On the Breakdown of the Stokes−Einstein Law in Supercooled Liquids

Nanoscopic Interfacial Hydrogel Viscoelasticity Revealed from Comparison of Macroscopic and Microscopic Rheology

Deviations between macrorheological and particle-based microrheological measurements are often considered to be a nuisance and neglected. We study aqueous poly(ethylene oxide) (PEO) hydrogels for varying PEO concentrations and chain lengths that contain microscopic tracer particles and show that these deviations reveal the nanoscopic viscoelastic properties of the particle-hydrogel interface. Based on the transient Stokes equation, we first demonstrate that the deviations are not due to finite particle radius, compressibility, or surface-slip effects. Small-angle neutron scattering rules out hydrogel heterogeneities. Instead, we show that a generalized Stokes-Einstein relation, accounting for an interfacial shell around tracers with viscoelastic properties that deviate from bulk, consistently explains our macrorheological and microrheological measurements. The extracted shell diameter is comparable to the PEO end-to-end distance, indicating the importance of dangling chain ends. Our methodology reveals the nanoscopic interfacial rheology of hydrogels and is applicable to different kinds of viscoelastic fluids and particles.

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.

Spatial Dimensionality Dependence of Heterogeneity, Breakdown of the Stokes-Einstein Relation, and Fragility of a Model Glass-Forming Liquid

We investigate the heterogeneity of dynamics, the breakdown of the Stokes-Einstein relation and fragility in a model glass forming liquid, a binary mixture of soft spheres with a harmonic interaction potential for spatial dimensions from 3 to 8. The dynamical heterogeneity is quantified through the dynamical susceptibility χ₄ and the non-Gaussian parameter α₂. We find that the fragility, the degree of breakdown of the Stokes-Einstein relation, and the heterogeneity of the dynamics decrease with increasing spatial dimensionality. We briefly describe the dependence of fragility on the density and use it to resolve an apparent inconsistency with previous results.

Dependence of the atomic level Green-Kubo stress correlation function on wavevector and frequency: molecular dynamics results from a model liquid

We report on a further investigation of a new method that can be used to address vibrational dynamics and propagation of stress waves in liquids. The method is based on the decomposition of the macroscopic Green-Kubo stress correlation function into the atomic level stress correlation functions. This decomposition, as was demonstrated previously for a model liquid studied in molecular dynamics simulations, reveals the presence of stress waves propagating over large distances and a structure that resembles the pair density function. In this paper, by performing the Fourier transforms of the atomic level stress correlation functions, we elucidate how the lifetimes of the stress waves and the ranges of their propagation depend on their frequency, wavevector, and temperature. These results relate frequency and wavevector dependence of the generalized viscosity to the character of propagation of the shear stress waves. In particular, the results suggest that an increase in the value of the frequency dependent viscosity at low frequencies with decrease of temperature is related to the increase in the ranges of propagation of the stress waves of the corresponding low frequencies. We found that the ranges of propagation of the shear stress waves of frequencies less than half of the Einstein frequency extend well beyond the nearest neighbor shell even above the melting temperature. The results also show that the crossover from quasilocalized to propagating behavior occurs at frequencies usually associated with the Boson peak.

Simple picture of supercooled liquid dynamics: dynamic scaling and phenomenology based on clusters

Although it is now well established that in glassy liquids, slow structural relaxation accompanies a correlated structural rearrangement, the role of such a correlation in the transport anomaly, and thus in the slow dynamics, remains unclear. In this paper, we argue from a hydrodynamic viewpoint that a correlated structure (cluster) with a characteristic size ξ sustains the long-lived stress and dynamically couples with the hydrodynamic fluctuations; therefore, the dynamics of this cluster is the origin of the mesoscopic nature of anomalous hydrodynamic transport. Based on this argument, we derive a dynamic scaling law for τ(α) (or η, where η is the macroscopic shear viscosity) as a function of ξ: τ(α)([proportionality]η)[proportionality]ξ(4). We provide a simple explanation for basic features of anomalous transport, such as the breakdown of the Stokes-Einstein relation and the length-scale-dependent decoupling between viscosity and diffusion. The present study further suggests a different physical picture: Through the coarse graining of smaller-scale fluctuations (</~ξ), the supercooled liquid dynamics can be regarded as the dynamics of normal (cluster) liquids composed of units with a typical size of ξ. Although the correlation length of hydrodynamic transport ξ and the dynamic heterogeneity size ξ(DH), which is determined by the usual four-point correlation function, reflect some aspects of the cooperative effects, the correspondence between ξ and ξ(DH) is not one to one. We highlight the possibility that ξ(DH) overestimates the actual collective transport range at a low degree of supercooling.

Generalized bond order parameters to characterize transient crystals

Higher order parameters in the hard disk fluid are computed to investigate the number, the lifetime, and size of transient crystal nuclei in the pre-freezing phase. The methodology introduces further neighbor shells bond orientational order parameters and coarse-grains the correlation functions needed for the evaluation of the stress autocorrelation function for the viscosity. We successfully reproduce results by the previous collision method for the pair orientational correlation function, but some two orders of magnitude faster. This speed-up allows calculating the time dependent four body orientational correlation between two different pairs of particles as a function of their separation, needed to characterize the size of the transient crystals. The result is that the slow decay of the stress autocorrelation function near freezing is due to a large number of rather small crystal nuclei lasting long enough to lead to the molasses tail.

Bond orientational order in liquids: Towards a unified description of water-like anomalies, liquid-liquid transition, glass transition, and crystallization: Bond orientational order in liquids

There are at least three fundamental states of matter, depending upon temperature and pressure: gas, liquid, and solid (crystal). These states are separated by first-order phase transitions between them. In both gas and liquid phases a complete translational and rotational symmetry exist, whereas in a solid phase both symmetries are broken. In intermediate phases between liquid and solid, which include liquid crystal and plastic crystal phases, only one of the two symmetries is preserved. Among the fundamental states of matter, the liquid state is the most poorly understood. We argue that it is crucial for a better understanding of liquids to recognize that a liquid generally has the tendency to have a local structural order and its presence is intrinsic and universal to any liquid. Such structural ordering is a consequence of many-body correlations, more specifically, bond angle correlations, which we believe are crucial for the description of the liquid state. We show that this physical picture may naturally explain difficult unsolved problems associated with the liquid state, such as anomalies of water-type liquids (water, Si, Ge, ...), liquid-liquid transition, liquid-glass transition, crystallization and quasicrystal formation, in a unified manner. In other words, we need a new order parameter representing a low local free-energy configuration, which is a bond orientational order parameter in many cases, in addition to a density order parameter for the physical description of these phenomena. Here we review our two-order-parameter model of liquid and consider how transient local structural ordering is linked to all of the above-mentioned phenomena. The relationship between these phenomena is also discussed.

Dynamic scaling for anomalous transport in supercooled liquids

The anomalous mesoscopic transport in supercooled liquids was investigated using three-dimensional molecular dynamics simulation. We show that the dynamic correlation length, ξ, can be identified as a crossover length between the microscopic and macroscopic transports. We also find that in highly supercooled states, where a clear stress plateau is observed in the stress autocorrelation, cooperative transport, in both the (longitudinal) density diffusion and the (transverse) viscous relaxation, can be characterized by the single length scale, ξ. The present identification of the dynamic correlation length has an advantage over other characterization methods in that it directly interprets the anomalous hydrodynamic transport in terms of the growing length scale. In the context of the present study, we provide a simple explanation for the long-standing problem of the breakdown of the Stokes-Einstein relation.

Manifestations of probe presence on probe dynamics in supercooled liquids

Experimental studies that follow behavior of single probes embedded in heterogeneous systems are increasingly common. The presence of probes may perturb the system, and such perturbations may or may not affect interpretation of host behavior from the probe observables typically measured. In this study, the manifestations of potential probe-induced changes to host dynamics in supercooled liquids are investigated via molecular dynamics simulations. It is found that probe dynamics do not necessarily mirror host dynamics as they exist either in the probe-free or probe-bearing systems. In particular, for a binary supercooled liquid, we find that smooth probes larger than the host particles induce increased translational diffusion in the host system; however, the diffusion is anisotropic and enhances caging of the probe, suppressing probe translational diffusion. This in turn may lead experiments that follow probe diffusion to suggest Stokes-Einstein behavior of the system even while both the probe-free and probe-bearing systems exhibit deviations from that behavior.

Direct evidence of heterogeneous mechanical relaxation in supercooled liquids

Dynamic heterogeneity is now considered to be an intrinsic kinetic feature of a supercooled liquid. Here, we access the nonlocal complex modulus of a glass-forming liquid using molecular dynamics simulations. We find that the shear-stress relaxation exhibits a marked wave number dependence in a supercooled state. This dependence provides direct evidence that slow mechanical relaxation takes place heterogeneously in space. Its characteristic length ξ increases with decreasing temperature and becomes mesoscopic. The response involves particle rearrangements over ξ and takes a time comparable to the structural relaxation time τ(α). Our finding suggests that the heterogeneous structural relaxation is of fundamental importance in anomalous viscous transport and viscoelasticity in supercooled liquids.

Heterogeneities in confined water and protein hydration water

We report recent efforts to understand a broad range of experiments on confined water and protein hydration water, many initiated by a collaboration between workers at the University of Messina and MIT-the editors of this special issue. Preliminary calculations are not inconsistent with one tentative interpretation of these experiments as resulting from the system passing from the high-temperature high-pressure 'HDL' side of the Widom line (where the liquid might display non-Arrhenius behavior) to the low-temperature low-pressure 'LDL' side of the Widom line (where the liquid might display Arrhenius behavior). The Widom line-defined to be the line in the pressure-temperature plane where the correlation length has its maximum-arises if there is a critical point. Hence, interpreting the Messina-MIT experiments in terms of a Widom line is of potential relevance to testing, experimentally, the hypothesis that water displays a liquid-liquid critical point.

Understanding Static and Dynamic Heterogeneities in Confined Water

We report recent efforts to understand the new MIT-Messina experimental discovery of a dynamic crossover at low temperatures in confined water. Preliminary calculations are not inconsistent with one tentative interpretation of this dynamic crossover as resulting from the system passing from the high-temperature high-pressure "HDL" side of the Widom line (where the liquid might display fragile behavior) to the low-temperature low-pressure "LDL" side of the Widom line (where the liquid might display strong behavior). The Widom line - defined to be the line in the pressure-temperature plane where the correlation length has its maximum - arises only if there is a critical point. Hence interpreting the MIT experiments in terms of a Widom line is of potential relevance to testing experimentally, for confined water, the liquid-liquid critical point hypothesis.

Nonlocal nature of the viscous transport in supercooled liquids: complex fluid approach to supercooled liquids

Using molecular dynamics simulations, we show clear evidence for the nonlocal mesoscopic nature of the anomalous viscous transport in a supercooled liquid and its direct link to dynamic heterogeneity: (i) a distinct crossover from the microscopic to macroscopic viscosity at a mesoscopic length scale, which is comparable to the correlation length of dynamic heterogeneity and grows with an increase in the degree of supercooling; (ii) a strong anisotropic decay of the shear-stress autocorrelation at a finite wave number, which indicates intrinsic decoupling between the longitudinal and transverse dynamics. Our findings suggest the fundamental importance of the growing dynamic correlation in anomalous transport and shed new light on the nature of slow dynamics.

Connection of translational and rotational dynamical heterogeneities with the breakdown of the Stokes-Einstein and Stokes-Einstein-Debye relations in water

We study the Stokes-Einstein (SE) and the Stokes-Einstein-Debye (SED) relations, Dt=kBT/6pietaR and Dr=kBT/8pietaR3, where Dt and Dr are the translational and rotational diffusivity, respectively, T is the temperature, eta the viscosity, kB the Boltzmann constant, and R the "molecular" radius. Our results are based on molecular dynamics simulations of the extended simple point charge model of water. We find that both the SE and SED relations break down at low temperature. To explore the relationship between these breakdowns and dynamical heterogeneities (DHs), we also calculate the SE and SED relations for subsets of the 7% "fastest" and 7% "slowest" molecules. We find that the SE and SED relations break down in both subsets, and that the breakdowns occur on all scales of mobility. Thus these breakdowns appear to be generalized phenomena, in contrast with a view where only the most mobile molecules are the origin of the breakdown of the SE and SED relations, embedded in an inactive background where these relations hold. At low temperature, the SE and SED relations in both subsets of molecules are replaced with "fractional" SE and SED relations, Dt approximately (tau/T)-xit and Dr approximately (tau/T)-xir, where xit approximately 0.84(<1) and xir approximately 0.75(<1). We also find that there is a decoupling between rotational and translational motion, and that this decoupling occurs in both the fastest and slowest subsets of molecules. Further, we find that, the decoupling increases upon cooling, but that the probability of a molecule being classified as both translationally and rotationally fastest also increases. To study the effect of time scale for SE and SED breakdown and decoupling, we introduce a time-dependent version of the SE and SED relations, and a time-dependent function that measures the extent of decoupling. Our results suggest that both the decoupling and SE and SED breakdowns originate at a time scale corresponding to the end of the cage regime, when diffusion starts. This is also the time scale when the DHs are more relevant. Our work also demonstrates that selecting DHs on the basis of translational or rotational motion more strongly biases the calculation of diffusion constants than other dynamical properties such as relaxation times.

Noncontinuum effects in nanoparticle dynamics in polymers

We propose a continuum model for the dynamics of particles in polymer matrices which encompasses arbitrary size ratios of the polymer and particle. We present analytical and computer simulation results for the mobility of the particles and the viscosity of the suspension for the case of unentangled polymer melts. Our results indicate strong dependencies of the particle mobility upon the polymer-particle size ratios and much reduced intrinsic viscosities for the suspensions. These predictions rationalize some recent experimental observations on the dynamics of nanoparticles in polymer melts.