Coalescence of spheres

Coagulation and coalescence of the dispersed phases will change the apparent viscosity at any rate of shear. Coagulation without coalescence will tend to increase it whilst coalescence of spheres by... 

Mackenzie and Shuttleworth (MS) analyzed [24] the shrinkage rate of a spherical shell according to Frenkel s rnethod. The shell, shown in Fig. 6, can be used to represent the densification of a body containing spherical pores. The dimensions of the shell are chosen so that the central void occupies the same volume fraction as the pores in the sintering body. This is a much more elegant treatment than the analysis of the coalescence of spheres, because the shell remains spherical as it shrinks. Exact expressions can be written for the change in surface area and the energy dissipated in viscous flow as the shell contracts. The result is... 

Figure 4.7 Schematic representation for the deterministic spherical coalescence to recover the original cylinder. Parts (a)-(e) show the process from cylinders to spheres via undulating cylinders. The long lifetime of poles on the spherical microdomains in the direction parallel to the original cylinder axis is relevant. Here, the dots specify the on-interface distribution of chemical junctions between A and B block chains in A-B diblock molecules. The sketches in (f)-(i) explain the role of poles in the deterministic coalescence of spheres. (Reproduced from K. Kimishima et al. (2000) Macromolecules 33 968-977, Copyright (2000) with permission from the American Chemical Society.)...

Suspension polymerization of VDE in water are batch processes in autoclaves designed to limit scale formation (91). Most systems operate from 30 to 100°C and are initiated with monomer-soluble organic free-radical initiators such as diisopropyl peroxydicarbonate (92—96), tert-huty peroxypivalate (97), or / fZ-amyl peroxypivalate (98). Usually water-soluble polymers, eg, cellulose derivatives or poly(vinyl alcohol), are used as suspending agents to reduce coalescence of polymer particles. Organic solvents that may act as a reaction accelerator or chain-transfer agent are often employed. The reactor product is a slurry of suspended polymer particles, usually spheres of 30—100 pm in diameter they are separated from the water phase thoroughly washed and dried. Size and internal stmcture of beads, ie, porosity, and dispersant residues affect how the resin performs in appHcations. 

Fig. 12. A, Schematic representation of parallel arrays of polynuclear aromatic hydrocarbon molecules in a mesophase sphere. B, a) isolated mesophasc spheres in an isotropic fluid pitch matrix b) coalescence of mesophase c) structure of semi-coke after phase inversion and solidification.

The mechanism of coalescence of solid spheres (Figure 9.6c and Figure 9.6d) differs from that considered above. In this case the viscous coalescence is impossible, but the mass transfer occurs through the surface diffusion or solubility [39], According to Equation 9.11, a motive force for this process is... 

Figure 9.6 Coalescence of fluid drops is accompanied with merging (a)->(b), and coalescence of solid spheres is accompanied with accretion without rapprochement of their centers (c)->(d).

FIGURE 2.23 (a) Formation, (b) growth and (c) coalescence of mesophase spheres in a pitch. 

Attraction of Two Spheres—An expression for the attraction between two or more particles, based on collision theory rather than attraction due to the motion of spheres, may be developed by the methods of dimensional analysis. Let the force of attraction between two particles of diameters d and d2 be F and assume that when the particles are close enough so that the gaseous film enveloping each particle coalesces over a region about the point of contact, then the whole attraction is due to a free surface energy a. If the average surface of contact of the particles is denoted by Sc then... 

Coalescence of mesophase is often said to be determined by the mesophase viscosity. This aspect requires much further investigation. However, it is clear that, amongst other factors, the rheological behaviour (including viscoelastic effects) of each phase is important in mesophase growth and coalescence. Diffusion of molecular species through the isotropic pitch to the mesophase spheres is likely to be related to the viscosity of the isotropic med i urn. 

Imafuku et al.46 measured the gas holdup in a batch (i.e., no liquid flow) three-phase fluidized-bed column. They found that the presence of solids caused significant coalescence of bubbles. They correlated the gas holdup with the slip velocity between the gas and liquid. They found that the gas holdup does not depend upon the type of gas distributor or the shape of the bottom of the column when solid particles are completely suspended. Kato et al.53 found that the gas holdup in an air-water-glass sphere system was somewhat less than that of the air-water system and that the larger solid particles showed a somewhat smaller... 

The sol-gel process is used to prepare dense, spherical particles of ThOa and (Th,U)02 for sphere-pac and coated-particle fuels. The thoria is dispersed in water from nitrate solutions by slow heating and steam denitration to form a stable sol from which spherical particles are produced. The sol droplets are injected at the top of a tapered glass column containing an upward flow of 2-ethylhexarol (2-EH). The water from the sol particles is slowly extracted by suspension in the 2-EH, and the gelled spheres drop out of the column. Coalescence of the particles is prevented with surfactants in the 2-EH. The sol-gel spheres are dried in steam and Ar at 220°C and sintered in H2 at ISOO C. 

A straightforward transition process [137] from the C to S was proposed (Fig. 46) The cylinder breaks up into spheres with the original cylindrical axis parallel to the [111] direction of the bcc sphere lattice induced by thermodynamic instability of the cylindrical interface at T > Tqot- The process may proceed via four subsequent steps (i) undulation (as shown from part a to b), (ii) break-up of cylinders into ellipsoids (from part b to c), (iii) relaxation of domains from the ellipsoids into spheres (from part c to d), and (iv) relaxation in a junction distribution to attain a uniform distribution (from part d to e). These processes involve only local rearrangements of the block copolymer molecules rather than large length-scale rearrangements of block copolymer molecules or microdomains. The reverse transition involves deformation and elongation of the spheres into ellipsoids and coalescence of... 

The coalescence of the two small droplets in contact is driven by the tendency for the doublet to approach a spherical shape corresponding to minimum surface free energy for the doublet volume. For liquid ptulicles including glassy materials like. silica, coalescence takes place by viscous flow. For a Newtonian liquid, during the initial stages of coale.scence of two liquid spheres of equal diameter, a neck forms at the contact point with a radius that grows as r(Frenkel, 1945). After a short initial period in which coalescence is very rapid, the rate of decrease in the surface area becomes linear in the deviation of the doublet surface area from the sphere of the same volume as the doublet (Koch and Friedlander, 1990) ... 

The coalescence of crystallites in contact is much more complex than the coalescence of a Newtonian liquid. For coalescing liquids, the equilibrium shape i.s a sphere. For crystallites in contact, the equilibrium form is presumably determined by a Wulff construction exact calculations for such complex configurations have not been made. To estimate Vf for crystallites, it is customary to assume a simple geometry— for example, two spheres in contact as in the case of liquid droplets in contact. The particle properties are assumed to be isotropic. 

If the aggregates of rod-shaped bacteria cells are assumed to coalesce into spheres, then the foregoing equations predict the variation of scattering eflBciency with the size-refractive index parameter p. Experimentally,... 

Lichtenbelt et al. (19) made a detailed comparison of extinction cross sections for real doublets and for hypothetical doublets coalesced into spheres. Using the Rayleigh-Debye theory for a up to three, Lichtenbelt et al. determined that the coalescence assumption leads to 10% larger values of the scattering cross section than would be found for real doublets created in coagulation. This is because a real doublet is less compact than a coalesced doublet of the same volume therefore, the interference between light waves, scattered by different parts of the real... 

Following a related approach, Castelvetro et al. reported the formation and properties of hybrid latex films resulting from the coalescence of low 7 poly(BA-co-MMA-co-MPTMS) terpolymer latex particles coated by a silica shell [78], The latex was synthesized at neutral pH by semi-continuous emulsion polymerization under starved-feed conditions in order to protect the MPTMS monomer from premature hydrolysis and condensation reactions. A substantial amount of free silanols were therefore available for further reaction with the silica precursor. In order to avoid the formation of a densely crosslinked silica network around the latex core, which may significantly alter film formation, the pH was kept at around 2 (at this pH, hydrolysis is promoted and condensation is significantly retarded). TEM and AFM studies of the nanocomposite film indicated that the silica shell formed a continuous percolating network throughout the polymer matrix. A porous film of interconnected hollow silica spheres was next elaborated by thermo-oxidative decomposition of the organic phase. 

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