Constant velocity drift

Hirao et al. (1995, 1995a) and Hirao and Nishizawa (1996) measured hole mobilities and diffusion coefficients of 4-dibenzylamino-2-methylbenzaldehyde-1,1-diphenylhydrazone (BMH) and DEH doped PC. The experimental technique involved the fitting of a theoretical expression to the photocurrent transients. The theoretical equation was derived by assuming that a carrier packet is spread by diffusion and that the carriers drift with a constant velocity. The photocurrent is given by the expression... 

Figure 8 A schematic representation of the motion of charge carriers in a semicondnctor lattice. (a) A charge carrier in the absence of any external field. The thermal motion is random, and wiU not lead to any motion of the charge carrier on a macroscopic level, (h) A charge carrier in the presence of an electric field. The charge carrier motion dne to the electric field is imposed npon its thermal motion, (c) The carrier drift shown at the macroscopic level. The random thermal motion of the charge carrier again leads to no motion on the macroscopic level. The charge carriers are propelled by the electric field at a constant velocity, v, either parallel or antiparallel to the applied field...

Diffusion is the macroscopic result of the sum of all molecular motions involved in the sample studied. Molecular motions are described by the general equation of dynamics. However, because of the enormous difference in the orders of magnitude of the masses, sizes, and forces that characterize molecules and macroscopic solids, it can be shown [1] that, when a force field (e.g., an electric field to an ionic solution) is applied to a chemical system, the acceleration of the molecules or ions is nearly instantaneous, molecules drift at a constant velocity, and, in the absence of an external field and of internal forces acting on the feed components, which is the case in chromatography, the diffusional flux, /, of a chemical species i in a gradient of chemical potential is given by... 

Extensive data of this type are necessary in order to verify the stability of a spectrometer and identify the potential sources of drift. The stability displayed in the figure is not sufficient for the thermal studies described above because the assignment of the isomer shift depends on the determination of the Debye temperature from the small changes in the slope of the thermal shift curve. The data presented in Figures 3,4, and 5 have been monitored with a constant velocity spectrometer with a verified long-term stability of 0.001 mm/s, which was integrated into a fully automated data-acquisition system. 

The direct substitution of the pgp (27) into the GLE (23) leads to Brownian motion affected by a constant force. On large time and space scales the steady-state solution is a drift with constant velocity [74-77]... 

In this regime, ions effectively have no inertia and stop virtually instantaneously if the field is switched off. Thus the dynamics in IMS complies with Aristotle s view that force exerted on an object produces constant velocity and the motion ceases once the force is removed Of course, the Galileo mechanics still applies and ions in IMS are constandy accelerated according to Equation 1.5. However, they are periodically decelerated by molecular collisions. When those are frequent enough, the stop-and-go motion appears as macroscopic steady-state drift (Figure 1.3c). The deceleration after a collision is incomplete, with the velocity loss dependent on the m/M ratio. Hence v is also a function of M, leading to ... 

The utility of the general formulas is illustrated by treating a simple drift-diffusion process with a constant diffusion coefficient and a constant velocity. The limits of drift-domination and diffusion-domination, and the crossover behavior between these limits are discussed for various quantities of interest in the context of translocation. We shall use the key formulas provided here in later chapters dealing with polymer capture by a pore and polymer translocation through a pore. 

If the initial distance between the centres of two spirals is not very large, a bound state of these two spirals is formed. It was noted in [21] that the properties of a bound pair do not depend on the initial conditions. The pair slowly drifts at a constant velocity along its symmetry line while the distance between the two centres remains constant. Such stable bound pairs of oppositely rotating spirals were found in the numerical simulations [21] not for any parameters of the excitable medium. A necessary condition was that a spiral should be loosely coiled, i.e. its pitch should be significantly larger than the recovery length in the medium. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

These simple models are based on the assumption of constant drift velocity i.e., particles are assumed to achieve their final charge instantaneously. This is a reasonable assumption in the case of large particles, the charging of which is governed by field-driven ion motion. The characteristic distance, x% corresponding to the time constant in Eq. (13,53) is given by... 

One could assume that this characteristic behavior of the mobility of the polymers is also reflected by the typical relaxation times r of the driven chains. Indeed, in Fig. 28 we show the relaxation time T2, determined from the condition g2( Z2) = - g/3 in dependence on the field B evidently, while for B < B t2 is nearly constant (or rises very slowly), for B > Be it grows dramatically. This result, as well as the characteristic variation of with B (cf. Figs. 27(a-c)), may be explained, at least phenomenologically, if the motion of a polymer chain through the host matrix is considered as consisting of (i) nearly free drift from one obstacle to another, and (ii) a period of trapping, r, of the molecule at the next obstacle. If the mean distance between obstacles is denoted by ( and the time needed by the chain to travel this distance is /, then - (/ t + /), whereby from Eq. (57) / = /Vq — k T/ DqBN). This gives a somewhat better approximation for the drift velocity... 

In the binary-electrolyte experiments carried out at large, constant cell potentials, the cell current is ohmically limited. If the conductivity of the solution is proportional to the concentration of electrolyte, the current density at a given overpotential is then proportional to Cb. Under this regime, the concentration cancels out of Eq. (2.3), and the velocity is proportional to the applied potential. For this special case, the velocity can be expressed in terms of the anion drift velocity [27, 28]. For a binary solution, this is equivalent to replacing (1 — t+) by t and i by the ohmically limited current density. 

A record of the axial velocity component vx for steady turbulent flow in a pipe would look like the trace shown in Figure 1.22. The trace exhibits rapid fluctuations about the mean value, which is determined by averaging the instantaneous velocity over a sufficiently long period of time. Figure 1.22 shows the case in which the mean velocity remains constant this is therefore known as steady turbulent flow. In unsteady turbulent flow, the mean value changes with time but it is still possible to define a mean value because, in practice, the mean will drift slowly compared with the frequency of the fluctuations. 

Time of Flight The principle of time-of-flight mass spectrometry (TOFMS) is very simple and makes use of the energy equation E = mv2/2 where m is mass and v is velocity. As illustrated in Figure 11.1, all ions are accelerated to constant energy E. The ions then enter a drift tube where they travel at velocity v. Because each ion mass m travels at a different velocity, the ions of different mass separate the lighter ones run ahead of the heavier ones. This separation means that each ion mass hits the detector at a different time. Ions of the same... 

A chromatogram without noise and drift is composed of a number of approximately bell-shaped peaks, resolved and unresolved. It is obvious that the most realistic model of a single peak shape or even the complete chromatogram could be obtained by the solution of mass transport models, based on conservation laws. However, the often used plug flow with constant flow velocity and axial diffusion, resulting in real Gaussian peak shape, is hardly realistic. Even a slightly more complicated transport equation... 

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