Mean-square displacement motion

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

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.

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. 

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. 

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. 

Fig. 18. Mean square displacement of a chain segment undergoing reptational motion as a function of time...

The correlation factor, for any mechanism, is given by the ratio of the values of the mean square displacement of the atom (often the tracer) moving in a correlated motion to that of the atom (or vacancy) moving by a random-walk process. If the number of jumps considered is large, the correlation factor/can be written as... 

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) ... 

In this way one obtains the mean value of P averaged over vibrational motion in terms of mean displacements, mean square displacements, and so on. This approach has been long used to discuss isotope effects on electrical properties of molecules. 

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. 

K for myoglobin (Parak et al., 1981). Thus, measurements of (x ) at temperatures below this value should show a much less steep temperature dependence than measurements above, if nonharmonic or collective motions (whose mean-square displacement is denoted (x )c) are a significant component of the total (x ). Figure 21 illustrates the expected behavior of (x )v, x, and their sum for a simple model system in which a small number of substates are separated by relatively large barriers. In practice, the relative contributions of simple harmonic vibrations and coUective modes will vary from residue to residue within a given protein. 

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... 

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... 

Table 65(a) Comparison of the Nitrogen Atom Thermal Motion for MI2MII[M(N02)IiJ Systems, Root-mean-square Displacements (A)540... 

Atoms taking part in diffusive transport perform more or less random thermal motions superposed on a drift resulting from field forces (V//,-, Vrj VT, etc.). Since these forces are small on the atomic length scale, kinetic parameters established under equilibrium conditions (i.e., vanishing forces) can be used to describe the atomic drift and transport, The movements of atomic particles under equilibrium conditions are Brownian motions. We can measure them by mean square displacements of tagged atoms (often radioactive isotopes) which are chemically identical but different in mass. If this difference is relatively small, the kinetic behavior is... 

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