Contact line lateral

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. 

FIGURE 5.20 Special types of immersion capillary forces (a) The contact line attachment to an irregular edge on the particle surface produces undulations in the surrounding fluid interface, which give rise to lateral capillary force between the particles, (b) When the size of particles entrapped in a hquid film is much greater than the nonperturbed fihn thickness, the meniscus surfaces meet at a finite distance, r in this case, the capillary interaction begins at L < 2rp. 

At the edges of the coated layer its free surface ends in ordinary static contact lines. These lateral contact lines necessarily bend round upstream and connect with the wetting line. Thus at each edge of the layer where it is being delivered to the substrate there must be a curved segment of dynamic contact line, and the apparent slip of the liquid... 

For small holes (R < Aq), viscous dissipation within the film due to radial and ortho-radial deformations dominate. This leads to an exponential growth of the hole, a consequence of a continuously increasing length of the three-phase contact line, i.e., the perimeter of the hole. During this regime, no rim is formed. For R > Aq interfacial friction dominates. At this later stage, the relative increase in hole perimeter is small and so the increase in driving force is smaller than the increase in frictional force. Thus, a rim of width Aq starts to buUd up. 

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... 

Multicomponent systems may also involve the selective adsorption of one component at the SL interface. Since the component that lowers the interfacial tension will be preferentially adsorbed, the rate of the adsorption process can affect the local tension and the contact angle. In many systems, the rate of adsorption at the solid surface is found to be quite slow compared to the rate of movement of the SLV contact line. As a result, the system does not have time for the various interfacial tensions to achieve their equilibrium values. Most surfactants, for example, require several seconds to attain adsorption equihbrium at a LV interface, and longer times at the SL interface. Therefore, if the hquid is flowing across fresh solid surface, or over any surface at a rate faster than the SL adsorption rate, the effective values of olv and osl (and therefore 6) will not be the equilibrium values one might obtain from more static measurements. More will be said about dynamic contact angles in later chapters. 

Two types of boundary conditions at the wall are analyzed theoretically [37,344] fixed contact line (Figure 4.25) or, alternatively, fixed contact angle. In particular, the lateral capillary force exerted on the particle depicted in Figure 4.25 is given by the following asymptotic expression [37,344] ... 

An interesting observation in Figure 3.14 is that for the heavy particle the horizontal force intersects the x-axis, which means there is a zero force point located close to the wall. For the fixed contact angle, the zero force point is not stable (the particle has tendency to move away from the zero force point), while for the fixed contact line, this point is stable (the particle has tendency to stay at the zero force point). The measurements (Velev et at [20]) of the zero force point for the fixed contact line agree with the theory well. In summary, when the separation is not too small, the lateral capillary force is attractive if the slope angles at the wall and on the floating particle have the same sign, otherwise, the force is repulsive. When the separation is too small, the situation is much more complicated. 

They observed that while the central dry zone increases, a bump is built up between the receding contact line and the liquid film the latter remains static and the receding contact line moves at constant dewetting velocity, V. They investigate the dependence of this velocity with the diffetent parameters of the system. The main results are that V4 does not depend on the film thickness (for h he) and that for viscous and non-polar liquids and small static contact angles (up to 50°) the dewetting capillary number Caj = i VdlV scales as the cube of (P while the prefactor varies weakly with the studied system. This result, that resembles Tanners law, was explainedby means of a simple hydrodynamic model that assumes a circular cross section for the bump and symmetrical dissipation at both of it ends. Following later observations of asymmetries in the bump s profile this last assumption was modified. 

If we place a second particle on the interface, it will notice the deformed liquid surface (Figure 5.16c) and vice versa [588, 589]. As a result, the three-phase contact line is not radially symmetric anymore. The asymmetry of the contact line leads to a lateral force between the two floating particles. It is called lateral flotation force (reviewed in Ref [590]). 

Lateral capillary forces also occur when the particles are partially immersed in a thin liquid film on a solid support (Figure 5.18) [590,595]. In this case, they are called immersion forces. Immersion forces on solid surfaces occur always when suspensions of solid particles are dried. In the last state of evaporation, the particles are only partially immersed in the thinning liquid film and attractive immersion forces lead to an aggregation of particles (Figure 5.19). Immersion forces can be used to self-assemble particles in two-dimensional arrays [595, 596]. The deformation of the liquid surface is related to the wetting properties of the particles, that is, to the position of the contact line and the contact angle rather than the weight. For this reason, also very small particles such as proteins are affected. 

---