Nonexponential Kinetics Research Articles

Physical Aging of Zr65Cu17.5Ni10Al7.5 Glassy Alloy

Physical aging of Zr65Cu17.5Ni10Al7.5 glassy alloy has been studied after holding its samples with variation of time and temperature of annealing. The results showed that the enthalpy and entropy decreased with time according to a non-exponential kinetics. The Differential Scanning Calorimetry (DSC) heating scan after annealing was analyzed using Tool-Narayanaswamy-Moynihan (TNM) model. The analysis showed that a set of parameters that fitted the heating scan data for one annealing time did not fit the heating scan data for another annealing time. This is explained due to approximations used in the model and the effect of Johari-Goldstein (JG) relaxation to the physical aging.

Populations and Dynamics of Guanine Radicals in DNA strands-Direct versus Indirect Generation.

Guanine radicals, known to be involved in the damage of the genetic code and aging, are studied by nanosecond transient absorption spectroscopy. They are generated in single, double and four-stranded structures (G-quadruplexes) by one and two-photon ionization at 266 nm, corresponding to a photon energy lower than the ionization potential of nucleobases. The quantum yield of the one-photon process determined for telomeric G-quadruplexes (TEL25/Na+) is (5.2 ± 0.3) × 10−3, significantly higher than that found for duplexes containing in their structure GGG and GG sequences, (2.1 ± 0.4) × 10−3. The radical population is quantified in respect of the ejected electrons. Deprotonation of radical cations gives rise to (G-H1)• and (G-H2)• radicals for duplexes and G-quadruplexes, respectively. The lifetimes of deprotonated radicals determined for a given secondary structure strongly depend on the base sequence. The multiscale non-exponential dynamics of these radicals are discussed in terms of inhomogeneity of the reaction space and continuous conformational motions. The deviation from classical kinetic models developed for homogeneous reaction conditions could also be one reason for discrepancies between the results obtained by photoionization and indirect oxidation, involving a bi-molecular reaction between an oxidant and the nucleic acid.

Three-stage phase separation kinetics in a model liquid binary mixture: A computational study

We study microscopic aspects of initial phase separation through atomistic molecular dynamics simulation of a structure breaking liquid binary mixture. We find that the phase separation kinetics in a fluid binary mixture model system can indeed be unusual. It can be fast, with a crossover from a pronounced exponential to non-exponential and non-linear dynamics. An important outcome of this work is the quantification of time scales involved in phase separation kinetics at an early stage. The initial exponential phase separation is complete within ∼100 ps. The initial phase separation involves aggregation of small droplets that form rapidly after the quench. This is followed by segregation that gives rise to pattern formation with multiple bands of segregated species. During this initial phase, a particle is found to have moved only about ∼5 molecular diameters. The next stage is slower and characterized by break-up and disappearance of small islands of species trapped inside the domains of other species of the binary mixture. The phase separation in this second stage is highly non-exponential and power-law-like. We identify a new feature in the very late stage of phase separation kinetics that seems to have eluded previous attention, the smoothing of the rugged interface between the two species. This is opposite to the roughening transition one finds on the surface of solids in contact with its vapor phase. The present atomistic simulation provides a molecular picture in terms of molecular motions and displacements.

Enzyme functioning: Along the lines of nonequilibrium phase transitions

In recent decades the mechanisms of regulation of enzyme activity attract growing attention. Even the cornerstone schemes and notions of enzymology, such as the Michaelis-Menten scheme or cooperativity, undergo permanent refinements and revisions. Theoretical interpretations of new data, however, remain within the standard chemical kinetics framework. Although this admits qualitative explanations of some significant regulation phenomena like performance of ‘hysteretic’ enzymes, many characteristic features of enzymatic reactions (essentially nonexponential kinetics, dependence of rate constants on substrate concentration, and others) can hardly be accommodated. On the other hand, the inherent factors of enzyme functioning – flux conditions, slow structural changeability, nonlinear feedback between fast reaction acts and slow conformational rearrangements – provide a natural basis for self-organization phenomena. While their typical manifestations are related to macroscopic systems, the enzyme functioning can serve as an example of self-organization on the lowest possible level, down to single molecules. We put forward this concept of dynamical molecular self-organization in early 90’s and then developed its theoretical and experimental substantiations with the example of primary reactions of photosynthesis. In this work, I apply it to the Michaelis-Menten scheme. Within this paradigm, suitable for either ensemble or single molecule conditions, the stable functional regimes of enzyme operation emerge and interchange/coexist due to analogues of nonequilibrium phase transitions determined by the strength of substrate-conformation interaction and controlled by substrate concentration. In particular, the phenomena like ‘monomeric cooperativity’ or self-inhibition naturally follow from even the simplest two-state model of an enzyme with an only binding site.

A reactive force field molecular dynamics simulation of the dynamic properties of hydrogen bonding in supercritical water

Dynamic properties especially lifetime and connectivity of hydrogen bonds in supercritical water play essential roles in physical and chemical properties in aqueous systems. Although average lifetime of hydrogen bonds in supercritical water derived using computer simulation has been reported, detailed distribution and survival analyses have yet to be performed. Here, we performed classical molecular dynamics (MD) simulations with a recently developed reactive force field method to investigate the detailed dynamic process of hydrogen bond forming, cracking, re-bonding and exchanging in supercritical water. We analyzed the MD-calculated trajectories with a sample interval equal to the MD time-step to obtain detailed lifetime information. The calculated lifetime distributions indicate that hydrogen bonds in supercritical water exhibit the newborn instability which is independent of temperature. Supercritical water shows non-exponential kinetics with a gradually decreasing failure rate. We also investigated the connectivity of hydrogen bonding by analyzing the fractions of water molecules involved in a certain number of hydrogen bonds and found that the fractions at high densities remarkably deviate from the widely applied simple binomial distribution. On the basis of the assumption that the maximal coordination number can be a non-integer w, we proposed a non-integer binomial distribution with a maximum coordination number of 4.4 for an enhanced prediction.

What can the use of well-defined statistical functions of pollutants sorption kinetics teach us? A case study of cyanide sorption onto LTA zeolite nanoparticles

Abstract This study demonstrates that the introduction of a fractal time yields a more clear description of the kinetics in the case of the aqueous phase sorption of cyanide onto LTA zeolite nanoparticles. The two best-known fractal models, the Brouers-Sotolongo ( B S f ( n , α ) ) fractal model and kinetics formula based on the Riemann–Liouville fractional derivative theory, are compared. These two models give very similar results and we present arguments for our preference for the former one. Another commonly employed model, the Elovich equation, is discussed. We justify that its continued use in its approximate form is currently not required, as it is inaccurate particularly for small uptake. Its exact expression produces results similar to the two former ones only for part of the kinetics curve, but it is unsuitable as it asymptotically yields an infinite value of the uptake. Besides, an extension of the B S f ( n , α ) is proposed by introducing a time-dependent fractal coefficient, which more clearly explains the evolution of the kinetics in detail and enables a more accurate calculation of the time-dependent rate, typical of non-exponential kinetics. The results reveal that, in this particular case of cyanide sorption onto zeolite, this model is the best fit to the experimental kinetics data.

Photophysical Model for Non-Exponential Relaxation Dynamics in Hybrid Perovskite Semiconductors

The photoluminescence (PL) decay of hybrid halide perovskite single crystals (MAPbX3, MA = CH3NH3+, Pb = Pb2+, X = Br–, and I–) is measured over 4 orders of magnitude in intensity over the time scales of 100s of nanoseconds to a few microseconds. This long PL decay is non-exponential, suggesting the presence of a distribution of carrier relaxation times. Spectro-temporal studies show that the emission peak red-shifts with increasing time. The physics of this problem is closely related to donor–acceptor pair recombination in crystalline semiconductors and recombination in a-Si:H. Based on these models, we present a simple model to account for the recombination dynamics in the perovskite systems. This model also accounts for the fluence dependence of the recombination kinetics. In this model, a fraction of the photogenerated electrons and holes are trapped in localized states. The electrons tunnel to the hole sites for recombination. The broad distribution of lifetimes is a consequence of the fact that the ...

Stretched-to-compressed-exponential crossover observed in the electrical degradation kinetics of some spinel-metallic screen-printed structures

Thermally-induced (170 °C) degradation-relaxation kinetics is examined in screen-printed structures composed of spinel Cu0.1Ni0.1Co1.6Mn1.2O4ceramics with conductive Ag or Ag-Pd layered electrodes. Structural inhomogeneities due to Ag and Ag-Pd diffusants in spinel phase environment play a decisive role in non-exponential kinetics of negative relative resistance drift. If Ag migration in spinel is inhibited by Pd addition due to Ag-Pd alloy, the kinetics attains stretched exponential behavior with ∼0.58 exponent, typical for one-stage diffusion in structurally-dispersive media. Under deep Ag penetration into spinel ceramics, as for thick films with Ag-layered electrodes, the degradation kinetics drastically changes, attaining features of two-step diffusing process governed by compressed-exponential dependence with power index of ∼1.68. Crossover from stretched- to compressed-exponential kinetics in spinel-metallic structures is mapped on free energy landscape of non-barrier multi-well system under strong perturbation from equilibrium, showing transition with a character downhill scenario resulting in faster than exponential decaying.

Nonexponential Kinetics of Loop Formation in Proteins and Peptides: A Signature of Rugged Free Energy Landscapes?

The kinetics of loop formation, i.e., the occurrence of contact between two atoms of a polypeptide, remains the focus of continuing interest. One of the reasons is that contact formation is the elementary event underlying processes such as folding and binding. More importantly, it is experimentally measurable and can be predicted theoretically for ideal polymers. Deviations from single exponential kinetics have sometimes been interpreted as a signature of rugged, protein-like, free energy landscapes. Here we present simulations, with different atomistic models, of short peptides with varied structural propensity, and of a structured protein. Results show exponential contact formation kinetics (or relaxation) at long times, and a power law relaxation at very short times. At intermediate times, a deviation from either power law or simple exponential kinetics is observed that appears to be characteristic of polypeptides with either specific or nonspecific attractive interactions but disappears if attractive interactions are absent. Our results agree with recent experimental measurements on peptides and proteins and offer a comprehensive interpretation for them.

Role of Proton Diffusion in the Nonexponential Kinetics of Proton-Coupled Electron Transfer from Photoreduced ZnO Nanocrystals.

Experiments have suggested that photoreduced ZnO nanocrystals transfer an electron and a proton to organic radicals through a concerted proton-coupled electron transfer (PCET) mechanism. The kinetics of this process was studied by monitoring the decay of the absorbance that reflects the concentration of electrons in the conduction bands of the nanocrystals. Interestingly, this absorbance exhibited nonexponential decay kinetics that could not be explained by heterogeneities of the nanoparticles or electron content. To determine if proton diffusion from inside the nanocrystal to reactive sites on the surface could lead to such nonexponential kinetics, herein this process is modeled using kinetic Monte Carlo simulations. These simulations provide the survival probability of a proton hopping among bulk, subsurface, and surface sites within the nanocrystal until it reaches a reactive surface site where it transfers to an organic radical. Using activation barriers predominantly obtained from periodic density functional theory, the simulations reproduce the nonexponential decay kinetics. This nonexponential behavior is found to arise from the broad distribution of lifetimes caused by different types of subsurface and surface sites. The longer lifetimes are associated with the proton becoming temporarily trapped in a subsurface site that does not have direct access to a reactive surface site due to capping ligands. These calculations suggest that movement of the protons rather than the electrons dominate the nonexponential kinetics of the PCET reaction. Thus, the impact of both bulk and surface properties of metal-oxide nanoparticles on proton conductivity should be considered when designing heterogeneous catalysts.

Observation of First-Order, Marcus Normal, Interfacial Electron Transfer at Dye-Sensitized TiO2 Interfaces

Dye-sensitized solar cells combine the robustness of nanocrystalline TiO2 with molecular dyes or “sensitizers” that absorb broadly across the visible. The surface anchored sensitizer undergoes a photo-induced electron transfer to the TiO2 acceptor states and is subsequently regenerated by a redox mediator present in the external electrolyte. Efficient light-to-energy conversion requires that the injected electron be transported to the external circuit prior to recombination with the oxidized redox mediator. Despite the importance of this unwanted recombination reaction the mechanism remains elusive. This is due in part to complex kinetics that usually cannot be described with first- or second order kinetic models, but satisfactory fitting instead require sums of exponentials or stretched exponential functions. The non-exponential kinetics are likely due to transport of the injected electron and/or the electron acceptor prior to recombination. Here we report charge recombination studies of charge recombination to a family of oxidized amines that provide compelling evidence for first-order kinetics with abstracted rate constants that show a driving force dependence. The recombination reaction between nanocrystalline anatase TiO2 and a series of symmetrically substituted triphenylamine (TPA) redox mediators that possessed electron donating or withdrawing groups at the para-position of the phenyl rings was quantified on nanosecond and longer time scales. The groups utilized afforded an ~0.5 V change in TPA+/0 reduction potential. The nanocrystalline TiO2 thin films were sensitized by a ruthenium polypyridyl complex bearing either a carboxylic or a phosphonic acid functional group for surface anchoring. Pulsed light excitation of the film resulted in excited-state injection to yield TiO2(e-)|S+. Sensitizer regeneration and the subsequent charge recombination reaction TiO2(e-) + TPA+ → TiO2 + TPA were then monitored spectroscopically. Concentration dependent data revealed that recombination could be described by a distribution of first-order rate constants that became strictly first-order when the driving force was larger. The reaction rate increased with more positive TPA+/0 reduction potentials, consistent with electron transfer in the Marcus normal region. The recombination reaction was monitored as a function of temperature, and an Arrhenius analysis provided activation energies. Interestingly, the same activation barrier was abstracted for all the TPA derivatives, suggesting that a common rate limiting step exists for all these reactions. A model for recombination consistent with both the thermodynamics and activational parameters will be presented.

Mechanochemical Kinetics in Elastomeric Polymer Networks: Heterogeneity of Local Forces Results in Nonexponential Kinetics.

A common approach to inducing selective mechanochemical transformations relies on embedding the target molecules (called mechanophores) within elastomeric polymer networks. Mechanical properties of such elastomers can also be modulated through the mechanochemical response of the constituent polymer chains. The inherent randomness in the molecular structure of such materials leads to heterogeneity of the local forces exerted on individual mechanophores. Here we use coarse-grained simulations to study the force distributions within random elastomeric networks and show that those distributions are close to exponential regardless of the applied macroscopic load, entanglement effects, or network parameters. Exponential form of the distribution allows one to completely characterize the mechanophore kinetics in terms of the mean value of the force. At the same time, heterogeneity of the local force affects the kinetics qualitatively: While a narrow force distribution around the mean would lead to exponential kinetics, exponential force distribution results in highly nonexponential kinetics, with a fast kinetic phase involving highly loaded molecules, followed by a slow phase dominated by unloaded molecules.

Single-Molecule Fluorescence Spectroscopy of Perylene Diimide Dyes in a γ-Cyclodextrin Film: Manifestation of Photoinduced H-Atom Transfer via Higher Triplet (n, π*) Excited States.

Supramolecular complexation of γ-cyclodextrin (γ-CD) with N,N'-bis(2,6-dimethylphenyl)perylene-3,4,9,10-tetracarboxylic diimide (DMP-PDI) or N,N'-bis(2,6-dioctyl)perylene-3,4,9,10-tetracarboxylic diimide (C8-PDI) dye in an aqueous solution and in a γ-CD solid film were investigated via ensemble and single-molecule fluorescence spectroscopy. These two perylene diimide derivatives possess almost the same electronic structure but have different terminal functional groups. This structural difference leads to formation of an inclusion complex of γ-CD with DMP-PDI but not with C8-PDI in aqueous solution. In a γ-CD solid film, the distributions of the wavelengths of emission maximum (λmaxem) are strikingly different between these two dyes; a much narrower and blue-shifted λmaxem distribution was observed for C8-PDI relative to DMP-PDI. This difference is attributed to the fact that the C8-PDI molecules are bound at the γ-CD/glass interface as a result of spin-coating of the sample solution, whereas the DMP-PDI molecules form 1:1 and 1:2 inclusion complexes with conformational heterogeneities in the film. In comparison to the case for C8-PDI, more frequent on-off blinking events were observed for DMP-PDI. The blinking statistics of DMP-PDI in the γ-CD film exhibit both single-exponential and nonexponential (i.e., dispersive) kinetics, revealed by robust statistical analysis. Energetic consideration with the aid of theoretical calculations suggests that the underlying photophysics most probably involves hydrogen atom transfer (HAT) between the DMP-PDI guest and γ-CD host via higher excited (n, π*) triplet states. The hypothesis of HAT in the inclusion complex reasonably explains the experimental results; however, a charge transfer hypothesis cannot explain the results. The dispersive kinetics is attributable to the effect of thermal fluctuation in the forward and backward HAT reactions.

Nonparametric analysis of nonexponential and multidimensional kinetics. I. Quantifying rate dispersion, rate heterogeneity, and exchange dynamics.

The quantification of nonexponential (dispersed) kinetics has relied on empirical functions, which yield parameters that are neither unique nor easily related to the underlying mechanism. Multidimensional kinetics provide more information on dispersed processes, but a good approach to their analysis is even less clear than for standard, one-dimensional kinetics. This paper is the first in a series that analyzes kinetic data in one or many dimensions with a scheme that is nonparametric: it quantifies nonexponential decays without relying on a specific functional form. The quantities obtained are directly related to properties of the mechanism causing the rate dispersion. Log-moments of decays, which parallel the standard moments of distributions (mean, standard deviation, etc.), are introduced for both one- and multi-dimensional decays. Kinetic spectra are defined to visualize the data. The utility of this approach is demonstrated on a simple, but general, model of dispersed kinetics-a nonexponential homogeneous decay combined with slowly exchanging rate heterogeneity. The first log-moments give a geometric-mean relaxation time. Second log-moments quantify the magnitude of rate dispersion, the fraction of the dispersion due to heterogeneity, and the dynamics of exchange between different rate subensembles. A suitable combination of these moments isolates exchange dynamics from three-dimensional kinetics without contamination by the rate-filtering effects that were identified in a recent paper [M. A. Berg and J. R. Darvin, J. Chem. Phys. 145, 054119 (2016)].

Thermodynamics and Kinetics of Globular Polymers under an Applied Force

We examine the problem of a polymer chain, folded into a globule in poor solvent, subjected to a constant tensile force. This system represents a Gibbs thermodynamic ensemble, and is useful for analysing force-clamp AFM measurements - now common in molecular biophysics. Using a basic Flory mean-field theory, we account for surface interactions of monomers with solvent. Under an increasing tensile force a first-order phase transition occurs. The compact globule ruptures and fully extends, in an ‘all-or-nothing’ unfolding event. This contrasts with the regime of imposed extension, first studied by Halperin and Zhulina, where there is a regime of coexistence of a partial globule with an extended chain segment. We relate the transition forces in this problem to the solvent quality and degree of polymerisation, and find analytical expressions for the energy barriers present in the problem. From these, we analyse the kinetic problem of a force-ramp experiment, showing that the rupture force depends on the rate of loading. We then change the system by inserting a super-hydrophobic ‘core’ at a given point in the polymer chain. This acts as a crude model for a large class of folded biomolecules with hydrophobic or hydrogen-bonded cores. Again, we consider applying a constant tensile force. Introducing a ‘core’ leads to an intrinsic (quenched) stochastic variation in the unfolding rate, even when the positions of the added monomers are fixed along the sequence. This gives rise to non-exponential population dynamics, which is consistent with a variety of experimental data. It does not need any structural disorder of the type thought to be at the origin of non-exponential relaxation laws.

Nonexponential kinetics of ion pair dissociation in electrofreezing water.

Temporally- or spatially-heterogeneous environments can participate in many kinetic processes, from chemical reactions and self-assembly to the forced dissociation of biomolecules. Here, we simulate the molecular dynamics of a model ion pair forced to dissociate in an explicit, aqueous solution. Triggering dissociation with an external electric field causes the surrounding water to electrofreeze and the ion pair population to decay nonexponentially. To further probe the role of the aqueous environment in the kinetics, we also simulate dissociation events under a purely mechanical force on the ion pair. In this case, regardless of whether the surrounding water is a liquid or already electrofrozen, the ion pair population decays exponentially with a well-defined rate constant that is specific to the medium and applied force. These simulation data, and the rate parameters we extract, suggest the disordered kinetics in an electrofreezing medium are a result of the comparable time scales of two concurrent processes, electrofreezing and dissociation.

Vibrational population dynamics of methyl formate in CCl4 solution

The vibrational relaxation of the CO stretching mode of HCO2CH3 in CCl4 solution was measured using femtosecond infrared spectroscopy. Time-resolved spectra after excitation of the CO stretching mode revealed a new absorption other than that of the excited CO mode. Transient signals for the fundamental band and the absorption of the excited CO mode showed nonexponential kinetics having a subpicosecond component. A three-state kinetic scheme showed that the excited CO mode decayed via the v1 state of the CO mode. The subpicosecond decay component might be due to fast equilibration of the excited state with the adjacent 2nd excited state of the OCH3 stretching mode. Or it might be caused by solvent memory effects, which may cause unusually rapid nonexponential absorption decay. As demonstrated herein, time-resolved infrared spectra with high spectral resolution and sensitivity can provide detailed relaxation pathways of vibrational energy.

Non-exponential kinetics of unfolding under a constant force.

We examine the population dynamics of naturally folded globular polymers, with a super-hydrophobic "core" inserted at a prescribed point in the polymer chain, unfolding under an application of external force, as in AFM force-clamp spectroscopy. This acts as a crude model for a large class of folded biomolecules with hydrophobic or hydrogen-bonded cores. We find that the introduction of super-hydrophobic units leads to a stochastic variation in the unfolding rate, even when the positions of the added monomers are fixed. This leads to the average non-exponential population dynamics, which is consistent with a variety of experimental data and does not require any intrinsic quenched disorder that was traditionally thought to be at the origin of non-exponential relaxation laws.

Relaxation and diffusion models with non-singular kernels

Abstract Anomalous relaxation and diffusion processes have been widely quantified by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to its limitation in describing different kinds of non-exponential decays (e.g. stretched exponential decay). Meanwhile, many efforts by mathematicians and engineers have been made to overcome the singularity of power function kernel in its definition. This study first explores physical properties of relaxation and diffusion models where the temporal derivative was defined recently using an exponential kernel. Analytical analysis shows that the Caputo type derivative model with an exponential kernel cannot characterize non-exponential dynamics well-documented in anomalous relaxation and diffusion. A legitimate extension of the previous derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel. Numerical tests show that the Caputo type derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel, implying the potential applicability of the new derivative in quantifying real-world, anomalous relaxation and diffusion processes.

Dynamic disorder can explain non-exponential kinetics of fast protein mechanical unfolding.

Protein unfolding often does not obey a simple two-state behavior. Previous single molecule force spectroscopy studies demonstrated stretched exponential kinetics of protein unfolding under a constant pulling force, the molecular origin of which remains subject to debate. We here set out to extensively sample the mechanical unfolding of ubiquitin and NuG2 by Molecular Dynamics (MD) simulations. Both proteins show kinetics best fit by stretched exponentials, with stretching exponents similar to those found in experiments, even though static disorder is absent in our short MD simulations. Instead, we can ascribe non-exponential kinetics to dynamic disorder, due to conformational fluctuations on the nanosecond timescale. Our study highlights the general role of dynamic disorder in protein kinetics on a broad range of time scales even including those probed in MD simulations.