Minimum primary

Coagulation a relatively irreversible aggregation often associated with the primary minimum of a potential energy diagram of two approaching particles. Particles are held together closely. 

As two particles approach in a liquid their charge fields may interact and form two minima as depicted in Figure 6.8. If the particles approach to a distance Li, known as the primary minimum they aggregate to form a configuration with minimum energy - and rapid coagulation is said to take place. On the other hand, if the particles remain separated at a distance L2, the secondary minimum, loose clusters form which do not touch. This is known as slow coagulation and is the more easily reversed. 

At a finite distance, where the surface does not come into molecular contact, equilibrium is reached between electrodynamic attractive and electrostatic repulsive forces (secondary minimum). At smaller distance there is a net energy barrier. Once overcome, the combination of strong short-range electrostatic repulsive forces and van der Waals attractive forces leads to a deep primary minimum. Both the height of the barrier and secondary minimum depend on the ionic strength and electrostatic charges. The energy barrier is decreased in the presence of electrolytes (monovalent < divalent [Pg.355]

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

Besides the resuspension of particles, the perfect sink model also neglects the effect of deposited particles on incoming particles. To overcome these limitations, recent models [72, 97-99] assume that particles accumulate within a thin adsorption layer adjacent to the collector surface, and replace the perfect sink conditions with the boundary condition that particles cannot penetrate the collector. General continuity equations are formulated both for the mobile phase and for the immobilized particles in which the immobilization reaction term is decomposed in an accumulation and a removal term, respectively. Through such equations, one can keep track of the particles which arrive at the primary minimum distance and account for their normal and tangential motion. These equations were solved both approximately, and by numerical integration of the governing non-stationary transport equations. 

Fig. 19 Typical free energy profile for the deposition of a colloid. Note the logarithmic distance scale and the divided energy scale for the relatively shallow secondary minimum and the much deeper primary minimum.

PVA and TaM -for the 88%-hydrolyzed PVA. The same dependence was found for the adsorbed layer thickness measured by viscosity and photon correlation spectroscopy. Extension of the adsorption isotherms to higher concentrations gave a second rise in surface concentration, which was attributed to multilayer adsorption and incipient phase separation at the interface. The latex particle size had no effect on the adsorption density however, the thickness of the adsorbed layer increased with increasing particle size, which was attributed to changes in the configuration of the adsorbed polymer molecules. The electrolyte stability of the bare and PVA-covered particles showed that the bare particles coagulated in the primary minimum and the PVA-covered particles flocculated in the secondary minimum and the larger particles were less stable than the smaller particles. 

Both secondary and primary minimum coagulation are observed in practice and the rate of coagulation is dependent on the height of the barrier. In general, coagulation into a primary minimum is difficult to reverse, whereas coagulation into a secondary minimum is often easily reversed, for example, by diluting the electrolyte. DLVO theory tells us... 

Fig. 1.6 DLVO interactions showing the energetics of colloidal particles as a competition between electrostatic double-layer repulsion and van der Waals attractions. The primary minimum is due to strong short-range electron overlap repulsion (shown in Figure 1.4... 

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. 

W Ursae Majoris stars can be understood as contact binary stars with a common envelope (Lucy 1968). They subdivide into two types The A-type are earlier in spectral class than about F5, are believed to have radiative envelopes, and associate primary (deeper) eclipse minimum with transit eclipse. The W-type have spectral classes later than F5, are believed to have convective envelopes, and associate primary minimum with occultation eclipse. Controversy has surrounded the explanation of W-type light curves. 

The VDW interactions seem to have little effect on the rate of aggregation of small vesicles in a primary minimum. However, this statement may be made only because the magnitudes of Hamaker coefficients are less than 10 13 erg (5 X 10 14 erg), in contrast to much higher values frequently used in treatments in colloid science (3). Our estimates of VDW parameters for phospholipid vesicles are based on the analysis of a significant amount of recent data (33,42). 

An expression for the critical coagulation concentration (c.c.c.) of an indifferent electrolyte can be derived by assuming that a potential energy curve such as V(2) in Figure 8.2 can be taken to represent the transition between stability and coagulation into the primary minimum. For such a curve, the conditions V = 0 and dV/dH = 0 hold for the same value of H. If Vr and VA are expressed as in equations (8.7) and (8.10), respectively,... 

Both specific adsorption (particularly, solvation) at the particle surfaces and the difficulty with which dispersion medium flows from the narrow gap between the particles may hinder particle approach to the small separation which corresponds to the primary minimum. 

These effects have been observed for both aqueous and non-aqueous media and good correlation between the point of incipient flocculation and the 0-temperature is well established112. The transition from stability to instability usually occurs over a very narrow temperature range (1 or 2 K). Enthalpic stabilisation tends to be the more common in aqueous media and entropic stabilisation the more common in non-aqueous media. Owing to the elastic effect, aggregation into a deep primary minimum does not take place (as is possible with lyophobic sols) and redispersion takes place readily on reverting to better than 0-solvent conditions. 

Provided that v > v for most values of h then the form of curve shown in Figure 1 is obtained. When the magnitude of is substantial, say >> 10 kT, a stable dispersion is obtained. The form of the potential energy curve obtained by this approach shows immediately that the stability of a dispersion to electrolyte is kinetic in origin rather than thermodynamic, that is, the lowest free energy state is in the primary minimum and entry into this is prevented by the presence of the large activation energy represented by AV. A more sophisticated and detailed representation of these ideas can be found elsewhere (12,15,16). 

The Onset of Instability - The Critical Coagulation Concentration. Provided that the magnitude of the primary maximum is substantial, then the probability of the transition of the approaching particle into the primary minimum is small. However,... 

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