Elastic adherence force

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. 

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). 

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 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 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. 

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

As previously discussed, many, if not most, cases of particles adhering to substrates involve at least one of the contacting materials deforming plastically, rather than elastically. Under such circumstances, it would be expected that the extent of the contact should increase with time and, with it, the force needed to detach a particle from a substrate. Moreover, material flow can occur, resulting in the engulfment or encapsulation of the particles. 

Modulus. The measure of the stress of a sealant at a specific strain is referred to as the modulus of elasticity, sometimes called the secant modulus. This important sealant property describes the force exerted by a sealant as it is stressed. Because a pnmary funcuon of sealants is to adhere to the substrates it is in contact with, the forces generated by a joint opening or closing are transmitted by the sealant to the substrate-sealant bond line. For this reason, it is important to know the modulus of the sealant and also the strength of (lie substrate. 

When attempting to relate the adhesion force obtained with the SFA to surface energies measured by cleavage, several problems occur. First, in cleavage experiments the two split layers have a precisely defined orientation with respect to each other. In the SFA the orientation is arbitrary. Second, surface deformations become important. The reason is that the surfaces attract each other, deform, and adhere in order to reduce the total surface tension. This is opposed by the stiffness of the material. The net effect is always a finite contact area. Depending on the elasticity and geometry this effect can be described by the JKR 65 or the DMT 1661 model. Theoretically, the pull-off force F between two ideally elastic cylinders is related to the surface tension of the solid and the radius of curvature by... 

Fig. 15.2. Single-molecule force spectroscopy. The natural TK protein construct is non-specifically adhered to a gold surface and contacted with the tip of an AFM cantilever. During retraction, force-extension traces are recorded. After each rupture, the total free contour length increases so that the force drops. If further stretched, a characteristic rise in force is observed, which is due to the polymer elasticity of the unfolded polypeptide chain. In the beginning, TK unfolds below 50 pN. After complete unfolding of TK, the five structural Ig/Fn domains unfold with regular contour length increments AL of about 30nm (Li-Le)...

For the transport of heavy ions to a solid surface coated with an adherent water film, like aluminium oxide, the visco-elastic properties of electric field forces and the concentration of heavy ions may be important for the rate of adsorption. For this reason we need information not only on relaxations restricted to a surface of an extended liquid, but also on the adherent water layer at the adsorbents. The last issue may be a bridge to the thermodynamics and flow properties of thin liquid films have been studied by some excellent research groups. 

Another technique widely used in the estimation of the properties of cells and their components is atomic force microscopy, where the sample is probed by a sharp tip located at the end of a cantilever of a prescribed stiffness, and the corresponding indentation is tracked with a laser. The force/indentation relationship is a characterization of the cell (cellular component) properties. A traditional interpretation of this experiment is based on Hertz theory of a frictionless contact of a rigid tip with an elastic isotropic half-space [Radmacher et al, 1996]. The finite thickness of the cell can be taken into account by considering an elastic layer adhered to a substrate. More details of cell geometry and rheology can be considered by using the finite element method. 

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