Rigidity in Condensed Matter and Its Origin in Configurational Constraint

A dynamical system approach to relaxation in glass-forming liquids

The "classical" thermodynamic and statistical mechanical theories of Gibbs and Boltzmann are both predicated on axiomatic assumptions whose applicability is hard to ascertain. Theoretical objections and an increasing number of observed deviations from these theories have led to sustained efforts to develop an improved mathematical and physical foundation for them, and the search for appropriate extensions that are generally applicable to condensed materials at low temperatures (T) and high material densities where the assumptions of these theories start to become particularly questionable. These theoretical efforts have largely focused on minimal models of condensed material systems, such as the Fermi-Ulam-Pasta-Tsingou model, and other simplified models of condensed materials that are amenable to numerical and analytic treatments and that can serve to illuminate essential features of relaxation processes in condensed materials under conditions approaching integrable dynamics where clear departures from classical thermodynamics and dynamics can be generally expected. These studies indicate an apparently general multi-step relaxation process, corresponding to an initial "fast" relaxation process (termed the fast β-relaxation in the context of cooled liquids), followed by a longer "equipartition time", namely, the α-relaxation time τα in the context of cooled liquids. This relaxation timescale can be enormously longer than the fast β-relaxation time τβ so that τα is the primary parameter governing the rate at which the material comes into equilibrium, and thus is a natural focus of theoretical attention. Since the dynamics of these simplified dynamical systems, originally intended as simplified models of real crystalline materials exhibiting anharmonic interactions, greatly resemble the observed relaxation dynamics of both heated crystals and cooled liquids, we adapt this dynamical system approach to the practical matter of estimating relaxation times in both cooled liquids and crystals at elevated temperatures, which we identify as weakly non-integrable dynamical systems.

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.

Direct measurement of the structural change associated with amorphous solidification using static scattering of coherent radiation

In this paper, we demonstrate that the weak temperature dependence of the structure factor of supercooled liquids, a defining feature of the glass transition, is a consequence of the averaging of the scattering intensity due to angular averaging. We show that the speckle at individual wavevectors, calculated from a simulated glass former, exhibits a Debye-Waller factor with a sufficiently large temperature dependence to represent a structural order parameter capable of distinguishing liquid from glass. We also extract from the speckle intensities a quantity proportional to the variance of the local restraint, i.e., a direct experimental measure of the amplitude of structural heterogeneity.

Activation free energy gradient controls interfacial mobility gradient in thin polymer films

We examine the mobility gradient in the interfacial region of substrate-supported polymer films using molecular dynamics simulations and interpret these gradients within the string model of glass-formation. No large gradients in the extent of collective motion exist in these simulated films, and an analysis of the mobility gradient on a layer-by-layer basis indicates that the string model provides a quantitative description of the relaxation time gradient. Consequently, the string model indicates that the interfacial mobility gradient derives mainly from a gradient in the high-temperature activation enthalpy ΔH₀ and entropy ΔS₀ as a function of depth z, an effect that exists even in the high-temperature Arrhenius relaxation regime far above the glass transition temperature. To gain insight into the interfacial mobility gradient, we examined various material properties suggested previously to influence ΔH₀ in condensed materials, including density, potential and cohesive energy density, and a local measure of stiffness or u²(z)^-3/2, where u²(z) is the average mean squared particle displacement at a caging time (on the order of a ps). We find that changes in local stiffness best correlate with changes in ΔH₀(z) and that ΔS₀(z) also contributes significantly to the interfacial mobility gradient, so it must not be neglected.

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.

Shear Softening in a Metallic Glass: First-Principles Local-Stress Analysis

Metallic glasses deform elastically under stress. However, the atomic-level origin of elastic properties of metallic glasses remain unclear. In this Letter using ab initio molecular dynamics simulations of the Cu_{50}Zr_{50} metallic glass under shear strain, we show that the heterogeneous stress relaxation results in the increased charge transfer from Zr to Cu atoms, enhancing the softening of the shear modulus. Changes in compositional short-range order and atomic position shifts due to the nonaffine deformation are discussed. It is shown that the Zr subsystem exhibits a stiff behavior, whereas the displacements of Cu atoms from their initial positions, induced by the strain, provide the stress drop and softening.

Theory of applying shear strains from boundary walls: Linear response in glasses

We construct a linear response theory of applying shear deformations from boundary walls in the film geometry in Kubo's theoretical scheme. Our method is applicable to any solids and fluids. For glasses, we assume quasi-equilibrium around a fixed inherent state. Then, we obtain linear-response expressions for any variables including the stress and the particle displacements, even though the glass interior is elastically inhomogeneous. In particular, the shear modulus can be expressed in terms of the correlations between the interior stress and the forces from the walls. It can also be expressed in terms of the inter-particle correlations, as has been shown in the previous literature. Our stress relaxation function includes the effect of the boundary walls and can be used for inhomogeneous flow response. We show the presence of long-ranged, long-lived correlations among the fluctuations of the forces from the walls and the displacements of all the particles in the cell. We confirm these theoretical results numerically in a two-dimensional model glass. As an application, we describe emission and propagation of transverse sounds after boundary wall motions using these time-correlation functions. We also find resonant sound amplification when the frequency of an oscillatory shear approaches that of the first transverse sound mode.

Assessing the utility of structure in amorphous materials

This paper presents a set of general strategies for the analysis of structure in amorphous materials and a general approach to assessing the utility of any selected structural description. Two measures of structure are defined, "diversity" and "utility," and applied to two model glass forming binary atomic alloys, Cu₅₀Zr₅₀ and a Lennard-Jones A₈₀B₂₀ mixture. We show that the change in diversity associated with selecting Voronoi structures with high localization or low energy, while real, is too weak to support claims that specific structures are the prime cause of these local physical properties. In addition, a new structure-free measure of incipient crystal-like organization in mixtures is introduced, suitable for cases where the stable crystal is a compound structure.

Precursors of fluidisation in the creep response of a soft glass

Via extensive numerical simulations, we study the fluidisation process of dense amorphous materials subjected to an external shear stress, using a three-dimensional colloidal glass model. In order to disentangle possible boundary effects from finite size effects in the process of fluidisation, we implement a novel geometry-constrained protocol with periodic boundary conditions. We show that this protocol is well controlled and that the longtime fluidisation dynamics is, to a great extent, independent of the details of the protocol parameters. Our protocol, therefore, provides an ideal tool to investigate the bulk dynamics prior to yielding and to study finite size effects regarding the fluidisation process. Our study reveals the existence of precursors to fluidisation observed as a peak in the strain-rate fluctuations, that allows for a robust definition of a fluidisation time. Although the exponents in the power-law creep dynamics seem not to depend significantly on the system size, we reveal strong finite size effects for the onset of fluidisation.

Structure-property relationships for wet dentin adhesive polymers

Dentin adhesive systems for composite tooth restorations are composed of hydrophilic/hydrophobic monomers, solvents, and photoinitiators. The adhesives undergo phase separation and concomitant compositional change during their application in the wet oral environment; phase separation compromises the quality of the hybrid layer in the adhesive/dentin interface. In this work, the adhesive composition in the hybrid layer can be represented using the phase boundaries of a ternary phase diagram for the hydrophobic monomer/hydrophilic monomer/water system. The polymer phases, previously unaccounted for, play an important role in determining the mechanical behavior of the bulk adhesive, and the chemomechanical properties of the phases are intimately related to the effects produced by differences in the hydrophobic-hydrophilic composition. As the composition of the polymer phases varies from hydrophobic-rich to hydrophilic-rich, the amount of the adsorbed water and the nature of polymer-water interaction vary nonlinearly and strongly correlate with the change in elastic moduli under wet conditions. The failure strain, loss modulus, and glass transition temperature vary nonmonotonically with composition and are explained based upon primary and secondary transitions observed in dynamic mechanical testing. Due to the variability in composition, the assignment of mechanical properties and the choice of suitable constitutive models for polymer phases in the hybrid layer are not straightforward. This work investigates the relationship between composition and chemomechanical properties of the polymer phases formed on the water-adhesive phase boundary using quasistatic and dynamic mechanical testing, mass transfer experiments, and vibrational spectroscopy.

Stress auto-correlation tensor in glass-forming isothermal fluids: From viscous to elastic response

We develop a generalized hydrodynamic theory, which can account for the build-up of long-ranged and long-lived shear stress correlations in supercooled liquids as the glass transition is approached. Our theory is based on the decomposition of tensorial stress relaxation into fast microscopic processes and slow dynamics due to conservation laws. In the fluid, anisotropic shear stress correlations arise from the tensorial nature of stress. By approximating the fast microscopic processes by a single relaxation time in the spirit of Maxwell, we find viscoelastic precursors of the Eshelby-type correlations familiar in an elastic medium. The spatial extent of shear stress fluctuations is characterized by a correlation length ξ which grows like the viscosity η or time scale τ ∼ η, whose divergence signals the glass transition. In the solid, the correlation length is infinite and stress correlations decay algebraically as r-d in d dimensions.

On the existence of thermodynamically stable rigid solids

Customarily, crystalline solids are defined to be rigid since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the metastable state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then used to obtain the equilibrium free energy as a function of this variable. Our results lead to a unique view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable when an external field, which penalizes particle rearrangements, is switched on. This produces a line of first-order phase transitions in the field-strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

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.

Emergence of Long-Ranged Stress Correlations at the Liquid to Glass Transition

A theory for the nonlocal shear stress correlations in supercooled liquids is derived from first principles. It captures the crossover from viscous to elastic dynamics at an idealized liquid to glass transition and explains the emergence of long-ranged stress correlations in glass, as expected from classical continuum elasticity. The long-ranged stress correlations can be traced to the coupling of shear stress to transverse momentum, which is ignored in the classic Maxwell model. To rescue this widely used model, we suggest a generalization in terms of a single relaxation time τ for the fast degrees of freedom only. This generalized Maxwell model implies a divergent correlation length ξ∝τ as well as dynamic critical scaling and correctly accounts for the far-field stress correlations. It can be rephrased in terms of generalized hydrodynamic equations, which naturally couple stress and momentum and furthermore allow us to connect to fluidity and elastoplastic models.

Thermodynamic scaling of relaxation: insights from anharmonic elasticity

Using molecular dynamics simulations of a molecular liquid, we investigate the thermodynamic scaling (TS) of the structural relaxation time [Formula: see text] in terms of the quantity [Formula: see text], where T and ρ are the temperature and density, respectively. The liquid does not exhibit strong virial-energy correlations. We propose a method for evaluating both the characteristic exponent [Formula: see text] and the TS master curve that uses experimentally accessible quantities that characterise the anharmonic elasticity and does not use details about the microscopic interactions. In particular, we express the TS characteristic exponent [Formula: see text] in terms of the lattice Grüneisen parameter [Formula: see text] and the isochoric anharmonicity [Formula: see text]. An analytic expression of the TS master curve of [Formula: see text] with [Formula: see text] as the key adjustable parameter is found. The comparison with the experimental TS master curves and the isochoric fragilities of 34 glassformers is satisfying. In a few cases, where thermodynamic data are available, we test (i) the predicted characteristic exponent [Formula: see text] and (ii) the isochoric anharmonicity [Formula: see text], as drawn by the best fit of the TS of the structural relaxation, against the available thermodynamic data. A linear relation between the isochoric fragility and the isochoric anharmonicity [Formula: see text] is found and compared favourably with the results of experiments with no adjustable parameters. A relation between the increase of the isochoric vibrational heat capacity due to anharmonicity and the isochoric fragility is derived.

Polymer glass transition occurs at the marginal rigidity point with connectivity z* = 4

We re-examine the physical origin of the polymer glass transition from the point of view of marginal rigidity, which is achieved at a certain average number of mechanically active intermolecular contacts per monomer. In the case of polymer chains in a melt/poor solvent, each monomer has two neighbors bound by covalent bonds and also a number of central-force contacts modelled by the Lennard-Jones (LJ) potential. We find that when the average number of contacts per monomer (covalent and non-covalent) exceeds the critical value z* ≈ 4, the system becomes solid and the dynamics arrested - a state that we declare the glass. Coarse-grained Brownian dynamics simulations show that at sufficient strength of LJ attraction (which effectively represents the depth of quenching, or the quality of solvent) the polymer globule indeed crosses the threshold of z*, and becomes a glass with a finite zero-frequency shear modulus, G∝ (z-z*). We verify this by showing the distinction between the 'liquid' polymer droplet at z < z*, which changes shape and adopts the spherical conformation in equilibrium, and the glassy 'solid' droplet at z > z*, which retains its shape frozen at the moment of z* crossover. These results provide a robust microscopic criterion to tell the liquid apart from the glass for the linear polymers.

Nonaffine displacements and the nonlinear response of a strained amorphous solid

We demonstrate that irreversible structural reorganization is not necessary for the observation of yield behavior in an amorphous solid. While the majority of solids strained to their yield point do indeed undergo an irreversible reorganization, we find that a significant fraction of solids exhibits yield via a reversible strain. We also demonstrate that large instantaneous strains in excess of the yield stress can result in complete stress relaxation, a result of the large nonaffine motions driven by the applied strain. The empirical similarity of the dependence of the ratio of stress over strain on the nonaffine mean-square displacement to that for the shear modulus obtained from quiescent liquid at nonzero temperature supports the proposition that rigidity depends on the size of the sampled configurational space only and is insensitive to how this space is sampled.