Squared displacement

Now for a particle undergoing difflision, it is also known that its mean square displacement grows linearly in time, for long times, as... 

For the system in thennal equilibrium, one can compute the time-dependent mean square displacement (ICr)... 

The relationship between mean squared phase shift and mean squared displacement can be modelled in a simple way as follows This motion is mediated by small, random jumps in position occurring with a mean interval ij. If the jump size in the gradient direction is e, then after n jumps at time the displacement of a spin is... 

Fig. 6.9 Variation in mean squared displacement during the initial steps of a molecular dynamics simulation of argon.

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. 

Fig. 6.10 Mean squared displacement for Li ions in Li N for motion parallel (xy) and perpendicular (z) to the LijN layers [Wolfet al. 1984],...

Table 1 Mean-Square Displacements of Hydrogen Atoms m Crystalline Acetanilide at 15 K ...

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]. 

Figure 13 Center-of-mass mean-square displacements computed from MD simulations at 323 K. (a) DPPC motion in the plane of a lipid bilayer averaged over 10 ps (b) DPPC motion in the plane of a lipid bilayer averaged over 100 ps (c) comparison of the DPPC m-plane mean-square displacement to linear and power law functions of time (d) comparison of the center-of-mass mean-square displacement from an MD simulation of liquid tetradecane to a linear function of time.

Figure 15 (a) Mean-square displacements of water molecules m three dimensions (see text for... 

FIG. 14 Mean square displacement (Ar )io after 10 ps as a function of the initial distance p of a water molecule from the pore axis for the two runs with 96% (full line) and 74% filling. (Taken with permission from Ref. 24.)... 

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],... 

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

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. 

Actually, it is also useful to introduce the mean-square displacement of inner monomers measured in the center of mass coordinate system of the chain... 

FIG. 18 Mean-square displacements gi t) vs time for chains with 7 = 128 in a narrow non-adsorbing sUt (Z)= l,e = 0) at density cp = 1.5. Straight lines show effective exponents = 0.56, and = 0.84, respectively. Broken horizontal lines show (above) and (below), (b) Log-log plot of the relaxation time r 3 vs N for the case (D = 1, e = 0) and various densities (p as indicated. Straight lines show interpretations in terms of effective exponents Zgff (r oc [16]. 

The simulations to investigate electro-osmosis were carried out using the molecular dynamics method of Murad and Powles [22] described earher. For nonionic polar fluids the solvent molecule was modeled as a rigid homo-nuclear diatomic with charges q and —q on the two active LJ sites. The solute molecules were modeled as spherical LJ particles [26], as were the molecules that constituted the single molecular layer membrane. The effect of uniform external fields with directions either perpendicular to the membrane or along the diagonal direction (i.e. Ex = Ey = E ) was monitored. The simulation system is shown in Fig. 2. The density profiles, mean squared displacement, and movement of the solvent molecules across the membrane were examined, with and without an external held, to establish whether electro-osmosis can take place in polar systems. The results clearly estab-hshed that electro-osmosis can indeed take place in such solutions. 

FIG. 5 Mean squared displacement of solvent molecules in a direction perpendicular to the plane of the membrane plotted against time [26]. 

FIG. 10 Effect of an electric field (Eg) on the mean-squared displacement perpendicular to the plane of the membrane for a 4.67 mole percent aqueous LiCl solution at 25°C and 1 bar [25]. 

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. 

The temperature dependence of the mean-square-displacements of Au adatom in the normal to the surface direction is shown in Figure 4 for the three low-index faces of Cu. We note that up to 500"K the MSD s on the three different faces are almost equal, while at higher temperatures the vibrational amplitudes of Au on Cu(llO) present enhanced anharmonicity and become much larger than on the other faces. These results denote that... 

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