Mean square displacement observation time

At the moment the noise in the mean squared displacement versus time curves does not permit easy observation of this type of time-dependent behavior, and it remains a matter for future investigation. Also of interest is the question of whether, under the highly diffusive conditions of computer simulations, the systems very near equilibrium relax exponentially ( single relaxation time ) or otherwise, as for laboratory glasses. ... 

Another way of observing the anomalous nature of diffusion on a fractal is to find out what becomes of the diffusion coefficient D t), defined from the classic Einstein relation between mean squared displacement and time ... 

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

The mean square segment displacements, which are the key ingredient for a calculation of the dynamic structure factor, are obtained from a calculation of the eigenfunctions of the differential Eq. 5.13. After retransformation from Fourier space to real space B k,t) is given by Eq. 41 of [213]. For short chains the integral over the mode variable q has to be replaced by the appropriate sum. Finally, for observation times mean square displacements can be expressed in... 

Aging behavior observed in the mean square displacement, (Ax ), as a function of time for different ages. The colloidal system reorganizes slower as it becomes older, (c) y = (Ax )/3 (upper curve) and (Ax ) (lower curve) as a function of the age measured over a fixed time window At = 10 min. For a diffusive dynamics both curves should coincide, however these measurements show deviations from diffusive dynamics as well as intermittent behavior. Panels (a) and (b) from http // www.physics.emory.edu/ weeks/lab/aging.html and Panel (c) from Refill. 

From the considerations above, a further semi-quantitative observation can be made about molecular motion. Over short times, the velocity of motion of a molecule is correlated with its velocity a little earlier. Under these circumstances, the mean square displacement is of the form... 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

The asymptotic (xdj l) evaluations for the mean square displacement of a particle were found by Kokorin and Pokrovskii (1990, 1993). For a short time of observation, t more than the mobility of the macromolecule... 

The mean-square displacement of the chain segments of BR swollen with deuterated benzene was observed to be independent of diffusion time, indicating restricted diffusion around an attractive centre. The mean-squared displacement decreased with increasing crosslinking density and was approximately equal to the mean-squared collective fluctuations calculated for these polymers. 

Independent of whether or not a well-defined crossover temperature can be observed in NS data above Tg, it has been well known for a considerable time that on heating a glass from low temperatures a strong decrease of the Debye-Waller factor, respectively Mossbauer-Lamb factor, is observed close to Tg [360,361], and more recent studies have confirmed this observation [147,148,233]. Thus, in addition to contributions from harmonic dynamics, an anomalously strong delocalization of the molecules sets in around Tg due to some very fast precursor of the a-process and increases the mean square displacement. Regarding the free volume as probed by positron annihilation lifetime spectroscopy (PALS), for example, qualitatively similar results were reported [362-364]. 

The Einstein-Smoluchowski equation, = 2Dt, gives a measure of the mean-square displacements of a diffusing particle in a time t. There is the mean-square distance traveled by most of the ions. Common observation using dyes or scents shows that diffusion of some particles occurs far ahead of the diffusion front represented by the = 2Dt equation. Determine the distance of this Einstein-Smoluchowski diffusion front for a colored ion diffusing into a solution for 24 hr (D = 3.8 x 10 cm s ). Determine for the same solution how far the farthest 1% of the total diffused material diffused in the same time. Discuss how it is possible that one detects perfume across the space of a room in (say) 30 s. 

According to Eqs. (4)-(6), the molecular mean square displacements and thus the self-diffusion coefficients may be determined from the slope of a semilogarithmic plot of the PFG NMR signal F versus (Sg) The observation time of self-diffusion is the separation between the two field gradient pulses, t. Owing to their relatively large gyromagnetic ratio and to their natural abundance of 1, protons provide very suitable conditions for NMR self-diffusion studies, but C 44), F 45), and Xe 46-48) resonances have also been used successfully in recent PFG NMR studies of zeolites. 

The molecular root mean square displacement, r t)), of the diffusing molecules during the observation time, t, has to be much smaller than the crystal radius, R, in order to guarantee that the measured r.m.s. displacement reflects the undisturbed intracrystalline self-diffusion. Assuming... 

Figures 42 and 43 show scanning electron micrographs of two ZSM-5 samples of different configurations coffin-shaped crystals and polycrystalline grains. To emphasize the relationship between sample dimensions and diffusion paths followed during the PFG NMR experiment, magnifications are referenced against typical root mean square displacements for methane and propane molecules during typical PFG NMR observation times.

Technically, self-diffusion describes the displacement of a labeled molecule in a fluid of unlabeled but otherwise identical molecules. If this motion is chaotic, the mean square displacement will eventually obey the prediction of equation 13 and one can calculate the diffusion constant Dq for motion in direction g. This particular motion is difficult to observe in real adsorption systems so that simulation becomes of particular interest here. Before reviewing the literature, it is useful to consider the mean square displacement of a particle at short time rather than in the long time diffiisional limit. In the short time limit, one can carry out a Taylor series expansion to show that, after averaging, the mean square displacement in the q th direction q = x, y, z) is [60] ... 

Figure 27.6b shows the trajectory of an individual synthetic virus during such an internalization process [29] (Movie, see supplementary material of [29]). Three different phases can be identified In phase I, binding to the plasma membrane is followed by a slow movement with drift, which can be deduced from the quadratic dependence of the mean square displacement as a function of time. Furthermore, a strong correlation between neighboring particles is seen and subsequent internalization is observed, and can be proven by quenching experiments. During this phase, the particles are subjected to actin-driven processes mediated by transmembrane proteins. Phase II is characterized by a sudden increase in particle velocity and random movement, often followed by confined movement. 

Another deviation from the pattern of ordinary diffusion must be expected if the reactant and product molecules are subjected to single-file conditions, i.e. if (i) the zeolite pore system consists of an array of parallel channels and if (ii) the molecules are too big to pass each other. In this case, the molecular mean-square displacement z t)) is found to be proportional to the square root of the observation time, rather than to the observation time itself. First PFG NMR studies of such systems are in agreement with this prediction [8]. By introducing a mobility factor F, in analogy to the Einstein relation for ordinary diffusion. 

Figure 12. Time series of the mean-square displacement of the phases aj t). N = 100,1000, and 10,000 from top to bottom. The vertical axis is the original scale only for N = 10,000, and is multiplied by 103 and 106 for N = 1000 and 100, respectively, just for a graphical reason. In Stage I where the system is quasi-stationary, the numerical results are approximated by solid curves that are obtained from Eq. (12) using functions in Eq. (19). After the system reaches equilibrium, diffusion becomes normal. Anomaly in diffusion is observed only in Stage II. The two short vertical lines on each curve show the end of Stages I and II, which correspond to the ones found in Fig. 2. [Reproduced with permission from Y. Y. Yamaguchi, Phys. Rev. E 68, 066210 (2003). Copyright 2004 by the American Physical Society.]...

Figure 3.2 shows the mean square displacement (MSD) for water in the glassy and supercooled glucose solutions in a log-log scale. The most remarkable result is that water diffuses at 220 K, in the glass, as observed in the experiments. The diffusion at 220 K, however, occurs in a scale comparable with the /rs of the simulation and cannot be quantified from the data in Figure 3.2. For the five supercooled solutions, T = 250 to 365 K, we computed the diffusion coefficient from the long time dependence of...

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