Cubing

Marching Cubes A High Resolution 3D Surface Construction Algorithm, Computer Graphics 21(4), pp 163-169 (1987)... 

Small drops or bubbles will tend to be spherical because surface forces depend on the area, which decreases as the square of the linear dimension, whereas distortions due to gravitational effects depend on the volume, which decreases as the cube of the linear dimension. Likewise, too, a drop of liquid in a second liquid of equal density will be spherical. However, when gravitational and surface tensional effects are comparable, then one can determine in principle the surface tension from measurements of the shape of the drop or bubble. The variations situations to which Eq. 11-16 applies are shown in Fig. 11-16. 

It is thus seen that the dipole-induced dipole propagation gives an exponential rather than an inverse x cube dependence of U x) with x. As with the dispersion potential, the interaction depends on the polarizability, but unlike the dispersion case, it is only the polarizability of the adsorbed species that is involved. The application of Eq. VI-43 to physical adsoiption is considered in Section XVII-7D. For the moment, the treatment illustrates how a long-range interaction can arise as a propagation of short-range interactions. 

It might be noted that only for particles smaller than about 1 /ig or of surface area greater than a few square meters per gram does the surface energy become significant. Only for very small particles does the edge energy become important, at least with the assumption of perfect cubes. 

A monolayer can be regarded as a special case in which the potential is a square well however, the potential well may take other forms. Of particular interest now is the case of multilayer adsorption, and a reasonable assumption is that the principal interaction between the solid and the adsorbate is of the dispersion type, so that for a plane solid surface the potential should decrease with the inverse cube of the distance (see Section VI-3A). To avoid having an infinite potential at the surface, the potential function may be written... 

The physical model is thus that of a liquid him, condensed in the inverse cube potential held, whose thickness increases to inhnity as P approaches... 

Equation XVII-78 turns out to ht type II adsorption isotherms quite well—generally better than does the BET equation. Furthermore, the exact form of the potential function is not very critical if an inverse square dependence is used, the ht tends to be about as good as with the inverse-cube law, and the equation now resembles that for a condensed him in Table XVII-2. Here again, quite similar equations have resulted from deductions based on rather different models. 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

Since solids do not exist as truly infinite systems, there are issues related to their temiination (i.e. surfaces). However, in most cases, the existence of a surface does not strongly affect the properties of the crystal as a whole. The number of atoms in the interior of a cluster scale as the cube of the size of the specimen while the number of surface atoms scale as the square of the size of the specimen. For a sample of macroscopic size, the number of interior atoms vastly exceeds the number of atoms at the surface. On the other hand, there are interesting properties of the surface of condensed matter systems that have no analogue in atomic or molecular systems. For example, electronic states can exist that trap electrons at the interface between a solid and the vacuum [1]. 

Slater was one of the first to propose that one replace V m equation A 1.3.18 by a tenn that depends only on the cube root of the charge density [T7,18 and 19]. In analogy to equation A1.3.32, he suggested that V be replaced by... 

In general, a point group synnnetry operation is defined as a rotation or reflection of a macroscopic object such that, after the operation has been carried out, the object looks the same as it did originally. The macroscopic objects we consider here are models of molecules in their equilibrium configuration we could also consider idealized objects such as cubes, pyramids, spheres, cones, tetraliedra etc. in order to define the various possible point groups. 

The point groups T, and /j. consist of all rotation, reflection and rotation-reflection synnnetry operations of a regular tetrahedron, cube and icosahedron, respectively. 

The argument is sometimes given that equation (Al.6,29) implies that the ratio of spontaneous to stimulated emission goes as the cube of the emitted photon frequency. This argument must be used with some care recall that for light at thennal equilibrium, goes as BP, and hence the rate of stimulated emission has a factor... 

One of the main uses of these wet cells is to investigate surface electrochemistry [94, 95]. In these experiments, a single-crystal surface is prepared by UFIV teclmiqiies and then transferred into an electrochemical cell. An electrochemical reaction is then run and characterized using cyclic voltaimnetry, with the sample itself being one of the electrodes. In order to be sure that the electrochemical measurements all involved the same crystal face, for some experiments a single-crystal cube was actually oriented and polished on all six sides Following surface modification by electrochemistry, the sample is returned to UFIV for... 

In an irreversible process the temperature and pressure of the system (and other properties such as the chemical potentials to be defined later) are not necessarily definable at some intemiediate time between the equilibrium initial state and the equilibrium final state they may vary greatly from one point to another. One can usually define T and p for each small volume element. (These volume elements must not be too small e.g. for gases, it is impossible to define T, p, S, etc for volume elements smaller than the cube of the mean free... 

Approximating the real potential by a square well and infinitely hard repulsive wall, as shown in figure A3.9.2 we obtain the hard cube model. For a well depth of W, conservation of energy and momentum lead [H, 12] to the very usefiil Baule fomuila for the translational energy loss, 5 , to the substrate... 

Of course the real projectile-surface interaction potential is not infinitely hard (cf figure A3,9,2. As E increases, the projectile can penetrate deeper into the surface, so that at its turning point (where it momentarily stops before reversing direction to return to the gas phase), an energetic projectile interacts with fewer surface atoms, thus making the effective cube mass smaller. Thus, we expect bE/E to increase with E (and also with W since the well accelerates the projectile towards the surface). 

This interpretation is largely based on the results of cube models for the surface motion. It may also be that... 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

---