Directed walks

The fact FIM only images a small area at the apex of the tip makes it insensitive to vibration and to thermal drift. Because of this, FIM has been used to study the motion of a single atom on a tip over a long period of time. By tracing the trajectory of a single atom, the surface diffusion coefficient and the rate of directional walk of single atoms was directly measured. 

The directional walk of a single atom on the tip surface is not only an interesting observation, but is also relevant to STM experiments. Many tip treatment methods for STM rely on the application of a high electric field near the tip end, which will induce directional walks of atoms at the tip surface, among other effects. Figure 1.35 is an example of the directional walk of a W atom on the surface of a W tip (Tsong, 1990). 

Fig. 1.35. Observation of directional walk of a W atom by FIM. In the absence of an electric field, a W atom on a W surface walks randomly. Under the influence of an electric field, the W atom is polarized and walks toward the direction of high field intensity. (Reproduced from Tsong and Kellogg, 197.5, with permission.)...

Tsong, T. T., and Kellogg, G. (1975). Direct observation of the direct walk of single adatoms and the adatom polarizability. Phys. Rev. B 12, 1343-1353. 

Directional walk of atoms 42 DNA 341 Dominant pole 265 Eddy-current damper 248 Elasticity theory 365—376 Electrochemical tip etching 282—285 Electrochemistry 323... 

Field evaporation 287 Field-emission microscopy 43 Field-ion microscopy 39—43 comparison with STM 41 directional walk of single atoms 42 resolution 40 Force and deformation 37 Force in tunneling experiments 49, 53, 171... 

The effective polarizability of surface atoms can be determined with different methods. In Section 2.2.4(a) a method was described on a measurement of the field evaporation rate as a function of field of kink site atoms and adsorbed atoms. The polarizability is derived from the coefficient of F2 term in the rate vs. field curve. From the rate measurements, polarizabilities of kink site W atoms and W adatoms on the W (110) surface are determined to be 4.6 0.6 and 6.8 1.0 A3, respectively. The dipole moment and polarizability of an adatom can also be measured from a field dependence of random walk diffusion under the influence of a chemical potential gradient, usually referred as a directional walk, produced by the applied electric field gradient, as reported by Tsong et a/.150,198,203 This study is a good example of random walk under the influence of a chemical potential gradient and will therefore be discussed in some detail. 

One can obtain values of both pQ and a from the intercept and the slope of a plot of (2A T/0.867/J7) sinh (/(p) V2(p2 0) against Fc, for brevity referred as a r-plot. All the parameters in the equation can be measured field gradient from desorption voltages of adatoms at different locations on the plane, (p2)0 from a random walk diffusion experiment, and [Pg.272]

Fig. 4.42 Room temperature, argon promoted helium field ion images showing the directional walk of an adatom from the center of the plane toward the edge of the plane. The adatom drifts along the direction of the maximum field gradient, or in the radial direction of the plane.

The sum of all the row and column elements is identical with the out- or indegree, respectivdy. The abscdute value of the elements of the power matrix A indicates the number of directed walks of length uiaV, Since in V no directed walk of length > 3 exists, (A(Pi)) = 0. 

The concept of the length of a path or cyde has already been introduced in Section 2. It was applied to undirected and directed walks in the last two sections. In each case, the length indicates how many edges (arcs) belong to a path, a cyde or a walk In eq.(51) a procedure is outlined for the determinatioii of the length of shortest path. 

Prom chemical point of view, what these motor proteins do is they catalyze the reaction of hydrolysis of ATP each act of hydrolysis yields energy about 14 ksT. Motor proteins use this energy to perform directed walk in one particular chosen direction (instead of random walk in random direction in thermal equilibrimn). 

This directed model consists of fully directed walks for the paths in the chain. These are clearly self-avoiding, and only take steps in positive directions. The corresponding loops are staircase polygons, which consist of two fully directed walks, which do not intersect or touch, but have a common starting point and end point. Paths are attached to these points. We distinguish the two strands of a loop. 

As shown in [89], this PS model of fully directed walks and loops is exactly solvable in arbitrary dimension. The phase transition is found to be first order for d > 6, to have a continuous phase transition in dimensions 2 < d < 5, and to occur at finite temperature in d > 4 only. 

In [96] exactly solvable models based on Dyck paths and Motzkin paths in two dimensions, and a partially directed walk model in three dimensions are given. Orlandini et al. [96] observe re-entrant behaviour in three dimensions, but not in two. Reentrant behaviour is shown in figure 10 below ... 

The values of the exponent in Eqs. (5), (6) and (7) imply that, for a fixed path-length L, wandering is minimum for paths that are in the universcJity class of ordinary random walks (C = 1/2) and maximum for paths that cire in the universality class of directed walks (C = 1). On the contrary, for a given end-to-end distance, paths in the universality class of ordinary random walks are much longer than those in the universcdity class of directed polymers. 

Since the crossover distance diverges asT Tc from below the asymptotic form (i.e., for large R) of the most probable paths undergo a transition from the directed walk phase for T random walk phase at T. ... 

There is another neat scheme (based on the linear recursion) applicable for hand computation on polyhex benzenoids of up to a dozen or so hexagonal rings. This John-Sachs scheme is based on a one-to-one correspondence between Kekule structures and sets of mutually self-avoiding directed walks on the graph, and indeed this correspondence was (in a special context) utilized in a statistical mechanical context in modelling collections of partly disordered polymer chains. 

Fig. 4.4 Illustration of the repton idea, from Ref. 28. (a) A polymer in an entangled net is confined to a tube. Filled circles divide the chain into segments of stored length, (b) Repton model representation of conformation in part (a). Cells of the entanglement net along the confining tube are represented by a one-dimensional lattice. Sections of chain length stored in these cells are modeled by reptons on the lattice sites, (c) Directed walk representation of the same conformation.

Suggestive examples that show the generality of such a model include the case of a general lattice random walk in (1 + 1) dimension, Figure 1.4(A), and the case of a directed walk in 1 + d dimension, that is the process (n, S ) =o,i,..., with S, like before, the partial sums of an IID sequence X, but this time Xi is a discrete random variable taking values in Z , with P(Xi = 0) > 0. Also in these cases we define if ( ) as the distribution of the returns to the origin of course it is very well possible that J if(n) < 1, like for d > 3 or if the walk is asymmetric. 

Rigorous results on copol3miers based on self-avoiding non-directed walks can be found in [Madras and Whittington (2003)]. There is also a considerable amount of numerical work in this direction, see e.g. [Causo and Whittington (2003)]. 

Another research direction that we will not take into account is the study of non-flat interfaces or of polymers in multi-interface environments. Models for such situations based on directed walks and rigorously analyzed may be found for example in [den Hollander and Wiithrich (2004)], [den Hollander and Whittington (2006)] and [Petrelis (2006)]... 

Fig. 6.4 A d-dimensional copolymer model in absence of self-interactions, that is a non-directed walk in a d-dimensional space filled with two solvents (above and below one of the (hyper-)planes of the reference system), may be reduced to a (1 -t l)-directed walk model, since the energy depends only on the sign of the component of S perpendicular to the (hyper-)plane. In the figure we draw on the left a walk in a two-dimensional space the increments are uniformly distributed on a circumference of fixed radius. The walk is then mapped on the right to a directed walk, by projecting on the vertical component. Of course the distribution of the increments of the directed walk is easily computed. We stress that the substantial difference between the resulting model and the one we have treated is that the walk crosses the interface without touching it, so in order to mimic the procedure we have used up to now, one has to introduce a more complex renewal structure, for example the sequence of successive visits to the opposite half-plane and the corresponding overshoot variables.

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