Self-diffusion mean-square displacement

Self-diffusion coefficients are dynamic properties that can be easily obtained by molecular dynamics simulation. The properties are obtained from mean-square displacement by the Einstein equation ... 

This equation, called the Kubo equation, is equivalent to the Einstein equation. However, it is easier to estimate self-diffusion coefficients from the slope of the mean-square displacements. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

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...

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

In Eq. (19) we used the fact that the mean square displacement of the center-of-mass provides the diffusion constant according to DR = (l/6t)<(x0(t) — Xo(0))2>. For the special case of the self correlation function (n = m) Arnm(t) reveals the mean square displacement of a polymer segment. For t < xR we obtain... 

The self-correlation function leads directly to the mean square displacement of the diffusing segments Ar2n(t) = <(rn(t) — rn(0))2>. Inserting Eq. (20) into the expression for Sinc(Q,t) [Eq. (4b)] the incoherent dynamic structure factor is obtained... 

Thus, for unbounded molecules, the mean-square displacement changes linearly with time. It is well known that the self-diffusion coefficient D in infinitely dilute solution is related to molecular size according to equation ... 

Translational motion is the change in location of the entire molecule in three-dimensional space. Figure 11 illustrates the translational motion of a few water molecules. Translational motion is also referred to as self-diffusion or Brownian motion. Translational diffusion of a molecule can be described by a random walk, in which x is the net distance traveled by the molecule in time At (Figure 12). The mean-square displacement (x2) covered by a molecule in a given direction follows the Einstein-derived relationship (Eisenberg and Crothers, 1979) ... 

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. 

Solution. During self-diffusion, the fraction of the time that a diffusing atom spends in dislocation cores is equal to the fraction of all available sites that are located in the dislocation cores. This fraction will be 77 = pn52/A. The mean-square displacement due to self-diffusion along the dislocations is then DDr)t, while the corresponding displacement in the crystal is DXL( 1 — g)t. Therefore,... 

If the tracer is composed of the same species as that of the solid host, then the diffusion coefficient is named the tracer self-diffusion coefficient, where DA is the tracer self-diffusion coefficient. It is necessary to clarify that self-diffusion is a particle transport process that takes place in the absence of a chemical potential gradient [13]. This process is described, as explained later, by following the molecular trajectories of a large number of molecules, and determining their mean square displacement (MSD). 

Some of the pioneering work has focused on the behavior of abstract Len-nard-Jones oligomers in nanoscopic confinements and consists of studies of the relaxation times of different modes [38a,b,d], of the transport properties like the self-diffusion coefficients and the mean square displacements [38c,d],... 

In the time scale of f, the self-diffusion coefficient D can be expressed by the mean-square displacement (z2) in the z direction from its starting point after the diffusing time A, followed by the Gaussian distribution as follows ... 

The classical MD simulations performed in task I provide self-diffusion coefficients for water and also for hydronium ions, which is strictly the vehicular component of the proton diffusivity. These diffusion coefficients are calculated from the mean square displacement of H2O and HsO using the Einstein relation. The numerical values for Nation and SSC membranes at the four hydration levels are hsted in Table 5 along with the experimental values. ... 

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... 

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] ... 

The molecular dynamics calculations were used to calculate the self-diffusion coefficient (D) of the guest molecule using the mean square displacements (MSD) of the center of mass with respect to time (Figure 6). Using the Einstein relation... 

The randomizaton takes place by a a diffusion of atoms that is implicit in our earlier description of the initial randomization process as being akin to melting [1]. Later it was shown that the root-mean-square displacement of each atom must be of the order of the nearest-neighbor distance in order that the network lose all memory of the original crystal structure as measured by the structure factor S q) [21]. In this context, the melting point can be defined as that temperature for which the mean square displacement increases linearly with time. It appears, though, that a sequence of bond switches as illustrated in Fig. 1 is not the primary mechanism for self-diffusion in silicon [31,32]... 

Taking the hard sphere colloids as a reference state, the mean-square displacement (MSD) in dilute suspensions is associated with the particle self-diffusion whereas at finite volume fractions the onset of interactions marks the alteration of the dynamics. The latter can be probed by the intermediate scattering function C(, t) which measures the spatiotemporal correlations of the thermal volume fraction fluctuations [91]. Figure depicts two representations (lower inset and main plot) of the non-exponential for a nondilute hard sphere colloidal... 

Figure The time dependence of the mean square displacements for adsorbed methane at T = 300 K in the direction parallel to the pore axis is shown for a range of time sufficient for the squares of the molecular displacement to become linear functions of time. NP denotes the total number of molecules in the pore. The slopes of the linear portions of the plots give the self-diffusion constant D = 2x slope.

According to these equations, the self-diffusion coefficients, as well as the molecular mean square displacements, may be determined from the slope of a semi-logarithmic plot of the echo attenuation f vs (.5 g)1 . 

For root mean square displacements 1 much less than the mean crystallite diameters,the thus determined self-diffusion coefficient exclusively refers to migration in the intracrystalline space. In the opposite limiting case of large molecular displacements one obtains the coefficient of long-range self-diffusion (Di.v.). This quantity is related to the relative... 

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