Forces between particles and surfaces

In colloid and surface literature it is customary to use H for the distance instead of r and this nomenclature is 

In the derivation of the equations shown in Table 2.2, it is assumed that the intermolecular distance dependency is that shown in Equations 2.3 and 2.4 (i.e. the one used in the Lennard-Jones potential). 

The key property in these calculations is the so-called Hamaker constant (A), which is directly linked to the C parameter of Equations 2.3 and 2.4 via the so-called microscopic (London) approach  

Two spherical particles of unequal/equal radii, valid at all separations. In the equation, user= R + R2 + H. 

Two flat plates/surfaces of infinite thickness (per unit 

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There are numerous modern developments that have made atomic-scale resolution possible in recent years. In fact, some of these developments in instruments can also be used to measure forces between particles and surfaces. These developments for force measurements are discussed briefly in Section 1.6c and in Vignette 1.8. In this section, we review electron and scanning probe microscopies (SPMs), which allow atomic-scale visualization of surfaces and particles. 

Podczeck F, Newton JM. James MB. Assessment of adhesion and autoadhesion forces between particles and surfaces. I. The investigation of autoadhesion phenomena of salme-terol xinafoate and lactose monohydrate particles using compacted powder surfaces. J Adhesion Sci Techn 1994 8 1459-1472. 

After adsorption the adhesion forces between particle and surface will keep the particle adsorbed on the surface until hydrodynamic forces or thermal motion are strong enough to overcome the adsorption barrier. This can be described by the equation... 

Thus we see that the evaluation of adhesion on the basis of the detachment force at identical adhesion numbers (i.e., on the basis of F, in, F ax or so) can lead to errors. In exactly the same manner, the evaluation of adhesion on the basis of adhesion numbers at equal detachment forces can serve only as a relative characterization of the interaction forces between particles and surface. 

Lubricants. Lubricants (36,38) are added to lower frictional forces between particles, and between particles and die surfaces to improve compaction and minimize die wear. Typically <1 wt % of a lubricant is required for forming, and usually only with hard binders. Stearic and oleic acids are good lubricants for ceramics. 

Increased contact angle and attachment force between particles and bubbles through tiny bubbles frosted on particle surfaces, and increased recovery of fine and coarse particles at reduced reagent consumption... 

FORCES BETWEEN PARTICLES AND BETWEEN PARTICLES AND SURFACES... 

A whole range of phenomena in interface science revolve around the effect of surface forces. Many practical applications in colloid science come down to the problem of controlling the force between colloidal particles, between particles and surfaces, and between two surfaces. For this reason scientists have devoted considerable effort to understanding surface forces and being able to influence them. 

The rate of deposition of Brownian particles is predicted by taking into account the effects of diffusion and convection of single particles and interaction forces between particles and collector [2.1] -[2.6]. It is demonstrated that the interaction forces can be incorporated into a boundary condition that has the form of a first order chemical reaction which takes place on the collector [2.1], and an expression is derived for the rate constant The rate of deposition is obtained by solving the convective diffusion equation subject to that boundary condition. The procedure developed for deposition is extended to the case when both deposition and desorption occur. In the latter case, the interaction potential contains the Bom repulsion, in addition to the London and double-layer interactions [2.2]-[2.7]. Paper [2.7] differs from [2.2] because it considers the deposition at both primary and secondary minima. Papers [2.8], [2.9] and [2.10] treat the deposition of cancer cells or platelets on surfaces. 

The objective of the present research is to predict the rate of deposition of Brownian particles by considering the effects of diffusion, convection, and interaction forces between particle and collector. It will be shown that, when the repulsion due to the double-layer is sufficiently large, the interaction forces can be incorporated into a boundary condition for the convective-diffusion equation. This boundary condition takes the form of a virtual first-order chemical reaction which occurs on the surface of the collector. 

The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. 

Adhesion — (a) When two compact materials, be they solid or liquid, are in intimate contact, attractive forces may act between their surface atoms or molecules. These forces are typically - van der Waals forces and electrostatic forces. The work of adhesion W (b)b(a) between the two phases (denoted A and B) is WAB = yA+yB -yAB> where yA and yg are the - interfacial tensions of A and B when each is interfaced only with the vapor phase, and yAB is the interfacial tension of the interface between A and B. In a more rigorous treatment (at thermodynamic equilibrium) each phase is regarded as saturated with the other phase [i]. In the case of liquid phases the equation for the work of adhesion is referred to as the -> Dupre equation. Adhesion forces between particles, or between particles and surfaces, dominate gravity for small particle sizes (pm and sub-pm range). In electrochemistry, increasing attention is being given to various phenomena related to the adhesion of vesicles [ii], particles [iii], droplets [iv], cells [v], etc. to electrode surfaces. 

DLVO theory [2,3] can be used to calculate the interaction forces between the slurry particle and the wafer surface to be polished. The interaction forces between particles and between particles and surfaces could provide important information on the stability of slurry and the degree of particle contamination on surfaces after CMP. 

Unlike adsorbed moisture bonding, once a liquid bridge is formed, it exerts an attractive force between particles and does not require physical contact between solid surfaces. The forces can be large enough to cause a restructuring of a static powder and can be one of the causes of lumping and crust formation in stored powders. 

By analogy with friction, we may distinguish static and kinetic adhesion. Static adhesion is measured by the force of resistance to the onset of detachment, and kinetic adhesion is measured by the interaction between particles and surface in the course of detachment. In order to detach particles, the force of static adhesion is the primary barrier that must be overcome since the kinetic adhesion is always smaller than the static. This situation attracted the attention of G. I. Fuks, who pointed out that static friction is measured by the force directed tangential to the substrate [12]. 

Such a method has certain advantages. In the first place, it eliminates the inaccuracy involved in the indeterminacy of contact area (it should be noted that no determination has yet been made of the true contact area between particle and surface) and in the second place, it enables us to compare the adhesive force of a powder layer with the adhesive force of a monolayer, i.e., to compare the two cases of adhesion. 

As can be seen from these data, Eq. (11.66) is followed satisfactorily (the force of adhesion is directly proportional to the square of particle diameter), and all calculations involved in the determination of the actual contact area between particles and surface are valid. 

Determination of Interaction in Liquid Medium as a Function of Width of Gap between Particles and Surface. In evaluating the interaction of a particle with a surface in a liquid medium through the use of Eq. (11.52) or (11.53), it is necessary to know how the force of adhesion varies with the quantity H characterizing the gap between the contiguous surfaces. Here we are speaking of a determination of the exponent n in Eq. (11.51). 

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