Separation between the spheres

Lund et al. (1999) reported finite element simulations of conduction between two contacting spheres without fluid flow. They modeled a small separation between the spheres to allow for interparticle micro-asperity gaps, and then the... 

We have seen that the local constraint on the surface curvatures, set by the surfactant parameter, can be treated within the context of differential geometry, which deals with the intrinsic geometry of the surface. In contrast, the global constraint, set by the composition of the mixture, is dependent upon the extrinsic properties of the surface, which need not be related to its intrinsic characteristics. (For example, the surface to volume ratio of a set of parallel planes can assume any value by suitably tuning the spacing bebveen the planes. Similarly, the ratio of surface area to external volume i.e. the volume of space outside each sphere closer to that sphere than any other) of a lattice of spheres depends upon the separation between the spheres.)... 

Once more, the EXEDOS method can be applied to find the potential of mean force AF(s) as a function of the separation of the two spheres. For convenience, we define the reaction coordinate s as the separation between the spheres surfaces... 

Figure 12. Radius of the three-ring structure as a function of the separation between the spheres. The radius of the equatorial ring remadns constant and the third ring shrinks as the separation increases.

The distance of separation between the spheres is h = 2[rsinP - a(l - cosa)] and the angles are related by... 

The first molecular dynamics simulation of a condensed phase system was performed by Alder and Wainwright in 1957 using a hard-sphere model [Alder and Wainwright 1957]. In this model, the spheres move at constant velocity in straight lines between collisions. All collisions are perfectly elastic and occur when the separation between the centres of... 

Figure 8.12 Schematic illustration showing with the solid line how the probability of placement varies with the distance of separation between the centers of the coils. The broken line is the equivalent result for hard spheres.

FIG. 4 Schematic of a liquid meniscus between the sphere and plane at finite separation. 

Every gas consists of particles, whether as atoms (such as neon) or as molecules (such as methane). To a relatively good first approximation, any atom can be regarded as a small, incompressible sphere. The reason why we can compress a gas relates to the large separation between the gas particles. The first effect of compressing a gas is to decrease these interparticle distances. 

The simplest models view the interacting bodies as hard spheres (e.g., billiard balls). Mathematically, if r is the separation between the center of two molecules, we write the potential energy of interaction between them as ... 

Since the masses are equal, the only way / tot,finai can vanish is if Ui,finai = — 2,final-Since the speeds are then equal, Equation 7.4 shows that both speeds must be 10 m s 1, but the direction is unknown. For hard spheres the additional information needed to determine the direction comes from the impact parameter b, defined as the minimum distance of separation between the centers of the two balls if they were to follow their initial trajectories. In Figure 2.4, the impact parameter is the distance between the two dotted lines. 

FIGURE 4.1. The interaction between a rigid particle and a liquid/Iiquid interface. A repulsive colloidal force causes the interface to bend as a function of radial position, r, thus separation distance between the sphere and interface is also a function of radial position, h r). 

The use of AFM has been expanded to measure the forces acting on an AFM tip as a function of separation between the tip and surface, but the tip geometry is often difficult to characterize and inconvenient to compare to theoretical predictions of force measurements. By attaching a colloidal size sphere (on the order of 5-10 (im in radius a) to the end of a cantilever (first accomplished by Ducker [26]), it is possible to measure the colloidal interaction between surfaces in system with a well-characterized geometry (shown schematically in Figure 4.2). The measurement of a force-distance cycle is the record of the deflection of the cantilever as the two surfaces approach through motion of the piezo stage. The separation distance between the surfaces is not measured directly... 

FIGURE lOS View of thermodynainic system of two spheres for interaction energy calculation, n is the normal vector at the midi ane, h is the separation between the two spheres of radius r. 

For two identical spheres 1 and 2 of radius a carrying unperturbed surface potential i/ o separated by R, one may choose the intermediate plane at z = 0 between the spheres as an arbitrary plane enclosing sphere 1 (Fig. 8.8). Here we use the cylindrical coordinate system (r, z) and take the z-axis to be the axis connecting the centers of the spheres and r to be the radial distance from the z-axis. Equation (8.34) can be rewritten by using the cylindrical coordinate (r, z) as... 

The adhesive force between a neutral particulate contaminant and the wafer is expected to be due to the attractive Van der Waal s interaction between molecules.This is a macroscopic force found by averaging over the force between all the molecules of a particle and the neighboring surface. For a spherical particle sitting on a flat wafer, it is known that surface roughness will cause the mean distance of separation between the particle and the wafer to be nonzero. The attractive force between these two entities acts along the normal between the sphere and the wafer, and is given by ... 

---