Diffusion, time scale

So far we studied the first passage of Markov processes such as described by the Smoluchowski equation (1.9). On a finer time scale, diffusion is described by the Kramers equation (VIII.7.4) for the joint probability of the position X and the velocity V. One may still ask for the time at which X reaches for the first time a given value R, but X by itself is not Markovian. That causes two complications, which make it necessary to specify the first-passage problem in more detail than for diffusion. 

Femtosecond lasers represent the state-of-the-art in laser teclmology. These lasers can have pulse widths of the order of 100 fm s. This is the same time scale as many processes that occur on surfaces, such as desorption or diffusion. Thus, femtosecond lasers can be used to directly measure surface dynamics tlirough teclmiques such as two-photon photoemission [85]. Femtochemistry occurs when the laser imparts energy over an extremely short time period so as to directly induce a surface chemical reaction [86]. 

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. 

Figure C3.1.7. Time-resolved optical absorjDtion data for the Soret band of photo lysed haemoglobin-CO showing six first-order (or pseudo-first-order) relaxation phases, I-VI, on a logaritlimic time scale extending from nanoseconds to seconds. Relaxations correspond to geminate and diffusive CO rebinding and to intramolecular relaxations of tertiary and quaternary protein stmcture. (From Goldbeck R A, Paquette S J, Bjorling S C and Kliger D S 1996 Biochemistry 35 8628-39.)...

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

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. 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

The thermal diffusivity for aluminum is = 5.2 x 10 m s [50]. Use this value to determine the time necessary for substantial temperature change over the length scale of 10 following creation of shear bands in the shock front. Should the temperature evolution of the shear band be included in a constitutive description on time scales of compression and release ... 

The Q and ft) dependence of neutron scattering structure factors contains infonnation on the geometry, amplitudes, and time scales of all the motions in which the scatterers participate that are resolved by the instrument. Motions that are slow relative to the time scale of the measurement give rise to a 8-function elastic peak at ft) = 0, whereas diffusive motions lead to quasielastic broadening of the central peak and vibrational motions attenuate the intensity of the spectrum. It is useful to express the structure factors in a form that permits the contributions from vibrational and diffusive motions to be isolated. Assuming that vibrational and diffusive motions are decoupled, we can write the measured structure factor as... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

We finish this section by comparing our results with NMR and incoherent neutron scattering experiments on water dynamics. Self-diffusion constants on the millisecond time scale have been measured by NMR with the pulsed field gradient spin echo (PFGSE) method. Applying this technique to oriented egg phosphatidylcholine bilayers, Wassail [68] demonstrated that the water motion was highly anisotropic, with diffusion in the plane of the bilayers hundreds of times greater than out of the plane. The anisotropy of... 

In the polar pores, the diffusion coefficient of all ions is strongly reduced relative to the bulk values. No counterion dependence is observed for the SDC of CP. A more detailed analysis shows that the ion SDC depends on the ion s relative position in the pore [174]. In the case of the K ion, this dependence is particularly strong. K ions forming contact pairs with the surface charges are almost completely immobilized on the time scale of the simulations. The few remaining ions in the center of the pore are almost unaffected by the (screened) surface charges. The fact that most of the K ions form contact pairs substantially reduces the average value of the normalized K SDC to 0.2. The behavior of CP is similar to that of K. The SDC of sodium ions, which... 

We consider desorption from an adsorbate where surface diffusion is so fast (on the time scale of desorption) that the adsorbate is maintained in equilibrium throughout the desorption process. That is to say that, at the remaining coverage 9 t) at temperature T t), all correlation functions attain... 

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 ... 

Molecular Diffusion If time scales are sufficiently long, dispersion results from molecular diffusion. 

Hierarchical Structures Huberman and Kerzberg [huber85c] show that 1// noise can result from certain hierarchical structures, the basic idea being that diffusion between different levels of the hierarchy yields a hierarchy of time scales. Since the hierarchical dynamics approach appears to be (on the surface, least) very different from the sandpile CA model, it is an intriguing challenge to see if the two approaches are related on a more fundamental level. 

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