Image boundary

As a general rule, all orthogonal wavelets lack symmetry. This becomes an issue in applications such as image processing where symmetric wavelets are preferable. The symmetric wavelets also facilitate the handling of image boundaries. 

Figure 1 Sideways view of a fluid-carbon nanotube system. The solid lines represent the nanotube. The dashedlines represent either minimum image boundaries (shock wave simulations) or initial locations of driving and driven membranes (pressure wave simulations). Fluid atoms are enclosed within...

Two types of simulations, namely shock and pressure wave, are reported. In the shock wave simulations, Zi and Zj are fixed at the tenth and tenth to last rings of the nanotube and represent locations of minimum image boundaries described previously. A 10 A plug of fluid is given an initial z velocity. Subsequent fluid motion is tracked very carefully through visualization and through extensive analysis. 

Molecular dynamics can be used to simulate the properties of bulk materials. This is done by using periodic boundary conditions to eliminate the system-vacuum interface. This complicated subject is treated elsewhere in this encyclopedia. Suffice it to say that for systems with charges one should not truncate the interactions but should rather use Ewald boundary conditions. For systems with only short-range interactions it is usually a good approximation to use minimum image boundary conditions or spherical truncation. Of course, the properties of small clusters can be simulated without the use of boundary conditions but then one risks losing some of the molecules in the system due to evaporation, and the cluster will eventually disappear. In such circumstances it is useful to invent a potential that binds the particles weakly to the center of the cluster, It is also possible to simulate systems with one or more interfaces. 

Figure Bl.17.9. A CoSi grain boundary as visualized in a spherical-aberration-corrected TEM (Haider et a/ 1998). (a) Individual images recorded at different defocus with and without correction of C(b) CTFs in the case of the uncorrected TEM at higher defocus (c) CTF for the corrected TEM at only 14 nm underfocus. Pictures by courtesy of M Haider and Elsevier.

Figure B3.3.3. Periodic boundary conditions. As a particle moves out of the simulation box, an image particle moves in to replace it. In calculating particle interactions within the cutoff range, both real and image neighbours are included.

Figure C 1.5.13. Schematic diagram of an experimental set-up for imaging 3D single-molecule orientations. The excitation laser with either s- or p-polarization is reflected from the polymer/water boundary. Molecular fluorescence is imaged through an aberrating thin water layer, collected with an inverted microscope and imaged onto a CCD array. Aberrated and unaberrated emission patterns are observed for z- and xr-orientated molecules, respectively. Reprinted with pennission from Bartko and Dickson [148]. Copyright 1999 American Chemical Society.

A striking feature of the images is the nonunifonnity of the distribution of the adsorbed species. The reaction between O and CO takes place at the boundaries between the surface domains and it was possible to detennine reaction rates by measuring the change in length L of the boundaries of the O islands. The kinetics is represented by the rate equation... 

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

Fig. 1. Periodic boundary conditions protect the inner simulation cell from disturbing effects of having all its particles close to the surface. With PBCs in force, as a particle moves out of the box on one side, one of its images will move back into the box on the opposite side.

Hor the periodic boundary conditions described below, the ctitoff distance is fixed by the nearest image approximation to be less than h alf th e sm allest box len gth. W ith a cutoff an y larger, more than nearest images would be included. 

Choose the nonbonded cutoff carefully when using periodic boundary conditions. The cutoff must be small enough to prevent an atom from interacting simultaneously with another atom and with that atom s virtual image. 

You can choose to calculate all nonbonded interactions or to truncate (cut off) the nonbonded interaction calculations using a switched or shifted function. Computing time for molecular mechanics calculations is largely a function of the number of nonbonded interactions, so truncating nonbonded interactions reduces computing time. You must also truncate nonbonded interactions for periodic boundary conditions to prevent interaction problems between nearest neighbor images. 

---