When we lower the temperature of a liquid, at some point we meet a first order phase transition to the crystal. Yet, under certain conditions it is possible to keep the system in its metastable phase and to avoid crystallization. In this way the liquid enters in the supercooled phase. Supercooled liquids have a very rich phenomenology, which is still far from being completely understood. To begin with, there is the problem of how to prevent crystallization and how deeply the liquid can be supercooled before a metastability limit is hit. But by far the most interesting feature of supercooled liquids is the dynamic glass transition: when the temperature is decreased below a certain point, the relaxation time increases so much that a dramatic dynamical arrest intervenes and we are unable to equilibrate the system within reasonable experimental times. The glass transition is a phenomenon whose physical origin has stirred an enormous interest in the last hundred years. Why does it occur? Is it just a conventional reference point, or does it have a more profound physical meaning? Is it a purely dynamical event, or the manifestation of a true thermodynamic transition? What is the correlation length associated to the sharp increase of the relaxation time? Can we define a new kind of amorphous order? A shared theory of supercooled liquids and the glass transition does not yet exist and these questions are still largely open. Here, I will illustrate in the most elementary fashion the main phenomenological traits of supercooled liquids and discuss in a very partial way a few theoretical ideas on the subject.
Lecture notes for LES HOUCHES 2002 SUMMER SCHOOL - SESSION LXXVII on SLOW RELAXATIONS AND NONEQUILIBRIUM DYNAMICS IN CONDENSED MATTER; Les Houches: July 1-26, 2002
The dynamical heterogeneity in supercooled liquids measured by a molecular dynamics simulation has been quantified on the basis of the multifractal formalism. The singularity spectrum becomes broader as the glass transition is approached. This behavior is similar to that observed in the case of the Anderson-localization transition.
In this paper, we discuss theoretically the behavior of the four point nonlinear susceptibility and its associated correlation length for supercooled liquids close to the Mode Coupling instability temperature $T_c$. We work in the theoretical framework of the glass transition as described by mean field theory of disordered systems, and the hypernetted chain approximation. Our results give an interpretation framework for recent numerical findings on heterogeneities in supercooled liquid dynamics.
In these lectures, which were presented at "Soft and Fragile Matter, Nonequilibrium Dynamics, Metastability and Flow" University of St. Andrews, 8 July - 22 July, 1999, I give an introduction to the physics of supercooled liquids and glasses and discuss some computer simulations done to investigate these systems.
For a deeply supercooled liquid just above its glass transition temperature, we present a simple thermodynamic model, where the deeply supercooled liquid is assumed to be a mixture of solid-like and liquid-like micro regions. The mole fraction of the liquid-like regions controls the thermodynamic properties of the supercooled liquid while that of the solid-like regions controls its relaxation time or viscosity. From the universal temperature dependence of the molar excess entropy for the deeply supercooled liquids, we derive the temperature dependence of the mole fraction of the liquid-like re…
We present a quantitative description of the thermodynamics in a supercooled binary Lennard Jones liquid via the evaluation of the degeneracy of the inherent structures, i.e. of the number of potential energy basins in configuration space. We find that for supercooled states, the contribution of the inherent structures to the free energy of the liquid almost completely decouples from the vibrational contribution. An important byproduct of the presented analysis is the determination of the Kauzmann temperature for the studied system. The resulting quantitative picture of the thermodynamics of t…
In two recent interesting letters evidence was presented for the existence of a growing dynamic correlation length when we approach the glass transition from the liquid phase (a similar divergence is present also in the off-equilibrium dynamics below $T_c$). Here we would like to point out that this phenomenon can be easily predicted using the replica approach.
Liquid is represented as complicated system of disclinations according to defect description of liquids and glasses. The expressions for the linear disclination field of an arbitrary form and energy of inter-disclination interaction are derived in the framework of gauge theory of defects. It allows us to describe liquid as a disordered system of topological moments and reduce this model to the Edwards--Anderson model with large-range interaction. Within the framework of this approach vitrifying is represented as a "hierarchical" phase transition. The suggested model allows us to explain the pr…
We combine Creutz energy conservation with Kawasaki spin exchange to simulate the microcanonical dynamics of a system of interacting particles. Relaxation occurs via Glauber spin-flip activation using a self-consistent temperature. Heterogeneity in the dynamics comes from finite-size constraints on the spin exchange that yield a distribution of correlated regions. The simulation produces a high-frequency response that can be identified with the boson peak, and a lower-frequency peak that contains non-Debye relaxation and non-Arrhenius activation, similar to the primary response of supercooled …
Using molecular dynamics simulations we investigate the relaxation dynamics of a supercooled liquid close to a rough as well as close to a smooth wall. For the former situation the relaxation times increase strongly with decreasing distance from the wall whereas in the second case they strongly decrease. We use this dependence to extract various dynamical length scales and show that they grow with decreasing temperature. By calculating the frequency dependent average susceptibility of such confined systems we show that the experimental interpretation of such data is very difficult.
We propose a quantitative measure of a time scale on which Fickian diffusion sets in for supercooled liquids and use Brownian Dynamics computer simulations to determine the temperature dependence of this onset time in a Lennard-Jones binary mixture. The time for the onset of Fickian diffusion ranges between 6.5 and 31 times the $α$ relaxation time (the $α$ relaxation time is the characteristic relaxation time of the incoherent intermediate scattering function). The onset time increases faster with decreasing temperature than the $α$ relaxation time. Mean squared displacement at the onset time …
Using molecular dynamics computer simulations we investigate how the relaxation dynamics of a simple supercooled liquid with Newtonian dynamics differs from the one with a stochastic dynamics. We find that, apart from the early beta-relaxation regime, the two dynamics give rise to the same relaxation behavior. The increase of the relaxation times of the system upon cooling, the details of the alpha-relaxation, as well as the wave vector dependence of the Edwards-Anderson-parameters are independent of the microscopic dynamics.
We present molecular dynamics results for a two component, two-dimensional Lennard-Jones supercooled liquid near the glass transition. We find that the supercooled liquid is spatially heterogeneous and that there are long-lived clusters whose size distribution satisfies a scaling relation up to a cutoff. The similarity of several properties of the supercooled liquid to those of a mean-field glass-forming fluid near the spinodal suggests that the glass transition in the supercooled liquid is associated with an underlying thermodynamic instability.
We propose a new method to measure time-dependent linear susceptibilities in molecular simulations, which does not require the use of nonequilibrium simulations, subtraction techniques, or fluctuation-dissipation theorems. The main idea is an exact reformulation of linearly perturbed quantities in terms of observables accessible in unperturbed trajectories. We have applied these ideas to two supercooled liquids in their nonequilibrium aging regime. We show that previous work had underestimated deviations from fluctuation-dissipation relations in the case of a Lennard-Jones system, while our re…
We study the specific heat of a model supercooled liquid confined in a spherical cavity with amorphous boundary conditions. We find the equilibrium specific heat has a cavity-size-dependent peak as a function of temperature. The cavity allows us to perform a finite-size scaling (FSS) analysis, which indicates that the peak persists at a finite temperature in the thermodynamic limit. We attempt to collapse the data onto a FSS curve according to different theoretical scenarios, obtaining reasonable results in two cases: a "not-so-simple" liquid with nonstandard values of the exponents α and ν, a…
We unveil the existence of non-affinely rearranging regions in the inherent structures (IS) of supercooled liquids by numerical simulations of two- and three-dimensional model glass formers subject to static shear deformations combined with local energy minimizations. In the liquid state IS, we find a broad distribution of rather large rearrangements which are correlated only over small distances. At low temperatures, the onset of the cooperative dynamics corresponds to much smaller displacements correlated over larger distances. This finding indicates the presence of non-affinely rearranging …
The dynamics of supercooled liquid and glassy systems are usually studied within the Lagrangian representation, in which the positions and velocities of distinguishable interacting particles are followed. Within this representation, however, it is difficult to define measures of spatial heterogeneities in the dynamics, as particles move in and out of any one given region within long enough times. It is also non-transparent how to make connections between the structural glass and the spin glass problems within the Lagrangian formulation. We propose an Eulerian formulation of supercooled liquids…
We propose deformations of inherent structures as a suitable tool for detecting structural changes underlying the onset of cooperativity in supercooled liquids. The non-affine displacement (NAD) field resulting from the applied deformation shows characteristic differences between the high temperature liquid and supercooled state, that are typically observed in dynamic quantities. The average magnitude of the NAD is very sensitive to temperature changes in the supercooled regime and is found to be strongly correlated with the inherent structure energy. In addition, the NAD field is characterize…
A formally exact set of equations is derived for the description of nonequilibrium phenomena in classical liquids and glasses. With the help of a non equilibrium projection operator formalism, the correlation functions and fluctuation propagators are expressed in terms of memory functions and time dependent collective frequencies. This formally exact set of equations is approximated by applying mode coupling approximations to the memory functions. The resulting set of equations for wavevector dependent correlation functions, fluctuation propagators and one-time structure factors $S_q(t)$ gener…
We compute critical exponents governing universal features of supercooled liquids through the effective theory of an overlap field. The correlation length diverges with the Ising exponent; the size of dynamically heterogeneous patches grows more rapidly; and the relaxation time obeys a generalized Vogel-Fulcher-Tammann relation.
For a model glass-former we demonstrate via computer simulations how macroscopic dynamic quantities can be inferred from a PEL analysis. The essential step is to consider whole superstructures of many PEL minima, called metabasins, rather than single minima. We show that two types of metabasins exist: some allowing for quasi-free motion on the PEL (liquid-like), the others acting as traps (solid-like). The activated, multi-step escapes from the latter metabasins are found to dictate the slowing down of dynamics upon cooling over a much broader temperature range than is currently assumed.
A general identification of the {\em positional specific heat} as the thermodynamic response function associated with the {\em static relaxation length} is proposed, and a phenomenological description for the thermal dependence of the static relaxation length in supercooled liquids is presented. Accordingly, through a phenomenological determination of positional specific heat of supercooled liquids, we arrive at the thermal variation of the static relaxation length $ξ$, which is found to vary in accordance with $ξ\sim (T-T_0)^{-ν}$ in the quasi-equilibrium supercooled temperature regime, where…
Despite the absence of consensus on a theory of the transition from supercooled liquids to glasses, the experimental observations suggest that a detail-independent theory should exist.
We introduce a supercooled liquid model and obtain parameter-free quantitative predictions that are in excellent agreement with numerical simulations, notably in the hard low-temperature region characterized by strong deviations from Mode-Coupling-Theory behavior. The model is the Fredrickson-Andersen Kinetically-Constrained-Model on the three-dimensional $M$-layer lattice. The agreement has implications beyond the specific model considered because the theory is potentially valid for many more systems, including realistic models and actual supercooled liquids.
A model for the slow dynamics of the supercooled liquid is formulated in terms of the standard equations of fluctuating nonlinear hydrodynamics (FNH) with the inclusion of an extra diffusive mode for the collective density fluctuations. If the compressible nature of the liquid is completely ignored, this diffusive mode sets the longest relaxation times in the supercooled state and smooths off a possible sharp ergodicity-nonergodicity (ENE) transition predicted in a mode coupling theory. The scenario changes when the complete dynamics is considered with the inclusion of $1/ρ$ nonlinearities in …
In recent years it has become widely accepted that a dynamical length scale ξ_α plays an important role in supercooled liquids near the glass transition. We examine the implications of the interplay between the growing ξ_α and the size of the crystal nucleus, ξ_M, which shrinks on cooling. We argue that at low temperatures where ξ_α > ξ_M a new crystallization mechanism emerges enabling rapid development of a large scale web of sparsely connected crystallinity. Though we predict this web percolates the system at too low a temperature to be easily seen in the laboratory, there are noticeable re…
Starting from a microscopic model of liquids, we construct an effective theory of an overlap field through duplication of the system and coarse-graining. We then propose a recipe to extract a relaxation time and two characteristic length scales of a supercooled liquid from this effective field theory. Appealing to the Ginzburg-Landau-Wilson paradigm near the putative critical point, we further conclude that this effective field theory resides within the Ising universality class.
We numerically obtain the first quantitative demonstration that development of spatial correlations of mobility as temperature is lowered is responsible for the ``decoupling'' of transport properties of supercooled liquids. This result further demonstrates the necessity of a spatial description of the glass formation and therefore seriously challenges a number of popular alternative theoretical descriptions.
The use of the isoconfigurational ensemble to explore structure-dynamic correlations in supercooled liquids is examined. The statistical error of the dynamic propensity and its spatial distribution are determined. The authors present the spatial distribution of the particle non-Gaussian parameter as a measure of the intermittency with which particles exhibit their propensity for motion. The ensemble average of the direction of particle motion is introduced to establish the anisotropy of the dynamic propensity.
We study supercooled dynamics in quantum hard-sphere liquid using quantum mode-coupling formulation. In the moderate quantum regime, classical cage effects lead to slower dynamics compared to strongly quantum regime, where tunneling overcomes classical caging, leading to faster relaxation. As a result, the glass transition critical density can become significantly higher than for the classical liquids. Perturbative approach is used to solve time dependent quantum mode-coupling equations to study in detail the dynamics of the supercooled liquid in moderate quantum regime. Similar to the classic…
The thermodynamics and dynamics of supercooled liquids correlate with their elasticity. In particular for covalent networks, the jump of specific heat is small and the liquid is {\it strong} near the threshold valence where the network acquires rigidity. By contrast, the jump of specific heat and the fragility are large away from this threshold valence. In a previous work [Proc. Natl. Acad. Sci. U.S.A., 110, 6307 (2013)], we could explain these behaviors by introducing a model of supercooled liquids in which local rearrangements interact via elasticity. However, in that model the disorder char…
First principles molecular dynamics simulations reveal a liquid-liquid phase transition in supercooled elemental silicon. Two phases coexist below $T_c\approx 1232K$. The low density phase is nearly tetra-coordinated, with a pseudogap at the Fermi surface, while the high density phase is more highly coordinated and metallic in nature. The transition is observed through the formation of van der Waals loops in pressure-volume isotherms below $T_c$.
We have studied the relation between the structural relaxation and the frequency dependent thermal response or the specific heat, $c_p(ω)$, in a supercooled liquid. The Mode Coupling Theory (MCT) results are used to obtain $c_p(ω)$ corresponding to different wavevectors. Due to the two-step relaxation process present in the MCT, an extra peak, in addition to the low frequency peak, is predicted in specific heat at high frequency.
Stochastic-Beta-Relaxation (SBR) provides a characterisation of the glass crossover in discontinuous Spin-Glasses and Supercoooled liquid. Notably it can be derived through a rigorous computation from a dynamical Landau theory. In this paper I will discuss the precise meaning of this connection in a language that does not require familiarity with statistical field theory. I will discuss finite-size corrections in mean-field Spin-Glass models and loop corrections in finite-dimensional models that are both described by the dynamical Landau theory considered. Then I will argue that the same Landa…
We study the transport and the relaxation properties of a molecular supercooled liquid by molecular-dynamics numerical simulations. The focus is on the translational motion. Jump motion is detected. At lower temperature the Stokes-Einstein law breaks down.
It is proposed that the rate of relaxation in a liquid is better described by the geometric mean of the van Hove distribution function, rather than the standard arithmetic mean used to obtain the mean squared displacement. The difference between the two means is shown to increase significantly with an increase in the non-Gaussian character of the displacement distribution. Preliminary results indicate that the geometric diffusion constant results in a substantial reduction of the deviation from Stokes-Einstein scaling.
Reply to "Comment on 'Fickian non-Gaussian diffusion in glass-forming liquids' ". In [ArXiv:2210.07119v1], Berthier et al. questioned the findings of our letter [Phys. Rev. Lett. 128, 168001 (2022)], concerning the existence and the features of Fickian non-Gaussian diffusion in glass-forming liquids. Here we demonstrate that their arguments are either wrong, or not meaningful to our scope. Thus, we fully confirm the validity and novelty of our results.
The rotational dynamics of a supercooled molecular liquid is investigated by a molecular-dynamics numerical study. We detect rotational jumps with a waiting-time distribution which is well fitted by a truncated power law. At lower temperatures the rotational correlation times are decoupled from the viscosity.
The interplay between self-diffusion and excitation lines in space-time was recently studied in kinetically constrained models to explain the breakdown of the Stokes-Einstein law in supercooled liquids. Here, we further examine this interplay and its manifestation in incoherent scattering functions. In particular, we establish a dynamic length scale below which Fickian diffusion breaks down, as is observed in experiments and simulations. We describe the temperature dependence of this length scale in liquids of various fragilities, and provide analytical estimates for the van Hove and self-inte…
The light scattering spectra of molecular liquids are derived within a generalized hydrodynamics. The wave vector and scattering angle dependences are given in the most general case and the change of the spectral features from liquid to solidlike is discussed without phenomenological model assumptions for (general) dielectric systems without long-ranged order. Exact microscopic expressions are derived for the frequency-dependent transport kernels, generalized thermodynamic derivatives and the background spectra.
The constitutive equations for the orientational dynamics of a liquid formed of linear molecules are derived microscopically. The resulting generalised Langevin equations coincide with the phenomenological approach of Dreyfus et al. Formally exact expressions are given for the phenomenological coefficients and various constraints are shown to be consequences of this microscopic approach.
The possible superfluid transition of supercooled liquid parahydrogen is investigated by quantum Monte Carlo simulations. The cooling protocol adopted here allows for the investigation of a fluid phase down to a temperature T=0.25 K. No evidence of superfluidity is found, as exchanges of identical particles are strongly suppressed even at the lowest temperature. Is shown that, contrary to a commonly held belief, it is not the well depth of the pair-wise interaction but rather its relatively large hard core diameter that physically hinders superfluidity in parahydrogen.
Because of the isotropic and disordered nature of liquids, the anisotropy hidden in intermolecular interactions are often neglected. Accordingly, the order parameter describing a simple liquid has so far been believed to be only density. In contrast to this common sense, we propose that two order parameters, namely, density and bond order parameters, are required to describe the phase behavior of liquids since they intrinsically tend to form local bonds. This model gives us clear physical explanations for two poorly-understood phenomena in supercooled liquids: (i) large-scale density fluctuati…
We investigate to what extend the replica-exchange Monte Carlo method is able to equilibrate a simple liquid in its supercooled state. We find that this method does indeed allow to generate accurately the canonical distribution function even at low temperatures and that its efficiency is about 10-100 times higher than the usual canonical molecular dynamics simulation.
We extend the conventional mode-coupling theory of supercooled liquids to systems under stationary shear flow. Starting from generalized fluctuating hydrodynamics, a nonlinear equation for the intermediate scattering function is constructed. We evaluate the solution numerically for a model of a two dimensional colloidal suspension and find that the structural relaxation time decreases as $\dotγ^{-ν}$ with an exponent $ν\leq 1$, where $\dotγ$ is the shear rate. The results are in qualitative agreement with recent molecular dynamics simulations. We discuss the physical implications of the result…
In this paper we study the four point correlation function $χ_4$ of collective density fluctuations in a nonequilibrium liquid. The equilibration is controlled by a modified stretched exponential behavior ( $\exp [-{(t_\mathrm{w}/τ)}^β]$ ) having the relaxation time $τ$ dependent on the aging time $t_\mathrm{w}$. Similar aging behavior has been seen experimentally in supercooled liquids. The basic equations of fluctuating nonlinear hydrodynamics (FNH) are solved here numerically to obtain $χ_4$ for equilibrium and non equilibrium states. We also identify a dynamic length scale $ξ$ from the equ…
We propose that the salient feature to be explained about the glass transition of supercooled liquids is the temperature-controlled superArrhenius activated nature of the viscous slowing down, more strikingly seen in weakly-bonded, fragile systems. In the light of this observation, the relevance of simple models of spherically interacting particles and that of models based on free-volume congested dynamics are questioned. Finally, we discuss how the main aspects of the phenomenology of supercooled liquids, including the crossover from Arrhenius to superArrhenius activated behavior and the hete…
We present mode-coupling equations for the description of the slow dynamics observed in supercooled molecular liquids close to the glass transition. The mode-coupling theory (MCT) originally formulated to study the slow relaxation in simple atomic liquids, and then extended to the analysis of liquids composed by linear molecules, is here generalized to systems of arbitrarily shaped, rigid molecules. We compare the predictions of the theory for the $q$-vector dependence of the molecular nonergodicity parameters, calculated by solving numerically the molecular MCT equations in two different appr…
We determine the phase diagram of liquid supercooled water by extensive computer simulations using the TIP5P-E model [J. Chem. Phys. {\bf 120}, 6085 (2004)]. We find that the transformation of water into a low density liquid in the supercooled range strongly enhances the solubility of hydrophobic particles. The transformation of water into a tetrahedrally structured liquid is accompanied by a minimum in the hydration entropy and enthalpy. The corresponding change in sign of the solvation heat capacity indicates a loss of one characteristic signature of hydrophobic hydration. The observed behav…
An analysis of diffusion in a supercooled liquid based solely in the density of diffusive directions and the value of energy barriers shows how the potential energy landscape (PEL) approach is capable of explaining the $α$ and $β$ relaxations and the fragility of a glassy system. We find that the $β$ relaxation is directly related to the search for diffusive directions. Our analysis shows how in strong liquids diffusion is mainly energy activated, and how in fragile liquids the diffusion is governed by the density of diffusive directions. We describe the fragile-to-strong crossover as a change…
In this work, we study the nature of transitions between inherent structures of a two-dimensional model supercooled liquid. We demonstrate that these transitions occur predominately along a small number of directions on the energy landscape. Moreover, we show that the number of such directions decreases as the temperature of the liquid is decreased in the mildly supercooled regime, in concert with earlier studies on an athermal jamming system. We show that this decrease happens in parallel with a change in character of the transitions as dynamics in the system become more heterogeneous and loc…
Metastable liquid mixtures of parahydrogen and orthodeuterium are studied theoretically by means of computer simulations. No reduced propensity of the mixture to undergo crystallization is observed, compared to that of pure liquid parahydrogen. Demixing of the two species as a precursor of crystallization is not observed either.
We study dynamical phase transitions in a model supercooled liquid. These transitions occur in ensembles of trajectories that are biased towards low (or high) dynamical activity. We compare two different measures of activity that were introduced in recent papers and we find that they are anti-correlated with each other. To interpret this result, we show that the two measures couple to motion on different length and time scales. We find that inactive states with very slow structural relaxation nevertheless have increased molecular motion on short scales. We discuss these results in terms of the…
Using a novel thermodynamic integration scheme, we compute the excess free energy, $γ$, of a glass-forming, binary Lennard-Jones liquid in contact with a frozen amorphous wall, formed by particles frozen into a similar structure as the liquid. We find that $γ$ is non-zero, becoming negative at low temperature. This indicates that the thermodynamics of the system is perturbed by the effect of the amorphous wall.
The dynamics of Lennard-Jones fluid is studied through extended mode coupling theory (MCT) with the inclusion of the slow mode of defect density. Inclusion of defect density facilitates the liquid like state for temperatures much lower than predicted from ideal MCT. From the present model the temperature dependence of the power law exponent is obtained at a {\em constant pressure}. We have also computed the wave number dependence of the power law exponent.
A theoretical treatment of deeply supercooled liquids is difficult because their properties emerge from spatial inhomogeneities that are self-induced, transient, and nanoscopic. I use computer simulations to analyse self-induced static and dynamic heterogeneity in equilibrium systems approaching the experimental glass transition. I characterise the broad sample-to-sample fluctuations of salient dynamic and thermodynamic properties in elementary mesoscopic systems. Findings regarding local lifetimes and distributions of dynamic heterogeneity are in excellent agreement with recent single molecul…
