Mesoscopic two-point collective dynamics of glass-forming liquids

The collective density-density and hydrostatic pressure-pressure correlations of glass-forming liquids are spatiotemporally mapped out using molecular dynamics simulations. It is shown that the sharp rise of structural relaxation time below the Arrhenius temperature coincides with the emergence of slow, nonhydrodynamic collective dynamics on mesoscopic scales. The observed long-range, nonhydrodynamic mode is independent of wave numbers and closely coupled to the local structural dynamics. Below the Arrhenius temperature, it dominates the slow collective dynamics on length scales immediately beyond the first structural peak in contrast to the well-known behavior at high temperatures. These results highlight a key connection between the qualitative change in mesoscopic two-point collective dynamics and the dynamic crossover phenomenon.

Coarse graining atomistic simulations of plastically deforming amorphous solids

The primary mode of failure in disordered solids results from the formation and persistence of highly localized regions of large plastic strains known as shear bands. Continuum-level field theories capable of predicting this mechanical response rely upon an accurate representation of the initial and evolving states of the amorphous structure. We perform molecular dynamics simulations of a metallic glass and propose a methodology for coarse graining discrete, atomistic quantities, such as the potential energies of the elemental constituents. A strain criterion is established and used to distinguish the coarse-grained degrees-of-freedom inside the emerging shear band from those of the surrounding material. A signal-to-noise ratio provides a means of evaluating the strength of the signal of the shear band as a function of the coarse graining. Finally, we investigate the effect of different coarse graining length scales by comparing a two-dimensional, numerical implementation of the effective-temperature description in the shear transformation zone (STZ) theory with direct molecular dynamics simulations. These comparisons indicate the coarse graining length scale has a lower bound, above which there is a high level of agreement between the atomistics and the STZ theory, and below which the concept of effective temperature breaks down.

Distribution of thermally activated plastic events in a flowing glass

The potential energy landscape of a flowing metallic glass is revealed using the activation-relaxation technique. For a two-dimensional Lennard-Jones system initially deformed into a steady-state condition through quasistatic shear, the distribution of activation energies is shown to contain a large fraction of low-energy barriers, consistent with a highly nonequilibrium flow state. The distribution of plastic strains has a fundamentally different shape than that obtained during quasistatic simulations, exhibiting a peak at finite strain and, after elastic unloading, a nonzero mean plastic strain that evidences a polarization of the flow state. No significant correlation is found between the activation energy of a plastic event and its associated plastic strain.

Statistical mechanics and dynamics of a three-dimensional glass-forming system

In the context of a classical example of glass formation in three dimensions, we exemplify how to construct a statistical-mechanical theory of the glass transition. At the heart of the approach is a simple criterion for verifying a proper choice of upscaled quasispecies that allow the construction of a theory with a finite number of "states." Once constructed, the theory identifies a typical scale xi that increases rapidly with lowering the temperature and which determines the alpha-relaxation time tau(alpha) as tau(alpha) approximately exp(muxi/T), with mu a typical chemical potential. The theory can predict relaxation times at temperatures that are inaccessible to numerical simulations.

Nonuniversality of the specific heat in glass forming systems

We present new simulation results for the specific heat in a classical model of a binary mixture glass former in two dimensions. We show that in addition to the formerly observed specific heat peak, there is a second peak at lower temperatures which was not observable in earlier simulations. This is a surprise, as most texts on the glass transition expect a single specific heat peak. We explain the physics of the two specific heat peaks by the micromelting of two types of clusters. While this physics is easily accessible, the consequences are that one should not expect any universality in the temperature dependence of the specific heat in glass formers.

Theory of specific heat in glass-forming systems

Experimental measurements of the specific heat in glass-forming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called "specific heat peak" in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same alpha -relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formers-unfortunately, one needs to understand each case by itself.

Quantitative theory of a relaxation function in a glass-forming system

We present a quantitative theory for a relaxation function in a simple glass-forming model (binary mixture of particles with different interaction parameters). It is shown that the slowing down is caused by the competition between locally favored regions (clusters) that are long-lived but each of which relaxes as a simple function of time. Without the clusters, the relaxation of the background is simply determined by one typical length, which we deduce from an elementary statistical mechanical argument. The total relaxation function (which depends on time in a nontrivial manner) is quantitatively determined as a weighted sum over the clusters and the background. The "fragility" in this system can be understood quantitatively since it is determined by the temperature dependence of the number fractions of the locally favored regions.

Ergodicity and slowing down in glass-forming systems with soft potentials: no finite-temperature singularities

The aim of this paper is to discuss some basic notions regarding generic glass-forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction, we discuss the so-called glass transition in which a supercooled amorphous state is formed, accompanied by a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length scales, we show the following. (i) The system remains ergodic at all temperatures. (ii) The number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the small world property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0 .

Aging and relaxation in glass-forming systems

We propose that there exists a generic class of glass-forming systems that have competing states (of crystalline order or not) which are locally close in energy to the ground state (which is typically unique). Upon cooling, such systems exhibit patches (or clusters) of these competing states which become locally stable in the sense of having a relatively high local shear modulus. It is in between these clusters where aging, relaxation, and plasticity under strain can take place. We demonstrate explicitly that relaxation events that lead to aging occur where the local shear modulus is low (even negative) and result in an increase in the size of local patches of relative order. We examine the aging events closely from two points of view. On the one hand we show that they are very localized in real space, taking place outside the patches of relative order, and from the other point of view we show that they represent transitions from one local minimum in the potential surface to another. This picture offers a direct relation between structure and dynamics, ascribing the slowing down in glass-forming systems to the reduction in relative volume of the amorphous material which is liquidlike. While we agree with the well-known Adam-Gibbs proposition that the slowing down is due to an entropic squeeze (a dramatic decrease in the number of available configurations), we do not agree with the Adam-Gibbs (or the Volger-Fulcher) formulas that predict an infinite relaxation time at a finite temperature. Rather, we propose that generically there should be no singular crisis at any finite temperature: the relaxation time and the associated correlation length (average cluster size) increase at most superexponentially when the temperature is lowered.

Mechanical properties of glass forming systems

We address the interesting temperature range of a glass forming system where the mechanical properties are intermediate between those of a liquid and a solid. We employ an efficient Monte Carlo method to calculate the elastic moduli, and show that in this range of temperatures the moduli are finite for short times and vanish for long times, where short and long depend on the temperature. By invoking some exact results from statistical mechanics we offer an alternative method to compute shear moduli using molecular dynamics simulations, and compare those to the Monte Carlo method. The final conclusion is that these systems are not "viscous fluids" in the usual sense, as their actual time-dependence concatenates solid-like materials with varying local shear moduli.

Constitutive model for plasticity in an amorphous polycarbonate

A constitutive model for describing the mechanical response of an amorphous glassy polycarbonate is proposed. The model is based on an isotropic elastic phase surrounded by an SO(3) continuum of plastic phases onto which the elastic phase can collapse under strain. An approximate relaxed energy is developed for this model on the basis of physical considerations and extensive numerical testing, and it is shown that it corresponds to an ideal elastic-plastic behavior. Kinetic effects are introduced as rate-independent viscoplasticity, and a comparison with experimental data is presented, showing that the proposed model is able to capture the main features of the plastic behavior of amophous glassy polycarbonate.

Statistical mechanics of the glass transition in one-component liquids with an anisotropic potential

We study a recently introduced model of one-component glass-forming liquids whose constituents interact with an anisotropic potential. This system is interesting per se and as a model of liquids such as glycerol (interacting via hydrogen bonds) which are excellent glass formers. We work out the statistical mechanics of this system, encoding the liquid and glass disorder using appropriate quasiparticles (36 of them). The theory provides a full explanation of the glass transition phenomenology, including the identification of a diverging length scale and a relation between the structural changes and the diverging relaxation times.

Plastic response of a two-dimensional amorphous solid to quasistatic shear: transverse particle diffusion and phenomenology of dissipative events

We perform extensive simulations of a two-dimensional Lennard-Jones glass subjected to quasistatic shear deformation at T=0. We analyze the distribution of nonaffine displacements in terms of contributions of plastic, irreversible events, and elastic, reversible motions. From this, we extract information about correlations between plastic events and about the elastic nonaffine noise. Moreover, we find that nonaffine motion is essentially diffusive, with a clearly size-dependent diffusion constant. These results, supplemented by close inspection of the evolving patterns of the nonaffine tangent displacement field, lead us to propose a phenomenology of plasticity in such amorphous media. It can be schematized in terms of elastic loading and irreversible flips of small, randomly located shear transformation zones, elastically coupled via their quadrupolar fields.

Statistical mechanics of the glass transition as revealed by a Voronoi tesselation

The statistical mechanics of simple glass forming systems in two dimensions is worked out. The glass disorder is encoded via a Voronoi tesselation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, T(g) and T(c) , the first associated with jamming and the second associated with crystallization at very low temperatures.

Athermal shear-transformation-zone theory of amorphous plastic deformation. I. Basic principles

We develop an athermal version of the shear-transformation-zone (STZ) theory of amorphous plasticity in materials where thermal activation of irreversible molecular rearrangements is negligible or nonexistent. In many respects, this theory has broader applicability and yet is simpler than its thermal predecessors. For example, it needs no special effort to assure consistency with the laws of thermodynamics, and the interpretation of yielding as an exchange of dynamic stability between jammed and flowing states is clearer than before. The athermal theory presented here incorporates an explicit distribution of STZ transition thresholds. Although this theory contains no conventional thermal fluctuations, the concept of an effective temperature is essential for understanding how the STZ density is related to the state of disorder of the system.

Voronoi neighbor statistics of homogeneously sheared inelastic hard disks and hard spheres

In this work we extend our earlier Voronoi neighbor analysis [J. Chem. Phys. 123, 074502 (2005)] to homogeneously sheared inelastic hard-particle structures, the simplest model for rapid granular matter. The pair distribution function is partitioned into the nth neighbor coordination number (Cn), and the nth neighbor position distribution [fn(r)]. The distribution of the number of Voronoi faces (Pn) is also considered since C1 is its mean. We report the Cn and Pn for the homogeneously sheared inelastic hard-disk and hard-sphere structures. These statistics are sensitive to shear ordering transition, the nonequilibrium analogue of the freezing transition. In the near-elastic limit, the sheared fluid statistics approach that of the thermodynamic fluid. On shear ordering, due to the onset of order, the Cn for sheared structures approach that of the thermodynamic solid phase. The suppression of nucleation by the homogeneous shear is evident in these statistics. As inelasticity increases, the shear ordering packing fraction increases. Due to the topological instability of the isotropically perturbed face-centered cubic lattice, polyhedra with faces 12 to 18, with a mean at 14, coexist even in the regular close packed limit for the thermodynamic hard-sphere solid. In shear ordered inelastic hard-sphere structures there is a high incidence of 14-faceted polyhedra and a consequent depletion of polyhedra with faces 12, 13, 15-18, due to the formation of body-centered-tetragonal (bct) structures. These bct structures leave a body-centered-cubic-like signature in the Cn and Pn data. On shear ordering, close-packed layers slide past each other. However, with a velocity-dependent coefficient of restitution, at a critical shear rate these layers get disordered or amorphized. We find that the critical shear rate for amorphization is inversely proportional to the particle diameter, as compared to the inverse square scaling observed in dense colloidal suspensions.

Numerical tests of constitutive laws for dense granular flows

We numerically and theoretically study the macroscopic properties of dense, sheared granular materials. In this process we first consider an invariance in Newton's equations, explain how it leads to Bagnold's scaling, and discuss how it relates to the dynamics of granular temperature. Next we implement numerical simulations of granular materials in two different geometries--simple shear and flow down an incline--and show that measurements can be extrapolated from one geometry to the other. Then we observe nonaffine rearrangements of clusters of grains in response to shear strain and show that fundamental observations, which served as a basis for the shear transformation zone (STZ) theory of amorphous solids [M. L. Falk and J. S. Langer, Phys. Rev. E. 57, 7192 (1998); M.R.S. Bull 25, 40 (2000)], can be reproduced in granular materials. Finally we present constitutive equations for granular materials as proposed by Lemaître [Phys. Rev. Lett. 89, 064303 (2002)], based on the dynamics of granular temperature and STZ theory, and show that they match remarkably well with our numerical data from both geometries.

Dynamics of shear-transformation zones in amorphous plasticity: nonlinear theory at low temperatures

We use considerations of energy balance and dissipation to derive a self-consistent version of the shear-transformation-zone (STZ) theory of plastic deformation in amorphous solids. The theory is generalized to include arbitrary spatial orientations of STZs. Continuum equations for elasto-plastic material and their energy balance properties are discussed.

High-density liquidlike component facilitates plastic flow in a model amorphous silicon system

Molecular dynamics simulations show that plastic deformation of amorphous Si modeled by the Stillinger-Weber potential is very sensitive to the density of the initial unstressed state. Low-density systems exhibit a pronounced yield phenomenon, strain softening, and a dramatic drop in pressure during deformation at a constant volume. This behavior is explained by the presence in every system of a certain concentration of solidlike and liquidlike components, the latter being denser and more amenable to plastic flow.

Thermal effects in the shear-transformation-zone theory of amorphous plasticity: comparisons to metallic glass data

We extend our earlier shear-transformation-zone theory of amorphous plasticity to include the effects of thermally assisted molecular rearrangements. This version of our theory is a substantial revision and generalization of conventional theories of flow in noncrystalline solids. As in our earlier work, it predicts a dynamic transition between jammed and flowing states at a yield stress. Below that yield stress, it now describes thermally assisted creep. We show that this theory accounts for the experimentally observed strain-rate dependence of the viscosity of metallic glasses, and that it also captures many of the details of the transient stress-strain behavior of those materials during loading. In particular, it explains the apparent onset of superplasticity at sufficiently high stress as a transition between creep at low stresses and plastic flow near the yield stress. We also argue that there are internal inconsistencies in the conventional theories of these deformation processes, and suggest ways in which further experimentation as well as theoretical analysis may lead to better understanding of a broad range of nonequilibrium phenomena.

Fracture behavior of Lennard-Jones glasses

The fracture behavior of binary Lennard-Jones (LJ) glasses is studied by extensive molecular dynamics simulations. These LJ glasses represent a nonbond limit of polymer network glasses. We determine that the low strain behavior of the LJ and polymer glasses is similar. Two different LJ glasses are fractured under tensile strain without any preexisting crack. Void formation and resulting growth as strain increases is the mechanism through which the system fails. Void formation initiates at the yield strain of varepsilon (y) approximately 0.09, which is approximately the same strain at which the yielding behavior is first observed in cross-linked network models of polymer adhesives. The yield stress increased only by small amounts with increased strain rate and with increased system size (from N=30 000 atoms to 120 000 atoms). Within the ranges tested, the stress-strain behavior of these systems is independent of the temperature drop during quench and the initial molecular configuration.

Crystal phases of a glass-forming Lennard-Jones mixture

We compare the potential energy at zero temperature of a range of crystal structures for a glass-forming binary mixture of Lennard-Jones particles. The lowest-energy ordered state consists of coexisting phases of a single component face centered cubic structure and an equimolar cesium chloride structure. An infinite number of layered crystal structures are identified with energies close to this ground state. We demonstrate that the finite size increase of the energy of the coexisting crystal with incoherent interfaces is sufficient to destabilize this ordered phase in simulations of typical size. Two specific local coordination structures are identified as of possible structural significance in the amorphous state. We observe rapid crystal growth in the equimolar mixture.

Stability and structure of a supercooled liquid mixture in two dimensions

The structural and thermodynamic properties of a two-dimensional binary mixture of soft discs are reported over a range of temperatures down to large supercoolings using constant NPT molecular dynamics simulations. It is shown that the four orders of magnitude increase in the structural relaxation time is not accompanied by any significant increase in translational or orientational order. The phase diagram, calculated in the temperature/composition plane using thermodynamic integration, exhibits a deep eutectic point that is responsible for stabilizing the amorphous state. Voronoi analysis of the low-temperature ground state reveals a structure characterized by a network of linear arrays of fivefold and sevenfold sites. The heat capacity C(P) exhibits an asymmetric peak with a maximum at T(*)=0.55. It is argued that the initial rapid drop in C(P) for T*<0.55 is an equilibrium result and, hence, the peak in the heat capacity corresponds to the existence of an "enthalpy gap" with a characteristic temperature of T* approximately 0.35. This gap results from a minimum volume change associated with an anharmonic fluctuation.

Hysteresis and avalanches in two-dimensional foam rheology simulations

Foams have unique rheological properties that range from solidlike to fluidlike. We study two-dimensional noncoarsening foams of different disorder under shear in a Monte Carlo simulation, using a driven large-Q Potts model. Simulations of periodic shear on an ordered foam show several different response regimes. At small strain amplitudes, bubbles deform and recover their shapes elastically, and the macroscopic response is that of a linear elastic cellular material. For increasing strain amplitude, the energy-strain curve starts to exhibit hysteresis before any topological rearrangements occur, indicating a macroscopic viscoelastic response. When the applied strain amplitude exceeds a critical value, the yield strain, topological rearrangements occur, the foam starts to flow, and we observe macroscopic irreversibility. We find that the dynamics of topological rearrangements depend sensitively on the structural disorder. Structural disorder decreases the yield strain; sufficiently high disorder changes the macroscopic response of a foam from a viscoelastic solid to a viscoelastic fluid. This wide-ranging dynamical response and the associated history effects of foams result from avalanchelike rearrangement events. The spatiotemporal statistics of rearrangement events do not display long-range correlations for ordered foams or at low shear rates, consistent with experimental observations. As the shear rate or structural disorder increases, the topological events become more correlated and their power spectra change from that of white noise toward 1/f noise. Intriguingly, the power spectra of the total stored energy also exhibit this 1/f trend.

Simulation of plastic deformation in a two-dimensional atomic glass by molecular dynamics IV

Plastic deformation in a structurally well-relaxed two-dimensional atomic glass was simulated by a computer molecular dynamics approach. The simulation, which was carried through yielding and to substantial plastic strains, demonstrated that the principal mechanism of plastic strain production is by local partly dilatant shear transformations nucleated preferentially in the boundaries of liquid-like material separating the small quasi-ordered domains that form when the glass is well relaxed. Under imposed forward shear-strain increments, local shear transformations in atomic clusters were found to be mostly in the same direction as the applied stress. There were, however, substantial levels of shear transformations in other random directions, including many opposed to the applied stress. In all instances, however, nucleaton of shear transformations reduced the Gibbs free energy monotonically, which is governed largely by the locked-in excess enthalpies of the glassy state. At shear strains above 15%, localization of shear into bands was observed to begin. This steadily intensified and formed well-defined sharp shear bands into which all the shear strain became concentrated by the end of the simulation at a strain of 27%. A strong correlation was found between the tendency for shear localization and retained shear-induced dilatation.

Topological features of structural relaxations in a two-dimensional model atomic glass II

The topological features of atom motions in a high-temperature melt, a sub-cooled melt above T g , and a glass below T g , were analysed in detail by means of a two-dimensional molecular dynamics simulation. A striking analogy was observed between the structure and properties of the liquid-like material separating quasi-ordered domains of atom clusters, and high-angle grain boundaries. The main feature of the structural relaxation below the melting point, both above and below T g was the gradual dissolution and disappearance of the liquid-like material, permitting increasing order in the previously quasi-ordered domains and a growth in their sizes. In these processes, many sequences reminiscent of cancellation of dislocation pairs, or mutual reactions to give more stable sets, were observed.

Kinetics of structural relaxations in a two-dimensional model atomic glass III

The kinetics of structural relaxation in a two-dimensional model atomic glass quenched infinitely rapidly from the melt to 0.55 of the glass transition tem perature was simulated by the molecular dynamics methods to study the chronological ordering of the atomic kinematics associated with such relaxations. Over the very short periods of ageing ( ca .200 atomic fluctuations) accessible to the molecular dynamics (MD) method, a Williams-Watts form of relaxation with a fractional exponent of 0.5 was found to hold for excess enthalpy, free volume and site distortion parameter. The distribution of free energy barriers associated with the relaxation that resulted from the analysis could be scaled up to describe processes occurring on macroscopic timescales, and agrees well with experimental results in Cu x Zr 1-x glasses. Results on the clustering of relaxations and other topological features of the relaxation process are also reported.