Strain Pattern in Supercooled Liquids

Solid-that-Flows Picture of Glass-Forming Liquids

This perspective article reviews arguments that glass-forming liquids are different from those of standard liquid-state theory, which typically have a viscosity in the mPa·s range and relaxation times on the order of picoseconds. These numbers grow dramatically and become 1012 - 1015 times larger for liquids cooled toward the glass transition. This translates into a qualitative difference, and below the "solidity length" which is roughly one micron at the glass transition, a glass-forming liquid behaves much like a solid. Recent numerical evidence for the solidity of ultraviscous liquids is reviewed, and experimental consequences are discussed in relation to dynamic heterogeneity, frequency-dependent linear-response functions, and the temperature dependence of the average relaxation time.

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

Molecular simulations and hydrodynamic theory of nonlocal shear stresscorrelations in supercooled fluids

A supercooled fluid close to the glass transition develops nonlocal shear stress correlations that anticipate the emergence of elasticity. We performed molecular dynamics simulations of a binary Lennard-Jones mixture at different temperatures and investigated the spatiotemporal autocorrelation function of the shear stressfor different wavevectors, q, from a locally measured and Fourier-transformed stress tensor. Anisotropic correlations are observed at non-zero wavevectors, exhibiting strongly damped oscillations with a characteristic frequency ω(q). A comparison with a recently developed hydrodynamic theory [Maier et al., Phys. Rev. Lett. 119, 265701 (2017)] shows a remarkably good quantitative agreement between the particle-based simulations and the theoretical predictions.

Theory of length-scale dependent relaxation moduli and stress fluctuations in glass-forming and viscoelastic liquids

The spatiotemporal correlations of the local stress tensor in supercooled liquids are studied both theoretically and by molecular dynamics simulations of a two-dimensional (2D) polydisperse Lennard-Jones system. Asymptotically exact theoretical equations defining the dynamical structure factor and all components of the stress correlation tensor for low wave-vector q are presented in terms of the generalized (q-dependent) shear and longitudinal relaxation moduli, G(q, t) and K(q, t). We developed a rigorous approach (valid for low q) to calculate K(q, t) in terms of certain bulk correlation functions (for q = 0), the static structure factor S(q), and thermal conductivity κ. The proposed approach takes into account both the thermostatting effect and the effect of polydispersity. The theoretical results for the (q, t)-dependent stress correlation functions are compared with our simulation data, and an excellent agreement is found for qb̄≲0.5 (with b̄ being the mean particle diameter) both above and below the glass transition without any fitting parameters. Our data are consistent with recently predicted (both theoretically and by simulations) long-range correlations of the shear stress quenched in heterogeneous glassy structures.

Transverse Viscous Transport in Classical Solid States

The transverse velocity time correlation function C[over ˜]_{T}(k,ω) with k and ω being the wave number and the frequency, respectively, is a fundamental quantity in determining the transverse mechanical and transport properties of materials. In ordinary liquids, a nonzero value of C[over ˜]_{T}(k,0) is inevitably linked to viscous material flows. Even in solids where significant material flows are precluded due to almost frozen positional degrees of freedom, our molecular dynamics simulations reveal that C[over ˜]_{T}(k,0) takes a nonzero value, whereby the time integration of the velocity field shows definite diffusive behavior with diffusivity C[over ˜]_{T}(k,0)/3. This behavior is attributed to viscous transport accompanying a small random convection of the velocity field (the inertia effect), and the resultant viscosity is measurable in the Eulerian description: the constituent particles that substantially carry momenta fluctuate slightly around their reference positions. In the Eulerian description, the velocity field is explicitly associated with such fluctuating instantaneous particle positions, whereas in the Lagrangian description, this is not the case. The present study poses a fundamental problem for continuum mechanics: reconciling liquid and solid descriptions in the limit of the infinite structural relaxation time.

Stress hyperuniformity and transient oscillatory-exponential correlation decay as signatures of strength vs fragility in glasses

We examine and compare the local stress autocorrelation in the inherent states of a fragile and a strong glass: the Kob-Andersen (KA) binary mixture and the Beest-Kramer-Santen model of silica. For both systems, local (domain-averaged) stress fluctuations asymptotically reach the normal inverse-volume decay in the large domain limit; accordingly, the real-space stress autocorrelation presents long-range power law tails. However, in the case of silica, local stress fluctuations display a high degree of hyperuniformity, i.e., their asymptotic (normal) decay is disproportionately smaller than their bond level amplitude. This property causes the asymptotic power law tails of the real-space stress autocorrelation to be swamped, up to very large distances (several nanometers), by an intermediate oscillatory-exponential decay regime. Similar contributions exist in the KA stress autocorrelation, but they never can be considered as dominating the power law decay and fully disappear when stress is coarse-grained beyond one interatomic distance. Our observations document that the relevance of power-law stress correlation may constitute a key discriminating feature between strong and fragile glasses. Meanwhile, they highlight that the notion of local stress in atomistic systems involves by necessity a choice of observation (coarse-graining) scale, the relevant value of which depends, in principle, on both the model and the phenomenon studied.

Relaxation moduli of glass-forming systems: temperature effects and fluctuations

Equilibrium and dynamical properties of a two-dimensional polydisperse colloidal model system are characterized by means of molecular dynamics (MD) and Monte Carlo (MC) simulations. We employed several methods to prepare quasi-equilibrated systems: in particular, by slow cooling and tempering with MD (method SC-MD), and by tempering with MC dynamics involving swaps of particle diameters (methods Sw-MD, Sw-MC). It is revealed that the Sw-methods are much more efficient for equilibration below the glass transition temperature T_g leading to denser and more rigid systems which show much slower self-diffusion and shear-stress relaxation than their counterparts prepared with the SC-MD method. The shear-stress relaxation modulus G(t) is obtained based on the classical stress-fluctuation relation. We demonstrate that the α-relaxation time τ_α obtained using a time-temperature superposition of G(t) shows a super-Arrhenius behavior with the VFT temperature T₀ well below T_g. We also derive novel rigorous fluctuation relations providing isothermic and adiabatic compression relaxation moduli in the whole time range (including the short-time inertial regime) based on correlation data for thermostatted systems. It is also shown that: (i) the assumption of Gaussian statistics for stress fluctuations leads to accurate predictions of the variances of the fluctuation moduli for both shear (μ_F) and compression (η_F) at T⪆T_g. (ii) The long-time (quasi-static) isothermic and adiabatic moduli increase on cooling faster than the affine compression modulus η_A, and this leads to a monotonic temperature dependence of η_F which is qualitatively different from μ_F(T) showing a maximum near T_g.

Elastoplasticity Mediates Dynamical Heterogeneity Below the Mode Coupling Temperature

As liquids approach the glass transition temperature, dynamical heterogeneity emerges as a crucial universal feature of their behavior. Dynamic facilitation, where local motion triggers further motion nearby, plays a major role in this phenomenon. Here we show that long-ranged, elastically mediated facilitation appears below the mode coupling temperature, adding to the short-range component present at all temperatures. Our results suggest deep connections between the supercooled liquid and glass states, and pave the way for a deeper understanding of dynamical heterogeneity in glassy systems.

Stress correlation function and linear response of Brownian particles

We determine the non-local stress autocorrelation tensor in an homogeneous and isotropic system of interacting Brownian particles starting from the Smoluchowski equation of the configurational probability density. In order to relate stresses to particle displacements as appropriate in viscoelastic states, we go beyond the usual hydrodynamic description obtained in the Zwanzig-Mori projection-operator formalism by introducing the proper irreducible dynamics following Cichocki and Hess, and Kawasaki. Differently from these authors, we include transverse contributions as well. This recovers the expression for the stress autocorrelation including the elastic terms in solid states as found for Newtonian and Langevin systems, in case that those are evaluated in the overdamped limit. Finally, we argue that the found memory function reduces to the shear and bulk viscosity in the hydrodynamic limit of smooth and slow fluctuations and derive the corresponding hydrodynamic equations.

Emergent solidity of amorphous materials as a consequence of mechanical self-organisation

Amorphous solids have peculiar properties distinct from crystals. One of the most fundamental mysteries is the emergence of solidity in such nonequilibrium, disordered state without the protection by long-range translational order. A jammed system at zero temperature, although marginally stable, has solidity stemming from the space-spanning force network, which gives rise to the long-range stress correlation. Here, we show that such nonlocal correlation already appears at the nonequilibrium glass transition upon cooling. This is surprising since we also find that the system suffers from giant anharmonic fluctuations originated from the fractal-like potential energy landscape. We reveal that it is the percolation of the force-bearing network that allows long-range stress transmission even under such circumstance. Thus, the emergent solidity of amorphous materials is a consequence of nontrivial self-organisation of the disordered mechanical architecture. Our findings point to the significance of understanding amorphous solids and nonequilibrium glass transition from a mechanical perspective.

Connecting shear localization with the long-range correlated polarized stress fields in granular materials

One long-lasting puzzle in amorphous solids is shear localization, where local plastic deformation involves cooperative particle rearrangements in small regions of a few inter-particle distances, self-organizing into shear bands and eventually leading to the material failure. Understanding the connection between the structure and dynamics of amorphous solids is essential in physics, material sciences, geotechnical and civil engineering, and geophysics. Here we show a deep connection between shear localization and the intrinsic structures of internal stresses in an isotropically jammed granular material subject to shear. Specifically, we find strong (anti)correlations between the micro shear bands and two polarized stress fields along two directions of maximal shear. By exploring the tensorial characteristics and the rotational symmetry of force network, we reveal that such profound connection is a result of symmetry breaking by shear. Finally, we provide the solid experimental evidence of long-range correlated inherent shear stress in an isotropically jammed granular system.

Long-time self-diffusion in quasi-two-dimensional colloidal fluids of paramagnetic particles

The effect of hydrodynamic interactions (HI) on the long-time self-diffusion in quasi-two-dimensional fluids of paramagnetic colloidal particles is investigated using a combination of experiments and Brownian dynamics (BD) simulations. In the BD simulations, the direct interactions (DI) between the particles consist of a short-ranged repulsive part and a long-ranged part that is proportional to 1/r^{3}, with r the interparticle distance. By studying the equation of state, the simulations allow for the identification of the regime where the properties of the fluid are fully controlled by the long-ranged interactions, and the thermodynamic state solely depends on the dimensionless interaction strength Γ. In this regime, the radial distribution functions from the simulations are in quantitative agreement with those from the experiments for different fluid area fractions. This agreement confirms that the DI in the experiments and simulations are identical, which thus allows us to isolate the role of HI, as these are not taken into account in the BD simulations. Experiment and simulation fall onto a master curve with respect to the Γ dependence of D_{L}^{★}=D_{L}/(D_{0}Γ^{1/2}), with D_{0} the self-diffusion coefficient at infinite dilution and D_{L} the long-time self-diffusion coefficient. Our results thus show that, although HI affect the short-time self-diffusion, for a quasi-two-dimensional system with 1/r^{3} long-ranged DI, the reduced quantity D_{L}^{★} is effectively not affected by HI. Interestingly, this is in agreement with prior work on quasi-two-dimensional colloidal hard spheres.

Probing the Degree of Heterogeneity within a Shear Band of a Model Glass

Recent experiments provide evidence for density variations along shear bands in metallic glasses with a length scale of a few hundred nanometers. Via molecular dynamics simulations of a generic binary glass model, here we show that this is strongly correlated with variations of composition, coordination number, viscosity, and heat generation. Individual shear events along the shear band path show a mean distance of a few nanometers, comparable to recent experimental findings on medium range order. The aforementioned variations result from these localized perturbations, mediated by elasticity.

Local versus Global Stretched Mechanical Response in a Supercooled Liquid near the Glass Transition

Amorphous materials have a rich relaxation spectrum, which is usually described in terms of a hierarchy of relaxation mechanisms. In this work, we investigate the local dynamic modulus spectra in a model glass just above the glass transition temperature by performing a mechanical spectroscopy analysis with molecular dynamics simulations. We find that the spectra, at the local as well as on the global scale, can be well described by the Cole-Davidson formula in the frequency range explored with simulations. Surprisingly, the Cole-Davidson stretching exponent does not change with the size of the local region that is probed. The local relaxation time displays a broad distribution, as expected based on dynamic heterogeneity concepts, but the stretching is obtained independently of this distribution. We find that the size dependence of the local relaxation time and moduli can be well explained by the elastic shoving model.

Field Theory for Amorphous Solids

Glasses at low temperature fluctuate around their inherent states; glassy anomalies reflect the structure of these states. Recently, there have been numerous observations of long-range stress correlations in glassy materials, from supercooled liquids to colloids and granular materials, but without a common explanation. Herein it is shown, using a field theory of inherent states, that long-range stress correlations follow from mechanical equilibrium alone, with explicit predictions for stress correlations in two and three dimensions. "Equations of state" relating fluctuations to imposed stresses are derived, as well as field equations that fix the spatial structure of stresses in arbitrary geometries. Finally, a new holographic quantity in 3D amorphous systems is identified.

Long-range stress correlations in viscoelastic and glass-forming fluids

A simple and rigorous approach to obtain stress correlations in viscoelastic liquids (including supercooled liquid and equilibrium amorphous systems) is proposed. The long-range dynamical correlations of local shear stress are calculated and analyzed in 2-dimensional space. It is established how the long-range character of the stress correlations gradually emerges as the relevant dynamical correlation length l grows in time. The correlation range l is defined by momentum propagation due to acoustic waves and vorticity diffusion which are the basic mechanisms for transmission of shear stress perturbations. We obtain the general expression defining the time- and distance-dependent stress correlation tensor in terms of material functions (generalized relaxation moduli). The effect of liquid compressibility is quantitatively analyzed; it is shown to be important at large distances and/or short times. The revealed long-range stress correlation effect is shown to be dynamical in nature and unconnected with static structural correlations in liquids (correlation length ξ_s). Our approach is based on the assumption that ξ_s is small enough as reflected in weak wave-number dependencies of the generalized relaxation moduli. We provide a simple physical picture connecting the elucidated long-range fluctuation effect with anisotropic correlations of the (transient) inherent stress field, and discuss its implications.

Elastic moduli of a Brownian colloidal glass former

The static, dynamic and flow-dependent shear moduli of a binary mixture of Brownian hard disks are studied by an event-driven molecular dynamics simulation. Thereby, the emergence of rigidity close to the glass transition encoded in the static shear modulus [Formula: see text] is accessed by three methods. Results from shear stress auto-correlation functions, elastic dispersion relations, and the elastic response to strain deformations upon the start-up of shear flow are compared. This enables one to sample the time-dependent shear modulus [Formula: see text] consistently over several decades in time. By that a very precise specification of the glass transition point and of [Formula: see text] is feasible. Predictions by mode coupling theory of a finite shear modulus at the glass transition, of α-scaling in fluid states close to the transition, and of shear induced decay in yielding glass states are tested and broadly verified.

Inherent stress correlations in a quiescent two-dimensional liquid: Static analysis including finite-size effects

After constructing a formalism to analyze spatial stress correlations in two-dimensional equilibrated liquids, we show that the sole conjunction of mechanical balance and material isotropy demands all anisotropic components of the inherent state (IS) stress autocorrelation matrix to decay at long range as 1/r^{2} in the large system size limit. Furthermore, analyzing numerical simulation data for an equilibrated supercooled liquid, we bring evidence that, in finite-sized periodic systems, the autocorrelations of pressure and shear stresses present uniform backgrounds of amplitudes proportional to the inverse cell area. These backgrounds bring relevant contributions to macroscopic IS stress fluctuations, with the consequence that the latter scale as inverse area, yet in an anomalous way, inconsistent with viewing an IS as equivalent, in the thermodynamic limit, to an ensemble of independent finite-sized subsystems. In that sense, ISs are not spatially ergodic.

Transient inhomogeneous flow patterns in supercooled liquids under shear

Supercooled liquids and other soft glassy systems show characteristic spatial inhomogeneities in their local dynamical properties. Using detailed molecular dynamics simulations, we find that for sufficiently low temperatures and sufficiently high shear rates supercooled liquids also show transient inhomogeneous flow patterns (shear banding) in the start-up of steady shear flow, similar to what has already been observed for many other soft glassy systems. We verify that the onset of transient shear banding coincides quite well with the appearance of a stress overshoot for temperatures in the supercooled regime. We find that the slower bands adapt less well to the imposed deformation and therefore accumulate higher shear stresses compared to the fast bands at comparable local shear rates. Our results also indicate that the shear rates of the fast and slow bands are adjusted such that the local dissipation rate is approximately the same in both bands.