Smoluchowski smoothing

Elastic interaction occurs when the displacement fields from steps substantially superpose. Atoms located in the vicinity of steps tend to relax stronger compared to those farther away. The resulting displacements or lattice distortions decay with increasing distance perpendicular to the steps. Atoms situated in between two steps experience two opposite forces and cannot fully relax to an energetically more favorable position as would be the case with quasiisolated steps. The line dipoles at steps are due to Smoluchowski smoothing [160] and interact electronically. Only dipole components perpendicular to the vicinal surface lead to repulsion whereas parallel components would lead to attractive interaction. The dipole-dipole interaction seems to be weaker than the elastic one. For instance, steps on vicinal Ag(lll) have weak dipoles as was shown in a theoretical study [161]. Entropic interaction is due to the condition that steps may not cross and leads to an effective repulsive potential, the weakest interaction type. This contribution is always present and results from the assumption that cavities under the surface are unstable. Experiments and theory investigating steps on surfaces were recently reviewed [162]. 

For free surfaces the local work function is usually lowered at steps on metals due to Smoluchowski smoothing [160]. The spatial width of such line dipoles can be determined only from STS work function maps. With steps on Au and Cu surfaces a reduction of the work function in an approximately 8 A wide zone was observed [206]. Figure 59 shows an STM image and a work function map of 0.8 monolayers Au on Cu(lll). Nonlocal methods, for example, traditional photoelectron spectroscopy, simply yield a lowered average work function for stepped surfaces. 

Ta(llO) II -1.0 sat. so that differences to Cl smaller so that Smoluchowski smoothing and polarization effects take over. 

Figure 5.8 The Smoluchowski smoothing effect illustrated for an open surface. Additional electronic charge is piled up in the grooves of the surface, while the protruding Wigner-Seitz cells are electron deficient. The resulting dipole lowers the work function.

Figure 5.9 The Smoluchowski smoothing effect at a step and the corresponding dipole field. (Adapted from Ref. [8].]...

Figure 2.10 Illustration of the Finnis-Heine model for electron density redistribution (Smoluchowski smoothing) and interlayer relaxation at metal surfaces. See text for details. After Ref [77].

Fig. 7. Surface relaxation in the charge smoothing model of Smoluchowski. ...

Smoluchowski attempted to calculate the smoothing. For the most closely packed crystal faces, more complete smoothing is expected and the constant electron density contours are fairly flat this lowers the double-layer moment (negative corrections to values without smoothing). The amount by which the smoothing decreases the double-layer moment will be greater the rougher (on... 

Further Complications. The theory as discussed in this section, when applied to dilute suspensions (since at not very small cp the diffusion distance of the particles is smaller than presumed) of small particles (say, < 1 pm) under quiescent conditions (see Section 13.2.3), has been successful in predicting a number of observations, at least for the first few aggregation steps. This concerns (a) the reaction being second order (b) the value of t0.s being inversely proportional to N0, (c) the distributions of multiplets obtained and (d) the effect of viscosity on the aggregation rate. Also the absolute rate is often well predicted (within a factor of two or three) if it is sure that fast aggregation occurs this rate is often called the Smoluchowski limit. However, the values of the capture efficiency often deviate markedly from the prediction, even for smooth spherical particles. 

Certainly, no microscope would let you see the twists and turns of an individual molecule s path. However, the Einstein-Smoluchowski theory tells us how to spot the difference between a fuzzy line which consists of a great number of tiny random kinks, and an ordinary smooth curve, even though we cannot discern the individual kinks. (We do not always need to see everything, e.g. we can happily tell water from alcohol even though the individual molecules are invisible ) In the same way, a poljrmer chain looks nothing like a shape stretched in a certain direction. And the path of a man in a forest would depend quite noticeably on whether he is equipped with a compass or not ... 

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