Contact line development

The detailed history of contact line development and its implications with respect to the paper structure are reported elsewhere (3)). 

As indicated, an implicit assumption of the JKR theory is that there are no interactions outside the contact radius. More specifically, the energy arguments used in the development of the JKR theory do not allow specific locations of the adhesion forces to be determined except that they must be associated with the contact line where the two surfaces of the particle and substrate become joined. Adhesion-induced stresses act at the surface and not a result of action-at-a-distance interatomic forces. This results in a stress singularity at the circumference of the contact radius [41]. The validity of this assumption was first questioned by Derjaguin et al. [42], who proposed an alternative model of adhesion (commonly referred to as the DMT theory ). Needless to say, the predictions of the JKR and DMT models are vastly different, as discussed by Tabor [41]. 

Figure 4. Stereodiagrams of enone lg, before (above) and after (below) pyramidalization at the -carbon atom. The steric compression contacts which develop are shown by the dotted lines.

In the JKR theory it is assumed that surface forces are active only in the contact area. In reality, surface forces are active also outside of direct contact. This is, for instance, the case for van der Waals forces. Derjaguin, Muller, and Toporov took this effect into account and developed the so-called DMT theory [206], A consequence is that a kind of neck or meniscus forms at the contact line. As one example, the case of a hard sphere on a soft planar surface, is shown in Fig. 6.19. 

Also adhesion between the tip and sample can cause deformation of the sample. Several theories have been developed to include the effect of adhesive forces. In the JKR theory adhesion forces outside the contact area are neglected and elastic stresses at the contact line are infinite [80]. Even under zero load, the adhesion force results in a finite contact radius a=(9jtR2y/2 E)1/3 as obtained from Eq. 7 for F=0. For example, for a tip radius R=10 nm, E=lGPa, typical surface energy for polymers y=25 mN/m, and typical SFM load F=1 nN, the contact radius will be about a=9.5 nm and 8=9 nm, while under zero load the contact radius and the deformation become a=4.5 nm and 8=2 nm, respectively. The experiment shows that under zero load the contact radius for a 10 nm tungsten tip and an organic film in air is 2.4 nm [240]. The contact radius caused only by adhesion is almost five times larger than the Hertzian diameter calculated above. It means, that even at very small forces the surface deformation as well as the lateral resolution is determined by adhesion between the tip and sample. 

Fig. 31. Three-dimensional images of droplets of the carbosilane dendrimer with hydroxyl end groups obtained during spreading at ambient conditions on a mica substrate [320]. The images corresponds to different duration of spreading (a - 10 min and b - 24 h), where a precursor film of 2 nm in thickness develops at the contact line in (b). The thickness corresponds to the diameter of a dendrimer molecule...

The principle of determination of xrfrom the dependence of 6 on r has been realised in the diminishing bubble method developed by Platikanov et. al. [470,471]. From the optically measured values of the bubble radius Rb and the film contact line radius r, the values... 

Another theory of the linear energy of the contact line wetting film/bulk liquid drop on a solid surface has been developed by Churaev at al. [478]. These authors also considered both cases of negative and positive line tension. In their interpretation the transition region film/bulk can be presented [478] schematically as shown in Fig. 3.103. The dashed line 1 represents the idealised surface. The real surface is shown for two different cases in case 2 the... 

Prins and Clint et al. developed a method of contact angle measurement for macroscopic flat foam films formed in a glass frame in contact with a bulk liquid. They measured the jump in the force exerted on the film at the moment when the contact angle is formed. A similar experimental setup was used by Yamanaka for measurement of the velocity of motion of the three-phase contact line. 

Dynamic Sorption of Ink Jet Ink Drops on Commercial Papers. Drying times (t ) and various parameters measured or derived from contact line spreading, profile development and penetration data for the various systems studied are summarized in Table I. A wide variation occured in the values of tj among the various papers studied. The maximum deviations in tdetailed contact line and profile analysis suggest that these deviations result from the structural variability in paper rather than drop-to-drop volume variations. This was substantiated by the marked variations in the final image shape observed among several drops printed on the same paper sample. 

Independently and almost simultaneously the effect of molecular forces near a three-phase contact line was analyzed by Miller and Ruckenstein (1974) and Jameson and Garcia del Cerro (1976). While both papers point to the presence of asymmetric force fields, mainly generated by the presence of two dense phases (solid and liquid) and a gas phase, Miller and Ruckenstein (1974) developed the concept of a resulting force to explain the movement of a contact line and Jameson and Garcia del Cerro (1976) balanced the resultant force with an interfacial tension gradient. 

Hysteresis in contact angle values is also dependent on the drop size and this is attributed to a pseudo-line tension. Line tensions are intimately associated with the equilibrium of three phases and have been discussed extensively by Rowlinson and Widom (1982). When three phases are in equilibrium they will meet in a line of three-phase contact and there will be an excess free energy per unit length, the line tension, associated with this contact line. For a liquid drop on a polymer surface, this contact line is circular and if the line tension is positive a driving force to shrink the drop laterally will be developed. For a drop of radius R and a line tension of a, the Young-Dupre equation is modified to... 

Developing the analogy in our impact problem with a hydrophobic surface, we identify the wetting phase as air, so that the static contact angle with respect to the wetting phase is defined as o = (rt — %) and the triple contact line becomes no longer stable above u a (rr — 0o) - Furthermore, v is fixed by a critical capillary number Ca = u 7l/tlv that evolves like Ca l/9 . 

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