Diffusion time-dependent

Figure 4 Diffusion time dependence on the diffusion length. Solid and dashed lines correspond to the diffusion time when the respective diffusion coefficients are 1 x 10 9 and 1 x 10-10 mV.

For normal (free) diffusion, D should be independent of the diffusion time, A. For cases where the free diffusion of the Xe atoms in a material is restricted by, for example, crystalline barriers or pore walls, the diffusion coefficient can be diffusion time dependent. This will be discussed again in Section 13.5. 

Increasing the NaCl concentration to 100 mM gives rise to a further strong increase in the diffusion coefficient. Interestingly, the apparent coefficient depends now on the diffusion time set by the experiment, i. e. the time r between the two gradient pulses. Sueh a dependence is standardly interpreted as being due to restricted diffusion. Both, the absolute increase and the diffusion time dependence, hint to a further deformation of the multilameUar vesicles into planar lamellar stacks or aggregates, which are partially ordered within the NMR tube. 

Afterward the substance has to diffuse from the core to the surface in order to be dissolved. The diffusion time depends on the corresponding distribution ratio of the substance within the solid matrix and if adsorbed in it or not... 

Straub J E, Berne B J and Roux B 1990 Spatial dependence of time-dependent friction for pair diffusion in a simple fluid J. Chem. Phys. 93 6804... 

FIdthmann H, Beck C, Schinke R, Woywod C and Domcke W 1997 Photodissociation of ozone in the Chappuis band. II. Time-dependent wave packet calculations and interpretation of diffuse vibrational... 

Tao T 1969 Time-dependent fluorescence depolarization and Brownian rotational diffusion coefficients of macromolecules Biopolymers 8 609-32... 

Figure C2.1.18. Schematic representation of tire time dependence of tire concentration profile of a low-molecular-weight compound sorbed into a polymer for case I and case II diffusion. In botli diagrams, tire concentration profiles are calculated using a constant time increment starting from zero. The solvent concentration at tire surface of tire polymer, x = 0, is constant.

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

Fiber R and M Karplus 1990. Enhanced Sampling in Molecular Dynamics Use of the Time-Dependent Hartree Approximation for a Simulation of Carbon Monoxide Diffusion through Myoglobin. Journal of the American Chemical Society 112 9161-9175. 

One of the most important characteristics of micelles is their ability to take up all kinds of substances. Binding of these compounds to micelles is generally driven by hydrophobic and electrostatic interactions. The dynamics of solubilisation into micelles are similar to those observed for entrance and exit of individual surfactant molecules. Their uptake into micelles is close to diffusion controlled, whereas the residence time depends on the sttucture of the molecule and the solubilisate, and is usually in the order of 10 to 10" seconds . Hence, these processes are fast on the NMR time scale. 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

Concentration gradients for the analyte in the absence of convection, showing the time-dependent change in diffusion as a method of mass transport. 

In these early reactions the reactivities of the individual phases are important in determining the overall reaction rate. However, as the cement particles become more densely coated with reaction products, diffusion of water and ions in solution becomes increasingly impeded. The reactions then become diffusion-controUed at some time depending on various factors such as temperature and water—cement ratio. After about 1 or 2 days, ie, at ca 40% of complete reaction, the remaining unhydrated cement phases react more nearly uniformly. 

Diffusion in the bulk crystals may sometimes be short circuited by diffusion down grain boundaries or dislocation cores. The boundary acts as a planar channel, about two atoms wide, with a local diffusion rate which can be as much as 10 times greater than in the bulk (Figs. 18.8 and 10.4). The dislocation core, too, can act as a high conductivity wire of cross-section about (2b), where b is the atom size (Fig. 18.9). Of course, their contribution to the total diffusive flux depends also on how many grain boundaries or dislocations there are when grains are small or dislocations numerous, their contribution becomes important. 

The diffusion of carbon into the steel is described by the time-dependent diffusion equation... 

The diffusion stage, characterized by H(t), will be discussed in detail with respect to the instantaneous wetting condition. However, in the presence of a time dependent wetting function (p(t), Eq. 2.1, we see from Eig. 2 that diffusion will have progressed to different extents in different areas of the interface. If the intrinsic diffusion function, H(t) as given by Eq. 1.1 does not change its nature with time due to the other stages, then the net diffusion, H"(t) can be expressed as the convolution product... 

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