Long-time Kinetics Research Articles

Effect of an external electric field on the diffusion-influenced geminate reversible reaction of a neutral particle and a charged particle in three dimensions. IV. Excited-state ABCD reaction

In the presence of an external electric field, an excited-state A + B(*q) <−> C(*q) + D diffusion-influenced geminate reversible reaction of a neutral particle and a charged particle, with two unimolecular decay rates and contact quenching processes, is investigated in three dimensions. The probability density functions to find individual particles, rates of reactions, and survival probabilities are analytically derived in the Laplace domain and the long-time kinetics is resolved. The probability density functions to find the particles and the rates of reactions in a scaled form exhibit a kinetic transition behavior from a t(-3/2) power law to t(-3/2)e(t) increase with the increase of external fields. The scaled survival probabilities present a kinetic transition behavior of t(-3/2) → constant → exponential with the increase of field strengths. The critical fields are found to determine the kinetic transition behaviors.

Effect of an external electric field on the diffusion-influenced geminate reversible reaction of a neutral particle and a charged particle in three dimensions. III. Ground-state ABCD reaction

In the presence of an external electric field, the ground-state A+B(q)<->C(q)+D diffusion-influenced reversible reaction for a geminate pair, a neutral and a charged particle, is investigated in three dimensions. The probability density functions, the rates of reactions, and the survival probabilities of individual particles are analytically derived in the Laplace domain in terms of series solutions. The long-time kinetics of probability density functions and rates of reactions in rescaled forms shows a kinetic transition behavior from a t(-3∕2) power law to a t(-3∕2)e(t) increase when the condition D1F1 (2)≤D2F2 (2), which depends on the diffusivities of particles and the external electric fields, changes to D1F1 (2)>D2F2 (2). In the transition region D1F1 (2)=D2F2 (2), the long-time behavior also shows a t(-3∕2) power law decay but with a different value of the prefactor. The rescaled survival probabilities only exhibit an exponentially increasing behavior at long times with no dependence on the various values of parameters.

Computation of transit times using the milestoning method with applications to polymer translocation

Milestoning is an efficient approximation for computing long-time kinetics and thermodynamics of large molecular systems, which are inaccessible to brute-force molecular dynamics simulations. A common use of milestoning is to compute the mean first passage time (MFPT) for a conformational transition of interest. However, the MFPT is not always the experimentally observed timescale. In particular, the duration of the transition path, or the mean transit time, can be measured in single-molecule experiments, such as studies of polymers translocating through pores and fluorescence resonance energy transfer studies of protein folding. Here we show how to use milestoning to compute transit times and illustrate our approach by applying it to the translocation of a polymer through a narrow pore.

Long-time crystallization kinetics in zinc-neutralized ethylene–methacrylic acid ionomers

The classic paper by Flory [1] regarding the crystallization of random copolymers presents a theory where fractional crystallization depends only on molar substitution level if the comonomer cannot be incorporated into the crystal structure of the crystallizable comonomer. A rather large number of indirect experiments on ethylene copolymers since that publication nearly 60 years ago are consistent with this hypothesis; for example the crystallinity content in ethylene copolymers depends to a first approximation only on the level of molar substitution and not the identity of the comonomer. Ethylene–methacrylic acid copolymer ionomers offer a unique opportunity to test this hypothesis directly, since it is possible to vary crystallization kinetics simply by varying the neutralization cation or neutralization level, without changing the comonomer content or its distribution along the polymer backbone. After annealing times as long as 6 months, it was found that the constraints caused by the phase-separating ionic groups lead to significantly lower crystallinity levels versus the case where such constraints were not present. Although disproving Flory's theory in this particular case, the result was not surprising since Flory's original paper certainly did not consider the case where the other monomer can phase separate. One surprising result of this study was that the rates of formation of secondary crystals relative to the crystallization amount at infinite time was independent of neutralization level for the phase-separated ionomers; in other words the Avrami constants n and K were independent of neutralization level for the secondary crystals.

Long-time decay kinetics of geminate electron–hole pairs in donor–acceptor heterojunction systems

We present an analytical theory of geminate charge recombination in donor–acceptor heterojunction systems based on the Smoluchowski equation. We show that the differential equation describing this problem is mathematically equivalent to that describing geminate recombination in a four-dimensional anisotropic system. By combining this result with already available analytical results for n-dimensional homogeneous systems, we predict that the survival probability of electron–hole pairs in heterojunction systems decays as t−1 at long times. We prove this by analyzing the results of kinetic Monte Carlo simulations. We also explain clearly why the ultimate escape probability is increased by introducing the heterojunction.

METAGUI. A VMD interface for analyzing metadynamics and molecular dynamics simulations

We present a new computational tool, METAGUI, which extends the VMD program with a graphical user interface that allows constructing a thermodynamic and kinetic model of a given process simulated by large-scale molecular dynamics. The tool is specially designed for analyzing metadynamics based simulations. The huge amount of diverse structures generated during such a simulation is partitioned into a set of microstates (i.e. structures with similar values of the collective variables). Their relative free energies are then computed by a weighted-histogram procedure and the most relevant free energy wells are identified by diagonalization of the rate matrix followed by a commitor analysis. All this procedure leads to a convenient representation of the metastable states and long-time kinetics of the system which can be compared with experimental data. The tool allows to seamlessly switch between a collective variables space representation of microstates and their atomic structure representation, which greatly facilitates the set-up and analysis of molecular dynamics simulations. METAGUI is based on the output format of the PLUMED plugin, making it compatible with a number of different molecular dynamics packages like AMBER, NAMD, GROMACS and several others. The METAGUI source files can be downloaded from the PLUMED web site ( http://www.plumed-code.org). Program summary Program title: METAGUI Catalogue identifier: AEKH_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEKH_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: GNU General Public License version 3 No. of lines in distributed program, including test data, etc.: 117 545 No. of bytes in distributed program, including test data, etc.: 8 516 203 Distribution format: tar.gz Programming language: TK/TCL, Fortran Computer: Any computer with a VMD installation and capable of running an executable produced by a gfortran compiler Operating system: Linux, Unix OS-es RAM: 1 073 741 824 bytes Classification: 23 External routines: A VMD installation ( http://www.ks.uiuc.edu/Research/vmd/) Nature of problem: Extract thermodynamic data and build a kinetic model of a given process simulated by metadynamics or molecular dynamics simulations, and provide this information on a dual representation that allows navigating and exploring the molecular structures corresponding to each point along the multi-dimensional free energy hypersurface. Solution method: Graphical-user interface linked to VMD that 1. clusterizes the simulation trajectories in the space of a set of collective variables and assigns each frame to a given microstate, 2. determines the free energy of each microstate by a weighted histogram analysis method, and 3. identifies the most relevant free energy wells (kinetic basins) by diagonalization of the rate matrix followed by a commitor analysis. Restrictions: Input format files compatible with PLUMED and all the MD engines supported by PLUMED and VMD. Running time: A few minutes.

Boxed Molecular Dynamics: Decorrelation Time Scales and the Kinetic Master Equation.

A number of methods proposed in the past few years have been aimed at accelerating the sampling of rare events in molecular dynamics simulations. We recently introduced a method called Boxed Molecular Dynamics (BXD) for accelerating the calculation of thermodynamics and kinetics ( J. Phys. Chem. B 2009 , 113 , 16603 - 16611 ). BXD relies upon confining the system in a series of adjacent "boxes" by inverting the projection of the system velocities along the reaction coordinate. The potential of mean force along the reaction coordinate is obtained from the mean first passage times (MFPTs) for exchange between neighboring boxes, simultaneously providing both kinetics and thermodynamics. In this paper, we investigate BXD in the context of its natural relation to a kinetic master equation and show that the BXD first passage times (FPTs) include different time scales-a fast short time decay due to correlated dynamical motion and slower long time decay arising from phase space diffusion. Correcting the FPTs to remove the fast correlated motion yields accurate thermodynamics and master equation kinetics. We also discuss interrelations between BXD and a recently described Markovian milestoning technique and use a simple application to show that, despite each method producing distinct nonstatistical effects on time scales on the order of dynamical decorrelation, both yield similar long-time kinetics.

Long-time growth kinetics of first order phase transitions in the presence of a boundary layer

The late stage growth mechanism for a first order phase transition, either through nucleation growth or spinodal decomposition, is well understood to be an Ostwald ripening or coarsening process, in which larger domains grow at the expense of smaller ones. The growth kinetics in this regime was shown by Lifshitz and Slyozov to follow at(1/3) law. However, the kinetics is altered if there exists a barrier ahead of the growth front, irrespective of the physical origin of the boundary layer. We present an analytic calculation for the growth kinetics in the presence of a boundary layer, showing that in the limit of barrier-dominated growth, the domains grow with at(1/2) law. This result holds true in the dilute regime independent of whether the growing nuclei are spherical or cylindrical.

Self-similar assemblies of globular whey proteins at the air–water interface: Effect of the structure

We investigated the structure of heat-induced assemblies of whey globular proteins using small angle neutron scattering (SANS), static and dynamic light scattering (SLS and DLS), and cryogenic transmission electron microscopy (Cryo-TEM). Whey protein molecules self-assemble in fractal aggregates with a structure density depending on the electrostatic interactions. We determined the static and dynamic properties of interfacial layer formed by the protein assemblies, upon adsorption and spreading at the air–water interface using surface film balance and interfacial dilatational rheology. Upon spreading, all whey protein systems show a power-law scaling behavior of the surface pressure versus concentration in the semi-dilute surface concentration regime, with an exponent ranging from 5.5 to 9 depending on the electrostatic interactions and the aggregation state. The dilatational modulus derived from surface pressure isotherms shows a main peak at 6–8 mN/m, generally considered to be the onset of a conformational change in the monolayer, and a second peak or a shoulder at 15 mN/m. Long-time adsorption kinetics give similar results for both the native whey proteins and the corresponding self-similar assemblies, with a systematic effect of the ionic strength.

Mechanical Folding Kinetics of RNA Hairpins: a New Computational Approach

From the distribution of the low-lying states on the energy landscape, we recently developed a new computational method for the prediction of RNA folding kinetics [1]. The method can treat long sequences because it is based on a small ensemble of seed states. Additionally, the method is based on an analytic formalism and thus enables predictions for long-time folding kinetics. Applications of the new kinetic model to mechanical folding of RNA hairpins reveal distinct kinetic behaviors for a wide range of different force regimes, from zero force to forces much stronger than the critical force for the folding-unfolding transition.

Long-time kinetics of the excited-state association–dissociation reaction with differentlifetimes

We consider the long-time behaviour of the reversible diffusion-influenced reaction when A and C are immobile electronically excited states andB particles arein excess. A andC have differentlifetimes. When C decays faster than A, the two leading terms of the asymptotic behaviour are obtained analytically by mappingthis problem onto an effective irreversible reaction. The first term is exponential, and itcoincides with that obtained from conventional rate equations with steady-statediffusion-influenced rate constants. The second term in the exponent is , as in the Smoluchowski irreversible kinetics.

Majority rule dynamics in finite dimensions

We investigate the long-time behavior of a majority rule opinion dynamics model in finite spatial dimensions. Each site of the system is endowed with a two-state spin variable that evolves by majority rule. In a single update event, a group of spins with a fixed (odd) size is specified and all members of the group adopt the local majority state. Repeated application of this update step leads to a coarsening mosaic of spin domains and ultimate consensus in a finite system. The approach to consensus is governed by two disparate time scales, with the longer time scale arising from realizations in which spins organize into coherent single-opinion bands. The consequences of this geometrical organization on the long-time kinetics are explored.

Hopping photoconduction and its long-time kinetics in a heterosystem with Ge quantum dots in Si

The effect of interband-transition-inducing illumination on the hole hopping conduction along a two-dimensional array of Ge quantum dots in Si was studied. It is found that the photoconductance has either positive or negative sign depending on the initial filling of quantum dots with holes. In the course of illumination and after switching off the light, long-time photoconduction kinetics was observed (102−104s at T=4.2 K). The results are discussed in terms of a model based on the spatial separation of nonequilibrium electrons and holes in a potential relief formed by positively charged dots. The effect of equalization of potential barrier heights as a result of photohole capture by the charged quantum dots during the process of illumination and relaxation is suggested as an additional factor for explaining the phenomenon of persistent conduction.

Linking atomistic and mesoscale simulations of nanocrystalline materials: quantitative validation for the case of grain growth

Using grain growth in nanocrystalline palladium as a simple case study, we demonstrate how a novel mesoscale approach for simulating microstructural evolution in polycrystalline materials can be validated directly against atomic-level simulations of the same system. We first describe molecular dynamics simulations of grain growth in a columnar model microstructure. The atomic-level insights into the grain-growth mechanism gained from these simulations, particularly in the role of grain rotations, are captured theoretically for incorporation into the mesoscale approach, in which the objects evolving in space and time are the grain boundaries and grain junctions rather than the atoms. With all the input parameters to the mesoscale being physically well defined and obtained directly from the atomic-level simulations, the mesoscale simulations are fully prescribed. We find that the morphology of the mesoscale system evolves in an almost identical manner with that of the molecular dynamics simulation, demonstrating that the length- and time-scale linking has been performed correctly. When applied to systems containing large numbers of grains, the now validated mesoscale simulation approach allows the growth topology and long-time growth kinetics to be determined. As an outlook, we describe how the effects of applied stress can be incorporated.

The kinetic spherical model in a magnetic field

The long-time kinetics of the spherical model in an external magnetic field and below the equilibrium critical temperature is studied. The solution of the associated stochastic Langevin equation is reduced exactly to a single non-linear Volterra equation. For a sufficiently small external field, the kinetics of the magnetization reversal transition from the metastable to the ground state is compared to the ageing behaviour of coarsening systems quenched into the low-temperature phase. For an oscillating magnetic field and below the critical temperature, we find evidence for the absence of the frequency-dependent dynamic phase transition, which was observed previously to occur in Ising-like systems.

Competition between dynamic and thermal relaxation in non-equilibrium critical spin systems above the critical point

We study the long-time behaviour and the spatial correlations of a simple ferromagnetic spin system whose kinetics is governed by a thermal bath with a time-dependent temperature which is characterized by a given external relaxation time τ .E xact results are obtained in the framework of the spherical model in d dimensions. In the paramagnetic phase, the long-time kinetics is shown to depend crucially on the ratio between τ and the internal equilibration time τeq .I fτ τeq ,t he mode lr elaxes rapidly towards an equilibrium state but there appear transient and spatially oscillating contributions in the spin–spin correlation function. On the other hand, if τ � τeq the system is clamped and its time evolution may be delayed with respect to that of the heat bath. For waiting times s such that τ � s � τeq ,a quasi-stationary state is found where th efl uctuation–dissipation theorem does not hold.

Clustering under short-range finite interactions.

In this paper the aggregation of surface modified colloidal particles is presented, paying special attention to the cluster structure and growth. The surface was modified by adsorbing bovine serum albumin (BSA). The interaction potential develops a minimum of restricted depth, weakening the clusters which subsequently restructure and form more compact morphologies. This minimum is responsible for the reversibility of the aggregation processes (this is an important difference between diffusion-limited cluster aggregation and reaction-limited cluster aggregation). The energy minimum is associated with the presence of a steric term in the energy balance, which depends on the size of the adsorbed molecules. BSA molecules with different sizes were employed to test this point. In addition, the short-range interaction seems not to affect significantly the paths of approximating particles, since the aggregation of the clusters at long times is independent of the size of these particles. The long-time kinetics was interpreted in the frame of dynamic scaling concepts. A kinetics model, including surface-surface, protein-surface, and protein-protein aggregation, is used to determine the dominant mechanism controlling the aggregation.

Colloid Particle Adsorption on Partially Covered (Random) Surfaces

The random sequential adsorption (RSA) approach was used to model irreversible adsorption of colloid particles at surfaces precovered with smaller particles having the same sign of surface charge. Numerical simulations were performed to determine the initial flux of larger particles as a function of surface coverage of smaller particles θs at various size ratios λ=al/as. These numerical results were described by an analytical formula derived from scaled particle theory. Simulations of the long-time adsorption kinetics of larger particles have also been performed. This allowed one to determine upon extrapolation the jamming coverage θl∞ as a function of the λ parameter at fixed smaller particle coverage θs. It was found that the jamming coverage θl∞ was very sensitive to particle size ratios exceeding 4. Besides yielding θl∞, the numerical simulations allowed one to determine the structure of large particle monolayers at the jamming state which deviated significantly from that observed for monodisperse systems. The theoretical predictions suggested that surface heterogeneity, e.g., the presence of smaller sized contaminants or smaller particles invisible under microscope, can be quantitatively characterized by studying larger colloid particle adsorption kinetics and structure of the monolayer.

Adsorption of colloidal particles by Brownian dynamics simulation: Kinetics and surface structures

Careful control of the microstructure of an adsorbed monolayer of colloidal particles is important for creating nanostructured devices through self-assembly processes. We present a computational model study for self-assembly of colloidal or nanoscale particulate systems. We develop a new technique for simulating colloidal adsorption processes, and we examine the kinetics and the structure formation on the surface. The technique allows the simulation of a nonhomogeneous suspension with an open boundary that is in equilibrium with a bulk suspension of known volume fraction, including the mean-field forces from the bulk solution and particle flux between the simulation box and the bulk. Short-time kinetics follow a power law similar to the case of diffusion-limited adsorption. Long-time kinetics fit a 2/3-power law form [P. Schaaf, A. Johner, and J. Talbot, Phys. Rev. Lett. 66, 1603 (1991)] and kinetic coefficients are calculated. The zeta potential of the particles is the dominant parameter controlling the final surface coverage, but the zeta potential of the adsorbing surface is the dominant control for the ordering of the adsorbed system. Particles with larger Debye layers (lower salt concentrations) order more easily. Jamming limit coverages are compared to existing equivalent hard-disk models and an energetic model. Since the process is kinetically frustrated, particle exclusion effects play a major role in determining coverage as well as structure.

Long-time luminescence kinetics of localized excitons and conduction band edge smearing in ZnSe(1−c)Tec solid solutions

It is shown that the integrated luminescence intensity of localized excitons in solid solutions ZnSe(1−c)Tec has a component slowly decaying with time. After the excitation above the mobility threshold, the long-time intensity decreases exponentially, with a fractional exponent changing from a value corresponding to the critical index of anomalous diffusion to the index of normal diffusion as the temperature increases from 5 to 80 K. This change allows estimation of the energy scale for the fluctuation tail of the conduction band.