Mean square diffusion

The mean square diffusion distance x2 for a random-walk diffusion process is x2 = nr2 = 6Dt, where n is the number of jumps and r is the jump distance. Assume that the cascade volume can be modeled by a gas, then n = vt/r where v is the mean speed of the ions in the cascade. Assume that the energy deposited in the cascade is 1 eV atom-1 and that the cascade lifetime is 10-11 s... 

Mean-square diffusion distances assessed from OSSM tracking on individual particles as a function of temperature (Adapted from [9])... 

Therefore, the mean-square diffusion is proportional to the thermodynamic state of the fluid through and... 

Temperature Measurement, Methods, Figure 6 (a) Fluid temperature determined by micro-PIV assessment of Brownian motion versus the actual fluid temperature measured by a thermocouple (Adapted from [7]). (b) Mean-square diffusion distances assessed from OSSM tracking on individual particles as a function of temperature (Adapted from [9])... 

Brownian motion is a random thermal motion of a particle inside a fluid medium. The collision between the fluid molecules and suspended microparticles are responsible for the Brownian motion. The Brownian motion consists of high frequencies and is not possible to be resolved easily. Average particle displacement after many velocity fluctuations is used as a measure of Brownian motion. The mean square diffusion distance, is proportional to DAt, where D is diffusion coefficient of the particle given by Einstein relation as... 

For a fluid, with no underlying regular structure, the mecin squared displacement gradually increases with time (Figure 6.9). For a solid, however, the mean squared displacement typically oscillates about a mean value. Flowever, if there is diffusion within a solid then tliis can be detected from the mean squared displacement and may be restricted to fewer than three dimensions. For example. Figure 6.10 shows the mean squared displacement calculated for Li+ ions in Li3N at 400 K [Wolf et al. 1984]. This material contains layers of LiiN mobility of the Li" " ions is much greater within these planes than perpendicular to them. 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

To make contact with the diffusion-in-a-sphere model, we have defined the spherical radius as the root-mean-square fluctuation of the protons averaged over 100 ps. The varia-... 

FIG. 2 Mean-square displacement (MSD) of helium atoms dissolved in polyisobutylene. There is a regime of anomalous diffusion (MSD a followed by a crossover at 100 ps to normal (Einstein) diffusion (MSD a r) [24],... 

Thus, in order to reproduce the effect of an experimentally existing activation barrier for the scission/recombination process, one may introduce into the MC simulation the notion of frequency , lo, with which, every so many MC steps, an attempt for scission and/or recombination is undertaken. Clearly, as uj is reduced to zero, the average lifetime of the chains, which is proportional by detailed balance to Tbreak) will grow to infinity until the limit of conventional dead polymers is reached. In a computer experiment Lo can be easily controlled and various transport properties such as mean-square displacements (MSQ) and diffusion constants, which essentially depend on Tbreak) can be studied. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

It is thus seen that the kinematic viscosity, the thermal diffusivity, and the diffusivity for mass transfer are all proportional to the product of the mean free path and the root mean square velocity of the molecules, and that the expressions for the transfer of momentum, heat, and mass are of the same form. 

Lateral diffusion coefficients were determined by monitoring mean-square displacements in the xy-plane, vdiich are proportional to MDt, where D is the diffusion coefficient and t the time. The following values were found ... 

The self-diffusion coefficient parallel to the pore walls was computed form the mean square particle displacement,... 

C05-0075. Determine the root-mean-square speed of SFg molecules under the conditions of Problem 5.31. C05-0076. Determine the root-mean-square speed of H2 molecules under the conditions of Problem 5.32. C05-0077. If a gas line springs a leak, which will diffuse faster through the atmosphere and why, CH4 or... 

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. 

The fractal behavior of diffusion trajectories of ions has been studied in the molten phase of Agl as well as in the a-phase. The Devalues for an MD system with 250 Ag and 250 I" at 900 K were calculated from Fig. 21 to be 2 and 2.17, respectively. The mean-square displacements are shown in Fig. 22 in comparison with those of the a-phase at 670 K. As results of supplementary MD simulations, these authors obtained Dj = 1 for Ag and D = 2.17 at 1000 K and Df = 2 for both ions at 2000 K. Thus, they have concluded that (1) at an extremely high temperature above the melting point, the system is in a completely liquid state, which leads to a... 

Figure 22. Mean-square displacements of Ag and I" at 670 K and 900 K. (Reprinted from M. Kobayashi and F. Shimojo, Molecular Dynamics Studies of Molten Agl.ll. Fractal Behavior of Diffusion Trajectory, J. Phys. Soc. Jpn. 60 4076-4080, 1991, Fig. 8, with permission of the Physical Society of Japan.)...

When it is limited by slow diffusion of the reactants, the mean-square amplitude is... 

According to these equations, in kinetically controlled reactions the mean-square amplitude is about 10 V, while in reactions occurring under diffusion control it is almost an order of magnitude smaller. Thus, the size of electrochemical (thermal) equilibrium fluctuations is extremely small. 

Figure 13.3 (a-c) Trajectories of the diffusion motion of a gold nanoparticle probe on a planar lipid membrane, (d) Mean-square displacement plots forthe diffusion shown in (a-c). Adapted from Ref [31] with permission. 

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