Glass-forming Fluids Research Articles

Long-range spatial correlations of particle displacements and the emergence of elasticity.

We examine correlations of transverse particle displacements and their relationship to the shear modulus of a glass and the viscosity of a fluid. To this end we use computer simulations to calculate a correlation function of the displacements, S_{4}(q;t), which is similar to functions used to study heterogeneous dynamics in glass-forming fluids. We show that in the glass the shear modulus can be obtained from the long-time, small-q limit of S_{4}(q;t). By using scaling arguments, we argue that a four-point correlation length ξ_{4}(t) grows linearly in time in a glass and grows as sqrt[t] at long times in a fluid, and we verify these results by analyzing S_{4}(q;t) obtained from simulations. For a viscoelastic fluid, the simulation results suggest that the crossover to the long-time sqrt[t] growth of ξ_{4}(t) occurs at a characteristic decay time of the shear stress autocorrelation function. Using this observation, we show that the amplitude of the long-time sqrt[t] growth is proportional to sqrt[η] where η is the viscosity of the fluid.

Communication: Towards first principles theory of relaxation in supercooled liquids formulated in terms of cooperative motion.

A general theory of the long time, low temperature dynamics of glass-forming fluids remains elusive despite the almost 20 years since the famous pronouncement by the Nobel Laureate P. W. Anderson, "The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition" [Science 267, 1615 (1995)]. While recent work indicates that Adam-Gibbs theory (AGT) provides a framework for computing the structural relaxation time of supercooled fluids and for analyzing the properties of the cooperatively rearranging dynamical strings observed in low temperature molecular dynamics simulations, the heuristic nature of AGT has impeded general acceptance due to the lack of a first principles derivation [G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965)]. This deficiency is rectified here by a statistical mechanical derivation of AGT that uses transition state theory and the assumption that the transition state is composed of elementary excitations of a string-like form. The strings are assumed to form in equilibrium with the mobile particles in the fluid. Hence, transition state theory requires the strings to be in mutual equilibrium and thus to have the size distribution of a self-assembling system, in accord with the simulations and analyses of Douglas and co-workers. The average relaxation rate is computed as a grand canonical ensemble average over all string sizes, and use of the previously determined relation between configurational entropy and the average cluster size in several model equilibrium self-associating systems produces the AGT expression in a manner enabling further extensions and more fundamental tests of the assumptions.

Accumulating evidence for non-diverging time-scales in glass-forming fluids

One perceived important signature of the “ideal” glass transition and of the complex fluid nature of glass-forming liquids remains the apparent divergence of the dynamics at temperatures above zero Kelvin. Recently, however, this perception has been increasingly challenged both through experiment, large scale data analysis, and in some new theories of the dynamics of glass forming systems. Here we summarize early evidence suggesting that time scales actually do not diverge in glasses that are aged into equilibrium, perhaps 15°C below the conventional glass transition temperature Tg. We then show novel results from an extremely densified glass – 20 million year old Dominican amber – in which we were able to obtain the upper bounds to the relaxation times at temperatures below the glass transition temperature through a step-wise temperature scan in which the stress relaxation response of the amber was measured both below and above the fictive temperature Tf. We find that in the case of the upper bound responses at T>Tf, there is a strong deviation of the response from the Super-Arrhenius Vogel–Fulcher-Tammann (VFT) extrapolation and this persists to the fictive temperature which is some 43.6°C below Tg. Evidence is also presented suggesting that different dynamic properties may deviate from the VFT extrapolation differently and challenges for future measurement are presented.

Channel flow of a tensorial shear-thinning Maxwell model: Lattice Boltzmann simulations.

We discuss pressure-driven channel flow for a model of shear-thinning glass-forming fluids, employing a modified lattice-Boltzmann (LB) simulation scheme. The model is motivated by a recent microscopic approach to the nonlinear rheology of colloidal suspensions and captures a nonvanishing dynamical yield stress and the appearance of normal-stress differences and a flow-induced pressure contribution. The standard LB algorithm is extended to deal with tensorial, nonlinear constitutive equations of this class. The new LB scheme is tested in 2D pressure-driven channel flow and reproduces the analytical steady-state solution. The transient dynamics after startup and removal of the pressure gradient reproduce a finite stopping time for the cessation flow of yield-stress fluids in agreement with previous analytical estimates.

Dynamical clustering and a mechanism for raft-like structures in a model lipid membrane.

We use molecular dynamics simulations to examine the dynamical heterogeneity of a model single-component lipid membrane using a coarse-grained representation of lipid molecules. This model qualitatively reproduces the known phase transitions between disordered, ordered, and gel membrane phases, and the phase transitions are accompanied by significant changes in the nature of the lipid dynamics. In particular, lipid diffusion in the liquid-ordered phase is hindered by the transient trapping of molecules by their neighbors, similar to the dynamics of a liquid approaching its glass transition. This transient molecular caging gives rise to two distinct mobility groups within a single-component membrane: lipids that are transiently trapped, and lipids with displacements on the scale of the intermolecular spacing. Most significantly, lipids within these distinct mobility states spatially segregate, creating transient "islands" of enhanced mobility having a size and time scale compatible with lipid "rafts," dynamical structures thought to be important for cell membrane function. Although the dynamic lipid clusters that we observe do not themselves correspond to rafts (which are more complex, multicomponent structures), we hypothesize that such rafts may develop from the same universal mechanism, explaining why raft-like regions should arise, regardless of lipid structural or compositional details. These clusters are strikingly similar to the dynamical clusters found in glass-forming fluids, and distinct from phase-separation clusters. We also show that mobile lipid clusters can be dissected into smaller clusters of cooperatively rearranging molecules. The geometry of these clusters can be understood in the context of branched equilibrium polymers, related to percolation theory. We discuss how these dynamical structures relate to a range observations on the dynamics of lipid membranes.

Dynamical correlations in a glass former with randomly pinned particles

The effects of randomly pinning particles in a model glass-forming fluid are studied, with a focus on the dynamically heterogeneous relaxation in the presence of pinning. We show how four-point dynamical correlations can be analyzed in real space, allowing direct extraction of a length scale that characterizes dynamical heterogeneity. In the presence of pinning, the relaxation time of the glassy system increases by up to two decades, but there is almost no increase in either the four-point correlation length or the strength of the four-point correlations. We discuss the implications of these results for theories of the glass transition.

Ising model of a glass transition

Numerical simulations by Tanaka and co-workers indicate that glass-forming systems of moderately polydisperse hard-core particles, in both two and three dimensions, exhibit diverging correlation lengths. These correlations are described by Ising-like critical exponents, and are associated with diverging, Vogel-Fulcher-Tamann, structural relaxation times. Related simulations of thermalized hard disks indicate that the curves of pressure versus packing fraction for different polydispersities exhibit a sequence of transition points, starting with a liquid-hexatic transition for the monodisperse case, and crossing over with increasing polydispersity to glassy, Ising-like critical points. I propose to explain these observations by assuming that glass-forming fluids contain twofold degenerate, locally ordered clusters of particles, similar to the two-state systems that have been invoked to explain other glassy phenomena. This paper starts with a brief statistical derivation of the thermodynamics of thermalized, hard-core particles. It then discusses how a two-state, Ising-like model can be described within that framework in terms of a small number of statistically relevant, internal state variables. The resulting theory agrees accurately with the simulation data. I also propose a rationale for the observed relation between the Ising-like correlation lengths and the Vogel-Fulcher-Tamann formula.

Thermodynamic scaling of dynamics in polymer melts: Predictions from the generalized entropy theory

Many glass-forming fluids exhibit a remarkable thermodynamic scaling in which dynamic properties, such as the viscosity, the relaxation time, and the diffusion constant, can be described under different thermodynamic conditions in terms of a unique scaling function of the ratio ρ(γ)∕T, where ρ is the density, T is the temperature, and γ is a material dependent constant. Interest in the scaling is also heightened because the exponent γ enters prominently into considerations of the relative contributions to the dynamics from pressure effects (e.g., activation barriers) vs. volume effects (e.g., free volume). Although this scaling is clearly of great practical use, a molecular understanding of the scaling remains elusive. Providing this molecular understanding would greatly enhance the utility of the empirically observed scaling in assisting the rational design of materials by describing how controllable molecular factors, such as monomer structures, interactions, flexibility, etc., influence the scaling exponent γ and, hence, the dynamics. Given the successes of the generalized entropy theory in elucidating the influence of molecular details on the universal properties of glass-forming polymers, this theory is extended here to investigate the thermodynamic scaling in polymer melts. The predictions of theory are in accord with the appearance of thermodynamic scaling for pressures not in excess of ~50 MPa. (The failure at higher pressures arises due to inherent limitations of a lattice model.) In line with arguments relating the magnitude of γ to the steepness of the repulsive part of the intermolecular potential, the abrupt, square-well nature of the lattice model interactions lead, as expected, to much larger values of the scaling exponent. Nevertheless, the theory is employed to study how individual molecular parameters affect the scaling exponent in order to extract a molecular understanding of the information content contained in the exponent. The chain rigidity, cohesive energy, chain length, and the side group length are all found to significantly affect the magnitude of the scaling exponent, and the computed trends agree well with available experiments. The variations of γ with these molecular parameters are explained by establishing a correlation between the computed molecular dependence of the scaling exponent and the fragility. Thus, the efficiency of packing the polymers is established as the universal physical mechanism determining both the fragility and the scaling exponent γ.

Probing a liquid to glass transition in equilibrium.

We use computer simulations to investigate the static properties of a simple glass-forming fluid in which the positions of a finite fraction of the particles have been frozen. By probing the equilibrium statistics of the overlap between independent configurations of the liquid, we find strong evidence that this random pinning induces a glass transition. At low temperatures, our numerical findings are consistent with the existence of a random first-order phase transition rounded by finite size effects.

Decorrelation of the static and dynamic length scales in hard-sphere glass formers

We show that, in the equilibrium phase of glass-forming hard-sphere fluids in three dimensions, the static length scales tentatively associated with the dynamical slowdown and the dynamical length characterizing spatial heterogeneities in the dynamics unambiguously decorrelate. The former grow at a much slower rate than the latter when density increases. This observation is valid for the dynamical range that is accessible to computer simulations, which roughly corresponds to that accessible in colloidal experiments. We also find that, in this same range, no one-to-one correspondence between relaxation time and point-to-set correlation length exists. These results point to the coexistence of several relaxation mechanisms in the dynamically accessible regime of three-dimensional hard-sphere glass formers.

Dynamic heterogeneities above and below the mode-coupling temperature: Evidence of a dynamic crossover

We examine dynamic heterogeneities in a model glass-forming fluid, a binary harmonic sphere mixture, above and below the mode-coupling temperature T(c). We calculate the ensemble independent susceptibility χ4(τ(α)) and the dynamic correlation length ξ4(τ(α)) at the α-relaxation time τ(α). We also examine in detail the temperature dependence of τ(α) and the diffusion coefficient D. For higher temperatures, we find that the standard Stokes-Einstein relationship, D∼τ(α)(-1), holds, but at lower temperatures a fractional Stokes-Einstein relationship, D∼τ(α)(-σ) with σ = 0.65, emerges. By examining the relationships between τ(α), D, χ4(τ(α)), and ξ4(τ(α)) we determine that the emergence of the fractional Stokes-Einstein relationship is accompanied by a dynamic crossover from τ(α)∼e(k2ξ4) at higher temperatures to τ(α)∼e(k1ξ4(3/2)) at lower temperatures.

Density-Temperature-Softness Scaling of the Dynamics of Glass-Forming Soft-Sphere Liquids

We employ the principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E 68, 011405 (2003)] to describe the interplay of the effects of varying the density n, the temperature T, and the softness (characterized by a softness parameter ν(-1)) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is the existence of a dynamic universality class associated with the hard-sphere fluid, constituted by the soft-sphere systems whose dynamic parameters depend on n, T, and ν only through the reduced density n*≡nσ(HS)(T*,ν). A number of scaling properties observed in recent experiments and simulations involving glass-forming fluids with repulsive short-range interactions are found to be a direct manifestation of this general dynamic equivalence principle.

The role of the dynamic crossover temperature and the arrest in glass-forming fluids

We discuss the role of the dynamic glass-forming fragile-to-strong crossover (FSC) in supercooled liquids. In the FSC, significant dynamic changes such as the decoupling (the violation of the Stokes-Einstein relation) of homologous transport parameters, e.g., the density relaxation time τ and the viscosity η, occur at a characteristic temperature T(c). We study the FSC using a scaling law approach. In particular, we use both forms of the mode-coupling theory (MCT): the original (ideal) and the extended form, which explicitly describes energy hopping processes. We demonstrate that T(c) plays the most important physical role in understanding dynamic arrest processes.

Universal jamming phase diagram in the hard-sphere limit

We present a new formulation of the jamming phase diagram for a class of glass-forming fluids consisting of spheres interacting via finite-ranged repulsions at temperature T, packing fraction ϕ or pressure p, and applied shear stress Σ. We argue that the natural choice of axes for the phase diagram are the dimensionless quantities T/pσ³, pσ³/ε, and Σ/p, where T is the temperature, p is the pressure, Σ is the stress, σ is the sphere diameter, ε is the interaction energy scale, and m is the sphere mass. We demonstrate that the phase diagram is universal at low pσ³/ε; at low pressure, observables such as the relaxation time are insensitive to details of the interaction potential and collapse onto the values for hard spheres, provided the observables are nondimensionalized by the pressure. We determine the shape of the jamming surface in the jamming phase diagram, organize previous results in relation to the jamming phase diagram, and discuss the significance of various limits.

The Descent into Glass Formation in Polymer Fluids

Glassy materials have been fundamental to technology since the dawn of civilization and remain so to this day: novel glassy systems are currently being developed for applications in energy storage, electronics, food, drugs, and more. Glass-forming fluids exhibit a universal set of transitions beginning at temperatures often in excess of twice the glass transition temperature T(g) and extending down to T(g), below which relaxation becomes so slow that systems no longer equilibrate on experimental time scales. Despite the technological importance of glasses, no prior theory explains this universal behavior nor describes the huge variations in the properties of glass-forming fluids that result from differences in molecular structure. Not surprisingly, the glass transition is currently regarded by many as the deepest unsolved problem in solid state theory. In this Account, we describe our recently developed theory of glass formation in polymer fluids. Our theory explains the origin of four universal characteristic temperatures of glass formation and their dependence on monomer-monomer van der Waals energies, conformational energies, and pressure and, perhaps most importantly, on molecular details, such as monomer structure, molecular weight, size of side groups, and so forth. The theory also provides a molecular explanation for fragility, a parameter that quantifies the rate of change with temperature of the viscosity and other dynamic mechanical properties at T(g). The fragility reflects the fluid's thermal sensitivity and determines the manner in which glass-formers can be processed, such as by extrusion, casting, or inkjet spotting. Specifically, the theory describes the change in thermodynamic properties and fragility of polymer glasses with variations in the monomer structure, the rigidity of the backbone and side groups, the cohesive energy, and so forth. The dependence of the structural relaxation time at lower temperatures emerges from the theory as the Vogel-Fulcher equation, whereas pressure and concentration analogs of the Vogel-Fulcher expression follow naturally from the theory with no additional assumptions. The computed dependence of T(g) and fragility on the length of the side group in poly(α-olefins) agrees quite well with observed trends, demonstrating that the theory can be utilized, for instance, to guide the tailoring of T(g) and the fragility of glass-forming polymer fluids in the fabrication of new materials. Our calculations also elucidate the molecular characteristics of small-molecule diluents that promote antiplasticization, a lowering of T(g) and a toughening of the material.

Dynamic Heterogeneity in a Glass Forming Fluid: Susceptibility, Structure Factor, and Correlation Length

We investigate the growth of dynamic heterogeneity in a glassy hard-sphere mixture for volume fractions up to and including the mode-coupling transition. We use an 80,000 particle system to test a new procedure to evaluate a dynamic correlation length ξ(t): we determine the ensemble independent dynamic susceptibility χ(4)(t) and use it to facilitate evaluation of ξ(t) from the small wave vector behavior of the four-point structure factor. We analyze relations between the α relaxation time τ(α), χ(4)(τ(α)), and ξ(τ(α)). We find that mode-coupling-like power laws provide a reasonable description of the data over a restricted range of volume fractions, but the power laws' exponents differ from those predicted by the inhomogeneous mode-coupling theory. We find ξ(τ(α))~ln(τ(α)) over the full range of volume fractions studied, which is consistent with an Adam-Gibbs-type relation.

Direct observation of stringlike collective motion in a two-dimensional driven granular fluid

Dense ensembles of monodisperse hard spheres spontaneously assemble into regions of local crystalline order (grains) that are separated by relatively disordered grain-boundary (GB) regions. It is well known that grains and grain boundaries (GB) profoundly affect the material properties of polycrystalline materials and plastic deformation and transport in this ubiquitous class of materials can be particularly influenced by GB structure and dynamics. We experimentally investigate the nature of particle motion in the GB regions of a quasi-two-dimensional driven system of hard spheres and find, in accord with recent molecular-dynamics simulations of metallic polycrystalline materials, that particle motion in the GB regions involves a highly cooperative stringlike motion as in glass-forming liquids. Moreover, we show that the addition of small particles to a granular fluid allows the scale of this collective motion to be modulated, a phenomenon that is also observed in simulations and measurements on glass-forming fluids.

Following microscopic motion in a two-dimensional glass-forming binary fluid

The dynamics of a binary mixture of large and small discs is studied at temperaturesapproaching the glass transition using an analysis based on the topology of the Voronoipolygon surrounding each atom. At higher temperatures we find that the dynamics isdominated by fluid-like motion that involves particles entering and exiting the nearestneighbour shells of nearby particles. As the temperature is lowered, the rate of topologicalmoves decreases and motion becomes localized to regions of mixed pentagons andheptagons. In addition we find that in the low temperature state particles may translatesignificant distances without undergoing changes in their nearest neighbour shell. Theseresults have implications for dynamical heterogeneities in glass-forming liquids.

Thermal versus deformation-induced relaxation in a glass-forming fluid

Plastic yielding in glassy solids has been interpreted as a strain-biased relaxation process, or, equivalently, as a strain-induced glass transition. The similarity between the structural relaxation induced by plastic deformation and thermal equilibrium of glasses has led to the formulation of the strain-temperature superposition principle. In the present work, the atomic motions caused by athermal plastic deformation of a binary Lennard-Jones glass are compared to thermal motion in the liquid in terms of the self part of the intermediate structure factor. A new approach is presented that allows to study the interplay of deformation-induced and thermal relaxation. It is found that these two processes occur independently of each other over a wide range of strain rates.

Computer simulations of undercooled fluids and the glass transition

Two model studies are presented that attempt to describe the static and dynamic properties of glass-forming fluids via molecular dynamics simulations: The first model is an atomistically realistic model of SiO 2, the second model provides a coarse-grained description of polymer liquids, i.e., typical `fragile' glassformers, while SiO 2 is the prototype of a `strong glassformer'. For both models, attention is given to the questions as to which range of temperatures are properties in equilibrium, and whether such simulations can help to interpret experiments and/or check theoretical predictions. While in the simulation of SiO 2 using the potential of van Beest, Kramer and van Santen (`BKS potential') only temperatures T⩾2750 K can be equilibrated, nevertheless many properties can be extracted that are in almost quantitative agreement with experiment. Sodium silicate liquids investigated at temperatures T⩾1900 K are in agreement to within a few percent with experimental data, too. At very high temperatures (T⩾2750 K) , even for SiO 2 a temperature range is found where the mode coupling theory (MCT) of the glass transition is applicable (with T c =3330 K ). The polymer melt is described by flexible polymer chains represented by a continuum bead-spring model with a Lennard–Jones interaction between the effective monomers. The typical features of undercooled macromolecular fluids are reproduced qualitatively, with the properties at small length scales (≪1 nm) described by MCT, while at larger length scales a crossover to Rouse-like dynamics occurs.