Diffusion Differential Equations Research Articles

Effect of site energy fluctuation on charge transport in disordered organic molecules.

Effect of dynamics of site energy disorder on charge transport in organic molecular semiconductors is not yet well-established. In order to study the relationship between the dynamics of site energy disorder and charge transport, we have performed a multiscale study on dialkyl substituted thienothiophene capped benzobisthiazole (BDHTT-BBT) and methyl-substituted dicyanovinyl-capped quinquethiophene (DCV5T-Me) molecular solids. In this study, we explore the structural dynamics and correlated charge transport by electronic structure calculations, molecular dynamics, and kinetic Monte-Carlo simulations. We have also proposed the differential entropy dependent diffusion and charge density equations to study the electric field drifted diffusion property and carrier density. In this investigation, we have addressed the transformation mechanism from dynamic to static disorder in the extended stacked molecular units. Here, the decrease in the charge transfer rate due to site energy fluctuations reveals the dispersion transport along the extended π-stacked molecules. Furthermore, the calculated current density for a different set of site energy difference values shows the validity and the limitations of the Einstein relation. Based on the calculated ideality factor, we have classified the charge transport in these molecules as either the Langevin or the Shockley-Read-Hall type mechanism. Through the calculated mobility, current density, and ideality factor analysis, we categorize the applicability of molecules of interest for photovoltaic or light emitting diode applications.

MODELLING OF OILY RAW MATERIAL EXCRACTION PROCESS

This paper concentrates on approaches to mathematical modelling of oily raw material extraction process. «Evolution» of modeling hypothesis based on analysis of differential diffusion equation under the right boundary and initial conditions; on simplified model of Fick's equation and on material balance equation; on mass transfer model in adsorption pore volume; on model for the surface layer by analogy of Van der Waals equation; on Gibbs' model based on the abrupt change of phases due to the intermolecular forces; on the simplest Langmuir equation model; on lattice-based models of Guggenheim, Pryhozhyn, Everett, Ohm, Briukhovetskyi and others. The fact that we need to know a large number of micro - parameters makes these models difficult to use in practice. Under normal extraction conditions, the flow, which comes out of solid phase, collides with the resistance of the diffusion boundary layer, which presents a tangible obstacle affecting the duration and quantity of special-purpose component extraction. As the boundary layer thickness depends on the hydrodynamics of the process, under the influence of the microwave field his obstacle is almost insensible, as the intense movement of liquid reduces its thickness. The main factor acting on the quantity of extracted substance is a pressure difference in capillaries and in the flow of extraction agent and mass transfer coefficient. The effect of pulse microwave input during the extraction transfers the process of diffusion from the external environment to internal, because internal pressure diffusion dominates in this process, but not the convective diffusion with the influence of external agent.

Estimation of the influence of variable boundary conditions when using thin layer equations for grain dryer simulation

Most dryer design programs or dryer simulators use “thin layer equations” to predict mass transfer rates in each subdivision of grain beds in dryers. These equations work well under the constant boundary conditions for which their parameters were determined. Considering that the atmosphere surrounding the grains in dryers vary with time and position, a specific study was conducted here to analyse the behaviour of analytical solutions under such variable conditions. This work provides examples for wheat using parameters obtained in previous research conducted in our laboratory, and compares analytical predictions of the diffusion equation with those of a numerical (finite difference) solution of the partial differential equation of diffusion for one grain, which is naturally prepared to predict under either constant or variable boundary conditions. As expected, predictions of the analytical and numerical solutions coincided under constant boundary conditions, and this aspect was important to corroborate the behaviour of the numerical method in the program. Then, simulated data of exhaust air temperature and relative humidity from a dryer were produced by a drying simulator previously validated and, under this variable environment, predictions for a thin layer with the analytically-based model and the numerically-based solution tend to depart, especially after the derivative of the mean grain moisture with time changed sign. The preliminary conclusion is that, at least in positions near the outlet of dryer columns, predictions of the analytical solution-based model might be misleading, and this must be taken into account in future research aimed at developing grain dryer simulators.

Inter-tree competitive processes during early growth of an experimental plantation of Eucalyptus pilularis in sub-tropical Australia

Individual tree diameters, heights and crown dimensions were measured over four growth periods, from 2.5 to 6.6 years of age, in a plantation experiment with blackbutt (Eucalyptus pilularis Smith) on a highly productive site in sub-tropical, eastern Australia. The experiment compared several stocking densities at planting (ranging over 816–1667 stems ha−1) and rectangularities of tree espacement (ranging over 1–6). The effects of inter-tree competitive processes on individual tree diameter growth rates were examined using both a partial differential diffusion equation to describe the change with time of the frequency distribution of diameters and by examining which tree size classes were dominating growth. Over all four growth periods, it appeared that inter-tree competition for the light and soil resources necessary for growth was largely symmetric. That is, whilst larger trees would have been occupying more of the above- and below-ground space from which to obtain those resources than smaller trees, all trees would have been able to obtain quantities of each in amounts proportional to their sizes and grow at rates accordingly. However, even at these young ages, a tendency for larger trees to display growth rates disproportionately large relative to their sizes gave early evidence of asymmetric inter-tree competition developing; that involves taller trees denying light to shorter trees through shading, the reverse of which may not occur. These inter-tree competitive interactions became more pronounced both as tree sizes increased with age and when trees were planted more closely to each other. Asymmetric competition is expected to be a major determinant of individual tree size development for a large part of the subsequent growth of the forest. The results are consistent with the forest showing a transition from the first to the second phase of the Binkley et al. model of individual tree growth and development in forest stands.

Green’s functions for the fourth-order diffusion equation

Distinct physical processes are involved in the representation of diffusion phenomenon. On the other side, the more commonly employed differential equation for diffusion in the reviewed literature does not provide neither complete nor necessarily adequate mathematical representation of such processes what ends up to still be an ongoing unsolved challenge. In particular, this complex task of mathematical modeling is more visible in when, for instance, a set of diffusive particles are characterized by the simultaneous existence of two states of energy, possessing both a deceleration and temporary retention of particles. The fundamental hypothesis considered in this paper is that there is a predominant flux, which is the one represented by the potential defined by means of the Fick’s law. In addition, after a short period, other possible excitation microstates begin to exist, leading to the modeling of such physical phenomena using the fourth-order anomalous diffusion equation. Given the exposed framework, the present work contributes with the development of the analytical solution of a partial differential equation (PDE) that represents the diffusion process caused by a pulse of local and instantaneous energy. Green’s function is obtained through the solution of a specific PDE associated with the equation governing the original problem and in the same domain with the support of Fourier and Laplace transforms. These expressions were then validated using the finite difference numerical method. The successful proposed approach allowed us to produce completely new results for the diffusion and propagation processes, also taking into account retention processes or retardation effects, propagation disorders, acceleration or deceleration, or even fluid stagnation.

A novel Legendre operational matrix for distributed order fractional differential equations

In this paper, for the first time, the shifted Legendre operational matrix of distributed order fractional derivative has been derived. Also, this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time. Moreover, eight numerical examples are implemented in order to show the validity and reliability of the suggested methods.

Precision Numerical Modeling of the Decay of a Metastable State at High Temperatures

The quasistationary decay rate of a metastable state, RD, often is interpreted as the inverse of the average lifetime of this state, τa. There are two ways for finding the rate: to evaluate it using one of the approximate Kramers formulas or to extract the rate RD from numerical modeling. We study quantitatively to what extent the inverse decay rate can be identified with the mean lifetime and to what extent the approximate and numerical quasistationary rates agree with each other. This is done for the values of dissipation strength covering 4 orders of magnitude. Both the numerical rate and the average lifetime are obtained from computer modeling of the decay process using the stochastic differential equations for the phase space diffusion. It is shown that at lower temperature, the numerical rate is in agreement with the inverse lifetime whereas at weak friction and high temperature, τa exceeds $$ {R}_D^{-1} $$ by up to 50%. The Kramers formula for the decay rate at strong friction is found to be in a good agreement with the dynamical rate deviating typically by 5%. For weak friction, the agreement is substantially worse: the difference varies from 20% to a factor of 2.

Entransy dissipation analysis of interfacial convection enhancing gas–liquid mass transfer process based on field synergy principle

Abstract An exploration of the gas CO2 absorbed into liquid ethanol accompanied with Rayleigh convection is performed by analyzing the mass entransy dissipation; this new statistical quantity is introduced to describe the irreversibility of mass transfer potential capacity. Based on the general advection–diffusion differential equation for an unsteady mass transfer process, the variation of the included angle between the velocity vector and concentration gradient fields is investigated to reveal the underlying mechanism of interfacial convection enhancing mass transfer. Results show some identical characteristics with the qualitative analyses of the synergy effects generated by the concentration and velocity fields after interfacial convection occurring for a boundary condition of fixed surface concentration. And the equivalent mass resistance for convective mass transfer process presents the similar variation with the reciprocal of instantaneous mass transfer coefficient. Accordingly, it is reasonable to be seen that mass transfer dissipation rate could be provided to assess the convection strength and explain fundamentally how Rayleigh convection improves mass transfer performance through establishing a close relationship between the mass transfer capacity and field synergy principle from the view of mass transfer theory.

Simulation of batik cracks and cloth dying

This paper simulates two important processes of batik: crack generation and cloth dyeing. We use FIT algorithm and other methods to reduce the execution time and achieve good visual effects. We attempt to simulate the multi-colored dyeing process without considering the cloth structure. Dyeing of a batik plain weave cloth is simulated by a batik cloth model and a differential diffusion equation. Mottling is simulated by an ellipse model with Perlin noise. Experiments show that these methods can properly simulate the characteristics of batik.

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a continuum evolution law expressed by a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the nonlinear gradient flow of a convex, singular surface free energy with a certain exponential mobility in homoepitaxy. This evolution law is motivated both by an atomistic broken-bond model and a mesoscale model for crystal steps. By constructing an explicit solution to this PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the continuum law via a regularized surface free energy.

A new model for simulating heat, air and moisture transport in porous building materials

This work presents a detailed mathematical model combined with an innovative efficient numerical model to predict heat, air and moisture transfer through porous building materials. The model considers the transient effects of air transport and its impact on the heat and moisture transfer. The achievement of the mathematical model is detailed in the continuity of Luikov’s work. A system composed of two advection–diffusion differential equations plus one exclusively diffusion equation is derived. The main issue to take into account the transient air transfer arises in the very small characteristic time of the transfer, implying very fine discretisation. To circumvent these difficulties, the numerical model is based on the Du Fort–Frankel explicit and unconditionally stable scheme for the exclusively diffusion equation. It is combined with a two–step Runge–Kutta scheme in time with the Scharfetter–Gummel numerical scheme in space for the coupled advection–diffusion equations. At the end, the numerical model enables to relax the stability condition, and, therefore, to save important computational efforts. A validation case is considered to evaluate the efficiency of the model for a nonlinear problem. Results highlight a very accurate solution computed about 16 times faster than standard approaches. After this numerical validation, the reliability of the mathematical model is evaluated by comparing the numerical predictions to experimental observations. The latter is measured within a multi-layered wall submitted to a sudden increase of vapor pressure on the inner side and driven climate boundary conditions on the outer side. A very satisfactory agreement is noted between the numerical predictions and experimental observations indicating an overall good reliability of the proposed model.

Semi-empirical methods for estimation/prediction of metal matrices behavior during thermochemical processing

Abstract Series of solutions for estimations/predictions of the behavior of metal matrices during thermochemical treatments are described on basis on minimal number of experimental results under relatively random chosen processing conditions. Three methods were considered: Popov, Kazeev and Baram. The Popov method is based on the solutions of the second differential equation of diffusion (Fick’s second law), obtained by analytical and criterial solving under third boundary conditions; in this way, the kinetic parameters D, h*, and k are estimated. The Kazeev method is based on the correct hypothesis that besides the atomic diffusion takes place also other phenomena/interactions in the metallic matrix. All these complex processes can be described by specific equations whose solutions allow to determine certain parameters which change continuously during the diffusion process. The Baram method considers the generalized equation of kinetics of heterogeneous chemical reactions. This was resulted by taking into account the law of mass action and the phenomenological dependence of change over time of the inter-phases separation surfaces positions in solid materials subjected to thermochemical processing. It was thus possible to describe the phenomena occurred during thermochemical processing. The algorithms for the utilization of these methods are exemplified by experimental data resulted from the gaseous nitriding of the pure technical iron (ARMCO).

Метод сопряжённого уравнения в задаче об определении источника диффузии

The objects of the research are differential equations of diffusion (or thermal conductivity) kind. The subject of research is the algorithm for determining the function of the source or the initial conditions of the problem as per the experimentally measured values. The approach is based on a dual representation of functionals corresponding to experimentally observed quantities in the processes of mass and heat transfer. The inverse problem is formulated in the form of integral equations of the first kind, the core of which is the adjoint function (importance function) obtained as a solution of the adjoint (in the Lagrange sense) diffusion (thermal conductivity) equation with the detector sensitivity function in the right-hand side. Meanwhile, solving the adjoint equations by changing the variables is reduced to solving direct equations. To regularize the solution of the Volterra equation of the first kind corresponding to the problem of recovering the dependence of the boundary condition on time, the residual minimization for an overdetermined system of linear equations has been proposed to be used. The problem of reconstructing the dependence of the initial condition on the coordinate is formulated as a Fredholm equation of the first kind, the solution to which has been obtained using the Tikhonov regularization method. The results of model calculations are presented for the restoration of the time dependence of the sources set by a smooth function, a step function and a function with a harmonic component in the problem of one-dimensional diffusion in a homogeneous medium. These results prove that with the selected calculation parameters, the solutions obtained by the proposed method behave regularly and have quite an acceptable accuracy, even though the values of the sought-for function within the set search interval change by six orders of magnitude. This is seen by the authors as the main difference of their method from other approaches to solving this problem

Hyperbolicity of the heat equation

In this manuscript, a comparison between the parabolic and hyperbolic partial differential equations for heat diffusion is studied. First, numerical results and an eigenvalue analysis for these two types of equations are shown, which help to understand the difference in the system dynamics. Then, both equations are also considered in a Stefan problem for the melting of an ice block in a one-dimensional setting. The results show that the hyperbolic partial differential equation shows a finite speed of propagation of heat and can represent the system as properly as the parabolic equation.

Hopf Bifurcation of a Delayed Predator–Prey Model with Nonconstant Death Rate and Constant-Rate Prey Harvesting

This paper is concerned with a delayed predator–prey model with nonconstant death rate and constant-rate prey harvesting. We mainly study the impact of the time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively. By choosing time delay [Formula: see text] as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes some critical values. In addition, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to depict our theoretical results.

Rotating waves in a spatially nonlocal delayed feedback optical system with diffraction

We study self-organization in a nonlinear optical system with a feedback loop, where light is delayed and undergoes a spatial transformation within the thin annulus aperture. The dynamics of the system are modelled by a quasilinear functional differential diffusion equation on a circle that tracks the effects of diffraction by means of a coupled linear Schrodinger equation. We prove the existence of rotating wave solutions and study their stability by constructing the Faria normal form of a Hopf bifurcation.

Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions

We present a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to describe the time evolution of solutions. Nonlocality was modeled with a space-time fractional differential equation of order 1 ≤ α ≤ 2 in time, and order 1 ≤ β ≤ 2 in space. We adopted the notion of Caputo for the time derivative and the Riesz pseudo-differential operator for the space derivative. The corresponding Cauchy problem was solved for zero initial velocity and initial disturbance, represented by either the Dirac delta or the Gaussian distributions. Well-known results for the conventional partial differential equations of wave propagation, diffusion, and (modified) transport processes were recovered as particular cases. In addition, regular solutions were found for the partial differential equation that arises from α = 2 and β = 1 . Unlike the above conventional cases, the latter equation permits the presence of nodes in its solutions.

Numerical Solution of the Differential Diffusion Equation for a Steel Carburizing Process

The simplified algorithm of the numerical solution of the differential diffusion equation is presented. The solution is based on the one-dimensional diffusion model with the third kind boundary conditions and the finite difference method. The proposed approach allows for the quick and precise assessment of the carburizing process parameters – temperature and time.

Belief Dynamics in Social Networks: A Fluid-Based Analysis

The advent and proliferation of social media have led to the development of mathematical models describing the evolution of beliefs/opinions in an ecosystem composed of socially interacting users. The goal is to gain insights into collective dominant social beliefs and into the impact of different components of the system, such as users' interactions, while being able to predict users' opinions. Following this thread, in this paper we consider a fairly general dynamical model of social interactions, which captures all the main features exhibited by a social system. For such model, by embracing a mean-field approach, we derive a diffusion differential equation that represents asymptotic belief dynamics, as the number of users grows large. We then analyze the steady-state behavior as well as the time dependent (transient) behavior of the system. In particular, for the steady-state distribution, we obtain simple closed-form expressions for a relevant class of systems, while we propose efficient semi-analytical techniques in the most general cases. At last, we develop an efficient semi-analytical method to analyze the dynamics of the users' belief over time, which can be applied to a remarkably large class of systems.

Intermediate phase cone growth kinetics along dislocation pipes inside polycrystal grains

Dislocation-pipe diffusion (DPD) becomes a major contribution to device failure in microelectronic components at working temperatures. Usually, the simple random walk law for diffusion (Type C kinetics t1/2) is employed to calculate of DPD coefficients. The article presents an analytically solvable model of describing the diffusion phase cone growth along dislocation pipes inside polycrystal grains involving outflow from dislocation lines (Type B kinetics). Correlative analytical method to solve differential diffusion equations for such model is suggested. Competition between phase cone growth along dislocation lines involving outflow and phase wedge growth along grain boundaries (GBs) involving outflow is analyzed. It is shown that while phase wedge growth law along GBs is the Fisher regime t1/4, phase cone growth law along dislocation lines is another diffusion regime t1/6. Real experimental data are analyzed using such diffusion regime. It is shown that it is possible to calculate DPD coefficients not only for the phase cone formation, but for migration of atoms along dislocations and self-diffusion along dislocation pipes too.