Instantaneous diffusion simulations

Demonstration of Non-equilibrium (2f/Zp < 0.5). Data were gathered on the same PIB coated column as above at a tenrperature of 40 C. Under these conditions, diffusion of the probes into the polymer is not expected to be instantaneous. The simulation under these conditions predicts that = (1 + Zp) and t will be reduced by the effect of non-zero Zf. The magnitude of the reduction is given by the second term of the right side of Equation 15. Since Zf is dependent on flew rate, it is possible to estimate Zf from the dependence of the peak width on the flew rate and hence, determine the size of the correction and conpare it with experimental results. This comparison should be made bearing in mind that theoretically the simulation is applicable for capillary columns and not packed columns. 

It is assumed that the time required for electron transfer through the biofilm matrix is negligible compared to the time scale of the diffusion of dilute solutes in the biofilm electrons are conducted instantaneously. For simulations considering the isolated-dual and interacting-dual EET cases, the total current density, j (A m ), is the total current derived from both underlying EET mechanisms (diffusion- and conduction-based) ... 

Section 4.2.2 shows how to use the scaling method to obtain the error function solution for the one-dimensional diffusion of a step function in an infinite medium given by Eq. 4.31. The same solution can be obtained by superposing the onedimensional diffusion from a distribution of instantaneous local sources arrayed to simulate the initial step function. The boundary and initial conditions are... 

In trajectory models, the particle turbulent diffusion can be considered by calculating the instantaneous motion of particles in the turbulent flow field. In order to simulate the stochastic characteristics of the instantaneous gas velocity in a turbulent flow, it is required to generate random numbers in the calculation process. 

Transport control of flocculation is realized in an especially direct way in the process known as diffusion-limited cluster-cluster aggregation5 (aggregation as used in this term means flocculation in the present chapter). In this process, which is straightforward to simulate and visualize on a computer, particles undergo Brownian motion (i.e., diffusion) until they come together in close proximity, after which they coalesce instantaneously and irreversibly to form floccules (or clusters ). The clusters then diffuse until they contact one another and combine to form larger clusters, and so on, until gravitational... 

Fig. 6.1. Visualization sequence of diffusion limited cluster cluster flocculation, based on a two dimensional computer simulation of the random encounter and instantaneous combination of 2(XX)...

The liquid phase of molecular matter is usually isotropic at equilibrium but becomes birefringent in response to an externally applied torque. The computer can be used to simulate (1) the development of this birefringence —the rise transient (2) the properties of the liquid at equilibrium under the influence of an arbitrarily strong torque and (3) the return to equilibrium when the torques are removed instantaneously—the fall transient. Evans initially considered the general case of the asymmetric top (C2 symmetry) diffusing in three-dimensional space and made no assumptions about the nature of the rotational and translational motion other than those inherent in the simulation technique itself. A sample of 108 such molecules was taken, each molecule s orientation described by three unit vectors, e, Cg, and parallel to its principal moment-of-inertia axes. 

Fransaer J, Penner RM (1999) Brownian dynamics simulation of the growth of metal nanocrystal ensembles on electrode surfaces from solution. I. Instantaneous nucleation and diffusion-controlled growth. Phys Chem B 103 7643... 

In order to facilitate the analysis of the shape and position of elution curves in inverse gas chromatography, such curves were generated in a computer for many well-defined situations. The effects of diffusion in the gas phase, of slow diffusion in the polymer phase (compared to an instantaneous equilibration of the probe), and of surface adsorption (Langmuir type) were simulated. 

Diffusion of the probe in the gas and polymer phases, and adsorption on the support and on the polymer surface (both types of adsorption have nonlinear isotherms), simultaneously play an important role in IGC experiments and must be accounted for properly. An extensive computational program is planned to simulate the individual processes and to assess their influence on chromatographic behavior. In a recent paper, simulated behavior of three types of system was described (9). In the simplest case, only diffusion in the gas phase was operative. This case corresponds to elution of an ideal marker. Simultaneous effects of gaseous diffusion and partitioning of the probe between the phases were simulated next, assuming an instantaneous... 

For instantaneous equilibration, the simulation predicts that is as predicted by theory, Equation 10, and tg is given by equation 13. Thus tR should be slightly less than F due to the gaseous diffusion coefficient. Table I shows the experimental values of tR and FM at several flow rates. 

In particular they consider a set of simulations for a system of CX4 molecules at three different temperatures ( hot , intermediate and cool ) which bears a close resemblance to our computations made in the presence of a second rank interaction potential. Their hot case corresponds to our low potential coupling cases, whereas their cool simulation is related to our high potential results that is, a decrease in temperature corresponds in our rescaled coordinates to an increase in the potential coupling. One may note that the presence of a negative tail, assigned by Lynden-Bell to librational motion of the observed molecule in an instantaneous cage, causes the momentum correlation functions to behave differently in the cool state with respect to the purely diffusive decay observed for the hot state. This behavior is very similar to our 2BKM-SRLS case for ta, = 5 and V2 = 3 (cf. Fig. 13b). 

Three thousand separate trajectories were simulated for each receptor density and at each truncation height. In contrast to molecular dynamics, in BDS the solvent is treated as a viscous continuum and, hence, time steps on the order of 0.1 ps are appropriate. Diffusion-limited binding or instantaneous reaction upon ligand-receptor collision was assumed. Computationally, this condition was considered met when the center of the ligand particle and the center of the receptor were less than 20 A apart. Trajectories were truncated when the ligand s height above the membrane surpassed Q, the truncation distance. Sample trajectories of both a successful capture and an escape are shown in Figs. 16a and 16b. 

The application of H ID-MRI for the study of hydrocarbon diffusion gives two types of information. First, the variation with time of the surface area of the full signal (amount adsorbed as a function of time) during adsorption makes it possible to determine transport coefficients by simulation of the kinetic curves. For example, the intracrystallite diffusion coefficients of hexane and benzene in HZSM-5 determined by this technique are lO" and 10 m s ", respectively, in good agreement with data in the literature. Second, the shape of the instantaneous concentration... 

Keys et al. (2000) explored five approaches to modeling the pharmacokinetics of di- -butyl phthalate and mono- -butyl phthalate. In a flow-limited version of the model, transfers between blood and tissues are simulated as functions of blood flow, tissue concentrations of di- -butyl phthalate or mono-n-butyl phthalate, and tissue blood partition coefficients, assuming instantaneous partitioning of the compounds between tissue and blood (Ramsey and Anderson 1984). In an enterohepatic circulation version of the model, the transfer of mono-n-butyl phthalate from the liver to the small intestine is represented with a first order rate constant (diffusion-limited) and a time delay constant for the subsequent reabsorption of mono- -butyl phthalate from the small intestine. In a diffusion-limited version of the model, the tissue transfers include a first order rate term (referred to as the permeation constant) that relates the intracellular-to-extracellular concentration gradient to the rates of transfer. This model requires estimates of extracellular tissue volume (ECV) and intracellular volume (ICV) ECV is assumed to be equal to tissue blood volume and ICV is assumed to be equal to the difference between tissue blood volume and... 

Examining the various timescales involved shows that droplet evaporation is commonly diffusion limited. Since the characteristic vapor diffusion times are well below the typical droplet evaporation times [2], the vapor concentration is assumed to adapt instantaneously to a droplet deformation, and within the so-called quasisteady approach, the dynamics of the evaporation is controlled solely by the diffusional mass transport from the droplet surface into the ambient gas phase. A characteristic vapor distribution above the droplet, obtained by finite-element simulations, is shown in Fig. la. The... 

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