Contact interactions cohesive force

We will next give a more detailed description of the contact force, the cohesive force, and the integration of the equations of motion—Eqs. (20) and (21). The description of the forces resulting from interaction with the gas phase is given in Section III.D, whereas the dynamics of the gas phase itself is described in Section III.C. 

Equation (1.46) shows that the contact angle results from the competition of two types of forces cohesion forces responsible for crLv (= Wc/2) and adhesion forces responsible for W . Depending on the strength of S/L and L/L interactions, different contact angles can be obtained (Table 1.1). 

Third, for particles smaller than about 100 pm, cohesive forces (believed to be due to van der Waals interactions for intimate contacts, and to surface tension of adsorbed water layers for lubricated contacts) between particles becomes comparable to particle weights, and small particles can stick to one another in relatively rigid aggregates. Unless such aggregates are destroyed, the system will behave as if it had an effective particle size much larger than the primary particle size. 

The force of cohesion, i.e. the maximum value of attractive force between the particles, may be determined by a direct measurement of force, F required to separate macroscopic (sufficiently large) particles of radius r, brought into a contact with each other. Such a measurement yields the free energy of interaction (cohesion) in a direct contact, A (h0) = Ff n r,. Due to linear dependence of F on r, one can then use F, to evaluate the cohesive force F2 = (r2/r )Fx, acting between particles in real dispersions consisting of particles with the same physico-chemical properties but of much smaller size, e.g. with r2 10 8 m (i.e. in the cases when direct force measurements can not be carried out). At the same time, in agreement with the Derjaguin equation... 

Adhesion, by its definition, depends on the ability of two unlike phases to hold themselves together across a common interface. Physical adhesion must first take place before any other bonding processes such as chemical reaction can occur, and such physical adhesion depends on the strength of intermolecular force interaction, on the area of contact and on the distance separating the atoms forming the top layer of each surface (see Dispersion forces, polar forces). When both phases are undeformable, such as with two solids that are not atomically smooth, poor adhesion results because an insufficient area of each surface is in atomic contact with the other. When one phase is deformable, such as with a liquid of low viscosity, physical adhesion takes place at all parts of the surface. Physical adhesion with a liquid in contact with a solid leads to spreading and wetting processes that now depend on the competition of adhesion forces with cohesion forces within the liquid. 

While we will continue to address the subject of using surfactants to control contact interactions in Chapters 3 and 4, we would like to devote the remainder of this chapter to a discussion of the cohesive forces between anisometric particles—specifically, between the cellulose fibers. This study conducted by the authors and their colleagues is of importance in papermaking applications. 

A small radius of curvature in the fiber-fiber contact areas results in cohesive forces of a very small magnitude. This necessitates the use of highly precise methods and devices for measuring these small forces in contacts between individual fibers, under both normal rupture and shear. Such measurements provide one with the means for the evaluation of a contribution of molecular attraction to friction forces and allow one to obtain a thermodynamic value of the free energy of interaction. The latter is a general characteristic of interactions in a given medium, invariant with respect to the specimen geometry. 

The results of the contact force measurements, p, between cellulosic fibers in the presence of PEI are shown in Figure 2.23. The observed trends are similar to those shown in Figure 2.21 for the friction coefficient measurements and are also in good agreanent with the patch model of flocculation by the PEI, and the previously mentioned electrokinetic studies, as reflected by the maximum in the cohesive force at low concentrations of PEI. As expected, after reaching the maximum, the cohesive forces decrease with a further increase in the PEI concentration. This corresponds to the decrease in the fraction of available negatively charged patches for interaction with PEI and the increase in the electrostatic repulsion due to a continued adsorption of PEI. 

The cohesion between the hydrophobic part of the interfacial adsorption layer and the adjacent nonpolar phase can be modeled nsing the cohesion between model hydrophobic snrfaces in the same liqnid. In snch a simnlation, the hydrophobic solid snrfaces represent the hydrophobic tails of the snrfactant molecnles. This approach allows one to overcome the difficnlties associated with the mutual solubility of the components (see Chapter 1). For the solid/liqnid/solid interface, the main parameter characterizing the interactions is the free energy of interaction, F (or Aoj), which can be established experimentally nsing Derjagnin s theorem, that is, p = %RF, where p is the cohesive force in a direct contact between two spherical particles immersed in a liqnid medinm. Snitable model systems include spherical molecularly smooth glass beads with a radius R 1-1.5 mm and hydrophobized surfaces of different natures, namely, HS and HL, immersed into the hydrocarbon and fluorocarbon liquids, HL and FL. Only dispersion forces are present in such systems, which makes the quantitative description of their interaction well defined and not complicated by the presence of various polar components. 

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. 

In order to establish reasons for the limited sensitivity of Aa, (i.e., of the measured force / ,) to the presence of electrolytes in experiments with hydrophobic particles in hydrocarbon/alcohol mixtures, one can compare these results with the results of measuranents in the aqueous medium. The latter represent the main subject of DLVO theory. The final diseussion in this chapter is devoted to addressing the relationship between the contact interactions (i.e., cohesion forces at the primary potential energy minimum) and the results of DLVO theory (i.e., mainly long-range forces). Some of these experiments were conducted by Yaminskiy [30,50-52]. In addition to the contact forces, the Pi(fi) and Ao((fi) isotherms shown in Figure 4.46 were also determined. 

The adhesion between the IA L (its hydrophobic part) and an adjacent nonpolar phase can be modeled by the adhesion, in the same Uquids, of the modified soUd surfaces simulating the hydrophobic parts of corresponding surfactants. In this method, experimental complications coimected with the mutual solubility of components are prevented. For the solid/liquid interface, the principal quantitative characteristic of interaction, the free energy of interaction F (mj m , in the plain-parallel gap or film) can be established experimentally from Derjaguin s equation p = kRF, where p is the cohesive force in the immediate contact between two spherical particles immersed in the corresponding liquid medium [29, 30]. 

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