Mean square displacement dynamics

When a large rigid sphere is suspended in a solvent, its dynamical properties, such as the velocity autocorrelation function, mean square displacement, dynamic structure factor, etc., are well known, and form the basis of suspension theory from Einstein onwards. 

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

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]. 

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]. 

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. 

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. 

Molecular Dynamics Simulation (i) 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 ... 

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.)...

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]...

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... 

As mentioned in Section 3.1, the incoherent dynamic structure is easily calculated by inserting the expression for the mean square displacements [Eqs. (42), (43)] into Eq. (4b). On the other hand, for reptational motion, calculation of the pair-correlation function is rather difficult. We must bear in mind the problem on the basis of Fig. 19, presenting a diagrammatic representation of the reptation process during various characteristic time intervals. 

POLYELECTROLYTE DYNAMICS The mean-square displacement of a labeled monomer is... 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

Finally, we note that low-temperature crystallographic studies have been carried out on one nucleic acid, the 5-DNA dodecamer whose room-temperature structure was solved in Dickerson s laboratory (Dickerson, 1981). Refinement at 16 K revealed a large overall drop in B, but some of the atoms in the molecule still had very large B-factors even at this very low temperature. These large residual mean-square displacements were interpreted as demonstrating the presence of static disorder however, by analogy with the results on myoglobin, a disorder which is dynamic at room temperature but becomes frozen into a static distribution at low temperature is also consistent with the observations. It is also possible that the disorder in these atoms is dynamic even at 16 K this point has been considered by Hartmann et al. (1982). 

The exponent Mk depends on the mean square displacement of the atom from its equilibrium position and hence upon temperature. It is linear with (kT/m Xsin / where k is the Boltzmann constant, T the absolute temperature, the scattering angle, the wavelength and m the atomic mass (for a monatomic material). In addition there are complicated expressions dependent upon the crystal symmetry. As an example, for silicon at room temperature the /, are reduced by approximately 6%. With this correction all the equations of dynamical theory still apply. 

In the Gaussian approximation (Eq. 4.12) the mean squared displacement is given by (r (t))=3/[2a(t)], and a2(t) is zero of course. In the light of the above results obtained by neutron scattering (summarized in Eq. 4.14), the values of the non-Gaussian parameter for this process should be very small. However, this result is in apparent contradiction to recent molecular dynamics (MD)... 

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... 

When a body undergoes vibrations, the displacements vary with time, so time averages must be taken to derive the mean-square displacements, as we did to obtain the lattice-dynamical expression of Eq. (2.58). If the librational and translational motions are independent, the cross products between the two terms in Eq. (2.69) average to zero, and the elements of the mean-square displacement tensor of atom n, U"j, are given by... 

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. 

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