The time dependence of diffusion

R being the interaction distance and / the fraction of encounters that lead to generation of singlets. Thus time-dependent diffusion renders ynA also time dependent. The time dependence of diffusivity and y-nA are related to the hopping frequency via [159] ... 

Jf is the number density (the number of atoms per unit volume). Eiquation (7.85) refers to diffusion in the z direction. The minus sign indicates that flux increases in the direction of negative concentration gradient. The time dependence of diffusive behaviour (which applies if a distribution is established at some time and is then allowed to evolve) is governed by Fick s second law, which gives the rate of change of concentration with time ... 

Concentration gradients can be time dependent. Pick s second law describes the time dependence of diffusion... 

The release of T from neutron-irradiated monocrystalline particles (0.15 to 0.84mm) was measured during isothermal annealing. Diffusion data were deduced from the results by considering the time-dependence of diffusion in a solid sphere. The diffusivity of T at between 573 and 950K was found to be described by ... 

Modern theory differentiates two cases for the time dependence of diffusion. The steady state condition first described by de Gennes (Section 5.4.2) holds for times longer than T, ... 

This variable is the argument of the error function of the semi-infinite slab, it determines the standard deviation of the decaying pulse, and it is central to the time dependence of diffusion into the cylinder. In other words, it is a key to all the foregoing unsteady-state problems. Indeed, it can be easily isolated by dimensional analysis. 

Here the lattice positions i and j should be adjacent and the -function assures that one of the two lattice positions is occupied and the other one is free, r/j is a characteristic time scale for a diffusion jump. The time-dependence of the average si) is calculated by approximating the higher moments (siSj) [49]. In practice the analysis is rather involved, so we do not give further details here. An important result, for example, is the correction to the Wilson-Frenkel rate (33) at high temperatures ... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

Another virtue of the procedure is that it can explicitly take into account a partially diffusion-controlled recombination reaction in the form of Collins-Kimball radiation boundary condition—namely, j(R, t) = -m(R, t) where j(R, t) is the current density at the reaction radius and K is the reaction velocity k— < > implies a fully diffusion-controlled reaction. Thus, the time dependence of e-ion recombination in high-mobility liquids can also be calculated by the Hong-Noolandi treatment. 

The main feature of the quantum kicked rotor is the quantum localization phenomenon, which implies suppression of the diffusive growth of energy of the quantum kicked rotor compared to the energy of the classical rotor (Izrailev, 1990). The time dependence of the energy can be calculated as... 

For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

The time-dependence of the SSP process has to be viewed in connection with the temperature and the factors concerning diffusion. After a certain period into the reaction, no increase in the IV is observed and thermal decomposition becomes the dominant process. 

The major reasons for using intrinsic fluorescence and phosphorescence to study conformation are that these spectroscopies are extremely sensitive, they provide many specific parameters to correlate with physical structure, and they cover a wide time range, from picoseconds to seconds, which allows the study of a variety of different processes. The time scale of tyrosine fluorescence extends from picoseconds to a few nanoseconds, which is a good time window to obtain information about rotational diffusion, intermolecular association reactions, and conformational relaxation in the presence and absence of cofactors and substrates. Moreover, the time dependence of the fluorescence intensity and anisotropy decay can be used to test predictions from molecular dynamics.(167) In using tyrosine to study the dynamics of protein structure, it is particularly important that we begin to understand the basis for the anisotropy decay of tyrosine in terms of the potential motions of the phenol ring.(221) For example, the frequency of flips about the C -C bond of tyrosine appears to cover a time range from milliseconds to nanoseconds.(222)... 

The systematic representation of the time dependence of chemical reactions occurring at an electrode surface. These reactions can be analyzed as a succession of events (a) diffusion of reactants within the bulk solution to the more condensed layer of solution near the electrode surface (b) penetration of that layer to achieve adsorption on the electrode s surface (c) electron transfer to (i.e., reduction) or from oxidation) the adsorbed reactant(s) product desorption, penetration of the condensed layer, and diffusion into the bulk solution. 

In the literature there are several, mostly just slightly different, equations that describe the rate coefficient of the diffusion controlled reactions these equations are usually based on the solutions of Pick II diffusion law assuming that the reaction probability at contact distance is 1. Andre et al. [131] used the following equation to describe the time dependence of excited molecule concentration [RH ] produced by an infinite excitation pulse ... 

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