Inhomogeneous Mode-coupling Theory Research Articles

Dynamical susceptibilities near ideal glass transitions.

Building on the recently derived inhomogeneous mode-coupling theory, we extend the generalized mode-coupling theory of supercooled liquids to inhomogeneous environments. This provides a first-principles-based, systematic, and rigorous way of deriving high-point dynamical susceptibilities from variations of the many-body dynamic structure factors with respect to their conjugate field. This framework allows for a fully microscopic possibility to probe for collective relaxation mechanisms in supercooled liquids near the mode-coupling glass transition. The behavior of these dynamical susceptibilities is then studied in the context of simplified self-consistent relaxation models.

Revisiting the single-saddle model for the β-relaxation of supercooled liquids

The dynamics of glass-forming liquids display several outstanding features, such as two-step relaxation and dynamic heterogeneities, which are difficult to predict quantitatively from first principles. In this work, we revisit a simple theoretical model of the β-relaxation, i.e., the first step of the relaxation dynamics. The model, first introduced by Cavagna et al. [J. Phys. A: Math. Gen. 36, 10721 (2003)], describes the dynamics of the system in the neighborhood of a saddle point of the potential energy surface. We extend the model to account for density–density correlation functions and for the four-point dynamic susceptibility. We obtain analytical results for a simple schematic model, making contact with related results for p-spin models and with the predictions of inhomogeneous mode-coupling theory. Building on recent computational advances, we also explicitly compare the model predictions against overdamped Langevin dynamics simulations of a glass-forming liquid close to the mode-coupling crossover. The agreement is quantitative at the level of single-particle dynamic properties only up to the early β-regime. Due to its inherent harmonic approximation, however, the model is unable to predict the dynamics on the time scale relevant for structural relaxation. Nonetheless, our analysis suggests that the agreement with the simulations may be largely improved if the modes’ spatial localization is properly taken into account.

Dynamic Heterogeneities in Colloidal Supercooled Liquids: Experimental Tests of Inhomogeneous Mode Coupling Theory.

The dynamics in supercooled liquids slow enormously upon approaching the glass transition, albeit without significant change of liquid structure. This empirical observation has stimulated development of many theoretical models which attempt to elucidate microscopic mechanisms in glasses and glass precursors. Here, quasi-two-dimensional colloidal supercooled liquids and glasses are employed to experimentally test predictions of widely used models: mode coupling theory (MCT) and its important extension, inhomogeneous MCT (IMCT). We measure two-point dynamic correlation functions in the glass forming liquids to determine structural relaxation times, τα, and mode coupling exponents, a, b, and γ; these parameters are then used to extract the mode coupling dynamic crossover packing area-fraction, ϕ c. This information, along with our measurements of supercooled liquid spatiotemporal dynamics, permits characterization of dynamic heterogeneities in the samples and facilitates direct experimental tests of the scaling predictions of IMCT. The time scales at which dynamic heterogeneities are largest, and their spatial sizes, exhibit power law growth on approaching ϕ c. Within experimental error, the exponents of the measured power laws are close to the predictions of IMCT.

Spatially heterogeneous dynamics in a metallic glass forming liquid imaged by electron correlation microscopy

Supercooled liquids exhibit spatial heterogeneity in the dynamics of their fluctuating atomic arrangements. The length and time scales of the heterogeneous dynamics are central to the glass transition and influence nucleation and growth of crystals from the liquid. Here, we report direct experimental visualization of the spatially heterogeneous dynamics as a function of temperature in the supercooled liquid state of a Pt-based metallic glass, using electron correlation microscopy with sub-nanometer resolution. An experimental four-point space-time correlation function demonstrates a growing dynamic correlation length, ξ, upon cooling of the liquid toward the glass transition temperature. ξ as a function of the relaxation time τ are in good agreement with Adam-Gibbs theory, inhomogeneous mode-coupling theory and random first-order transition theory of the glass transition. The same experiments demonstrate the existence of a nanometer thickness near-surface layer with order of magnitude shorter relaxation time than inside the bulk.

Fredrickson-Andersen model on Bethe lattice with random pinning

We study the effects of random pinning on the Fredrickson-Andersen model on the Bethe lattice. We find that the nonergodic transition temperature rises as the fraction of the pinned spins increases and the transition line terminates at a critical point. The freezing behavior of the spins is analogous to that of a randomly pinned p-spin mean-field spin glass model which has been recently reported. The diverging behavior of correlation lengths in the vicinity of the terminal critical point is found to be identical to the prediction of the inhomogeneous mode-coupling theory at the A3 singularity point for the glass transition.

Critical dynamical heterogeneities close to continuous second-order glass transitions.

We analyze, using inhomogeneous mode-coupling theory, the critical scaling behavior of the dynamical susceptibility at a distance ε from continuous second-order glass transitions. We find that the dynamical correlation length ξ behaves generically as ε(-1/3) and that the upper critical dimension is equal to six. More surprisingly, we find that ξ grows with time as ln²t exactly at criticality. All of these results suggest a deep analogy between the glassy behavior of attractive colloids or randomly pinned supercooled liquids and that of the random field Ising model.

Dynamic Length Scales in Glass-Forming Liquids: An Inhomogeneous Molecular Dynamics Simulation Approach

In this work, we numerically investigate a new method for the characterization of growing length scales associated with spatially heterogeneous dynamics of glass-forming liquids. This approach, motivated by the formulation of the inhomogeneous mode-coupling theory (IMCT) [Biroli, G.; et al. Phys. Rev. Lett. 2006 97, 195701], utilizes inhomogeneous molecular dynamics simulations in which the system is perturbed by a spatially modulated external potential. We show that the response of the two-point correlation function to the external field allows one to probe dynamic correlations. We examine the critical properties shown by this function, in particular, the associated dynamic correlation length, that is found to be comparable to the one extracted from standardly employed four-point correlation functions. Our numerical results are in qualitative agreement with IMCT predictions but suggest that one has to take into account fluctuations not included in this mean-field approach to reach quantitative agreement. Advantages of our approach over the more conventional one based on four-point correlation functions are discussed.

Dynamic Heterogeneity in a Glass Forming Fluid: Susceptibility, Structure Factor, and Correlation Length

We investigate the growth of dynamic heterogeneity in a glassy hard-sphere mixture for volume fractions up to and including the mode-coupling transition. We use an 80,000 particle system to test a new procedure to evaluate a dynamic correlation length ξ(t): we determine the ensemble independent dynamic susceptibility χ(4)(t) and use it to facilitate evaluation of ξ(t) from the small wave vector behavior of the four-point structure factor. We analyze relations between the α relaxation time τ(α), χ(4)(τ(α)), and ξ(τ(α)). We find that mode-coupling-like power laws provide a reasonable description of the data over a restricted range of volume fractions, but the power laws' exponents differ from those predicted by the inhomogeneous mode-coupling theory. We find ξ(τ(α))~ln(τ(α)) over the full range of volume fractions studied, which is consistent with an Adam-Gibbs-type relation.

Analysis of Dynamic Heterogeneity in a Glass Former from the Spatial Correlations of Mobility

The growth of characteristic length scales associated with dynamic heterogeneity in glass-forming liquids is investigated in an extensive computational study of a four-point, time-dependent structure factor defined from spatial correlations of mobility, for a model liquid for system sizes extending up to 351 232 particles, in constant-energy and constant-temperature ensembles. Our estimates for dynamic correlation lengths and susceptibilities are consistent with previous results from finite size scaling. We find scaling exponents that are inconsistent with predictions from inhomogeneous mode coupling theory and a recent simulation confirmation of these predictions.

Diverging length scale of the inhomogeneous mode-coupling theory: A numerical investigation

Biroli 's extension of the standard mode-coupling theory to inhomogeneous equilibrium states [G. Biroli, J. P. Bouchaud, K. Miyazaki, and D. R. Reichman, Phys. Rev. Lett. 97, 195701 (2006)] allowed them to identify a characteristic length scale that diverges upon approaching the mode-coupling transition. We present a numerical investigation of this length scale. To this end we derive and numerically solve equations of motion for coefficients in the small q expansion of the dynamic susceptibility chiq(k;t) that describes the change of the system's dynamics due to an external inhomogeneous potential. We study the dependence of the characteristic length scale on time, wave vector, and on the distance from the mode-coupling transition. We verify scaling predictions of Biroli In addition, we find that the numerical value of the diverging length scale qualitatively agrees with lengths obtained from four-point correlation functions. We show that the diverging length scale has very weak k dependence, which contrasts with very strong k dependence of the q-->0 limit of the susceptibility, chiq=0(k;t) . Finally, we compare the diverging length obtained from the small q expansion to that resulting from an isotropic approximation applied to the equation of motion for the dynamic susceptibility chiq(k;t) .

Scaling Analysis of Dynamic Heterogeneity in a Supercooled Lennard-Jones Liquid

We have performed molecular dynamics computer simulations of a dense Lennard-Jones liquid mixture to study dynamic heterogeneity from normal liquid temperatures down to a supercooled temperature 15% above the previously identified mode-coupling temperature Tc of the model. A temperature-dependent correlation length associated with the correlation function of mobility fluctuations is calculated. The results are used to test two sets of scaling hypotheses for the dynamic heterogeneity. The results are in close agreement with the inhomogeneous mode-coupling theory of Biroli et al. [Phys. Rev. Lett. 97, 195701 (2006)] for both the alpha and beta relaxation regimes. Comparison with results for kinetically constrained models suggest that the Lennard-Jones mixture studied is more similar to models of fragile liquids than models of very strong liquids.

Inhomogeneous Mode-Coupling Theory and Growing Dynamic Length in Supercooled Liquids

We extend mode-coupling theory (MCT) to inhomogeneous situations, relevant for supercooled liquid in an external field. We compute the response of the dynamical structure factor to a static inhomogeneous external potential and provide the first direct evidence that the standard formulation of MCT is associated with a diverging length scale. We find that the so-called cages are, in fact, extended objects. Although close to the transition the dynamic length grows as |T-T(c)|(-1/4) in both the beta and alpha regimes, our results suggest that the fractal dimension of correlated clusters is larger in the alpha regime. We derive inhomogeneous MCT equations valid to second order in gradients.