Adhesion sphere

Despite experimental progress, the mechanisms and kinetics of acid milk gelation are still not fully understood. Theoretical approaches such as the adhesive sphere, percolation or fractal model applied to acid-induced milk gel formation can successfully explain specific aspects of the process (Tuinier and Kruif 1999), but all fail in rationalizing its kinetics (Home 1999). 

Figure 9.19. (a) Loose tree-like structure formed by high adhesion spheres, (b) compacted aggregate formed by low adhesion particles. 

JKR Elastic-Adhesive Normal Contact Model The theory of Johnson et al. [22], referred to as JKR model, assumes that the attractive forces are confined within the area of contact and are zero outside. In other words, the attractive interparticle forces are of infinitely short range. JKR model extends the Hertz model to two elastic-adhesive spheres by using an energy balance approach. The contact area predicted by the JKR model is larger than that by Hertz. Consequently, there is an outer annulus in the contact area that experiences tensile stresses. This annulus surrounds an inner circular region over which a Hertzian compressive distribution acts [23]. Figure 7.9 shows schematieally the force-overlap response of the JKR model. 

Henrich O., Puertas A.M., Sperl M., Baschnagel J., and Fuchs M. 2007. Bond formation and slow heterogeneous dynamics in adhesive spheres with long-range repulsion Quantitative test of mode coupling theory. Phys. Rev. E16 031404. 

Baxter R J 1968 Percus-Yevick equation for hard spheres with surface adhesion J. Chem. Phys. 49 2770... 

Rouw P W and de Kruif C G 1989 Adhesive hard-sphere colloidal dispersions fractal structures and fractal growth in silica dispersions Phys. Rev. A 39 5399-408... 

In the JKR experiments, a macroscopic spherical cap of a soft, elastic material is in contact with a planar surface. In these experiments, the contact radius is measured as a function of the applied load (a versus P) using an optical microscope, and the interfacial adhesion (W) is determined using Eqs. 11 and 16. In their original work, Johnson et al. [6] measured a versus P between a rubber-rubber interface, and the interface between crosslinked silicone rubber sphere and poly(methyl methacrylate) flat. The apparatus used for these measurements was fairly simple. The contact radius was measured using a simple optical microscope. This type of measurement is particularly suitable for soft elastic materials. 

Thin sheets of mica or polymer films, which are coated with silver on the back side, are adhered to two cylindrical quartz lenses using an adhesive. It may be noted that it is necessary to use an adhesive that deforms elastically. One of the lenses, with a polymer film adhered on it, is mounted on a weak cantilever spring, and the other is mounted on a rigid support. The axes of these lenses are aligned perpendicular to each other, and the geometry of two orthogonally crossed cylinders corresponds to a sphere on a flat surface. The back-silvered tbin films form an optical interferometer which makes it possible... 

The work of adhesion was determined from the a versus P measurements (see Eq. 11). The work of adhesion between two rubber spheres was found to be 71 4 mJ/m. The work of adhesion reduced to 6.8 0.4 mJ/m in the presence of 0.01 M solution of dodecyl sulfate. Using these measurements of adhesion between rubber in air and a surfactant solution, Johnson et al. [6] provided the first direct experimental verification of the Young s equation (Eq. 40). They also measured... 

In an attempt to determine the applicability of JKR and DMT theories, Lee [91] measured the no-load contact radius of crosslinked silicone rubber spheres in contact with a glass slide as a function of their radii of curvature (R) and elastic moduli (K). In these experiments, Lee found that a thin layer of silicone gel transferred onto the glass slide. From a plot of versus R, using Eq. 13 of the JKR theory, Lee determined that the work of adhesion was about 70 7 mJ/m". a value in clo.se agreement with that determined by Johnson and coworkers 6 using Eqs. 11 and 16. 

Fig. 18. Adhesive contact of elastic spheres. pH(r) and pa(r) are the Hertz pressure and adhesive tension distributions, (a) JKR model uses a Griffith crack with a stress singularity at the edge of contact (r = a) (b) Maugis model uses a Dugdale crack with a constant tension aa in a < r < c [1111.

Lest one be lulled into a false sense that, assuming that the JKR theory properly describes particle adhesion within its regime, DeMejo et al. [56] also reported that, for soda-lime glass particles with radii less than about 5 p.m, the contact radius varied, not as the predicted but, rather, as Similar results were reported for other systems including polystyrene spheres on polyurethane [58], as shown in Fig. 2, and for glass particles having radii between about 1 and 100 p,m on a highly compliant, plasticized polyurethane substrate [59] as illustrated in Fig. 3. 

Hertzian mechanics alone cannot be used to evaluate the force-distance curves, since adhesive contributions to the contact are not considered. Several theories, namely the JKR [4] model and the Derjaguin, Muller and Torporov (DMT) model [20], can be used to describe adhesion between a sphere and a flat. Briefly, the JKR model balances the elastic Hertzian pressure with attractive forces acting only within the contact area in the DMT theory attractive interactions are assumed to act outside the contact area. In both theories, the adhesive force is predicted to be a linear function of probe radius, R, and the work of adhesion, VFa, and is given by Eqs. 1 and 2 below. 

The failure of systems with dispersed fillers (exemplified by polystyrene plus glass spheres with different treatment) was studied by subjecting specimens to deformation in the microscope field [255,256]. Where adhesion was good the cracks were observed to be formed near the glass sphere pole, in regions corresponding to the maximum deformation, where adhesion was poor, anywhere between the pole and the equator. It was discovered that microcracks began to... 

Various continuum models have been developed to describe contact phenomena between solids. Over the years there has been much disagreement as to the appropriateness of these models (Derjaguin et al. [2 ] and Tabor [5-7]). Experimental verification can be complex due to uncertainties over the effects of contaminants and asperities dominating the contact. Models trying to include these effects are no longer solvable analytically. A range of models describing contact between both nondeformable and deformable solids in various environments are discussed in more detail later. In all cases, the system of a sphere on a plane is considered, for this is the most relevant to the experimental techniques used to measure nanoscale adhesion. 

This equation is useful in that it is applicable to any type of force law so long as the range of interaction and the separation are much less than the radius of the sphere. Thus the force to overcome the work of adhesion between a rigid sphere and a flat surface written in terms of the surface energy Ay is ... 

Erom the previous sections it is clear that there are a number of different possible models that can be applied to the contact of an elastic sphere and a flat surface. Depending on the scale of the objects, their elasticity and the load to which they are subjected, one particular model can be more suitably applied than the others. The evaluation of the combination of relevant parameters can be made via two nondimensional coordinates X and P [16]. The former can be interpreted as the ratio of elastic deformation resulting from adhesion to the effective range of the surface forces. The second parameter, P, is the load parameter and corresponds to the ratio of the applied load to the adhesive puU-off force. An adhesion map of model zones can be seen in Figure 2. 

Vakarelski et al. [88] also investigated the adhesive forces between a colloid particle and a flat surface in solution. In their case they investigated a sihca sphere and a mica surface in chloride solutions of monovalent cations CsCl, KCl, NaCl, and LiCl. The pH was kept at 5.6 for all the experiments. To obtain the adhesive force in the presence of an electrostatic interaction, they summed the repulsive force and the pull-off force (coined foe by the authors ) to obtain a value for the adhesive force that is independent of the electrostatic component. 

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