Microscopic jump time

Elliott (1987, 1988 and 1989) approached the relaxation problem differently. In his diffusion controlled relaxation (DCR) model, Elliott, like Charles (1961) considers ionic motion to occur by an interstitialcy mechanism. There is a local motion of cations (for example Li ion in a silicate glass) among equivalent positions located around a NBO ion. Motions of cations among these positions causes the primary relaxational event and it occurs with a characteristic microscopic relaxation time t. The process gives rise to a polarization current. However, when another Li ion hops into one of the nearby equivalent positions with a probability P(/), a double occupancy results around the anion and this makes the relaxation instantaneous. Since the latter process involves the diffusion of a Li ion, the process as a whole involves both polarization and diffusion currents. Thus the relaxation function can be written as [l-P(/)]exp(-t/r). [1-P(0] is a function of the jump distance and the diffusion constant. Making use of the Glarum-Bordewijk relation (Glarum, 1960 Bordewijk, 1975) for [1-/ (/)] Elliott (1987) has shown that... 

The term macroscopic description refers to the long-time and large-scale limit, t -> oo and x -> oo, of mesoscopic equations where the details of the microscopic movement are irrelevant. In particular, it refers to the diffusive limit where balance equations such as (3.13), (3.41), and (3.74) are approximated by the diffusion equation (2.1). The standard derivation of the diffusion equation involves the assumption that the typical microscopic jumps and times are small compared to the characteristic macroscopic space and time scales. Let us illustrate this using the mesoscopic transport equation (3.74). If the jump density w(z) is a rapidly decaying function for large z, one can expand p(x - z, t) in z and truncate the Taylor series at the second moment ... 

The main interest of the QNS technique lies, however, in the S (Q, to) behaviour at higher Q values, i.e. when the translational process is followed on molecular distances. A departure from Pick s law is generally observed ". This is indicative of the microscopic aspect of the motion and its interpretation needs the elaboration of more sophisticated models involving characteristic jump lengths, jump times, etc. 

Any multi-barrier process (a random-walk problem) has a relaxation time associated with each microscopic jump. Thus, the steady-state approach fails to predict the transients during the initial period. The appearance of induction periods is frequently observed in chain reactions. The induction period occurs also in the recr tal-lization of cold worked aluminum. These induction periods are understandable in terms of the multi-barrier processes. Further interest in the random-walk treatment lies in the fact that time-dependence of concentration changes in various systems can be predicted. 

QENS studies on the dynamics in PEMs suggest that water and protons attain microscopic mobilities that are similar to those in bulk water, with mean jump times of protons in the order of 1 ps. The experimental scattering data include two components, which correspond to fasf (At O (1 ps)) and slow (At 0 (100 ps)) motions. Observation of the latter type of motion suggests that hydronium ions exist as long-lived entities in Nation (Perrin et al., 2006). 

The relaxation time r of the mean length, = 2A Loo, gives a measure of the microscopic breaking rate k. In Fig. 16 the relaxation of the average length (L) with time after a quench from initial temperature Lq = 1.0 to a series of lower temperatures (those shown on the plot are = 0.35,0.37, and 0.40) is compared to the analytical result, Eq. (24). Despite some statistical fluctuations at late times after the quench it is evident from Fig. 16 that predictions (Eq. (24)) and measurements practically coincide. In the inset is also shown the reverse L-jump from Tq = 0.35 to = 1.00. Clearly, the relaxation in this case is much ( 20 times) faster and is also well reproduced by the non-exponential law, Eq. (24). In the absence of laboratory investigations so far, this appears the only unambiguous confirmation for the nonlinear relaxation of GM after a T-quench. 

An even more serious problem concerns the corresponding time scales on the most microscopic level, vibrations of bond lengths and bond angles have characteristic times of approx. rvib 10-13 s somewhat slower are the jumps over the barriers of the torsional potential (Fig. 1.3), which take place with a time constant of typically cj-1 10-11 s. On the semi-microscopic level, the time that a polymer coil needs to equilibrate its configuration is at least a factor of the order larger, where Np is the degree of polymerization, t = cj 1Np. This formula applies for the Rouse model [21,22], i. e., for non-... 

Thus, microscopically, the diffusion coefficient may be interpreted as one-sixth of the jumping distance squared times the overall jumping frequency. Since / is of the order 3 x 10 ° m (interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D m /s such as Mg diffusion in... 

By assuming an Arrhenius type temperature relation for both the diffusional jumps and r, we can use the asymptotic behavior of /(to) and T, as a function of temperature to determine the activation energy of motion (an example is given in the next section). We furthermore note that the interpretation of an NMR experiment in terms of diffusional motion requires the assumption of a defined microscopic model of atomic motion (migration) in order to obtain the correct relationships between the ensemble average of the molecular motion of the nuclear magnetic dipoles and both the spectral density and the spin-lattice relaxation time Tt. There are other relaxation times, such as the spin-spin relaxation time T2, which describes the... 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

Equation 7.35 is a fundamental relationship between the diffusivity and the mean-square displacement of a particle diffusing for a time r. Because diffusion processes in condensed matter are comprised of a sequence of jumps, the mean-square displacement in Eq. 7.31 should be equivalent to Eq. 7.35. This equivalence, as demonstrated below, results in relations between macroscopic and microscopic diffusion parameters. 

Einstein-Smoluchowski equation — Relationship between diffusion coefficient D, average width of a jump A of a microscopic species (atom, ion, molecule) involved in diffusion, and average time r between two jumps... 

Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a two-dimensional lattice then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant Ax, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation... 

Transient intermediates are most commonly observed by their absorption (transient absorption spectroscopy see ref. 185 for a compilation of absorption spectra of transient species). Various other methods for creating detectable amounts of reactive intermediates such as stopped flow, pulse radiolysis, temperature or pressure jump have been invented and novel, more informative, techniques for the detection and identification of reactive intermediates have been added, in particular EPR, IR and Raman spectroscopy (Section 3.8), mass spectrometry, electron microscopy and X-ray diffraction. The technique used for detection need not be fast, provided that the time of signal creation can be determined accurately (see Section 3.7.3). For example, the separation of ions in a mass spectrometer (time of flight) or electrons in an electron microscope may require microseconds or longer. Nevertheless, femtosecond time resolution has been achieved,186 187 because the ions or electrons are formed by a pulse of femtosecond duration (1 fs = 10 15 s). Several reports with recommended procedures for nanosecond flash photolysis,137,188-191 ultrafast electron diffraction and microscopy,192 crystallography193 and pump probe absorption spectroscopy194,195 are available and a general treatise on ultrafast intense laser chemistry is in preparation by IUPAC. 

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