Spherical particles in contact

The force of attraction, calculated from Coulomb s law, for a uniformly charged, spherical, particle in contact with a grounded, conducting substrate is simply... 

Figure 6.18 Schematic diagram of two spherical particles in contact.

The effect of the particle size distribution on shrinkage rates has been considered previously for the initial stage sintering [46]. The problem was treated for the linear shrinkage of a pair of different size spherical particles in contact. These results were extended to distributions of a few discrete sizes by a weighting of all possible contacts. For a continuous distribution in size, this pairwise interaction can be... 

There is an extensive literature on the rate at which spherical particles in contact coalesce as a result of various mechanisms for molecular transport in the contact region. This literature, developed largely for application to sintering in the ceramics field, has been summarized by Kingery et al. (1976, Chapter 10) and Brinker and Scherer (1990, Chapter 11). For liquids. 

Fig. 31 Schematics of a spherical particle in contact with a plane...

Coalescence of two particles in contact during sintering (a) two spherical particles in contact (b) formation of grain boundary. 

When we consider three spherical particles in contact, we can see that an open pore gets closed during sintering when the boundaries are formed. This is shown in Figure 14.4. The directions of migration of atoms by different mechanisms, except the plastic flow, are also shown. 

Figure 6.7 Spherical particle in contact with a planar surface with a linearly increasing fluid flow.

However, conductive elastomers have only ca <10 of the conductivity of soHd metals. Also, the contact resistance of elastomers changes with time when they are compressed. Therefore, elastomers are not used where significant currents must be carried or when low or stable resistance is required. Typical apphcations, which require a high density of contacts and easy disassembly for servicing, include connection between Hquid crystal display panels (see Liquid crystals) and between printed circuit boards in watches. Another type of elastomeric contact has a nonconducting silicone mbber core around which is wrapped metalized contacts that are separated from each other by insulating areas (25). A newer material has closely spaced strings of small spherical metal particles in contact, or fine soHd wires, which are oriented in the elastomer so that electrical conduction occurs only in the Z direction (26). 

Consider a gas-solid suspension which is in a state of steady dilute flow with no interparticle collision or contact. In this situation, the linear particle velocity is practically identical to the superficial particle velocity. The motion of a spherical particle in an oscillating flow field can thus be given by... 

Electrophoretic interactions between spherical particles with infinitely thin double layers can also be examined using the boundary collocation technique [16,54]. This method enables one not only to calculate the interactions among more than two particles, but also to deal with the case of particles in contact, for which the bispherical coordinate solution becomes singular. Analogous to the result for a pair of spheres, no interaction arises among the particles in electrophoresis as long as all the particles have an equal zeta potential. This important result is also confirmed by a potential-flow reasoning [10,55]. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

After burnout of the binder, we are left with a green body that is composed of ceramic particles in contact as shown in Figure 16.2. As the temperature is increased, material flows from various sources within the ceramic green body to the neck at the intersection between particles, as shown in Figure 16.3 [3]. This neck has a negative curvature, compared to the positive curvature of the spherical ceramic particle, and results in a energetically more favorable location for the material. A tabulation of the possible sources of material is given in Table... 

Experimental validation. When a model cannot be verified by experiments, it can be considered as an excellent exercise ivhich however does not have any practical significance. So as to validate the results of our stochastic model for a liquid flow in a MWPB, we will use data previously published [4.80, 4.82] for a model contacting bed of spherical particles, in which the gas and liquid fluids were respectively air and water. In these studies, the liquid residence time was estimated by measuring the response of a signal injected into the bed. The MWPB liquid hold-up was obtained by the procedure of instantly stopping the water and air at the bed input. 

Sphericity has been assumed in most studies for many reasons. In theoretical work and computer simulations it is easy to detect a collision of spherical particles, as particles are in contact whenever their centers are a distance of two radii apart. For non-spherical particles, the contact mechanisms become much more complicated, as the orientation of the particle, which changes as the particle rotates, must be taken into account. For this reason the assump>-tion of spherical particles are normally considered a fair approximation for catalyst pellets. 

To calculate the surface energies at the oxide-ionic melt interface boundary by means of equation (3.6.57) it is necessary to know the radii of the oxide particles in contact with the saturated solution in the melt-solvent. In this connection there arises a question whether one can correctly estimate the values of the oxide particles radii from the data on the molar surface area obtained using any experimental method of estimating the specific surface area, e.g. the BET treatment. In order to answer this question we first consider the relationship between the volume and surface area of a spherical particle. 

The strength of a coagulation contact, i.e., adhesive force between particles, is determined primarily by molecular forces. For spherical particles, in agreement with eq. (VII. 13), this force is given by... 

The charging process of a conducting spherical particle on contact with a flat electrode is described in ref [98]. In this work, the charge acquired by the particle was determined, and the force of particle interaction with the flat electrode was calculated. 

Although the microparticles circulate in the fluid bulk as a result of external agitation, it is assumed that the particles contained in the fluid elements in transient contact with the interface remain stationary. Thus it is possible to assume a Sherwood number of 2, a generally accepted value for mass transfer to and from a spherical particle in an infinite medium. 

The quantity p gw(0) is the density of molecular sites in contact with a hard wall. This method was employed recently by Dickman and Hall [36] to determine pressure from Monte Carlo simulations. Yethiraj and Hall [37] demonstrated that gw(r) can be approximately computed from PRISM theory by considering a mixture of polymer chains and spherical particles in the limit of zero concentration of particles with diameters approaching infinity. 

Amorphous substances have a liquid-like supramolecular stmcture. Like liquid droplets, two amorphous particles in contact with each other tend to adopt a spherical shape. Accordingly, molecules are transported to the contact point between the two particles. The capillary and vapor pressure gradients between the particle volumes and the contact point between the two particles are the driving force for these transport processes. Linked to these local differences in capillary and vapor pressure different molecular transport mechanisms are observed. The molecules can be transported via the surrounding gas phase (evaporation and sublimation), diffusion on the surface (surface diffusion/grain boundary diffusion) or diffusion... 

The free energy of interaction (free energy of cohesion for dispersion forces), AOf, is the main invariant with the respect to the surface geometry characteristic. Integrating U h) twice over the cross section of the spherical particles in the zone of contact yields the cohesive force between two particles, that is, a parameter that can be directly measured in experiments with macroscopic bodies with spherical or cylindrical surfaces, that is. 

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