Surface Mean-Square Displacements

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... 

Figure 4. Mean-square-displacements of Au adatom in the normal to the surface direction as a function of temperature, for the three low-index faces of Cu.

FIGURE 2-11 Video-enhanced Differential Interference Contrast (DIC) images of gold-labeled p opioid GPCR on the surface of a GPCR-transfected fibroblast. The white trace is the trajectory of one particle over 2 minutes at 25frames/s. The black trace is the mean square displacement of the particle as a function of time. Reproduced from Figure 1 of [30], with permission. 

For proteins the X-ray structures usually are not determined at high enough resolution to use anisotropic temperature factors. Average values for B in protein structures range from as low as a few A2 for well-ordered structures to 30 A2 for structures involving flexible surface loops. Using equation 3.6, one can calculate the root mean square displacement fu2 for a well-ordered protein structure at approximately 0.25 A (for B = 5 A2) and for a not-so-well-ordered structure at... 

Vibrations in the surface plane, however, will be rather similar to those in the bulk because the coordination in this plane is complete, at least for fee (111) and (100), hep (001) and bcc (110) surfaces. Thus the Debye temperature of a surface is lower than that of the bulk, because the perpendicular lattice vibrations are softer at the surface. A rule of thumb is that the surface Debye temperature varies between about 1/3 and 2/3 of the bulk value (see Table A.2). Also included in this table is the displacement ratio, the ratio of the mean squared displacements of surface and bulk atoms due to the lattice vibrations [1]. 

Finally, if the temperature increases, becomes larger until the crystal melts. The Lindemann criterion predicts that melting sets in when becomes about 0.25 a2, where a is the interatomic distance of the metal. Because the mean squared displacements of surface atoms is higher we expect that the surface melts at lower temperatures than the bulk does [2]. Indeed, evidence has been presented that the (110) surface of lead starts to melt at 560 K whereas the bulk melting temperature is about 600 K [13]. 

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

Adatom diffusion, at least under the low temperature of field ion microscope measurements, almost always follows the direction of the surface channels. Thus adatoms on the W (112) and Rh (110) surfaces diffuse in one direction along the closely packed atomic rows of the surface channels. Such one-dimensional surface channel structures and random walks can be directly seen in the field ion images, and thus the diffusion anisotropy is observed directly through FIM images. Unfortunately, for smoother surfaces such as the W (110) and the fee (111), no atomic or surface channel structures can be seen in field ion images. But even in such cases, diffusion anisotropy can be established through a measurement of the two-dimensional displacement distributions, as discussed in the last section. Because of the anisotropy of a surface channel structure, the mean square displacements along any two directions will be different. In fact this is how diffusion anisotropy on the W (110) surface was initially found in an FIM observation.120... 

Diffusion anisotropy on the W (110) surface was realized when the mean square displacements of a Re adatom on a W (110) plane along the [100] and [110] directions, or x- and y-directions shown in Fig. 4.28, were... 

Atomic jumps in random walk diffusion of closely bound atomic clusters on the W (110) surface cannot be seen. A diatomic cluster always lines up in either one of the two (111) surface channel directions. But even in such cases, theoretical models of the atomic jumps can be proposed and can be compared with experimental results. For diffusion of diatomic clusters on the W (110) surface, a two-jump mechanism has been proposed by Bassett151 and by Cowan.152 Experimental studies are reported by Bassett and by Tsong Casanova.153 Bassett measured the probability of cluster orientation changes as a function of the mean square displacement, and compared the data with those derived with a Monte Carlo simulation based on the two-jump mechanism. The two results agree well only for very small displacements. Tsong Casanova, on the other hand, measured two-dimensional displacement distributions. They also introduced a correlation factor for these two atomic jumps, which resulted in an excellent agreement between their experimental and simulated results. We now discuss briefly this latter study. 

Arnold and Hobert (231) studied the chemisorption of ferrocene, (C5H5)2Fe, on a silica surface from an alcohol solution. Ferrocene itself shows a symmetric quadrupole splitting. After chemisorption this doublet is no longer symmetric, and the authors explain this in terms of a Gol danskii-Karyagin effect, where the iron atoms in the adsorbed state have a larger mean square displacement perpendicular to the silica surface than parallel to it. 

As mentioned above, the diffusion process is thought to be a random walk across the surface. Then the mean-square displacement of the adparticles is related to the diffusion coefficient via the relation... 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

Figure 2 Mean square displacement versus time for H2O at Cap2 and CaC03 surfaces at 300 K...

NPT ensemble anti used the shell-model to describe polarizability. All simulation runs were performed at atmospheric pressure and in the temperature range 10 - 1100 K. For all three surfaces at both 300 and 1100 K it was found that the surface mean square displacements are generally larger for the oxide ions than for the cations and that the out-of-plane surface motion is usually larger than the in-plane surface motion. At room temperature, the oxygen mean square displacements at the (111) surface arc a factor 1.2 larger than in the bulk, a factor 1.6 for the (Oil) surface and approximately five limes larger at the metastable (001) surface compared to the bulk. The effect of the presence of a surface on the ion dynamics (and on the structure for (011)) persists all the way to the slab centers, even for these rather thick slabs. 

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