Random walks on fractals and stretched exponential relaxation

Dynamics of structural relaxation in bioactive 45S5 glass

Kinetics of physical aging in archetypic 45S5 bioactive silicate glass composition with different types of phase separation are studied in situ below the glass transition temperature (T g). The qualitative nature of aging is found to be almost independent of the structural differences on the micrometer scale. A well-expressed step-like behavior in the enthalpy recovery kinetics is observed for aging temperatures T a ∼ 0.90T g and T a ∼ 0.85T g, which, however, disappears when the aging occurs at T a ∼ 0.95T g. The overall kinetics are described by a stretched-exponential function with stretching exponent close to 3/7 at T a ∼ 0.95T g, and 1/3 when the aging temperature drops to ∼0.90T g and below. The values correlate well with the predictions of Phillips' diffusion-to-traps and percolating fractals models. Appearance of step-like behavior at larger departure from T g is attributed to the hierarchical scheme of approaching equilibrium based on an alignment-shrinkage mechanism of physical aging proposed earlier for chalcogenide glasses.

Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics

We consider the survival probability f(t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension d_{s}≤2, with spatially correlated traps. The traps form a sublattice with fractal dimension d_{a}<d and are characterized by the absorption rate w_{a} which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (w_{a}≪w), we find that f(t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α=1-(d-d_{a})/d_{w}, where d_{w} is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics f(t)∼t^{-α} with the same exponent α as for the stretched exponential regime. For strong absorption w_{a}≳w, including the limit of perfect traps w_{a}→∞, the stretched exponential regime is absent and the decay of f(t) follows, after a short transient, the aforementioned power law for all times.

Crossover between cooperative and fractal relaxation in complex glass-formers

Kinetics of physical aging at different temperatures is studied in situ in arsenic selenide glasses using high-precision differential scanning calorimetry technique. A well-expressed step-like behaviour in the enthalpy recovery kinetics is recorded for low aging temperatures. These fine features disappear when the aging temperature (T a) approaches the glass transition temperature (T g). The overall kinetics is described by stretched exponential function with stretching exponent close to 3/5 at T a  >  ~0.95 T g almost independent on glass composition, and 3/7 when the aging temperature drops to ~0.9 T g. These values are consistent with the prediction of Phillips' diffusion-to-traps model. Further decrease in aging temperature to ~0.85 T g leads to the appearance of step-like behaviour and stretching exponent of 1/3 for the overall kinetics, which is the limiting value predicted by random walk on the fractal model. Such behavior is explained as crossover from homogeneous cooperative relaxation of non-percolating structural units to high-dimensional fractal relaxation within hierarchically-arranged two-stage physical aging model.

Statistics of energy dissipation and stress relaxation in a crumpling network of randomly folded aluminum foils

We study stress relaxation in hand folded aluminum foils subjected to the uniaxial compression force F(λ). We found that once the compression ratio is fixed (λ=const) the compression force decreases in time as F∝F_{0}P(t), where P(t) is the survival probability time distribution belonging to the domain of attraction of max-stable distribution of the Fréchet type. This finding provides a general physical picture of energy dissipation in the crumpling network of a crushed elastoplastic foil. The difference between energy dissipation statistics in crushed viscoelastic papers and elastoplastic foils is outlined. Specifically, we argue that the dissipation of elastic energy in crushed aluminum foils is ruled by a multiplicative Poisson process governed by the maximum waiting time distribution. The mapping of this process into the problem of transient random walk on a fractal crumpling network is suggested.

(-)-Epicatechin gallate prevents alkali-salt mediated fibrillogenesis of hen egg white lysozyme

Green tea polyphenols (GTPs) are found to be potent inhibitors of amyloid fibril formation. We report the effective inhibitory property of (-)-epicatechin gallate (ECG) during the alkali-salt induced fibrillogenesis of hen egg white lysozyme (HEWL) at 37 °C. Spectroscopic techniques such as fluorescence, circular dichroism and microscopic images show that (-)-epigallocatechin (EGC), (-)-epicatechin gallate (ECG), and (-)-epigallocatechin gallate (EGCG) show moderate inhibition of fibrillation with ECG as the most potent polyphenol. Aromatic interactions, hydrophobic interactions, the radical scavenging activity and autoxidation of polyphenols are likely to be the major reasons for ECG being the most effective inhibitor.

Geometric theory for Weibull's distribution

Weibull's distribution is the principal phenomenological law of relaxation in the physical sciences and spans three different relaxation regimes: subexponential ("stretched exponential"), exponential, and superexponential. The probabilistic theory of extreme-value statistics asserts that the linear scaling limits of minima of ensembles of positive-valued random variables, which are independent and identically distributed, are universally governed by Weibull's distribution. However, this probabilistic theory does not take into account spatial geometry, which often plays a key role in the physical sciences. In this paper we present a general and versatile model of random reactions in random environments and establish a geometry-based theory for the universal emergence of Weibull's distribution.

Physicochemical characteristics of protein-NP bioconjugates: the role of particle curvature and solution conditions on human serum albumin conformation and fibrillogenesis inhibition

Gold nanoparticles (Au NPs) from 5 to 100 nm in size synthesized with HAuCl(4) and sodium citrate were complexed with the plasma protein human serum albumin (HSA). Size, surface charge, and surface plasmon bands of the Au NPs are largely modified by the formation of a protein corona via electrostatic interactions and hydrogen bonding as revealed by thermodynamic data. Negative values of the entropy of binding suggested a restriction in the biomolecule mobility upon adsorption. The structure of the adsorbed protein molecules is slightly affected by the interaction with the metal surface, but this effect is enhanced as the NP curvature decreases. Also, it is observed that the protein molecules adsorbed onto the NP surface are more resistant to complete thermal denaturation than free protein ones as deduced from the increases in the melting temperature of the adsorbed protein. Differences in the conformations of the adsorbed protein molecules onto small (<40 nm) and large NPs were observed on the basis of ζ-potential data and FTIR spectroscopy, also suggesting a better resistance of adsorbed protein molecules to thermal denaturing conditions. We think this enhanced protein stability is responsible for a reduced formation of HSA amyloid-like fibrils in the presence of small Au NPs under HSA fibrillation conditions.

Stretched-exponential behavior and random walks on diluted hypercubic lattices

Diffusion on a diluted hypercube has been proposed as a model for glassy relaxation and is an example of the more general class of stochastic processes on graphs. In this article we determine numerically through large-scale simulations the eigenvalue spectra for this stochastic process and calculate explicitly the time evolution for the autocorrelation function and for the return probability, all at criticality, with hypercube dimensions N up to N=28. We show that at long times both relaxation functions can be described by stretched exponentials with exponent 1/3 and a characteristic relaxation time which grows exponentially with dimension N. The numerical eigenvalue spectra are consistent with analytic predictions for a generic sparse network model.

Structure conserving correlation and the Kohlrausch-Williams-Watts decay of the incoherent intermediate scattering function in simulated Ni(0.5)Zr(0.5) melt

Results are presented about the origin of the Kohlrausch-Williams-Watts decay of the incoherent intermediate scattering function (ISF) in molecular dynamics simulated liquid Ni(0.5)Zr(0.5). By the concept of weakly effective particles (WEPs), we establish an interrelationship between ISF and particle dynamics. Temporal correlations in the action of WEPs act structure conserving, reflecting that immobile particles tend to remain immobile. Analysis of the related correlation function yields that these correlations account quantitatively for the stretched exponential like decay.

Existence of different structural intermediates on the fibrillation pathway of human serum albumin

The fibrillation propensity of the multidomain protein human serum albumin (HSA) was analyzed under different solution conditions. The aggregation kinetics, protein conformational changes upon self-assembly, and structure of the different intermediates on the fibrillation pathway were determined by means of thioflavin T (ThT) fluorescence and Congo Red absorbance; far- and near-ultraviolet circular dichroism; tryptophan fluorescence; Fourier transform infrared spectroscopy; x-ray diffraction; and transmission electron, scanning electron, atomic force, and microscopies. HSA fibrillation extends over several days of incubation without the presence of a lag phase, except for HSA samples incubated at acidic pH and room temperature in the absence of electrolyte. The absence of a lag phase occurs if the initial aggregation is a downhill process that does not require a highly organized and unstable nucleus. The fibrillation process is accompanied by a progressive increase in the beta-sheet (up to 26%) and unordered conformation at the expense of alpha-helical conformation, as revealed by ThT fluorescence and circular dichroism and Fourier transform infrared spectroscopies, but changes in the secondary structure contents depend on solution conditions. These changes also involve the presence of different structural intermediates in the aggregation pathway, such as oligomeric clusters (globules), bead-like structures, and ring-shaped aggregates. We suggest that fibril formation may take place through the role of association-competent oligomeric intermediates, resulting in a kinetic pathway via clustering of these oligomeric species to yield protofibrils and then fibrils. The resultant fibrils are elongated but curly, and differ in length depending on solution conditions. Under acidic conditions, circular fibrils are commonly observed if the fibrils are sufficiently flexible and long enough for the ends to find themselves regularly in close proximity to each other. These fibrils can be formed by an antiparallel arrangement of beta-strands forming the beta-sheet structure of the HSA fibrils as the most probable configuration. Very long incubation times lead to a more complex morphological variability of amyloid mature fibrils (i.e., long straight fibrils, flat-ribbon structures, laterally connected fibers, etc.). We also observed that mature straight fibrils can also grow by protein oligomers tending to align within the immediate vicinity of the fibers. This filament + monomers/oligomers scenario is an alternative pathway to the otherwise dominant filament + filament manner of the protein fibril's lateral growth. Conformational preferences for a certain pathway to become active may exist, and the influence of environmental conditions such as pH, temperature, and salt must be considered.

Criticality in the Burridge-Knopoff model

Criticality is a potential origin of the scale invariance observed in the Gutenberg-Richter law for earthquakes. In support of this hypothesis, the Burridge-Knopoff (BK) model of an earthquake fault system is known to exhibit a dynamic phase transition, but the critical nature of the transition is uncertain. Here it is shown that the BK model exhibits a dynamic transition from large-scale stick-slip to small-scale creep motion and through a finite size scaling analysis the critical nature of this transition is established. The order parameter describing the critical transition suggests that the Olami-Feder-Christensen model may be tuned to criticality through its assumptions describing the relaxation of the system.

Topological derivation of shape exponents for stretched exponential relaxation

In homogeneous (ideal) glasses, the important dimensionless stretched-exponential shape parameter beta is described by magic (not adjusted) simple fractions derived from fractal configuration spaces of effective dimension d* determined by different topological axioms (rules) in the presence (absence) of a forcing electric field. The rules are based on a new central principle for defining glassy states: equal a priori distributions of fractal residual configurational entropy. Our approach and its beta estimates are fully supported by the results of relaxation measurements involving many different glassy materials and probe methods. The present unique topological predictions for beta typically agree with observed values to approximately 1% and indicate that for field-forced conditions beta should be constant for appreciable ranges of such exogenous variables as temperature and ionic concentration, as indeed observed using appropriate frequency-domain data analysis. The present approach can also be inverted and used to test sample homogeneity and quality.

Constrained spin-dynamics description of random walks on hierarchical scale-free networks

We study a random walk problem on the hierarchical network which is a scale-free network grown deterministically. The random walk problem is mapped onto a dynamical Ising spin chain system in one dimension with a nonlocal spin update rule, which allows an analytic approach. We show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes N as T--N(z). From a scaling argument, we also show the power-law decay of the autocorrelation function C(sigma)(t)--t(-alpha), which is the probability to find the Ising spins in the initial state sigma after t time steps, with the state-dependent nonuniversal exponent alpha. It turns out that the power-law scaling behavior has its origin in a quasiultrametric structure of the configuration space.

Origin of stretched exponential relaxation for hopping-transport models

We propose a novel geometric approach to the description of the relaxation phenomena in complex condensed-matter systems. It is shown within a fairly general random site hopping model that the stretched exponential decay law, exp([-(t/tau)(beta)], originates from the simple and general geometric features of a random distribution of transport and trapping sites in the 3D space. The value of the variable stretching index beta is determined by the localization radius of hopping electrons. The possibilities for generalization of the obtained results and interpretation of the relevant experimental data are discussed.

Relationship between dynamical heterogeneities and stretched exponential relaxation

We identify dynamical heterogeneities as an essential prerequisite for stretched exponential relaxation in dynamically frustrated systems. This heterogeneity takes the form of ordered domains of finite but diverging lifetime for particles in atomic or molecular systems, or spin states in magnetic materials. At the onset of the dynamical heterogeneity, the distribution of time intervals spent in such domains or traps becomes stretched exponential at long times. We rigorously show that once this is the case the autocorrelation function of the renewal process formed by these time intervals is also a stretched exponential at long times.

A kinetic study of beta-lactoglobulin amyloid fibril formation promoted by urea

The formation of fibrillar aggregates by beta-lactoglobulin in the presence of urea has been monitored by using thioflavin T fluorescence and transmission electron microscopy (TEM). Large quantities of aggregated protein were formed by incubating beta-lactoglobulin in 3-5 M urea at 37 degrees C and pH 7.0 for 10-30 days. The TEM images of the aggregates in 3-5 M urea show the presence of fibrils with diameters of 8-10 nm, and increases in thioflavin T fluorescence are indicative of the formation of amyloid structures. The kinetics of spontaneous fibrillogenesis detected by thioflavin T fluorescence show sigmoidal behavior involving a clear lag phase. Moreover, addition of preformed fibrils into protein solutions containing urea shows that fibril formation can be accelerated by seeding processes that remove the lag phase. Both of these findings are indicative of nucleation-dependent fibril formation. The urea concentration where fibril formation is most rapid, both for seeded and unseeded solutions, is approximately 5.0 M, close to the concentration of urea corresponding to the midpoint of unfolding (5.3 M). This result indicates that efficient fibril formation involves a balance between the requirement of a significant population of unfolded or partially unfolded molecules and the need to avoid conditions that strongly destabilize intermolecular interactions.

Random geometric graphs

We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bipartitioning are included.

Slow dynamics and aging in a nonrandomly frustrated spin system

A simple, non-disordered spin model has been studied in an effort to understand the origin of the precipitous slowing down of dynamics observed in supercooled liquids approaching the glass transition. A combination of Monte Carlo simulations and exact calculations indicates that this model exhibits an entropy-vanishing transition accompanied by a rapid divergence of time scales. Measurements of various correlation functions show that the system displays a hierarchy of time scales associated with different degrees of freedom. Extended structures, arising from the frustration in the system, are identified as the source of the slow dynamics. In the simulations, the system falls out of equilibrium at a temperature T(g) higher than the entropy-vanishing transition temperature and the dynamics below T(g) exhibits aging as distinct from coarsening. The cooling rate dependence of the energy is also consistent with the usual glass formation scenario.

Stretched exponential relaxation in a diffusive lattice model

We studied the single dimer dynamics in a lattice diffusive model as a function of particle density in the high densification regime. The mean square displacement is found to be subdiffusive both in one and two dimensions. The spatial dependence of the self-part of the van Hove correlation function displays as a function of r a single peak and signals a dramatic slow down of the system for high density. The self-intermediate scattering function is fitted to the Kohlrausch-Williams-Watts law. The exponent beta extracted from the fits is density independent while the relaxation time tau follows a scaling law with an exponent 2.5.