Adherence force

As described in Chapter 3, at high current densities bubbles coalesce to form larger bubbles. It becomes possible to create a large enough structure on the electrode surface such that the buoyancy force can no longer overcome the adherence force of the coalesced bubble. Thus, a gas film forms at the electrode surface. 

Figure 9 is for spherical probe and shows that even at very low crosshead velocity the viscoelastic effects considerably increase the adherence force compared to the elastic (or quasistatic) adherence forces at fixed load (point C) or fixed grips (point D). 

Dwell times generally increase the adherence force, and Fig. 10 shows the increase of w with contact time between glass and polyurethane (26). Between two polymers (auto-hesion ) this effect is ascribed to interdiffusion of chains by reptation theory and experiments (27-29) lead to a (hence a Dupre energy of adhesion, see below)... 

In high temperature contact strong metallic bonds are easily developed and the separation generally occurs by creep rupture of the material (cohesive rupture), so that the adherence force, as in cold welding, can be written as... 

Due to roughness effects, adherence of metals at moderate temperature and pressure is difficult to analyze. When roughnesses undergo plastic deformation, the true area of contact is proportional to the applied load P, and the adherence force F is often proportional to the load (hence the definition of an adhesion coefficient a = F/P), and independent of the apparent area of contact. These two "Laws of adhesion (41) are similar to Amonton s laws of friction. As shown by Gilbreath (42) the adhesion coefficient is very sensitive to adsorption. More precise experiments by Buckley (43,44) on single crystals in ultrahigh vacuum have shown that the adherence force does not increase linearly with the load, and that the position of the knees depends on the adsorption as if the effectively applied load depended on adsorption. 

To avoid complications due to roughness a number of adhesion experiments have been performed on single asperity contact, but the situation is still complex. The contact of the tip can be elastic, elastoplastic or full plastic as in hardness experiments the separation can occur at the interface ("brittle" or adhesive rupture) or in the softer material ("ductile or cohesive rupture). In the latter case the adherence force is the force to rupture a deeply notched bar (45)... 

On an elastoplastic material the various adherence forces can be computed. If the contact is elastic, the... 

The adherence force between two disks of radius R separated by a liquid of thickness h is a function of the time t for separation ... 

We can siunmarize at this point that the forces measured in AFM adherence force (and frequently friction) measurements depend crucially on the chemistry of the AFM tip and the surface of interest, as well as the medium in which the contact takes place. Hence by systematic variation of the tip chemistry via chemical modification/functionahzation and of the medium chemical contrast spatially resolved chemical imaging becomes possible. 

The load P corresponding to the limit of stability will be called the adherence force of the two elastic bodies, and thus will generally depend on the stiffness of the measuring apparatus. In some geometries (unstable geometries), equilibrium is always unstable, and thus the criterion for adherence force is simply G = w. li is the case for the double cantilever beam at fixed load, or for a flat punch on an elastic half-space. It is also the classical Griffith case of a crack in an infinite solid. 

The equilibrium corresponding to G = w is unstable (as for a crack of radius a in an infinite body) and the load P given by G = w is the adherence force at both fixed load and fixed grips. Equation (23) could also be deduced from the value of Kj given by Paris and Sih(8) for a deeply notched bar. Stress and displacemeent at the edge of the contact are, of course, those of fracture mechanics in mode I. 

It is thus easy to obtain the adherence force for any punch whose profiley(r) is known. Insertion of f x) = a x llR gives directly the JKR results with the stress distribution and the surface profile. For a conical punch of semiangle tt/2 - one has at fixed load... 

It is noteworthy that in this crude cylindrical approximation for the volume, the adherence force at z = 0 is independent of the volume of the liquid bridge. In fact, the circle approximation for the meniscus or exact calculation(20) shows that it decreases with the volume. 

The recorded force first increases, then decreases. The maximum value, termed the tack force, is a measure of the adherence under this particular experimental condition and has no clear physical significance. The area under the curve, termed the tack energy, is equal to the work jGda of the cohesive stress at the crack tip. Tackiness refers to the ability of an elastomer to adhere instantaneously to a solid surface, or to itself, after a brief time of contact under low pressure. Probe tack testing can be analyzed by Eq. (54), and tack curves obtained by computer integration almost coincide with experimental ones. S) Figure 8 pertains to a spherical probe and shows that even at a very low cross-head velocity the viscoelastic effects considerably increase the adherence force compared to the elastic (or quasi-static) adherence force at fixed displacement (point D). 

The subj t of adhesive contact mechanics may be said to have started when Kendall (//), solving the problem of the adhesive contact of a rigid flat cylinder punch indenting the smooth plane surface of an elastic medium, demonstrated that the border of the contact area can be considered as a crack tip. The more complex problem of a spherical punch was solved in 1971 by Johnson, KendaU and Roberts (72). The JKR theory predicts the existence of contact area greater then that ven by the elastic contact Hertz s theory. The molecular attractive forces are responsible for this increase and, even in the absence of external compressive loading, the contact area has a finite size. Separating the two solids requires the application of an adherence force despite the existence of infinite normal stresses in the border of the contact area. 

For rubber-like materials, the viscoelastic losses vary with the strain rate and the temperature. The principle of time-temperature equivalence propose in 1955 by Williams et al (75) allows us to superimpose the experimental curves obtained at different temperatures through the known translation factor ar of the WLF transformation. As a consequence, at fixed geometry, the adherence forces provoking crack extensions at different speeds V can be studied as a function of the reduced parameter ajV. 

As expected, its right hand side is the difference of an Hertz s term and an adhesive term. This equilibrium can be stable, unstable according to the sign of (dG dA)f. The load Pc corresponding to the limit of stability (dGIdA), which may be called the elastic adherence force representing the critical tensile force for which the spontaneous rupture of the contact area just begins, and the associated half-width be of the ultimate contact area are equal to... 

G=kV with /i=0.55, which is in perfect agreement with previous kinetics of adherence of punches, rolling and rebound results obtained with the same rubber-like material (22,24J2J9,40y It should be pointed out that, as already observed, rolling upon and under the rubber surface occurs with the same speed for the same inclination, as in the case where the load per unit axial length of the cylinder is smaller than the absolute value of the elastic adherence force Pc. 

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