monday 18.10.2010

Tagged-particle motion in glassy systems under shear: Comparison of mode coupling theory and Brownian Dynamics simulations

We study the dynamics of a tagged particle in a glassy system under shear. The recently developed integration through transients approach based on mode coupling theory, is continued to arrive at the equations for the tagged particle correlators and the mean squared displacements. The equations are solved numerically for a two dimensional system, including a nonlinear stability analysis of the glass solution, the so called beta-analysis. We perform Brownian Dynamics simulations in 2-D and compare with theory. After switch on, transient glassy correlation functions show strong fingerprints of the stress overshoot scenario, including, additionally to previously studied superexponential decay, a shoulder-like slowing down after the overshoot. We also find a new type of Taylor dispersion in glassy states which has intriguing similarity to the known low density case. The theory qualitatively captures most features of the simulations with quantitative deviations concerning the shear induced timescales. We attribute these deviations to an underestimation of the overshoot scenario in the theory.Comment: 22 pages, 28 figures, Eur. Phys. J. E (in print

Overshoots in stress strain curves: Colloid experiments and schematic mode coupling theory

The stress versus strain curves in dense colloidal dispersions under start-up shear flow are investigated combining experiments on model core-shell microgels, computer simulations of hard disk mixtures, and mode coupling theory. In dense fluid and glassy states, the transient stresses exhibit first a linear increase with the accumulated strain, then a maximum ('stress overshoot') for strain values around 5%, before finally approaching the stationary value, which makes up the flow curve. These phenomena arise in well-equilibrated systems and for homogeneous flows, indicating that they are generic phenomena of the shear-driven transient structural relaxation. Microscopic mode coupling theory (generalized to flowing states by integration through the transients) derives them from the transient stress correlations, which first exhibit a plateau (corresponding to the solid-like elastic shear modulus) at intermediate times, and then negative stress correlations during the final decay. We introduce and validate a schematic model within mode coupling theory which captures all of these phenomena and handily can be used to jointly analyse linear and large-amplitude moduli, flow curves, and stress-strain curves. This is done by introducing a new strain- and time-dependent vertex into the relation between the the generalized shear modulus and the transient density correlator.Comment: 21 pages, 13 figure

From Equilibrium to Steady-State Dynamics after Switch-On of Shear

A relation between equilibrium, steady-state, and waiting-time dependent dynamical two-time correlation functions in dense glass-forming liquids subject to homogeneous steady shear flow is discussed. The systems under study show pronounced shear thinning, i.e., a significant speedup in their steady-state slow relaxation as compared to equilibrium. An approximate relation that recovers the exact limit for small waiting times is derived following the integration through transients (ITT) approach for the nonequilibrium Smoluchowski dynamics, and is exemplified within a schematic model in the framework of the mode-coupling theory of the glass transition (MCT). Computer simulation results for the tagged-particle density correlation functions corresponding to wave vectors in the shear-gradient directions from both event-driven stochastic dynamics of a two-dimensional hard-disk system and from previously published Newtonian-dynamics simulations of a three-dimensional soft-sphere mixture are analyzed and compared with the predictions of the ITT-based approximation. Good qualitative and semi-quantitative agreement is found. Furthermore, for short waiting times, the theoretical description of the waiting time dependence shows excellent quantitative agreement to the simulations. This confirms the accuracy of the central approximation used earlier to derive fluctuation dissipation ratios (Phys. Rev. Lett. 102, 135701). For intermediate waiting times, the correlation functions decay faster at long times than the stationary ones. This behavior is predicted by our theory and observed in simulations.Comment: 16 pages, 12 figures, submitted to Phys Rev

Rheology of Brownian Discs

In this thesis the properties of binary mixtures of hard discs, undergoing Brownian motion, have been studied. Two major cases have been considered: First the approach of the glass transition for a binary mixture along with the effect of changes in the composition within that mixture. For one selected mixture a detailed discussion of its transition followed. Second the rheological properties of this selected mixture were investigated by focusing on the distorted microstructure and the dynamical correlation functions of mainly tagged quantities. Finally time dependent shear flows were investigated. Performing simulations for different binary mixtures at the glass transition was motivated by recent results of MCT for two dimensions. With the simulations it was possible to confirm four mixing effects, predicted by MCT for these binary mixtures, by David Hajnal. It could be shown that for big radius ratios when we keep the total packing fraction constant, the increase in the concentration of the small particles leads to a melting of the glass, the plasticization. For small radius ratios the increase in the concentration of small particles at constant volume fraction leads to an even stronger glass. Both remaining effects, the increase of non-ergodicity parameters, and the slowing down of the relaxation towards the non-ergodicity parameters on increasing the small particles concentrations could be confirmed. These qualitative effects, especially the first two ones, are not only of theoretical interest, as industrial applications of so called plasticizers show. Going into detail for a selected mixture made it possible to determine the ideal MCT glass transition point. Along with the verification of the factorization close to the critical point, the α-scaling and even the increase of the plateaus according to a square root law could be validated. The simulation shows additional decay processes, leading to a decay of the correlation functions even above the ideal glass transition. This is not in accordance with MCT, however, the rise of the plateau values can be explained. This analysis closes a gap in the field of the glass transition as: First, a quantitative test for exactly the model system MCT uses was performed, and second to the knowledge of the author, no such analysis for a this two dimensional glass former has been performed before. With the characterization of the considered system the foundation for the rheological part was laid. That the simulation algorithm yields Brownian motion for the quiescent system was known before from the work of Erik Lange. This thesis goes one step further and shows, with the same arguments, that even for the sheared case the algorithm is still in accordance with short time expansions of the Smoluchowski equation for the shear modulus. The significance here is, that even on the two particle level used in the theoretical derivation, the simulation agrees in its short time asymptote with the theoretical findings. Having ensured that the simulation actually solves the Smoluchowski equation for Brownian motion under shear, and having characterized the glass transition for one special mixture in the preceding chapters, the rheological properties of exactly that mixture could be probed. In the framework of an extended MCT, MCT-ITT, which was developed by Matthias Fuchs and Michael Cates, the microscopic structure was investigated. The appearance of a yield stress, shear thinning and the distortion of the structure factor in the simulation can be qualitatively explained by MCT-ITT and the interplay of different relaxation time scales involved in the glass and in the liquid. Quantitatively we found that MCT-ITT overestimates the distortion of the structure and the resulting shear stresses and viscosities by about a factor of 10. However, it should be remarked that a simplified monodisperse MCT-ITT calculation was used, which could explain a part of the deviation. Future works should consider using a multi-component approach, as used for the analysis of the mixture effects in this thesis. Scrutinizing the tagged particle correlation functions, the interplay between purely structural relaxation and shear induced relaxation could be observed in detail in the simulation: For correlators in the liquid, the shear induced decay competes with the intrinsic decay, whereas in the glassy system the time scale is always set by the shear field. The relaxation in the shear melted glass thus follows a master function which could be found for very low shear rates in the simulation. The famous Taylor dispersion, expressing itself, for example, through a cubic growth in the mean squared displacement in the shear direction could be found in the simulations. The MCT-ITT prediction, that actually the shear and the gradient direction are connected via one single long time diffusion coefficient, could be confirmed. Furthermore the MCT-ITT scaling properties of these coefficients, stating that they scale linearly with the shear rate in the limit of vanishing shear, could be worked out with the simulation. Yet, the Taylor dispersion still leaves a riddle, as next-to-leading order asymptotes in MCT-ITT and the simulation seem to differ: A term proportional to the square of time couldn’t be found as proposed by MCT-ITT in the simulations, which propose a next-to-leading order dependence that is linear in time. Along with the discussion of the mean squared displacement, a connection between the superdiffusivity and the overshoot in the transient stress could be pointed out. Also the stress overshoot is connected to the anisotropy in the correlation functions for certain directions. Coherent correlators for the stationary state have been shown. In principle they show that simulating this quantities, although it requires a lot more numerical tricks and computational power, is possible. Calculating the transient correlators, which are the main quantity in MCT-ITT, will be a task for the near future, and will make further tests possible. Finally in the last chapter with a method, developed by Matthias Krüger, it was possible to close the gap between stationary and transient correlation functions. The simulation showed that for small waiting times the connection given between transient, stationary and equilibrium correlator is perfectly reproduced by Krüger’s relation. Furthermore, it could be shown, that even for large waiting times reasonable results can be achieved, although the shear modulus entered as a fitting parameter. Extending the simulation to oscillatory shear a connection between the yield stress of a glassy system and its non-linear reaction upon oscillatory shear for low frequencies could be inferred. The results support the theoretical results from a schematic MCT model invented by Joseph Brader. For time dependent stresses below the yield stress the system shows the elastic behavior of a solid state body. Above the yield stress the system shows the dissipative properties of a liquid accompanied by higher harmonics in the stress response. An upper bound for the stress can be estimated from the flow curves

Hard discs under steady shear: comparison of Brownian dynamics simulations and mode coupling theory

Brownian dynamics simulations of bidisperse hard discs moving in two dimensions in a given steady and homogeneous shear flow are presented close to and above the glass transition density. The stationary structure functions and stresses of shear-melted glass are compared quantitatively to parameter-free numerical calculations of monodisperse hard discs using mode coupling theory within the integration through transients framework. Theory qualitatively explains the properties of the yielding glass but quantitatively overestimates the shear-driven stresses and structural anisotropies

Shear moduli of two dimensional binary glasses

The shear moduli of two-component glasses in two dimensions are studied within mode coupling theory. Varying the concentration, strong mixing effects are observed along the glass transition lines for two interaction potentials. Nonoverlapping disks with size ratios between 0.3 and 0.9, and point particles interacting with (magnetic) dipoles of strength ratio between 0.1 and 0.6 are considered. Equilibrium structure factors (partially obtained from Monte Carlo simulations) and glass form factors, and perturbative calculations show that a softening of the elastic shear constant of glass upon adding another component arises from a dilution effect of the majority component. For very disparate mixtures, an anomalous elastic strenghtening results from what we interpret as clustering of the smaller particles in the voids between the larger ones. This might point to a glass–glass transition. We include simulation data on hard disk mixtures which show that the theory underestimates the moduli by around 50%, but otherwise captures the qualitative trends (within the rather large simulational error bars)