Surface elastic modulus using Hertz

Force-Curves Measurements In force curve measurements, a vertical displacement of the sample, z, is imposed and the subsequent tip displacement, d, is measured. The tip-sample interaction force, F, is deduced by means of the Hooke s relation, F = -kcd, where is the cantilever stiffness. Force curves arc generally divided into different regions (7). If the part where the electrostatic repulsion forces are dominant is only considered, with silicon tips much stiffer than polymers, tips penetrate the sample surface and an indentation depth, 5, equal to Z d, can be measured. The lower the sample elastic modulus, the greater will be the indentation depth. By using the Hertz mechanical model adapted to the geometry of the tip-sample system (8,9) surface elastic modulus could be deduced from the following equations corresponding respectively to a spherical, a paraboloid and a conical tip ... 

In figure 2, the results obtained in force modulation on various polymers are presented. As expected, the elastic response increases with the bulk modulus. Using Hertz models, the elastic modulus has been derived. For rigid polymers, the agreement between the surface modulus and the bulk modulus is quantitatively good. For softer polymers, a large discrepancy is observed, probably due to the fact that the adhesion force and the viscoelasticity are neglected. This could also be explained by... 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

Figure 2. (a) Variation of the dynamic elastic response, d /z, measured by local force modulation as a function of the bulk elastic modulus, (b) Comparison between the surface Young s modulus deduced from force modulation experiments and the volume modulus calculated using Hertz model. 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

The case of a perfectly elastic contact between the solid surface and the absolnte solid ball is known as the Hertz Problem of contact mechanics. The Hertz Problem has a rather cumbersome solution. With the application of dimensional analysis (Section 5.2), one can get a characteristic nonlinear dependence of the size of the impression on the indenting ball s diameter, the applied force and the Young s modulus of the material. In its reverse version, that is, for the case of a contact between a compliant sphere and a solid surface (bottom of a 15 g weight), this method was used for a long time to measure the internal eye pressure of the eye. 

Figure 1. Typical force-indentation curves obtained respectively (a) on a rigid polymer (E = 610 MPa) and (b) on a soft one(E = 27 MPa), (c) Comparison l tween the surface Young s modulus deduced from the analysis of the force-indentation curves and the volume modulus measured by dynamic mechanical analysis, DMA, using the Hertz elastic model ( ) and using the JKR model (A).

---