JKR analysis

OMCL-TR800PS A is designed for contact-mode operation and has a conical probe tip. The probe was scanned over a sapphire surface to estimate the effective radius of curvature, R, determined to be about 20 nm. This value is not valid for larger sample deformation. To utilize JKR analysis, the probe tip must approximate a spherical shape. Thus, we tried to keep the deformation value to be as small as possible. R150FM-10 is a commercially available spherical probe tip with R= 150 nm. 

It is possible to test the validity of JKR analysis using Tabor s equation [Tabor, 1977 Johnson and Greenwoods, 1997], The dimensionless parameter /u. is expressed... 

A similar observation could be made for the adhesive energy. The adhesive energy must be intrinsically constant, independent of any experimental parameters, as seen in the case of PDMS10. However, results for PDMS3 showed a small variation and those for HR showed a large variation. This is, of course, due to the failure of JKR analysis, but we suspect that these variations contain fruitful information about nanometer-scale rheological phenomena. 

We have discussed the recent progress in nanomechanical property evaluation using AFM based on JKR analysis. Sufficiently elastic materials such as PDMS 10 can be treated within the JKR framework, whereas viscoelastic character causes a large deviation from JKR theory, as observed in the case of PDMS3 or UR. 

The JKR theory, similar to the Hertz theory, is a continuum theory in which two elastic semi-infinite bodies are in a non-conforming contact. Recently, the contact of layered solids has been addressed within the framework of the JKR theory. In a fundamental study, Sridhar et al. [32] analyzed the adhesion of elastic layers used in the SFA and compared it with the JKR analysis for a homogeneous isotropic half-space. As mentioned previously and depicted in Fig. 5, in SFA thin films of mica or polymeric materials ( i, /ji) are put on an adhesive layer Ej, I12) coated onto quartz cylinders ( 3, /i3). Sridhar et al. followed two separate approaches. In the first approach, based on finite element analysis, it is assumed that the thickness of the layers and their individual elastic constants are known in advance, a case which is rare. The adhesion characteristics, including the pull-off force are shown to depend not only on the adhesion energy, but also on the ratios of elastic moduli and the layers thickness. In the second approach, a procedure is proposed for calibrating the apparatus in situ to find the effective modulus e as a function of contact radius a. In this approach, it is necessary to measure the load, contact area... 

For very compliant materials, use the largest probe radius R compatible with the goals of the measurement. This is primarily for convenience in analysis. The widely used analytical results of the Hertz, JKR and DMT models all assume that a R. This assumption is easily violated for compliant materials. Consider, for example, a material with E 2 MPa and a work of adhesion IV 20 mJ/m. At a load of only 2 nN, JKR analysis predicts that a/R = 1.5 for R = 50 nm, a typical value for a commercial SFM tip. This is clearly non-physical and shows that JKR analysis is not appropriate and a more realistic, non-analytical model such as that of Mau-gis (25) must be used. In this example, even at R = 1000 nm, JKR cannot be used Unfortunately, it appears that quantitative analysis of nanometer-scale contacts to compliant elastic or viscoelastic materials will typically be non-analytical. 

It is somewhat disconcerting that the MYD analysis seems to present a sharp transition between the JKR and DMT regimes. Specifieally, in light of the vastly different response predicted by these two theories, one must ponder if there would be a sharp demarcation around /x = 1. This topic was recently explored by Maugis and Gauthier-Manuel [46-48]. Basing their analysis on the Dugdale fracture mechanics model [49], they concluded that the JKR-DMT transition is smooth and continuous. 

An issue, at present unresolved, is that Derjaguin, Muller and Toporov [24,25] have put forward a different analysis of the contact mechanics from JKR. Maugis has described a theory which comprehends both the theories as special cases [26]. 

Exact analysis shows that the two models represent two extremes of the real situation [207-209], For large, soft solids the JKR model describes the situation realistically. For small, hard solids it is appropriate to use the DMT model. A criterion, which model is to be used, results from the height of the neck (Fig. 6.19)... 

In this section we describe our recent progress in nanomechanical analysis to make it applicable to conditions where we cannot ignore the adhesive and viscoelastic effects, and where JKR contact plays an important role, as explained in Section 3.3. Then we discuss the realistic applicable limit of this theory based on several experimental results. Furthermore, the viscoelastic effect is treated experimentally and theoretically to some extent, with the future goal of making nanomechanical mapping a nanorheological mapping technique. 

For nanoaerosol, e 1 because of the great rigidity. Several theoretical models of adhesion energy can be found in the literature. Two of the most well-known ones are JKR model and DMT model. These two models contradict each other because they represent two extremes in the Tabor parameter spectrum. JKR model is applied for soft material, large radius, compliant spheres, and large adhesion energy, and DMT model is for hard material, small radius, and low adhesion energy. Therefore, DMT model should be considered first for nanoaerosol thermal rebound analysis. 

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).

Tabor calculated that the pull-off force would approach the DMT value for radii of curvature of the order of 1 pm. This problem has been examined recently by Miller, Yushchenko and Derjaguin Their analysis suggests that the JKR theory... 

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