Four-point Dynamic Susceptibility Research Articles

Connection between the time distribution and length scale of dynamic heterogeneity explored by probe reorientations of different sizes.

The rotational dynamics of fluorescent probes of different sizes in glass-forming materials were examined to correlate the time distribution and length scale of the dynamic heterogeneity (ξhet). As the size of the probe increased, the temperature dependence of the rotation correlation time (τc) shifted to longer times, and from this shift, the length scale associated with the glass transition (ξα) was estimated through the Debye-Stokes-Einstein (DSE) relationship and the length scale of the probe (ξsDFT) estimated from quantum mechanical calculations. The estimated ξα values roughly matched with ξhet obtained from calorimetric analysis but were considerably smaller than those deduced from 4D NMR, boson peak, and four-point dynamic susceptibility measurements but with a similar trend of decrease in the length scale upon the increase in the stretching exponent (β) of the system. Because β of the glass formers represents the time distribution of the system, and τc is related to the weighted average of the distribution, the length-scale distribution of the glass transition can be deduced by adopting the DSE relationship and assuming ξα is the weighted average of this distribution at the glass transition temperature. In such a case, the upper bound of the length scale and trend matches the experimentally obtained ξhet from 4D NMR, boson peak, and four-point dynamic susceptibility measurements. Furthermore, at a given temperature, as the probe size increased, the β value reported by the probe increased, whereas the temperature dependence of β, which strongly correlates with the fragility of the system, was independent of the probe size.

Revisiting the single-saddle model for the β-relaxation of supercooled liquids

The dynamics of glass-forming liquids display several outstanding features, such as two-step relaxation and dynamic heterogeneities, which are difficult to predict quantitatively from first principles. In this work, we revisit a simple theoretical model of the β-relaxation, i.e., the first step of the relaxation dynamics. The model, first introduced by Cavagna et al. [J. Phys. A: Math. Gen. 36, 10721 (2003)], describes the dynamics of the system in the neighborhood of a saddle point of the potential energy surface. We extend the model to account for density–density correlation functions and for the four-point dynamic susceptibility. We obtain analytical results for a simple schematic model, making contact with related results for p-spin models and with the predictions of inhomogeneous mode-coupling theory. Building on recent computational advances, we also explicitly compare the model predictions against overdamped Langevin dynamics simulations of a glass-forming liquid close to the mode-coupling crossover. The agreement is quantitative at the level of single-particle dynamic properties only up to the early β-regime. Due to its inherent harmonic approximation, however, the model is unable to predict the dynamics on the time scale relevant for structural relaxation. Nonetheless, our analysis suggests that the agreement with the simulations may be largely improved if the modes’ spatial localization is properly taken into account.

Enhanced short time peak in four-point dynamic susceptibility in dense active glass-forming liquids

Active glassy systems are simple model systems that imitate complex biological processes. Sometimes, it becomes crucial to estimate the amount of activity present in such biological systems, such as predicting the progression rate of the cancer cells or the healing time of the wound, etc. In this work, we study a model active glassy system to quantify the degree of activity from the collective, long-wavelength fluctuations in the system. These long-wavelength fluctuations present themselves as an additional peak in the four-point dynamic susceptibility (χ4(t)) apart from the usual peak at structural relaxation time. We then show how the degree of the activity at such a small timescale can be obtained by measuring the variation in χ4(t) due to changing activity. A Detailed finite size analysis of the peak height of χ4(t) suggests the existence of an intrinsic dynamic length scale that grows with increasing activity. Finally, we show that this peak height is a unique function of effective activity across all system sizes, serving as a possible parameter for characterizing the degree of activity in a system.

Dynamic and thermodynamic crossover scenarios in the Kob-Andersen mixture: Insights from multi-CPU and multi-GPU simulations.

The physical behavior of glass-forming liquids presents complex features of both dynamic and thermodynamic nature. Some studies indicate the presence of thermodynamic anomalies and of crossovers in the dynamic properties, but their origin and degree of universality is difficult to assess. Moreover, conventional simulations are barely able to cover the range of temperatures at which these crossovers usually occur. To address these issues, we simulate the Kob-Andersen Lennard-Jones mixture using efficient protocols based on multi-CPU and multi-GPU parallel tempering. Our setup enables us to probe the thermodynamics and dynamics of the liquid at equilibrium well below the critical temperature of the mode-coupling theory, [Formula: see text]. We find that below [Formula: see text] the analysis is hampered by partial crystallization of the metastable liquid, which nucleates extended regions populated by large particles arranged in an fcc structure. By filtering out crystalline samples, we reveal that the specific heat grows in a regular manner down to [Formula: see text] . Possible thermodynamic anomalies suggested by previous studies can thus occur only in a region of the phase diagram where the system is highly metastable. Using the equilibrium configurations obtained from the parallel tempering simulations, we perform molecular dynamics and Monte Carlo simulations to probe the equilibrium dynamics down to [Formula: see text]. A temperature-derivative analysis of the relaxation time and diffusion data allows us to assess different dynamic scenarios around [Formula: see text]. Hints of a dynamic crossover come from analysis of the four-point dynamic susceptibility. Finally, we discuss possible future numerical strategies to clarify the nature of crossover phenomena in glass-forming liquids.

Block Analysis for the Calculation of Dynamic and Static Length Scales in Glass-Forming Liquids

We present block analysis, an efficient method of performing finite-size scaling for obtaining the length scale of dynamic heterogeneity and the point-to-set length scale for generic glass-forming liquids. This method involves considering blocks of varying sizes embedded in a system of a fixed (large) size. The length scale associated with dynamic heterogeneity is obtained from a finite-size scaling analysis of the dependence of the four-point dynamic susceptibility on the block size. The block size dependence of the variance of the α relaxation time yields the static point-to-set length scale. The values of the obtained length scales agree quantitatively with those obtained from other conventional methods. This method provides an efficient experimental tool for studying the growth of length scales in systems such as colloidal glasses for which performing finite-size scaling by carrying out experiments for varying system sizes may not be feasible.

Structural anomaly and dynamic heterogeneity in cycloether/water binary mixtures: Signatures from composition dependent dynamic fluorescence measurements and computer simulations

We have performed steady state UV-visible absorption and time-resolved fluorescence measurements and computer simulations to explore the cosolvent mole fraction induced changes in structural and dynamical properties of water/dioxane (Diox) and water/tetrahydrofuran (THF) binary mixtures. Diox is a quadrupolar solvent whereas THF is a dipolar one although both are cyclic molecules and represent cycloethers. The focus here is on whether these cycloethers can induce stiffening and transition of water H-bond network structure and, if they do, whether such structural modification differentiates the chemical nature (dipolar or quadrupolar) of the cosolvent molecules. Composition dependent measured fluorescence lifetimes and rotation times of a dissolved dipolar solute (Coumarin 153, C153) suggest cycloether mole-fraction (X(THF)/Diox) induced structural transition for both of these aqueous binary mixtures in the 0.1 ≤ X(THF)/Diox ≤ 0.2 regime with no specific dependence on the chemical nature. Interestingly, absorption measurements reveal stiffening of water H-bond structure in the presence of both the cycloethers at a nearly equal mole-fraction, X(THF)/Diox ∼ 0.05. Measurements near the critical solution temperature or concentration indicate no role for the solution criticality on the anomalous structural changes. Evidences for cycloether aggregation at very dilute concentrations have been found. Simulated radial distribution functions reflect abrupt changes in respective peak heights at those mixture compositions around which fluorescence measurements revealed structural transition. Simulated water coordination numbers (for a dissolved C153) and number of H-bonds also exhibit minima around these cosolvent concentrations. In addition, several dynamic heterogeneity parameters have been simulated for both the mixtures to explore the effects of structural transition and chemical nature of cosolvent on heterogeneous dynamics of these systems. Simulated four-point dynamic susceptibility suggests formation of clusters inducing local heterogeneity in the solution structure.

Composition Dependence of Dynamic Heterogeneity Time- and Length Scales in [Omim][BF4]/Water Binary Mixtures: Molecular Dynamics Simulation Study.

Composition dependence of four-point dynamic susceptibilities, overlap functions, and other dynamic heterogeneity (DH) parameters have been investigated by using all-atom molecular dynamics simulations for aqueous solutions of the ionic liquid (IL), 1-octyl-3-methyl imidazolium tetrafluoroborate ([Omim][BF4]) covering the pure-to-pure range. Upon addition of water in the IL, the DH time scales become faster and the four-point dynamic susceptibility time scale softens. Evidences for jump motions for both water and ions have been found from the simulated single particle displacements that show strong deviation from Gaussian distribution. Estimated dynamic correlation length for water reflects effects of IL, whereas those for ions remain largely insensitive to the mixture composition. Simulated structural aspects and DH time scales provide microscopic explanations to the existing experimental observations from time-resolved fluorescence and Kerr spectroscopic measurements.

Relaxation dynamics in a transient network fluid with competing gel and glass phases.

We use computer simulations to study the relaxation dynamics of a model for oil-in-water microemulsion droplets linked with telechelic polymers. This system exhibits both gel and glass phases and we show that the competition between these two arrest mechanisms can result in a complex, three-step decay of the time correlation functions, controlled by two different localization lengthscales. For certain combinations of the parameters, this competition gives rise to an anomalous logarithmic decay of the correlation functions and a subdiffusive particle motion, which can be understood as a simple crossover effect between the two relaxation processes. We establish a simple criterion for this logarithmic decay to be observed. We also find a further logarithmically slow relaxation related to the relaxation of floppy clusters of particles in a crowded environment, in agreement with recent findings in other models for dense chemical gels. Finally, we characterize how the competition of gel and glass arrest mechanisms affects the dynamical heterogeneities and show that for certain combination of parameters these heterogeneities can be unusually large. By measuring the four-point dynamical susceptibility, we probe the cooperativity of the motion and find that with increasing coupling this cooperativity shows a maximum before it decreases again, indicating the change in the nature of the relaxation dynamics. Our results suggest that compressing gels to large densities produces novel arrested phases that have a new and complex dynamics.

Dynamic correlation length scales under isochronal conditions.

The origin of the dramatic changes in the behavior of liquids as they approach their vitreous state-increases of many orders of magnitude in dynamic time scales and transport properties-is a major unsolved problem in condensed matter. These changes are accompanied by greater dynamic heterogeneity, which refers to both spatial variation and spatial correlation of molecular mobilities. The question is whether the changing dynamics are coupled to this heterogeneity; that is, does the latter cause the former? To address this, we carried out the first nonlinear dielectric experiments at elevated hydrostatic pressures on two liquids, to measure the third-order harmonic component of their susceptibilities. We extract from this the number of dynamically correlated molecules for various state points and find that the dynamic correlation volume for non-associated liquids depends primarily on the relaxation time, sensibly independent of temperature and pressure. We support this result by molecular dynamic simulations showing that the maximum in the four-point dynamic susceptibility of density fluctuations is essentially invariant along isochrones for molecules that do not form hydrogen bonds. Our findings are consistent with dynamic cooperativity serving as the principal control parameter for the slowing down of molecular motions in supercooled materials.

Investigating isomorphs with the topological cluster classification

Isomorphs are lines in the density-temperature plane of certain "strongly correlating" or "Roskilde simple" liquids where two-point structure and dynamics have been shown to be close to identical up to a scale transformation. Here we consider such a liquid, a Lennard-Jones glass former, and investigate the behavior along isomorphs of higher-order structural and dynamical correlations. We then consider an inverse power law reference system mapped to the Lennard-Jones system [Pedersen et al., Phys. Rev. Lett. 105, 157801 (2010)]. Using the topological cluster classification to identify higher-order structures, in both systems we find bicapped square antiprisms, which are known to be a locally favored structure in the Lennard-Jones glass former. The population of these locally favored structures is up to 80% higher in the Lennard-Jones system than the equivalent inverse power law system. The structural relaxation time of the two systems, on the other hand, is almost identical, and the four-point dynamical susceptibility is marginally higher in the inverse power law system. Upon cooling, the lifetime of the locally favored structures in the Lennard-Jones system is up to 40% higher relative to the reference system.

Manifestation of random first-order transition theory in Wigner glasses

We use Brownian dynamics simulations of a binary mixture of highly charged spherical colloidal particles to test some of the predictions of the random first-order transition (RFOT) theory [Phys. Rev. Lett. 58, 2091 (1987); Phys. Rev. A 40, 1045 (1989)]. In accord with mode-coupling theory and RFOT, we find that as the volume fraction of the colloidal particles ϕ approaches the dynamical transition value ϕ(A), three measures of dynamics show an effective ergodic to nonergodic transition. First, there is a dramatic slowing down of diffusion, with the translational diffusion constant decaying as a power law as ϕ→ϕ(A)(-). Second, the energy metric, a measure of ergodicity breaking in classical many-body systems, shows that the system becomes effectively nonergodic as ϕ(A) is approached. Finally, the time t(*), at which the four-point dynamical susceptibility achieves a maximum, also increases as a power law near ϕ(A). Remarkably, the translational diffusion coefficients, ergodic diffusion coefficient, and (t(*))(-) all vanish as (ϕ(-1)-ϕ(A)(-1))(γ) with both ϕ(A)(≈0.1) and γ being the roughly the same for all three quantities. Above ϕ(A), transport involves crossing free energy barriers. In this regime, the density-density correlation function decays as a stretched exponential [exp-(t/τ(α))(β)] with β≈0.45. The ϕ dependence of the relaxation time τ(α) could be fit using the Vogel-Tamman-Fulcher law with the ideal glass transition at ϕ(K)≈0.47. By using a local entropy measure, we show that the law of large numbers is not obeyed above ϕ(A), and gives rise to subsample to subsample fluctuations in all physical observables. We propose that dynamical heterogeneity is a consequence of violation of law of large numbers.

Distribution of local relaxation events in an aging three-dimensional glass: Spatiotemporal correlation and dynamical heterogeneity

We investigate the spatiotemporal distribution of microscopic relaxation events, defined through particle hops, in a model polymer glass using molecular dynamics simulations. We introduce an efficient algorithm to directly identify hops during the simulation, which allows the creation of a map of relaxation events for the whole system. Based on this map, we present density-density correlations between hops and directly extract correlation scales. These scales define collaboratively rearranging groups of particles and their size distributions are presented as a function of temperature and age. Dynamical heterogeneity is spatially resolved as the aggregation of hops into clusters, and we analyze their volume distribution and growth during aging. A direct comparison with the four-point dynamical susceptibility χ(4) reveals the formation of a single dominating cluster prior to the χ(4) peak, which indicates maximally correlated dynamics. An analysis of the fractal dimension of the hop clusters finds slightly noncompact shapes in excellent agreement with independent estimates from four-point correlations.

Relaxation Dynamics and Crystallization Study of Sildenafil in the Liquid and Glassy States

In this paper, the physical stability and molecular dynamics of amorphous sildenafil are investigated in both the liquid and glassy states. We have established that the amorphous sildenafil is resistant to recrystallization at temperatures below the glass transition temperature Tg during the experimental period of its storage (i.e., above 6 months), however, it easily undergoes cold crystallization at T > Tg. To determine the crystallization mechanism, the isothermal and non-isothermal studies of the cold crystallization kinetics of the drug are performed by using the broadband dielectric spectroscopy (BDS) and the differential scanning calorimetry (DSC), respectively. The cold crystallization mechanism has been found to be similar in both the isothermal and non-isothermal cases. This mechanism has been analyzed from the point of view of the molecular mobility of sildenafil investigated in the supercooled liquid and glassy states by using the BDS measurements in the wide temperature range. This analysis has been enriched with a new approach based on a recently reported measure of dynamic heterogeneity given by a four-point dynamic susceptibility function. No tendency to recrystallization of glassy sildenafil at T < Tg is also discussed in relation to molecular dynamics of sildenafil in the glassy state. The relatively small molecular mobility reflected in one secondary relaxation as well as the predicted large time scale of structural relaxation of glassy sildenafil suggests that amorphous sildenafil should not recrystallize during its long-term storage at room temperature.

Comparing dynamic correlation lengths from an approximation to the four-point dynamic susceptibility and from the picosecond vibrational dynamics

Recently an alternative approach to the determination of dynamic correlation lengths ξ for supercooled liquids, based on the properties of the slow (picosecond) vibrational dynamics, was carried out [Hong, Novikov, and Sokolov, Phys. Rev. E 83, 061508 (2011)]. Although these vibrational measurements are typically conducted well below the glass transition temperature, the liquid is frozen at T(g), whereby structural correlations, density variations, etc., manifested at low temperatures as spatial fluctuations of local elastic constants, can be related to a dynamic heterogeneity length scale for the liquid state. We compare ξ from this method to values calculated using an approximation to the four-point dynamic susceptibility. For 26 different materials we find good correlation between the two measures; moreover, the pressure dependences are consistent within the large experimental error. However, ξ from Boson peak measurements above T(g) have a different, and unrealistic, temperature dependence.

Dynamical heterogeneity in soft-particle suspensions under shear

We present experimental measurements of dynamical heterogeneities in a dense system of microgel spheres, sheared at different rates and at different packing fractions in a microfluidic channel, and visualized with high-speed digital video microscopy. A four-point dynamic susceptibility is deduced from video correlations, and is found to exhibit a peak that grows in height and shifts to longer times as the jamming transition is approached from two different directions. In particular, the time for particle-size root-mean square relative displacements is found to scale as τ*∼(γΔφ4)(-1), where γ is the strain rate and Δφ = |φ - φ(c)| is the distance from the random close-packing volume fraction. The typical number of particles in a dynamical heterogeneity is deduced from the susceptibility peak height and found to scale as n*∼(γΔφ4)(-0.3). Exponent uncertainties are less than ten percent. We emphasize that the same power-law behavior is found at packing fractions above and below φ(c). Thus our results considerably extend a previous observation of n*∼γ(-0.3) for granular heap flow at fixed packing below φ(c). Furthermore, the implied result n*∼(τ*)(0.3) compares well with the expectation from mode-coupling theory and with prior observations for driven granular systems.

Correlation of nonexponentiality with dynamic heterogeneity from four-point dynamic susceptibility χ4(t) and its approximation χT(t)

Various properties of vitrifying liquids are correlated with the dispersity of the dynamics, the latter reflected in the magnitude of the nonexponentiality parameter, β(K), describing the distribution of relaxation times. These properties include the mean relaxation time, τ(α), the fragility, and the dynamic crossover. The correlations with β(K) are observed in both experimental data and the results from molecular dynamics simulations on Lennard-Jones (LJ) type systems. Another, rather obvious property to correlate with β(K) is the dynamic heterogeneity, which can be quantified from the number of molecules, N(c), dynamically correlated over a time span τ(α). For a given LJ system, N(c) can be rigorously calculated and we find that it does indeed correlate with β(K) over a range of thermodynamic conditions. However, the analysis of experimental data for a broad range of real materials, wherein an approximation is required to obtain N(c), reveals the absence of any relationship between N(c) and β(K) among different materials.

Density scaling in viscous liquids: From relaxation times to four-point susceptibilities

We present numerical calculations of a four-point dynamic susceptibility, chi(4)(t), for the Kob-Andersen Lennard-Jones mixture as a function of temperature T and density rho. Over a relevant range of T and rho, the full t-dependence of chi(4)(t) and thus the maximum in chi(4)(t), which is proportional to the dynamic correlation volume, are invariant for state points for which the scaling variable rho(gamma)/T is constant. The value of the material constant gamma is the same as that which superposes the relaxation time tau of the system versus rho(gamma)/T. Thus, the dynamic correlation volume is a unique function of tau for any thermodynamic condition in the regime where density scaling holds. Finally, we examine the conditions under which the density scaling properties are related to the existence of strong correlations between pressure and energy fluctuations.

Spatial correlations in the dynamics of glassforming liquids: Experimental determination of their temperature dependence

We use recently introduced three-point dynamic susceptibilities to obtain an experimental determination of the temperature evolution of the number of molecules Ncorr that are dynamically correlated during the structural relaxation of supercooled liquids. We first discuss in detail the physical content of three-point functions that relate the sensitivity of the averaged two-time dynamics to external control parameters (such as temperature or density), as well as their connection to the more standard four-point dynamic susceptibility associated with dynamical heterogeneities. We then demonstrate that these functions can be experimentally determined with good precision. We gather available data to obtain the temperature dependence of Ncorr for a large number of supercooled liquids over a wide range of relaxation time scales from the glass transition up to the onset of slow dynamics. We find that Ncorr systematically grows when approaching the glass transition. It does so in a modest manner close to the glass transition, which is consistent with an activation-based picture of the dynamics in glassforming materials. For higher temperatures, there appears to be a regime where Ncorr behaves as a power-law of the relaxation time. Finally, we find that the dynamic response to density, while being smaller than the dynamic response to temperature, behaves similarly, in agreement with theoretical expectations.

Dynamical Heterogeneity and Nonlinear Susceptibility in Supercooled Liquids with Short-Range Attraction

Recent work has demonstrated the strong qualitative differences between the dynamics near a glass transition driven by short-ranged repulsion and one governed by short-ranged attraction. Here we study in detail the behavior of nonlinear, higher-order correlation functions that measure the growth of length scales associated with dynamical heterogeneity in both types of systems. We find that this measure is qualitatively different in the repulsive and attractive cases with regards to the wave vector dependence as well as the time dependence of the standard nonlinear four-point dynamical susceptibility. We discuss the implications of these results for the general understanding of dynamical heterogeneity in glass-forming liquids.

Topological persistence and dynamical heterogeneities near jamming

We introduce topological methods for quantifying spatially heterogeneous dynamics, and use these tools to analyze particle-tracking data for a quasi-two-dimensional granular system of air-fluidized beads on approach to jamming. In particular, we define two overlap order parameters, which quantify the correlation between particle configurations at different times, based on a Voronoi construction and the persistence in the resulting cells and nearest neighbors. Temporal fluctuations in the decay of the persistent area and bond order parameters define two alternative dynamic four-point susceptibilities chi(A)(tau) and chi(B)(tau), well suited for characterizing spatially heterogeneous dynamics. These are analogous to the standard four-point dynamic susceptibility chi4(l,tau), but where the space dependence is fixed uniquely by topology rather than by discretionary choice of cutoff function. While these three susceptibilities yield characteristic time scales that are somewhat different, they give domain sizes for the dynamical heterogeneities that are in good agreement and that diverge on approach to jamming.