Dynamics of Barrierless and Activated Chemical Reactions in a Dispersive Medium within the Fractional Diffusion Equation Approach

Barrierless chemical reactions have often been modeled as a Brownian motion on a one-dimensional harmonic potential energy surface with a position-dependent reaction sink or window located near the minimum of the surface. This simple (but highly successful) description leads to a nonexponential survival probability only at small to intermediate times but exponential decay in the long-time limit. However, in several reactive events involving proteins and glasses, the reactions are found to exhibit a strongly nonexponential (power law) decay kinetics even in the long time. In order to address such reactions, here, we introduce a model of barrierless chemical reaction where the motion along the reaction coordinate sustains dispersive diffusion. A complete analytical solution of the model can be obtained only in the frequency domain, but an asymptotic solution is obtained in the limit of long time. In this case, the asymptotic long-time decay of the survival probability is a power law of the Mittag−Leffler functional form. When the barrier height is increased, the decay of the survival probability still remains nonexponential, in contrast to the ordinary Brownian motion case where the rate is given by the Smoluchowski limit of the well-known Kramers' expression. Interestingly, the reaction under dispersive diffusion is shown to exhibit strong dependence on the initial state of the system, thus predicting a strong dependence on the excitation wavelength for photoisomerization reactions in a dispersive medium. The theory also predicts a fractional viscosity dependence of the rate, which is often observed in the reactions occurring in complex environments.

Dynamics of pulled desorption with effects of excluded-volume interaction: The p-Laplacian diffusion equation and its exact solution

We analyze the dynamics of desorption of a polymer molecule which is pulled at one of its ends with force f, trying to desorb it. We assume a monomer to desorb when the pulling force on it exceeds a critical value f(c). We formulate an equation for the average position of the n-th monomer, which takes into account excluded-volume interaction through the blob-picture of a polymer under external constraints. The approach leads to a diffusion equation with a p-Laplacian for the propagation of the stretching along the chain. This has to be solved subject to a moving boundary condition. Interestingly, within this approach, the problem can be solved exactly in the trumpet, stem-flower and stem regimes. In the trumpet regime, we get tau = tau(0)n(d)(2), where n(d) is the number of monomers that have desorbed at the time tau. tau(0) is known only numerically, but for f close to f(c), it is found to be tau(0) similar to f(c)/(f(2/3) - f(c)(2/3)) If one used simple Rouse dynamics, this result would change to tau similar to f(c)n(d)(2)/(f - f(c)). In the other regimes too, one can find exact solution, and interestingly, in all regimes tau similar to n(d)(2). Copyright (C) EPLA, 2011

An Exact Analysis of Unsteady Convective Diffusion in an Annular Pipe

An exact solution to the unsteady convective diffusion equation for the dispersion of a solute in a fully developed laminar flow in an annular pipe is obtained. Generalized dispersion model which is valid for all time after the injection of solute in the flow is used to evaluate the dispersion coefficients as functions of time. It is observed that the axial dispersion decreases with an increase in the radius of the inner cylinder.

Diffusion of localized optical excitations and trap emission in ruby

Using the critical percolation conductance method the energy-dependent diffusion coefficient associated with thermally assisted transfer of the R1 line excitation between single Cr3+ ions with strain-induced randomness has been calculated in the 4A2 to E(2E) transition energies. For localized states sufficiently far away from the mobility edge the energy transfer is dominated by dipolar interactions, while very close to the mobility edge it is determined by short-range exchange interactions. Using the above energy-dependent diffusion coefficient a macroscopic diffusion equation is solved for the rate of light emission by Cr3+ ion-pair traps to which single-ion excitations are transferred. The dipolar mechanism leads to good agreement with recent measurements of the pair emission rate by Koo et al. (Phys. Rev. Lett., vol.35, p.1669 (1975)) right up to the mobility edge.

The ratio of diffusion coefficient to mobility for slow electrons in dry air

Using Huxley's solution of the diffusion equation for electron-attaching gases, the ratio of diffusion coefficient D to mobility μ for electrons in dry air was measured over the range 3·06 × 10-17 diffusion equation for non-attaching gases.

Computationally efficient optical tomographic reconstructions through waveletizing the normalized quadratic perturbation equation - art. no.61640R

In this paper, we present a wavelet - based approach to solve the non-linear perturbation equation encountered in optical tomography. A particularly suitable data gathering geometry is used to gather a data set consisting of differential changes in intensity owing to the presence of the inhomogeneous regions. With this scheme, the unknown image, the data, as well as the weight matrix are all represented by wavelet expansions, thus yielding the representation of the original non - linear perturbation equation in the wavelet domain. The advantage in use of the non-linear perturbation equation is that there is no need to recompute the derivatives during the entire reconstruction process. Once the derivatives are computed, they are transformed into the wavelet domain. The purpose of going to the wavelet domain, is that, it has an inherent localization and de-noising property. The use of approximation coefficients, without the detail coefficients, is ideally suited for diffuse optical tomographic reconstructions, as the diffusion equation removes most of the high frequency information and the reconstruction appears low-pass filtered. We demonstrate through numerical simulations, that through solving merely the approximation coefficients one can reconstruct an image which has the same information content as the reconstruction from a non-waveletized procedure. In addition we demonstrate a better noise tolerance and much reduced computation time for reconstructions from this approach.

A Reliable method of obtaining Breakthrough Curves of ions in Soils using Transport Equation

Molecular diffusion plays a dominant role in transport of contaminants through fine-grained soils with low hydraulic conductivity. Attenuation processes occur while contaminants travel through the soils. Effective diffusion coefficient (De) is expected to take into consideration various attenuation processes. Effective diffusion coefficient has been considered to develop a general approach for modelling of contaminant transport in soils.The effective diffusion coefficient of sodium in presence of sulphate has been obtained using the column test.The reliability of De, has been checked by comparing theoretical breakthrough curves of sodium ion in soils obtained using advection diffusion equation with the experimental curve.

Generalized eigenvalue decomposition of the field autocorrelation in correlation diffusion of photons in turbid media

We propose a novel numerical method based on a generalized eigenvalue decomposition for solving the diffusion equation governing the correlation diffusion of photons in turbid media. Medical imaging modalities such as diffuse correlation tomography and ultrasound-modulated optical tomography have the (elliptic) diffusion equation parameterized by a time variable as the forward model. Hitherto, for the computation of the correlation function, the diffusion equation is solved repeatedly over the time parameter. We show that the use of a certain time-independent generalized eigenfunction basis results in the decoupling of the spatial and time dependence of the correlation function, thus allowing greater computational efficiency in arriving at the forward solution. Besides presenting the mathematical analysis of the generalized eigenvalue problem on the basis of spectral theory, we put forth the numerical results that compare the proposed numerical method with the standard technique for solving the diffusion equation.

Exponential compact higher order scheme and their stability analysis for unsteady convection-diffusion equations

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Peclet number 0 <= Pe <= 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.

Multiscale modeling of lamellar mesophases

The mesoscale simulation of a lamellar mesophase based on a free energy functional is examined with the objective of determining the relationship between the parameters in the model and molecular parameters. Attention is restricted to a symmetric lamellar phase with equal volumes of hydrophilic and hydrophobic components. Apart from the lamellar spacing, there are two parameters in the free energy functional. One of the parameters, r, determines the sharpness of the interface, and it is shown how this parameter can be obtained from the interface profile in a molecular simulation. The other parameter, A, provides an energy scale. Analytical expressions are derived to relate these parameters to r and A to the bending and compression moduli and the permeation constant in the macroscopic equation to the Onsager coefficient in the concentration diffusion equation. The linear hydrodynamic response predicted by the theory is verified by carrying out a mesoscale simulation using the lattice-Boltzmann technique and verifying that the analytical predictions are in agreement with simulation results. A macroscale model based on the layer thickness field and the layer normal field is proposed, and the relationship between the parameters in the macroscale model from the parameters in the mesoscale free energy functional is obtained.

Cooling of a composite slab

A mixed boundary value problem associated with the diffusion equation, that involves the physical problem of cooling of an infinite parallel-sided composite slab, is solved completely by using the Wiener-Hopf technique. An analytical expression is derived for the sputtering temperature at the quench front being created by a cold fluid moving on the upper surface of the slab at a constant speed v. The dependence of the various configurational parameters of the problem under consideration, on the sputtering temperature, is rather complicated and representative tables of numerical values of this important physical quantity are prepared for certain typical values of these parameters. Asymptotic results in their most simplified forms are also obtained when (i) the ratio of the thicknesses of the two materials comprising the slab is very much smaller than unity, and (ii) the quench-front speed v is very large, keeping the other parameters fixed, in both the cases.

Polymer melt dynamics: Microscopic roots of fractional viscoelasticity

The rheological properties of polymer melts and other complex macromolecular fluids are often successfully modeled by phenomenological constitutive equations containing fractional differential operators. We suggest a molecular basis for such fractional equations in terms of the generalized Langevin equation (GLE) that underlies the renormalized Rouse model developed by Schweizer [J. Chem. Phys. 91, 5802 (1989)]. The GLE describes the dynamics of the segments of a tagged chain under the action of random forces originating in the fast fluctuations of the surrounding polymer matrix. By representing these random forces as fractional Gaussian noise, and transforming the GLE into an equivalent diffusion equation for the density of the tagged chain segments, we obtain an analytical expression for the dynamic shear relaxation modulus G(t), which we then show decays as a power law in time. This power-law relaxation is the root of fractional viscoelastic behavior.

Interfacial tension model for catalytically driven nanorods

We present an analysis of the interfacial tension model for the movement of the catalytically driven nanorod. The model considers the convective reaction-diffusion equation for the production and diffusion of oxygen around the bimetallic nanorod. We solve the equation and find the concentration difference, which drives the nanorod. We use our expression to calculate the force on the nanorod and find that the result is within 20% of the results found earlier [ W. Paxton et al., J. Am. Chem. Soc. 128, 14881 (2006) ] by an approximate method. Unlike the earlier results, our results are valid from short to long lengths of the nanorod.

Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map

We describe a noniterative method for recovering optical absorption coefficient distribution from the absorbed energy map reconstructed using simulated and noisy boundary pressure measurements. The source reconstruction problem is first solved for the absorbed energy map corresponding to single- and multiple-source illuminations from the side of the imaging plane. It is shown that the absorbed energy map and the absorption coefficient distribution, recovered from the single-source illumination with a large variation in photon flux distribution, have signal-to-noise ratios comparable to those of the reconstructed parameters from a more uniform photon density distribution corresponding to multiple-source illuminations. The absorbed energy map is input as absorption coefficient times photon flux in the time-independent diffusion equation (DE) governing photon transport to recover the photon flux in a single step. The recovered photon flux is used to compute the optical absorption coefficient distribution from the absorbed energy map. In the absence of experimental data, we obtain the boundary measurements through Monte Carlo simulations, and we attempt to address the possible limitations of the DE model in the overall reconstruction procedure.

A Model of Anomalous Enzyme-Catalyzed Gel Degradation Kinetics

We show that a model of target location involving n noninteracting particles moving subdiffusively along a line segment (a generalization of a model introduced by Sokolov et al. [Biophys. J. 2005, 89, 895.]) provides a basis for understanding recent experiments by Pelta et al. [Phys. Rev. Lett. 2007, 98, 228302.] on the kinetics of diffusion-limited gel degradation. These experiments find that the time t(c) taken by the enzyme thermolysin to completely hydrolyze a gel varies inversely as roughly the 3/2 power of the initial enzyme concentration [E]. In general, however, this time would be expected to vary either as [E](-1) or as [E](-2), depending on whether the Brownian diffusion of the enzyme to the site of cleavage took place along the network chains (1-d diffusion) or through the pore spaces (3-d diffusion). In our model, the unusual dependence of t(c) on [E] is explained in terms of a reaction-diffusion equation that is formulated in terms of fractional rather than ordinary time derivatives.