Quasisaddles as relevant points of the potential energy surface in the dynamics of supercooled liquids

Heterogeneous-elasticity theory of instantaneous normal modes in liquids

Since decades, the concept of vibrational density of states in glasses has been mirrored in liquids by the instantaneous-normal-mode spectrum. In glasses instantaneous configurations are believed to be situated close to minima of the potential-energy hypersurface and all eigenvalues of the associated Hessian matrix are positive. In liquids this is no longer true, and modes corresponding to both positive and negative eigenvalues exist. The instantaneous-normal-mode spectrum has been numerically investigated in the past, and it has been demonstrated to bring important information on the liquid dynamics and transport properties. A systematic deeper theoretical understanding is now needed. Heterogeneous-elasticity theory has proven to be particularly successful in explaining many details of the low-frequency excitations in glasses, ranging from the thoroughly studied boson peak, to other anomalies related to the crossover between wave-like and random-matrix-like excitations. Here we present an extension of heterogeneous-elasticity theory to the liquid state, and show that the outcome of the theory agrees well to the results of extensive molecular-dynamics simulations of a model liquid at different temperatures. We find that the spectrum of eigenvalues [Formula: see text] has a sharp maximum close to (but not at) [Formula: see text], and decreases monotonically with [Formula: see text] on both its stable and unstable side. We show that the spectral shape strongly depends on temperature, being symmetric at high temperatures and becoming rather asymmetric at low temperatures, close to the dynamical critical temperature. Most importantly, we demonstrate that the theory naturally reproduces a surprising phenomenon, a zero-energy spectral singularity with a cusp-like character developing in the vibrational spectra upon cooling. This feature, known from a few previous numerical studies, has been generally overlooked in the past due to a misleading representation of the data. We provide a thorough analysis of this issue, based on both very accurate predictions of our theory, and computational studies of model liquid systems with extended size.

Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We present a theory, based on an unstable elasticity model with thermally fluctuating elastic constants, which explains the salient features of the instantaneous normal mode (INM) spectrum of a numerically simulated liquid over a large range of temperatures. The INM spectrum of a liquid is obtained in a molecular dynamics (MD) simulation by considering the harmonic eigenvalue spectrum of the potential energy, taken at a given instant (snapshot). Because the INM spectrum records the curvatures of the fluctuating potential landscape of the liquid, an understanding of this spectrum may pave the way toward understanding the liquid to glass transformation process.

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues ρ(λ) has a sharp maximum near (but not at) λ=0 and decreases monotonically with |λ| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for ρ(λ), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of ρ(λ) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T=0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation

Yielding behavior in amorphous solids has been investigated in computer simulations using uniform and cyclic shear deformation. Recent results characterize yielding as a discontinuous transition, with the degree of annealing of glasses being a significant parameter. Under uniform shear, discontinuous changes in stresses at yielding occur in the high annealing regime, separated from the poor annealing regime in which yielding is gradual. In cyclic shear simulations, relatively poorly annealed glasses become progressively better annealed as the yielding point is approached, with a relatively modest but clear discontinuous change at yielding. To understand better the role of annealing on yielding characteristics, we perform athermal quasistatic cyclic shear simulations of glasses prepared with a wide range of annealing in two qualitatively different systems-a model of silica (a network glass) and an atomic binary mixture glass. Two strikingly different regimes of behavior emerge. Energies of poorly annealed samples evolve toward a unique threshold energy as the strain amplitude increases, before yielding takes place. Well-annealed samples, in contrast, show no significant energy change with strain amplitude until they yield, accompanied by discontinuous energy changes that increase with the degree of annealing. Significantly, the threshold energy for both systems corresponds to dynamical cross-over temperatures associated with changes in the character of the energy landscape sampled by glass-forming liquids.

Analysis of the anomalous mean-field like properties of Gaussian core model in terms of entropy

Studies of the Gaussian core model (GCM) have shown that it behaves like a mean-field model and the properties are quite different from standard glass former. In this work, we investigate the entropies, namely, the excess entropy (S_ex) and the configurational entropy (S_c) and their different components to address these anomalies. Our study corroborates most of the earlier observations and also sheds new light on the high and low temperature dynamics. We find that unlike in standard glass former where high temperature dynamics is dominated by two-body correlation and low temperature by many-body correlations, in the GCM both high and low temperature dynamics are dominated by many-body correlations. We also find that the many-body entropy which is usually positive at low temperatures and is associated with activated dynamics is negative in the GCM suggesting suppression of activation. Interestingly despite the suppression of activation, the Adam-Gibbs (AG) relation that describes activated dynamics holds in the GCM, thus suggesting a non-activated contribution in AG relation. We also find an overlap between the AG relation and mode coupling power law regime leading to a power law behavior of S_c. From our analysis of this power law behavior, we predict that in the GCM the high temperature dynamics will disappear at dynamical transition temperature and below that there will be a transition to the activated regime. Our study further reveals that the activated regime in the GCM is quite narrow.

Role of the Pair Correlation Function in the Dynamical Transition Predicted by Mode Coupling Theory

In a recent study, we have found that for a large number of systems the configurational entropy at the pair level S_{c2}, which is primarily determined by the pair correlation function, vanishes at the dynamical transition temperature T_{c}. Thus, it appears that the information of the transition temperature is embedded in the structure of the liquid. In order to investigate this, we describe the dynamics of the system at the mean field level and, using the concepts of the dynamical density functional theory, show that the dynamical transition temperature depends only on the pair correlation function. Thus, this theory is similar in spirit to the microscopic mode coupling theory (MCT). However, unlike microscopic MCT, which predicts a very high transition temperature, the present theory predicts a transition temperature that is similar to T_{c}. This implies that the information of the dynamical transition temperature is embedded in the pair correlation function.

The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero

Rapid cooling of liquids below a certain temperature range can result in a transition to glassy states. The traditional understanding of glasses includes their thermodynamic metastability with respect to crystals. However, here we present specific examples of interactions that eliminate the possibilities of crystalline and quasicrystalline phases, while creating mechanically stable amorphous glasses down to absolute zero temperature. We show that this can be accomplished by introducing a new ideal state of matter called a "perfect glass". A perfect glass represents a soft-interaction analog of the maximally random jammed (MRJ) packings of hard particles. These latter states can be regarded as the epitome of a glass since they are out of equilibrium, maximally disordered, hyperuniform, mechanically rigid with infinite bulk and shear moduli, and can never crystallize due to configuration-space trapping. Our model perfect glass utilizes two-, three-, and four-body soft interactions while simultaneously retaining the salient attributes of the MRJ state. These models constitute a theoretical proof of concept for perfect glasses and broaden our fundamental understanding of glass physics. A novel feature of equilibrium systems of identical particles interacting with the perfect-glass potential at positive temperature is that they have a non-relativistic speed of sound that is infinite.

Unraveling the success and failure of mode coupling theory from consideration of entropy

We analyze the dynamics of model supercooled liquids in a temperature regime where predictions of mode coupling theory (MCT) are known to be valid qualitatively. In this regime, the Adam-Gibbs (AG) relation, based on an activation picture of dynamics, also describes the dynamics satisfactorily, and we explore the mutual consistency and interrelation of these descriptions. Although entropy and dynamics are related via phenomenological theories, the connection between MCT and entropy has not been argued for. In this work, we explore this connection and provide a microscopic derivation of the phenomenological Rosenfeld theory. At low temperatures, the overlap between the MCT power law regime and AG relation implies that the AG relation predicts an avoided divergence at Tc, the origin of which can be related to the vanishing of pair configurational entropy, which we find occurring at the same temperature. We also show that the residual multiparticle entropy plays an important role in describing the relaxation time.

Length scales and dynamics in the reorientational relaxation of tracers in molecular and polymeric glass formers via electron spin resonance spectroscopy

It is widely accepted that a temperature region exists above the glass transition temperature, playing a fundamental role in the physics of polymers and glass formers. In this region, several dynamic crossovers have been experimentally and numerically revealed in the past years and the onset of glassy behavior is generally located, because of cooperative and heterogeneous features exhibited by the dynamics. In this Article, the rotational dynamics of two different stiff molecular spin probes dissolved in poly(propylene glycol) has been investigated by electron spin resonance spectroscopy in a wide temperature range. Decoupling phenomena between macroscopic and microscopic transport properties have been observed in the crossover region. A comparison with previous studies carried out with the same tracers dissolved in different polymers and glass formers strongly supports the idea that the observed crossover signals the onset of spatial correlations of dynamics on the length scale probed by the tracers.

Dynamics and energy landscape in a tetrahedral network glass-former: direct comparison with models of fragile liquids

We report molecular dynamics simulations for a new model of tetrahedral network glass-former, based on short-range spherical potentials. Despite the simplicity of the forcefield employed, our model reproduces some essential physical properties of silica, an archetypal network-forming material. Structural and dynamical properties, including dynamic heterogeneities and the nature of local rearrangements, are investigated in detail and a direct comparison with models of close-packed, fragile glass-formers is performed. The outcome of this comparison is rationalized in terms of the properties of the potential energy surface, focusing on the unstable modes of the stationary points. Our results indicate that the weak degree of dynamic heterogeneity observed in network glass-formers may be attributed to an excess of localized unstable modes, associated with elementary dynamical events such as bond breaking and reformation. In contrast, the more fragile Lennard-Jones mixtures are characterized by a larger fraction of extended unstable modes, which lead to a more cooperative and heterogeneous dynamics.

Intermolecular forces and the glass transition

Random first-order transition theory is used to determine the role of attractive and repulsive interactions in the dynamics of supercooled liquids. Self-consistent phonon theory, an approximate mean field treatment consistent with random first-order transition theory, is used to treat individual glassy configurations, whereas the liquid phase is treated using common liquid-state approximations. Free energies are calculated using liquid-state perturbation theory. The transition temperature, T*A, the temperature where the onset of activated behavior is predicted by mean field theory; the lower crossover temperature, T*C, where barrierless motions actually occur through fractal or stringy motions (corresponding to the phenomenological mode coupling transition temperature); and T*K, the Kauzmann temperature (corresponding to an extrapolated entropy crisis), are calculated in addition to T*g, the glass transition temperature that corresponds to laboratory cooling rates. Relationships between these quantities agree well with existing experimental and simulation data on van der Waals liquids. Both the isobaric and isochoric behavior in the supercooled regime are studied, providing results for DeltaCV and DeltaCp that can be used to calculate the fragility as a function of density and pressure, respectively. The predicted variations in the alpha-relaxation time with temperature and density conform to the empirical density-temperature scaling relations found by Casalini and Roland. We thereby demonstrate the microscopic origin of their observations. Finally, the relationship first suggested by Sastry between the spinodal temperature and the Kauzmann temperatures, as a function of density, is examined. The present microscopic calculations support the existence of an intersection of these two temperatures at sufficiently low temperatures.

Understanding fragility in supercooled Lennard-Jones mixtures. II. Potential energy surface

The connection between isobaric fragility and the properties of high-order stationary points of the potential energy surface in different supercooled Lennard-Jones mixtures was investigated. The increase of effective activation energies upon supercooling appears to be driven by the increase of average potential energy barriers measured by the energy dependence of the fraction of unstable modes. Such an increase is sharper, the more fragile the mixture. Correlations between fragility and other properties of high-order stationary points, including the vibrational density of states and the localization features of unstable modes, are also discussed.

Diffusivity, excess entropy, and the potential-energy landscape of monatomic liquids

The connection between thermodynamic, transport, and potential-energy landscape features is studied for liquids with Lennard-Jones-type pair interactions using both microcanonical molecular-dynamics and isothermal-isobaric ensemble Monte Carlo simulations. Instantaneous normal-mode and saddle-point analyses of two variants of the monatomic Lennard-Jones liquid have been performed. The diffusivity is shown to depend linearly on several key properties of instantaneous and saddle configurations-the energy, the fraction of negative curvature directions, and the mean, maximum, and minimum eigenvalues of the Hessian. Since the Dzugutov scaling relationship also holds for such systems [Nature (London) 381, 137 (1996)], the exponential of the excess entropy, within the two-particle approximation, displays the same linear dependence on energy landscape properties as the diffusivity.

Determining landscape-based criteria for freezing of liquids

The correlation between statistical properties of the energy landscape and the number of accessible configurational states, as measured by the exponential of the excess entropy (e(Se)), are studied in the case of a simple Lennard-Jones-type liquid in the neighborhood of the thermodynamic freezing transition. The excess entropy Se is defined as the difference between the entropy of the liquid and that of the ideal gas under identical temperature and pressure conditions and is estimated using the pair correlation contribution, S2. Landscape properties associated with three categories of configurations are considered: instantaneous configurations, inherent saddles, and inherent minima. Landscape properties studied include the energy and the key parameters of the Hessian eigenvalue distribution as well as the mean distances between instantaneous configurations and the corresponding inherent saddles and minima. The signatures of the thermodynamic freezing transition are clearest in the case of inherent structure properties which show, as a function of e(S2), a pronounced change in slope in the vicinity of the solid-liquid coexistence. The mean distance between instantaneous and saddle configurations also shows a similar change in slope when the system crosses from the stable to the supercooled regime. In the case of inherent saddles, the minimum eigenvalue acts as a similar indicator of the thermodynamic freezing transition but the average and maximum eigenvalues do not carry similar signatures. In the case of instantaneous configurations, a weak indicator of the thermodynamic freezing transition is seen in the behavior of the fraction of negative curvature directions as a function of the exponential of the excess entropy.

Potential-energy landscape of a supercooled liquid and its resemblance to a collection of traps

It is analyzed whether the potential energy landscape of a glass-forming system can be effectively mapped on a random model which is described in statistical terms. For this purpose we generalize the simple trap model of Monthus and Bouchaud [J. Phys. A 29, 3847 (1996)] by dividing the total system into M weakly interacting identical subsystems, each being described in terms of a trap model. The distribution of traps in this extended trap model (ETM) is fully determined by the thermodynamics of the glass former. The dynamics is described by two adjustable parameters, one characterizing the common energy level of the barriers, the other the strength of the interaction. The comparison is performed for the standard binary mixture Lennard-Jones system with 65 particles. The metabasins, identified in our previous work, are chosen as traps. Comparing molecular dynamics simulations of the Lennard-Jones system with Monte Carlo calculations of the ETM allows one to determine the adjustable parameters. Analysis of the first moment of the waiting distribution yields an optimum agreement when choosing M approximately 3 subsystems. Comparison with the second moment of the waiting time distribution, reflecting dynamic heterogeneities, indicates that the sizes of the subsystems may fluctuate.

Relationship between phase transitions and topological changes in one-dimensional models

We address the question of the quantitative relationship between thermodynamic phase transitions and topological changes in the potential energy manifold analyzing two classes of one dimensional models, the Burkhardt solid-on-solid model and the Peyrard-Bishop model for DNA thermal denaturation, both in the confining and nonconfining version. These models, apparently, do not fit [M. Kastner, Phys. Rev. Lett. 93, 150601 (2004)] in the general idea that the phase transition is signaled by a topological discontinuity. We show that in both models the phase transition energy v(c) is actually noncoincident with, and always higher than, the energy v(theta) at which a topological change appears. However, applying a procedure already successfully employed in other cases as the mean field phi4 model, i.e., introducing a map M:v-->v(s) from levels of the energy hypersurface V to the level of the stationary points "visited" at temperature T, we find that M (v(c))=v(theta). This result enhances the relevance of the underlying stationary points in determining the thermodynamics of a system, and extends the validity of the topological approach to the study of phase transition to the elusive one-dimensional systems considered here.

Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics

The elsewhere surmized topological origin of phase transitions is given here important evidence through the analytic study of an exactly solvable model for which both topology of submanifolds of configuration space and thermodynamics are worked out. The model is a mean-field one with a k-body interaction. It undergoes a second-order phase transition for k=2 and a first-order one for k >2 . This opens a perspective for the understanding of the deep origin of first and second-order phase transitions, respectively. In particular, a remarkable theoretical result consists of a mathematical characterization of first-order transitions. Moreover, we show that a "reduced" configuration space can be defined in terms of collective variables, such that the correspondence between phase transitions and topology changes becomes one-to-one, for this model. Finally, an unusual relationship is worked out between the microscopic description of a classical N -body system and its macroscopic thermodynamic behavior. This consists of a functional dependence of thermodynamic entropy upon the Morse indexes of the critical points (saddles) of the constant energy hypersurfaces of the microscopic 2N-dimensional phase space. Thus phase space (and configuration space) topology is directly related to thermodynamics.

Topological properties of the mean-field phi4 model

We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a phi4 mean-field model. We compare the critical energy vc [i.e., the potential energy vT evaluated at the phase transition temperature Tc ] with the energy vtheta at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc >> vtheta, at variance to what has been found in different mean-field and short ranged systems, where the thermodynamic phase transitions take place at vc=vtheta [Phys. Rep. 337, 237 (2000)]]. By direct calculation of the energy vs T of the "inherent saddles," i.e., the saddles visited by the equilibrated system at temperature T , we find that vsTc approximately vtheta. Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather than to a change of the topology of the potential energy surface at T= Tc. Finally, we discuss the approximation involved in our analysis and the generality of our method.

Saddle index properties, singular topology, and its relation to thermodynamic singularities for a phi4 mean-field model

We investigate the potential energy surface of a phi(4) model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers alpha(+) , alpha(0) , alpha(-) with alpha(+) + alpha(0) + alpha(-) =1 , provided that the interaction strength mu is smaller than a critical value. The saddle index n(s) is equal to alpha(0) and its distribution function has a maximum at n(max)(s) =1/3 . The density p (e) of stationary points with energy per particle e , as well as the Euler characteristic chi (e) , are singular at a critical energy e(c) (mu) , if the external field H is zero. However, e(c) (mu) not equal upsilon(c) (mu) , where upsilon(c) (mu) is the mean potential energy per particle at the thermodynamic phase transition point T(c) . This proves that previous claims that the topological and thermodynamic transition points coincide is not valid, in general. Both types of singularities disappear for H not equal 0 . The average saddle index n (s) as function of e decreases monotonically with e and vanishes at the ground state energy, only. In contrast, the saddle index n(s) as function of the average energy e ( n(s) ) is given by n(s) ( e ) =1+4 e (for H=0 ) that vanishes at e =-1/4> upsilon(0) , the ground state energy.

Density-of-states Monte Carlo simulation of a binary glass

A newly proposed Monte Carlo formalism has been used to simulate a glass-forming liquid above and below the glass transition temperature. The heat capacity exhibits a sharp peak at a temperature lower than that reported from extensive molecular dynamics simulations. Its height is larger than that reported earlier. At temperatures below mode coupling, the average inherent-structure energy of the configurations generated in this work is significantly lower than that reported in the literature. The entropy of the supercooled liquid is calculated directly from our simulations and that of a disordered solid is calculated by a normal-mode analysis. We find that at low temperatures these two entropy curves become essentially parallel. They do not intersect each other, raising questions about the existence of a Kauzmann temperature in this glass-forming mixture.

Saddles in the energy landscape: extensivity and thermodynamic formalism

We formally extend the energy landscape approach for the thermodynamics of liquids to account for saddle points. By considering the extensive nature of macroscopic potential energies, we derive the scaling behavior of saddles with system size, as well as several approximations for the properties of low-order saddles (i.e., those with only a few unstable directions). We then cast the canonical partition function in a saddle-explicit form and develop, for the first time, a rigorous energy landscape approach capable of reproducing trends observed in simulations, in particular, the temperature dependence of the energy and fractional order of sampled saddles.

Real space origin of temperature crossovers in supercooled liquids

We show that the various crossovers between dynamical regimes observed in experiments and simulations of supercooled liquids can be explained in simple terms from the existence and statistical properties of dynamical heterogeneities. We confirm that dynamic heterogeneity is responsible for the slowing down of glass formers at temperatures well above the dynamic singularity Tc predicted by mode-coupling theory. Our results imply that activated processes govern the long-time dynamics even in the temperature regime where they are neglected by mode-coupling theory. We show that alternative interpretations based on topographic properties of the potential energy landscape are inefficient ways of describing simple physical features which are naturally accounted for within our approach. We show in particular that the reported links between mode coupling and landscape singularities do not exist.

Energy barriers and activated dynamics in a supercooled Lennard-Jones liquid

We study the relation of the potential energy landscape (PEL) topography to relaxation dynamics of a small model glass former of Lennard-Jones type. The mechanism under investigation is the hopping between superstructures of PEL minima, called metabasins (MBs). Guided by the idea that the mean durations <tau> of visits to MBs should reflect the local PEL structure, we first derive the effective depths of MBs from dynamics, by the relation E(app)=d ln<tau>/dbeta, where beta=1/k(B)T. Second, we establish a connection of E(app) to the barriers that surround MBs. As the consequence of a rugged PEL, it turns out that escapes from MBs do not happen by single hops between PEL minima, but correspond to complicated multiminima sequences. We introduce the concept of return probabilities to the bottom of the MBs in order to judge when the attraction range of a MB has been left. The energy barriers overcome can then be identified. These turn out to be in good agreement with the effective depths E(app), calculated from dynamics. We are thus able to relate MB lifetimes to their local structure. Moreover, we can trace back the overall diffusive dynamics to the population of MBs and to their local topology, i.e., to purely thermodynamic and structural quantities. Single energy barriers are identified with the help of a new method, which accurately performs a descent along the ridge between two minima. We analyze the population of transition regions between minima, called basin borders. No indication for the mechanism of diffusion to change around the mode-coupling temperature can be found. We discuss the question whether the one-dimensional reaction paths connecting two minima are relevant for the calculation of reaction rates at the temperatures under study.

Crossover between equilibrium and shear-controlled dynamics in sheared liquids

We present a numerical simulation study of a simple monatomic Lennard-Jones liquid under shear flow, as a function of both temperature T and shear rate .gamma. By investigating different observables we find that (i) there exists a line, T(.gamma), in the (T-(.gamma)) plane that sharply marks the border between an "equilibrium" and a "shear-controlled" region for both the dynamic and the thermodynamic quantities; and (ii) along this line the structural relaxation time, tau(alpha)(T(.gamma)), is proportional to .gamma(-1), i.e., to the typical time scale introduced by the shear flow. Above T(.gamma), the liquid dynamics is unaffected by the shear flow, while below T(.gamma) both T and .gamma control the particle motion.