Mode-coupling Crossover Research Articles

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.

Cooperatively rearranging regions change shape near the mode-coupling crossover for colloidal liquids on a sphere

The structure and dynamics of liquids on curved surfaces are often studied through the lens of frustration-based approaches to the glass transition. Competing glass transition theories, however, remain largely untested on such surfaces and moreover, studies hitherto have been entirely theoretical/numerical. Here we carry out single particle-resolved imaging of dynamics of bi-disperse colloidal liquids confined to the surface of a sphere. We find that mode-coupling theory well captures the slowing down of dynamics in the moderate to deeply supercooled regime. Strikingly, the morphology of cooperatively rearranging regions changed from string-like to compact near the mode-coupling crossover—a prediction unique to the random first-order theory of glasses. Further, we find that in the limit of strong curvature, Mermin–Wagner long-wavelength fluctuations are irrelevant and liquids on a sphere behave like three-dimensional liquids. A comparative evaluation of competing mechanisms is thus an essential step towards uncovering the true nature of the glass transition.

Dynamical phase transitions and their relation to structural and thermodynamic aspects of glass physics.

We review recent developments in structural-dynamical phase transitions in trajectory space based on dynamic facilitation theory. An open question is how the dynamic facilitation perspective on the glass transition may be reconciled with thermodynamic theories that posit collective reorganization accompanied by a growing static length scale and, eventually, a vanishing configurational entropy. In contrast, dynamic facilitation theory invokes a dynamical phase transition between an active phase (close to the normal liquid) and an inactive phase, which is glassy and whose order parameter is either a time-averaged dynamic or structural quantity. In particular, the dynamical phase transition in systems with non-trivial thermodynamics manifests signatures of a lower critical point that lies between the mode-coupling crossover and the putative Kauzmann temperature, at which a thermodynamic phase transition to an ideal glass state would occur. We review these findings and discuss such criticality in the context of the low-temperature decrease in configurational entropy predicted by thermodynamic theories of the glass transition.

Finite Dimensional Vestige of Spinodal Criticality above the Dynamical Glass Transition.

Finite dimensional signatures of spinodal criticality are notoriously difficult to come by. The dynamical transition of glass-forming liquids, first described by mode-coupling theory, is a spinodal instability preempted by thermally activated processes that also limit how close the instability can be approached. We combine numerical tools to directly observe vestiges of the spinodal criticality in finite dimensional glass formers. We use the swap MonteCarlo algorithm to efficiently thermalize configurations beyond the mode-coupling crossover, and analyze their dynamics using a scheme to screen out activated processes, in spatial dimensions ranging from d=3 to d=10. We observe a strong softening of the mean-field square-root singularity in d=3 that is progressively restored as d increases above d=8, in surprisingly good agreement with perturbation theory.

A localization transition underlies the mode-coupling crossover of glasses

We study the equilibrium statistical properties of the potential energy landscape of several glass models in a temperature regime so far inaccessible to computer simulations. We show that unstable modes of the stationary points undergo a localization transition in real space close to the mode-coupling crossover temperature determined from the dynamics. The concentration of localized unstable modes found at low temperature is a non-universal, finite dimensional feature not captured by mean-field glass theory. Our analysis reconciles, and considerably expands, previous conflicting numerical results and provides a characteristic temperature for glassy dynamics that unambiguously locates the mode-coupling crossover.

Nonmonotonic dynamic correlations in quasi-two-dimensional confined glass-forming liquids.

It has been broadly accepted that the behavior of glass-forming liquids, where their relaxation dynamics exhibit a pronounced slowdown as they are cooled toward the glass transition temperature, is caused by the increase in one or more correlation lengths. However, the role of length scales in the dynamics of glass-forming liquids is not clearly established, and past simulation work that suggests a surprising nonmonotonic temperature evolution of spatial dynamical correlations near the mode-coupling crossover temperature has been both questioned and supported by subsequent work. Here, using molecular dynamics simulation, we also show a striking maximum in the dynamic length scale ξ_{c}^{dyn} at a given temperature, but the temperature of this maximum is found to shift as the size of the confined system increases. Furthermore, we find that such a maximum disappears for all geometry sizes considered when a rough wall is replaced with a smooth, hard wall, suggesting that the nature of the nonmonotonic temperature dependence of ξ_{c}^{dyn} does not reflect an intrinsic property of bulk liquids, but originates from wall effects. Our results provide new insights into the dynamics of glass-forming liquids, particularly for quasi-two-dimensional systems.

Effect of gold nanoparticles on structure and dynamics of binary Lennard-Jones liquid: Wave-vector space analysis

Abstract Molecular dynamics simulations at constant ( N , V , T ) are used to study the mutual effects of gold nanoparticles on the structure and dynamics of Kob–Andersen binary Lennard-Jones (BLJ) liquid within the framework of mode coupling theory of dynamic glass transition in the reciprocal space. The results show the ‘softening’ effect of the gold nanoparticles on the liquid dynamics in terms of (i) reducing the mode coupling crossover temperature T c with respect to that of the bulk BLJ (i.e. BLJ without nanoparticles), (ii) decreasing the time interval of β -relaxation, and (iii) decreasing the exponent γ characterizing the power-law behavior of the α -relaxation time. This softening effect is explained in terms of the van der Waals attraction between the gold atoms comprising the nanoparticle and the BLJ host atoms, such that adsorption of host atoms onto the nanoparticle surface creates more space or free-volume for the other atoms to diffuse. By the same token interactions of purely excluded-volume-type are expected to result in the opposite effect. It is also noted that, much unlike BLJ host particles, the dynamics of gold nanoparticles is much less dependent on the wave-vector and that it exhibits a nearly exponential behavior in the α -relaxation regime.

Efficient measurement of point-to-set correlations and overlap fluctuations in glass-forming liquids.

Cavity point-to-set correlations are real-space tools to detect the roughening of the free-energy landscape that accompanies the dynamical slowdown of glass-forming liquids. Measuring these correlations in model glass formers remains, however, a major computational challenge. Here, we develop a general parallel-tempering method that provides orders-of-magnitude improvement for sampling and equilibrating configurations within cavities. We apply this improved scheme to the canonical Kob-Andersen binary Lennard-Jones model for temperatures down to the mode-coupling theory crossover. Most significant improvements are noted for small cavities, which have thus far been the most difficult to study. This methodological advance also enables us to study a broader range of physical observables associated with thermodynamic fluctuations. We measure the probability distribution of overlap fluctuations in cavities, which displays a non-trivial temperature evolution. The corresponding overlap susceptibility is found to provide a robust quantitative estimate of the point-to-set length scale requiring no fitting. By resolving spatial fluctuations of the overlap in the cavity, we also obtain quantitative information about the geometry of overlap fluctuations. We can thus examine in detail how the penetration length as well as its fluctuations evolve with temperature and cavity size.

Relation between the two-body entropy and the relaxation time in supercooled water.

The two-body excess entropy of supercooled water is calculated from the radial distribution functions obtained from computer simulation of the TIP4P model for different densities upon supercooling. This quantity is considered in connection with the relaxation time of the self intermediate scattering function. The relaxation time shows a mode coupling theory (MCT) behavior in the region of mild supercooling and a strong behavior in the deep supercooled region. We find here that the two-body entropy is connected to the relaxation time and shows a logarithmic behavior with an apparent asymptotic divergence at the mode coupling crossover temperature. There is also evidence of a change in behavior of the two-body entropy upon crossing from the fragile (hopping-free) state to the strong (hopping-dominated) state of supercooled water, and the relation that connects the two-body entropy and the relxation time in the MCT region no longer holds.

Dynamical transition in the D=3 Edwards-Anderson spin glass in an external magnetic field.

We study the off-equilibrium dynamics of the three-dimensional Ising spin glass in the presence of an external magnetic field. We have performed simulations both at fixed temperature and with an annealing protocol. Thanks to the Janus special-purpose computer, based on field-programmable gate array (FPGAs), we have been able to reach times equivalent to 0.01 s in experiments. We have studied the system relaxation both for high and for low temperatures, clearly identifying a dynamical transition point. This dynamical temperature is strictly positive and depends on the external applied magnetic field. We discuss different possibilities for the underlying physics, which include a thermodynamical spin-glass transition, a mode-coupling crossover, or an interpretation reminiscent of the random first-order picture of structural glasses.

Diffusive Dynamics of Binary Lennard-Jones Liquid in the Presence of Gold Nanoparticle: A Mode Coupling Theory Analysis

Molecular dynamics simulation has been performed to analyze the effect of the presence of gold nanoparticle on dynamics of Kob-Anderson binary Lennard-Jones mixture upon supercooling within the framework of the mode coupling theory of the dynamic glass transition. The presence of gold nanoparticle has a direct effect on the liquid structure and causes the peaks of the radial distribution functions to become shorter with respect to the bulk binary Lennard-Jones liquid. It is found that the dynamics of the liquid at a given density is consistent with the mode coupling theory (MCT) predictions. In accordance with the idealized MCT, the diffusion constants D(T) show power-law behavior at low temperatures for both types of binary Lennard-Jones (BLJ) particles as well as the 13 gold atoms comprising the nanoparticle. The mode coupling crossover temperature Tc is the same for all particle types, however, Tc = 0.4 is reduced with respect to that of the bulk BLJ liquid and the exponent is found to depend on the particle type. The existence of the nanoparticle causes the short-time β-relaxation regime to shorten and the range of validity of the MCT shrink with respect to the bulk BLJ. It is also found that the behavior of intermediate scattering function (ISF) is in agreement with MCT prediction and in spite of the presence of gold nanoparticle the time temperature superposition principle at intermediate and low temperatures is still valid and the curves of ISF vs. t/ (T) fall onto a master curve.

Effect of gold nanoparticles on structure and dynamics of binary Lennard-Jones liquid: Direct space analysis

We investigate the static structure and diffusive dynamics of binary Lennard-Jones mixture upon supercooling in the presence of gold nanoparticle within the framework of the mode-coupling theory of the dynamic glass transition in the direct space by means of constant-NVT molecular dynamics simulations. It is found that the presence of gold nanoparticle causes the energy per particle and the pressure of this system to decrease with respect to the bulk binary Lennard-Jones mixture. Furthermore, the presence of nanoparticle has a direct effect on the liquid structure and causes the peaks of the radial distribution functions to become shorter with respect to the bulk binary Lennard-Jones liquid. The dynamics of the liquid at a given density is found to be consistent with the mode-coupling theory (MCT) predictions in a certain range at low temperatures. In accordance with the idealized MCT, the diffusion constants D(T) show a power-law behavior at low temperatures for both types of binary Lennard-Jones (BLJ) particles as well as the gold atoms comprising the nanoparticle. The mode-coupling crossover temperature T(c) is the same for all particle types; however, T(c)=0.4 is reduced with respect to that of the bulk BLJ liquid, and the γ exponent is found to depend on the particle type. The existence of the nanoparticle causes the short-time β-relaxation regime to shorten and the range of validity of the MCT shrinks with respect to the bulk BLJ. It is also found that at intermediate and low temperatures the curves of the mean-squared displacements (MSDs) versus tD(T) fall onto a master curve. The MSDs follow the master curve in an identical time range with the long-time α-relaxation regime of the mode-coupling theory. By obtaining the viscosity, it is observed that the Stokes-Einstein relation remains valid at high and intermediate temperatures but breaks down as the temperatures approach T(c) as a result of the cooperative motion or activated processes.

Finite-size effects in the dynamics of glass-forming liquids

We present a comprehensive theoretical study of finite-size effects in the relaxation dynamics of glass-forming liquids. Our analysis is motivated by recent theoretical progress regarding the understanding of relevant correlation length scales in liquids approaching the glass transition. We obtain predictions both from general theoretical arguments and from a variety of specific perspectives: mode-coupling theory, kinetically constrained and defect models, and random first-order transition theory. In the last approach, we predict in particular a nonmonotonic evolution of finite-size effects across the mode-coupling crossover due to the competition between mode-coupling and activated relaxation. We study the role of competing relaxation mechanisms in giving rise to nonmonotonic finite-size effects by devising a kinetically constrained model where the proximity to the mode-coupling singularity can be continuously tuned by changing the lattice topology. We use our theoretical findings to interpret the results of extensive molecular dynamics studies of four model liquids with distinct structures and kinetic fragilities. While the less fragile model only displays modest finite-size effects, we find a more significant size dependence evolving with temperature for more fragile models, such as Lennard-Jones particles and soft spheres. Finally, for a binary mixture of harmonic spheres we observe the predicted nonmonotonic temperature evolution of finite-size effects near the fitted mode-coupling singularity, suggesting that the crossover from mode-coupling to activated dynamics is more pronounced for this model. Finally, we discuss the close connection between our results and the recent report of a nonmonotonic temperature evolution of a dynamic length scale near the mode-coupling crossover in harmonic spheres.

Spatial Correlations in Glass-Forming Liquids Across The Mode-Coupling Crossover

We discuss a novel approach, the point-to-set correlation functions, that allows to determine relevant static and dynamic length scales in glass-forming liquids. We find that static length scales increase monotonically when the temperature is lowered, whereas the measured dynamic length scale shows a maximum around the critical temperature of mode-coupling theory. We show that a similar non-monotonicity is found in the temperature evolution of certain finite size effects in the relaxation dynamics. These two independent sets of results demonstrate the existence of a change in the transport mechanism when the glass-former is cooled from moderately to deeply supercooled states across the mode-coupling crossover and clarify the status of the theoretical calculations done at the mean field level.

Non-monotonic temperature evolution of dynamic correlations in glass-forming liquids

The viscosity of glass-forming liquids increases by many orders of magnitude if their temperature is lowered by a mere factor of 2-3 [1,2]. Recent studies suggest that this widespread phenomenon is accompanied by spatially heterogeneous dynamics [3,4], and a growing dynamic correlation length quantifying the extent of correlated particle motion [5-7]. Here we use a novel numerical method to detect and quantify spatial correlations which reveal a surprising non-monotonic temperature evolution of spatial dynamical correlations, accompanied by a second length scale that grows monotonically and has a very different nature. Our results directly unveil a dramatic qualitative change in atomic motions near the mode-coupling crossover temperature [8] which involves no fitting or indirect theoretical interpretation. Our results impose severe new constraints on the theoretical description of the glass transition, and open several research perspectives, in particular for experiments, to confirm and quantify our observations in real materials.

Critical viscosity near the liquid–liquid phase transition in the solution of the ionic liquid 1-methyl-3-hexylimidazolium tetrafluoroborate in 1-pentanol

In this work, we report measurements of viscosity and density near the liquid–liquid phase transition of the binary solution of the room temperature ionic liquid (RTIL) 1-methyl-3-hexylimidazolium tetrafluoroborate (C6mim+BF4−) in 1-pentanol. The density and the shear viscosity have been determined as function of the temperature and of the concentration. The critical point is located at the temperature Tc = 314.91 K and the mass fraction wc = 0.31034 of the salt. For the critical concentration, the viscosity measurements yield an enhancement as expected for Ising criticality with a crossover to regular behaviour. The regular background is obtained by interpolation from the measurements at non-critical concentrations. The critical enhancement is well described by the mode-coupling crossover function of Bhattacharjee et al. (Phys. Rev. A, 1981, 24, 1469) .The closed form result of the limiting case qD/qc → 0 matches the data almost perfectly.

Slow dynamics of a confined supercooled binary mixture: direct space analysis.

Dynamical properties of a Lennard-Jones binary mixture embedded in an off-lattice matrix of soft spheres are studied in the direct space upon supercooling by molecular dynamics simulations. On lowering the temperature, the smaller particles tend to avoid the soft sphere interfaces and correspondingly their mobility decreases below one of the larger particles. The system displays a dynamic behavior, consistent with the mode coupling predictions. A decrease in the mode coupling crossover temperature with respect to the bulk is found. We however find that the range of validity of the theory shrinks with respect to the bulk. This is due to the change in the smaller particle mobility and to a substantial enhancement of hopping processes well above the crossover temperature upon confinement.

Supercooled confined water and the mode coupling scenario

Studies of single particle dynamics of water confined in silica pores performed with computer molecular dynamics and inelastic neutron scattering are presented. In the computer study for the highest hydrations two dynamical regimes are found: close to the hydrophilic substrate molecules are below the mode coupling crossover temperature, T C , already at ambient temperature (bound water). The water closer to the center of the pore (free water) approaches T C upon supercooling as predicted by mode coupling theories. Inelastic neutron scattering data are analyzed upon supercooling for hydration levels of 12% and 8%. Also these data are discussed in the framework of the mode coupling theory, in the region of the β relaxation. Strong deviations from the theoretical predictions are found and ascribed to the existence of a low-frequency scattering excess also visible in the simulation.

Dynamics of supercooled water in configuration space.

We study the potential energy surface (PES) sampled by a liquid modeled via the widely studied extended simple point charge (SPC/E) model for water. We characterize the curvature of the PES by calculating the instantaneous normal mode (INM) spectrum for a wide range of densities and temperatures. We discuss the information contained in the INM density of states, which requires additional processing to be unambiguously associated with the long-time dynamics. For the SPC/E model, we find that the slowing down of the dynamics in the supercooled region-where the ideal mode coupling theory has been used to describe the dynamics-is controlled by the reduction in the number of directions in configuration space that allow a structural change. We find that the fraction f(dw) of the double-well directions in configuration space determines the value of the diffusion constant D, thereby relating a property of the PES to a macroscopic dynamic quantity; specifically, it appears that square root D is approximately linear in f(dw). Our findings are consistent with the hypothesis that, at the mode coupling crossover temperature, dynamical processes based on the free exploration of configuration space vanish, and processes requiring activation dominate. Hence, the reduction of the number of directions allowing free exploration of configuration space is the mechanism of diffusion implicitly implemented in the ideal mode coupling theory. Additionally, we find a direct relationship between the number of basins sampled by the system and the number of free directions. In this picture, diffusion appears to be related to geometrical properties of the PES, and to be entropic in origin.

Supercooled confined water and the mode coupling crossover temperature

We present a molecular dynamics study of the single-particle dynamics of supercooled water confined in a silica pore. Two dynamical regimes are found. Close to the hydrophilic substrate molecules are below the mode coupling crossover temperature, T(C), already at ambient temperature. The water closer to the center of the pore (free water) approaches upon supercooling T(C) as predicted by mode coupling theories. For free water the crossover temperature and crossover exponent gamma are extracted from power-law fits to both the diffusion coefficient and the relaxation time of the late alpha region.