Two Crossed Cylinders

Similarly, for two crossed cylinders of radii ai and U2, respectively, at separation H, the interaction energy Vcy (H) is considered to be formed by the contributions of laths, each pair of laths (having a planar intersection area of dxdy) contributing to the interaction energy an amount equal to V i(Ji)dxdy[l6, 17] (Eig. 12.4) 

FIGURE 12.4 Deijaguin s approximation for the two interacting crossed cylinders 1 and 2 at separation H, having radii and U2, respectively. 

That is, the interaction energy between two crossed identical cylinders equals twice the interaction energy between two identical spheres. 

It follows from Eq. (12.25) that the interaction force between two 

The interaction energy (H) per unit area between two crossed cylinders with constant surface potentials i/ oi and J/o2 can be obtained by introducing Eq. (10.58) into Eq. (12.47), namely. 

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Fig. VI-6. The force between two crossed cylinders coated with mica and carrying adsorbed bilayers of phosphatidylcholine lipids at 22°C. The solid symbols are for 1.2 mM salt while the open circles are for 10.9 roM salt. The solid curves are the DLVO theoretical calculations. The inset shows the effect of the van der Waals force at small separations the Hamaker constant is estimated from this to be 7 1 x 10 erg. In the absence of salt there is no double-layer force and the adhesive force is -1.0 mN/m. (From Ref. 66.)...

From the discussion about the Deijaguin approximation for spheres and Eq. VI-26, show that the approximation for two crossed cylinders of radius R is... 

Recently, measurements in the non-retarded range have been made, the most notable being those of Tabor and co-workers95 108, on the attraction between cleaved layers of mica stuck to two crossed cylinders. In addition to providing successful tests of the distance dependence of the van der Waals attraction, the effects of adsorbed monolayers have also been studied, again giving reasonable agreement with theoretical predictions. 

The interaction forcep between two crossed cylinders of radius R, for which experimental data are available,34 can be obtained from the interaction free energy F between two parallel plates using the Deijaguin approximation... 

The same type of solution can be made for any type of curved bodies, where the results for two crossed cylinders is worth mentioning. In this case it can be shown that, the total force between the cylinders is... 

Consider next the case of two crossed soft cylinders (Fig. 15.8). Deijaguin s approximation for two crossed cylinders under condition (15.43) is given by Eq. (12.48), namely. 

FIGURE 19.12 Two crossed cylinders of radii ai and A2 separated by a distance ff. 19.11 TWO CROSSED CYLINDERS... 

With the help of Derjaguin s approximation (see Chapter 12), one can derive the van der Waals interaction energy between two crossed cylinders of radii oj and a2 at separation //between their surfaces (Fig. 19.12). By substituting Eq. (19.20) into Eq. (12.48), we obtain... 

Equation (553) shows the important simplification of the dispersion interaction potential between two crossed cylinders, and thus this geometry is preferred for use in the surface force apparatus, which we will see in Section 7.4. 

Figure 7.7 Interactions between two cylinder surfaces having the same length, L, but different radii, R, and R2. a. Between two parallel cylinders having a distance, D between them. b. Between two crossed cylinders.

Later, in 1973, Tabor and Israelachvili developed a surface force apparatus, SFA, to measure the interaction force in a vacuum at the 1.5 nm level for the first time. In this equipment, the interaction forces between two crossed cylinders coated with freshly cleaved mica sheets having atomically smooth surfaces were measured. One of the cylinders is mounted on a piezoelectric transducer, and the other cylinder is mounted on a spring of known and adjustable spring constant with a force resolution down to 1CT8N. SFA has been further developed for performing measurements in liquids and vapors. In these developed SFA versions, the separation distance between the cylinders can be measured interfero-... 

Another aspect of current interest associated with the lipid-water system is the hydration force problem.i -20 When certain lipid bilayers are brought closer than 20-30 A in water or other dipolar solvents, they experience large repulsive forces. This force is called solvation pressure and when the solvent is water, it is called hydration pressure. Experimentally, hydration forces are measured in an osmotic stress (OS) apparatus or surface force apparatus (SFA)2o at different hydration levels. In OS, the water in a multilamellar system is brought to thermodynamic equilibrium with water in a polymer solution of known osmotic pressure. The chemical potential of water in the polymer solution with which the water in the interlamellar water is equilibrated gives the net repulsive pressure between the bilayers. In the SEA, one measures the force between two crossed cylinders of mica coated with lipid bilayers and immersed in solvent. 

Figure 9.16, (a) Friction force measurements on crossed polylethylene tercphthalaic) fibers showing agreement with Equation (9.16) with friction coefficient = 0..T.3 and work of adhesion W = 0.01 m . (b) Contact between two crossed cylinders, equivalent to sphere on flat, showing the forces acting. 

The shape of radial dependence of the adhesion forces shown in Figure 1.92 for nanosilica in the aqueous medium differs significantly from similar dependences measured directly as forces between two crossed cylinders (Claesson et al. 1995, Atkins et al. 1997, Spalla and Kekicheff 1997). Clearly from these results (Figure 1.92), the adhesion force value decreases practically linearly with the distance (not far from the surface). At the same time, the interaction force between two cylinders changes in the sign at a short distance between them due to too much decrease in the liquid layer thickness and can be described as follows f=jr . When x is smaller than the adsorbed layer, the work is made against the adsorption forces, and the structure of this... 

Figure 1.5 Force measured between two crossed cylinders of mica Immersed in OMCTS (reproduced with permission from Ref 25, copyright 1981, American Institute of Physics).

Sphere of radius R and semi-infinite flat slab also valid for two crossed cylinders at 90 -AR 6t ... 

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