This dependence of ccc on 1/z 6 known as the Schultz-Hardy rule is consistent with the DLVO theory. When the zp is very high, the term P approaches unity and the critical coagulation concentration (ccc) becomes inversely proportional to the sixth power of the valency, z. Where K is a constant which depends only on the properties of the dispersion medium and A is the effective Hamaker constant. Accordingly, the colloidal particles come in contact and coagulate. When a trivalent electrolyte, AlCl 3, is added into system, the zp comes down to zero, that is called the isoelectric point owing to the charge neutralization on the silica surface the height of the energy barrier disappears and the van der Waals attractive forces become dominant in the system. Sometimes the van der Waals attractive forces may become dominant depending on the kinetic conditions and/or the existence of the non-DLVO forces such as hydration, hydrophobic, and steric. When KCl is added into water the zp of silica particles comes down to the -14 mV due to the double layer compression with a resultant decrease in repulsive energy and in turn in the height of the energy barrier. The total interaction energy (V T) curves for zp value of -30 mV exhibit dispersion and the height of the energy barrier is considerably high. For example, for negatively charged colloidal silica particles the zp values distilled water, in KCl and in AlCl 3 are −30, -14 and 0 mV, respectively. The EDL repulsive forces can be altered by changing the zp of particles through changing parameters such as the type and concentration of electrolyte and solution pH. These two cases can be realized by changing the EDL repulsive forces, as it is perhaps impossible to change the van der Waals forces. Conversely, for a good dispersion the height of energy barrier must be enlarged. If aggregation is required, the height of energy barrier shown in Fig.
![sto potential energy entangler sto potential energy entangler](https://www.mdpi.com/entropy/entropy-16-04101/article_deploy/html/images/entropy-16-04101f2.png)
If the particles are further away, van der Waals attraction forces decrease sharply because of the large exponent of inverse distance, and the EDL repulsion forces take over with an energy barrier occurring between the particles. Rheological properties such as thixotropy are closely related to coagulation at the secondary minimum. But the coagulation in this region is not stable and reversible with respect to the case in the primary minimum. There is also negative attraction energy usually beyond 3 nm known as the aggregation region or the secondary minimum. At contact state, the total interaction energy is known as the primary minimum. If the colloidal particles are very close, the van der Waals attractive forces take over with a resultant negative energy of interaction leading to the coagulation of particles. Repulsive and attractive forces as a function of distance of separation In fact, the Gibbs point process can be regarded as a pairwise interacting process.įigure 25.
![sto potential energy entangler sto potential energy entangler](https://image3.slideserve.com/6729198/slater-type-orbitals-sto-l.jpg)
One advantage of the Gibbs point process over the Poisson point process is that the Poisson point process is not able to account for interactions between points, whereas the Gibbs point process can. Also, they are more applicable when the system to be modeled contains only a finite number of points in a bounded region B. More importantly, they do not perform well in applications with strong regularity they are good for applications with some degree of regularity. Instead, their distributions are defined according to the application of interest. Gibbs processes are not universal models that apply to all situations. It enables the total potential energy associated with a given configuration of particles to be decomposed into terms representing the external force field on individual particles and terms representing interactions between particles taken in pairs, triplets, etc. The Gibbs process originated in statistical physics and is described in terms of forces acting on and between particles. Ibe, in Markov Processes for Stochastic Modeling (Second Edition), 2013 15.5.3 Spatial Gibbs Processes