Diffusion time integration

Since the diffusion coefficient is the infinite-time integral of the velocity correlation function, we have the Einstein relation, D = kBT/Q. 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

Such a decomposition of the diffusion coefficient has previously been noted by Pattle et al.(l ) Now we must evaluate >. The time-integrated velocity correlation function Aj j is due to the hydrodynamic interaction and can be described by the Oseen tensor. The Oseen tensor is related to the velocity perturbation caused by the hydrodynamic force, F. By checking units, we see that A is the Oseen tensor times the energy term, k T, or... 

Because the firn is ventilated by atmospheric air while the bubbles are forming over a period of time and ice depths, the air eventually trapped in the bubbles is a time-integrated sample that is younger than the snow deposit itself. For example, in one recent study (Smith et al., 1997), the air bubbles were, on average, 220-700 years younger than the ice in which they were embedded, but the difference can be as much as several thousand years (e.g., see Rommelaere et al., 1997). These exchange processes with the atmosphere, gas diffusion, and the porosity and tortuosity of the ice pores have to be taken into account in relating the depth of the core to the age of the trapped air. 

One can now approximate the current autocorrelation function in the diffusive limit. When the time integration is performed, the above expression reduces to... 

Excess Point Defects and Low-Thermal-Budget Annealing. Submicrometer VLSI (very-large-scale integration) technologies require low thermal budgets (the product of dopant diffusivity and diffusion time) to limit the diffusional motion of dopants. Two options exist to reduce the thermal... 

Mathematical analysis of the diffusion problem in this case for a rectangular parallelepiped adsorbent leads to an equation for the total moles of adsorptive that have entered the adsorbent pores by the elapsed time t (the time integral of the right side of Eq. 4.72) 39... 

Fig. 4. Frequency-domain modulated gradient NMR rf and gradient pulse sequences, showing the (actual) gradient modulation wave form Git), the time integral of the effective gradient wave form Fit), and the spectrum of Fit). H )P directly samples the diffusion spectrum. The wave forms and spectra are for (a) double lobe/dc rectangular modulation, (b) single lobe/ac rectangular modulation, and (c) single lobe/ac sawtooth-shaped phase modulation. Note that pulse sequences (b) and (c) sample the diffusion spectrum at a single frequency.

Various accessories were designed for recording diffuse reflectance spectra. Apart from special devices developed by different groups and described in the literature, several commercially available types must be noted. A few accessories are shown schematically in Fig. 1, which are representative of the diversity of optics. For a long time, integrating spheres have been in use, in particular for UV/VIS and near-infrared spectroscopy, although a few applications with sensitive MCT detectors also can be found within the mid-infrared. Usually, a baffle is placed within the sphere... 

A time correlation function that involves the same observable at two different times is called an autocorrelation function. We have found that the self-diffusion coefficient is the time integral of the velocity auto-correlation function... 

This system of coupled equations must be solved by integrating forward in time starting from the initial conditions mk,i(0). Eor stability, the time integration of the diffusion term can be treated implicitly using, for example, the Crank-Nicolson (CN) scheme. If we denote the volume-average moments at time t = n At by , a semi-implicit scheme for... 

The tip characteristics in Figure 2 are identical to an original treatment of the positive feedback response simulated using a Krylov integrator (19). At very short times (too short for the scale on Fig. 2), the normalized tip current is independent of d/a, since the diffusion field adjacent to the UME is small compared to the size of the gap and the tip shows the behavior predicted for a simple microdisk electrode under chronoamperometric control (23). With time, the diffusion field extends towards, and ultimately intercepts, the substrate causing a current flow at this detector electrode. Since the tip/substrate diffusion time is of the order d2ID, the smaller the value of d/a, the sooner the substrate current begins to flow [in normalized time Eq. (15)] and the more rapidly the currents at the tip and substrate attain steady-... 

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