Particles cubes

Fig. 2.22 Electrode resistance Ra versus the electrode thickness and the contact area ratio D = 50 nm, tp = t/3, gp = D, Ra — i t 1 fxm, f = 10%), where gp is the electrode density in terms of gap size between the inner particle cube. Reprinted from [Kim et al. (2007b)].

The number of moles of substance in the particle is n = AnR H V . The fraction reacted (a) is related to the ratio of the radii of the core and particle cubed. 

The regularly-shaped particles (cube, cuboid, cylinder, cone etc.) were individually made in the size range of between 1 mm and 6 mm. Regular bodies allow the average projected area and volume to be calculated from their linear dimensions. In the case of the average projected area, this can either be calculated numerically by a computer or with the aid of the Cauchy theorem ... 

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. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

The truncated octahedron and the rhombic dodecahedron provide periodic cells that are approximately spherical and so may be more appropriate for simulations of spherical molecules. The distance between adjacent cells in the truncated octahedron or the rhombic df)decahedron is larger than the conventional cube for a system with a given number of particles and so a simulation using one of the spherical cells will require fewer particles than a comparable simulation using a cubic cell. Of the two approximately spherical cells, the truncated octahedron is often preferred as it is somewhat easier to program. The hexagonal prism can be used to simulate molecules with a cylindrical shape such as DNA. 

Of the five possible shapes, the cube/parallelepiped and the truncated octahedron have been most widely used, with some simulations in the hexagonal prism. The formulae used to translate a particle back into the central simulation box for these three shapes are given in Appendix 6.4. It may be preferable to use one of the more common periodic cells even if there are aesthetic reasons for using an alternative. This is because the expressions for calculating the images may be difficult and inefficient to implement, even though the simulation would use fewer atoms. 

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

The surface area of a given mass of solid is inversely related to the size of the constituent particles. Thus, for the idealized case where these are equi-sized cubes of edge length /, the specific surface area A, being the surface area of I gram of solid, is given by (cf. p. 26)... 

We will now consider the dependence of specific surface on particle size for systems composed of particles of simple shape, and exhibiting a distribution of particle sizes. The shapes chosen will, in the first instance, be cubes and spheres, rods, and plates, and will be dealt with in turn. 

The numerical values of and a, for a particular sample, which will depend on the kind of linear dimension chosen, cannot be calculated a priori except in the very simplest of cases. In practice one nearly always has to be satisfied with an approximate estimate of their values. For this purpose X is best taken as the mean projected diameter d, i.e. the diameter of a circle having the same area as the projected image of the particle, when viewed in a direction normal to the plane of greatest stability is determined microscopically, and it includes no contributions from the thickness of the particle, i.e. from the dimension normal to the plane of greatest stability. For perfect cubes and spheres, the value of the ratio x,/a ( = K, say) is of course equal to 6. For sand. Fair and Hatch found, with rounded particles 6T, with worn particles 6-4, and with sharp particles 7-7. For crushed quartz, Cartwright reports values of K ranging from 14 to 18, but since the specific surface was determined by nitrogen adsorption (p. 61) some internal surface was probably included. f... 

Figure 9.2 (a) Schematic representation of a unit cube containing a suspension of spherical particles at volume fraction [Pg.589]

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

The aspect ratio E/IE refers to the shape of the particles in the discontinuous phase. It is the average dimension of this phase parallel to the plane of the film E divided by the average dimension perpendicular to the film W. Plates in the plane of the film would have a high aspect ratio. Spheres or cubes would have an aspect ratio equal to 1. 

Irregular-shaped particles exhibit greater surface area than regular-shapea cubes and spheres, the amount of this increase being possibly 25 percent. The effect of particle size and size distribution on effective surface, in a shaft employed for calcination of limestone, is shown in... 

Heywood [Heywood, Symposium on Paiticle Size Analysis, lust. Chem. Engrs. (1 7), Suppl. 25, 14] recognized that the word shape refers to two distinc t charac teiistics of a particle—form and proportion. The first defines the degree to which the particle approaches a definite form such as cube, tetr edron, or sphere, and the second by the relative proportions of the particle which distinguish one cuboid, tetrahedron, or spheroid from another in the same class. He replaced historical quahtative definitions of shape by numerical shape coefficients. 

To include the volume as a dynamic variable, the equations of motion are determined in the analysis of a system in which the positions and momenta of all particles are scaled by a factor proportional to the cube root of the volume of the system. Andersen [23] originally proposed a method for constant-pressure MD that involves coupling the system to an external variable, V, the volume of the simulation box. This coupling mimics the action of a piston on a real system. The piston has a mass [which has units of (mass)(length) ]. From the Fagrangian for this extended system, the equations of motion for the particles and the volume of the cube are... 

The equivalent diameter can be calculated from the dimensions of regular particles, such as cubes, pyramids. 

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

One general benefit of subunit association is a favorable reduction of the protein s surface-to-volume ratio. The surface-to-volume ratio becomes smaller as the radius of any particle or object becomes larger. (This is because surface area is a function of the radius squared and volume is a function of the radius cubed.) Because interactions within the protein usually tend to stabilize the protein energetically and because the interaction of the protein surface with... 

The electrons are treated as independent particles constrained to a three-dimensional box, treated here for simplicity as a cube of side L. The box contains the metallic sample. The potential U is infinite outside the box, and a constant Uq inside the box. We focus attention on a single electron whose electronic Schrodinger equation is... 

Photo flash powders are loose mixts of powdered oxidizers such as Ba nitrate and K perchlorates with metallic fuels, principally Mg, A1 and Zr. These ingredients have such small particle sizes that they bum with expl violence for durations of less than 0.1 sec. At present photoflash powders are used exclusively in military aerial photography, whereas civilian applications are served by electrically ignited Zr or Hf wire containing flashbulbs. Since 1970. non-electric, pyrotechnically functioned, flash cubes have appeared on the market (USPs 3535063,3540813 3674411)... 

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