Particles overlapping

This means that particle configurations where at least two particles overlap, i.e., have a distance r smaller than the diameter cr, are forbidden. They are forbidden because the Boltzmann factor contains a term, exp(—oo) 0, that leads to a vanishing statistical weight. Hence we have an ensemble of... 

Two mechanisms of steric stabilization can be distinguished entropic stabilization and osmotic repulsion. Entropic stabilization arises when two opposing adsorbed polymer layers of adjacent particles overlap, resulting in compression and interpenetration of their... 

Semi-dilute solution p2 - As the concentration rises the rods will begin to interact and their diffusion will become restricted. However, we have not allowed for the excluded volume of the rod and have treated it as a line with no thickness. The excluded volume is of the order of bL2 and until the concentration of rods is such that the particles overlap into this excluded volume the spatial distribution of rods is relatively unaffected ... 

The lattice theory deals with rod-like particles which do not have interactions with their neighbors except, of course, repulsions occur when the particles overlap. Above a certain criticsd concentration (V2 ) that depends on the axi d ratio x the theory predicts the system will adopt a state of partial order (biphasic region). Below V2 the system... 

Repulsive forces between Fe oxide particles can be established by adsorption of suitable polymers such as proteins (Johnson and Matijevic, 1992), starches, non-ionic detergents and polyelectrolytes. Adsorption of such polymers stabilizes the particles at electrolyte concentrations otherwise high enough for coagulation to occur. Such stabilization is termed protective action or steric stabilization. It arises when particles approach each other closely enough for repulsive forces to develop. This repulsion has two sources. 1) The volume restriction effect where the ends of the polymer chains interpenetrate as the particles approach and lose some of their available conformations. This leads to a decrease in the free energy of the system which may be sufficient to produce a large repulsive force between particles. 2) The osmotic effect where the polymer chains from two particles overlap and produce a repulsion which prevents closer approach of the particles. 

As we saw in Chapter 11, surfaces of colloidal particles typically acquire charges for a number of reasons. The electrostatic force that results when the electrical double layers of two particles overlap, if repulsive, serves to counteract the attraction due to van der Waals force. The stability in this case is known as electrostatic stability, and our task is to understand how it depends on the relevant parameters. 

FIG. 13.13 Interaction between polymer-coated particles. Overlap of adsorbed polymer layers on close approach of dispersed solid particles (parts a and b). The figure also illustrates the repulsive interaction energy due to the overlap of the polymer layers (dark line in part c). Depending on the nature of the particles, a strong van der Waals attraction and perhaps electrostatic repulsion may exist between the particles in the absence of polymer layers (dashed line in part c), and the steric repulsion stabilizes the dispersion against coagulation in the primary minimum in the interaction potential. 

Fig. 10.6. Percolation cluster model of tunnel current in composite film containing M/SC nanoparticles (a) two-sphere model of spherical M/SC nanoparticle of radius Rq surrounded by outer sphere (radius Rd) that is defined by a degree of electron delocalization extending the nanoparticle and characterizes electron tunneling (see text) (b) the distribution of conductivity G(r) over the two-sphere particle (c) two-dimensional pattern of cluster from overlapping two-sphere particles (overlapping areas of outer spheres are shown).

The consequences for suspended particles can be understood from either a mechanical or a thermodynamic standpoint. A particle immersed in a polymer solution experiences an osmotic pressure acting normal to its surface. For an isolated particle, the integral of the pressure over the entire surface nets zero force. But when the depletion layers of two particles overlap, polymer will be excluded from a portion of the gap (Fig. 30). Consequently, the pressure due to the polymer solution becomes unbalanced, resulting in an attraction. The same conclusion follows from consideration of the Helmholtz free-energy. Overlap of the depletion layers reduces the total volume depleted of polymer, thereby diluting the bulk solution and decreasing the free energy. 

Typically, colloidal suspensions are deposited on polycarbonate filters immediately following sampling to reduce post-collection precipitation or aggregation artifacts and mimic the operational procedures used to define dissolved and particulate sample components. Obviously, a well-dispersed dilute sample should be used to avoid artifactual particle overlap. Filter sections are then mounted on SEM stubs and carbon or metal coated (Au/Pd) for compositional analysis and micrographic imaging, respectively. The tedious nature of operator controlled analysis of particulate samples in the SEM and the TEM has often precluded the acquisition of statistically valid information regarding suspension composition. 

At mtermediate electrolyte concentrations ( 10 mol dm" ) and at low volume fractions of the dispersed phase, the charged particles occupy random positions in the system and undergo continuous Brownian motion with transient repulsive contacts when the particles approach each other. The range of the electrostatic repulsive forces is represented by the dashed circle in Fig. 1. which implies that when a similar circle on another particle overlaps with it on a collision trajectory, a transient electrostatic repulsion occurs and the particles move out of range. With most latices the particles. have a real refractive index and their visual appearance is milky white. 

In the bubbling bed model of Kunii and Levenspiel (K24), there are two transfer steps for the bubble mass transfer, namely, the transfer between bubble void and cloud-particle overlap region kbOb and that between the cloud-particle overlap region and the emulsion phase keOr,. They further assume that the cloud-particle overlap region and the bubble wake are mixed perfectly, and contact freely with the cloud gas. Their basic equations in the present notation are (for their case 2) ... 

When calculating continuous phase volume fraction, it is very important to account for the possibility of dispersed phase particles overlapping with more than one computational cell. The volume occupied by such particles needs to be distributed over the respective cells. Calculation of the exact distribution of the volume of the dispersed phase particles to the respective cells may become computationally intensive when several particles are considered. Equations to distribute the volume of dispersed phase particles to different computational cells are illustrated in Fig. 7.15 for the two-dimensional case. Delnoij (1999) proposed some approximations based on the lengths... 

This approach is very general. For example, it is not restricted to monodis-perse systems, and Krauth and co-workers have applied it successfully to binary [17] and polydisperse [18] mixtures. Indeed, conventional simulations of size-asymmetric mixtures typically suffer from jamming problems, in which a very large fraction of all trial moves is rejected because of particle overlaps. In the geometric cluster algorithm particles are moved in a nonlocal fashion, yet overlaps are avoided. 

At last, let us consider another type of capillary interactions — between particles snrronnded hy finite menisci. Such interactions appear when micrometer-sized or snb-micrometer particles are captured in a liquid film of much smaller thickness (Fignre 5.20b). If such particles are approaching each other, the interaction begins when the meiusci aronnd the two particles overlap, L < 2rp in Figure 5.20b. The capillary force in this case is nonmonotonic initially the attractive force increases with the increase of interparticle distance then it reaches a maximum and further decays. In addition, there are hysteresis effects the force is different on approach and separation at distances aronnd L = 2rp. ... 

In this case, as indicated above, the colloid stability is controlled by the form of the interaction free-energy curve as a function of particle separation. The DLVO theory in its original form considers just two contributions to this energy, namely the attractive van der Waals potential and the repulsive potential that arises when the diffuse double layers round the two particles overlap. To put this in a quantitative form we need to examine more closely the origin of the curves shown in Figures 3.6 and 3.7. 

The behavior of various quantities is shown for an electron approaching a node or another particle, either a nucleus or an electron. The singularity in the local energy at particle overlap is only present for a that fails to satisfy the cusp conditions. 

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