JKR model

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

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.

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

It is apparent that, in order to satisfy Eq. 30, the JKR model requires that detachment occurs, not when the contact radius vanishes, as might at first be thought, but rather at a finite value 0.63a(0). 

Whereas the JKR model approached the topic of particle adhesion from a contact mechanics viewpoint, the DMT theory simply assumes that the adhesion-induced contact has the same shape as a Hertzian indentor. The normal pressure distribution Ph(p) for the Hertzian indentor is related to the repulsive force and the distance from the center of the contact circle to the point represented by r according to the relationship [49]... 

Upon comparison of Eqs. 29 and 36, it is readily apparent that both theories predict the same power law dependence of the contact radius on particle radius and elastic moduli. However, the actual value of the contact radius predicted by the JKR theory is that predicted by the DMT model. This implies that, for a given contact radius, the work of adhesion would have to be six times as great in the DMT theory than in the JKR model. It should be apparent that it is both necessary and important to establish which theory correctly describes a system. 

There have been several theories proposed to explain the anomalous 3/4 power-law dependence of the contact radius on particle radius in what should be simple JKR systems. Maugis [60], proposed that the problem with using the JKR model, per se, is that the JKR model assumes small deformations in order to approximate the shape of the contact as a parabola. In his model, Maugis re-solved the JKR problem using the exact shape of the contact. According to his calculations, o should vary as / , where 2/3 < y < 1, depending on the ratio a/R. 

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. 

Fig. 6. Lateral stiffness vs. load data for a silicon nitride tip vs. mica surface in ultra-high vacuum. Solid line is fit of the JKR model to the data. Reprinted with pennission from ref. [67].

We have recently been exploring this technique to evaluate the adhesive and mechanical properties of compliant polymers in the form of a nanoscale JKR test. The force and stiffness data from a force-displacement curve can be plotted simultaneously (Fig. 13). For these contacts, the stiffness response appears to follow the true contact stiffness, and the curve was fit (see [70]) to a JKR model. Both the surface energy and modulus can be determined from the curve. Using JKR analyses, the maximum pull off force, surface energy and tip radius are... 

Mapping of the elastic modulus of the glassy and rubbery blocks and clay regions was probed by employing Hertzian and Johnson-Kendall-Roberts (JKR) models from both approaching and retracting parts of the force-distance curves. In order to determine the elastic properties of SEBS nanocomposites in its different constituting zones, the corrected force-distance curve was fitted to the Hertz model ... 

Due to adhesive interaction in the retracting portion of the force-distance (f-d) curve, the JKR model registered better insight into nano-mechanical measurements... 

Block and clay regions of SEBS nanocomposite Modulus from Hertz model ( Sample) MPa Localized sample deformation (d), nm Modulus from JKR model ( Sample) MP Bulk modulus3 of SEBS/clay nanocomposite, MPa... 

From the calculation in (7), the softer PEB region was shown to have maximum adhesive force in nature with the calculated modulus in the range of 15 1 MPa (Table 2). The harder PS domains found to have modulus in the range of 24 1 MPa in the SEBS/clay nanocomposite. The non attractive clay regions generally did not fit the JKR model. This was the reason for obtaining much less modulus than that of the literature values for clays in the GPa range. The discussion infers that the bulk modulus of the SEBS/clay nanocomposite (26 1 MPa as shown in Table 2) was dictated by the contribution from PS domains in the matrix. 

In the JKR model a force is necessary to separate two solids. This force is called the adhesion force. The adhesion force is... 

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)... 

If the neck height is larger than some atomic distances, the JKR model is more favorable. With shorter neck heights the DMT model is more suitable. 

The contact radius is estimated using the JKR model. Without external forces we have... 

Figure 11.5 Dependence of friction on load for a single microcontact. The friction force between a silica sphere of 5 //in diameter and an oxidized silicon wafer is shown (filled symbols). Different symbols correspond to different silica particles. The solid line is a fitted friction force using a constant shear strength and the JKR model to calculate the true contact area (based on Eq. (6.68)). Results obtained with five different silanized particles (using hexamethylsililazane) on silanized silica are shown as open symbols. Redrawn after Ref. [467].

JKR model In contrast to the Hertz model where the minimal load is zero, here we can apply negative loads, that is we can even pull on the particles. The greatest negative load is equal to the adhesion force 3 7SR = 0.471 p,N. 

In analogy to indentation experiments, measurements of the lateral contact stiffness were used for determining the contact radius [114]. For achieving this, the finite stiffness of tip and cantilever have to be taken into account, which imposes considerable calibration issues. The lateral stiffness of the tip was determined by means of a finite element simulation [143]. As noted by Dedkov [95], the agreement of the experimental friction-load curves of Carpick et al. [115] with the JKR model is rather unexpected when considering the low value of the transition parameter A(0.2Further work seems to be necessary in order to clarify the limits of validity of the particular contact mechanics models, especially with regard to nanoscale contacts. 

Here a is the work of adhesion. In most cases, the JKR model is adequate to describe the experimental conditions found in analyzing soft and very adhesive materials. 

Near the contact, the vertical arrows at the dashed contour schematically represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from aH (Hertz) to aJKR (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius of the contacting sphere (or the reduced radii of the contacting spheres, if two spheres are in contact) and the elastic constant K (a combination of the Young s moduli and the Poisson s ratios of the contacting materials), defined as... 

Carpick et al [84] used AFM, with a Pt-coated tip on a mica substrate in ultrahigh vacuum, to show that if the deformation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model... 

In this context, Mazur has corrunented [32b] that for films dried at T > MFT, fire extent of deformation for an elastic particle should be predicted by the Hertz model, not the JKR model. Thus the reduced neck diameter (x/a) should be proportional to fl (Equation 14.3), and not rfirectly dependent on particle size... 

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