Assumptions and Limits

Applying the circular approximation allowed us to calculate capillary forces between axisymmetric objects analytically. What are the limits and errors involved. One limit 

Even in the absence of gravitation, the Laplace equation predicts a nodoid or an unduloid [513] rather than a circle. Numerical calculations of the precise shape showed that the difference can usually be neglected. For example, for spheres the errors in the capillary force and the volume of the meniscus are below 5% for (3 40° and0 90° [513,515,517,518]. For p 20° and 40°, the errors are even below 2%. Considering all the following effects, such an error is negligible. 

The circular approximation fails at vapor pressures close to saturation. For P/Po O, the total curvature and thus the Laplace pressure goes to zero. In addition, the circumference of the meniscus might become large. As a result, the contribution of the capillary pressure term in the total force might become insignificant and the direct action of the surface tension dominates. For two equal spheres in contact, this leads to an adhesion [584] of 

Several aspects are often neglected in the calculation and discussion of capillary forces, although they are intensely debated with respect to wetting or adsorption [535, 536, 805]  

Different crystal surfaces are exposed, and defects and variations in the chemical composition lead to local changes in the contact angle. This can influence the capillary force. 

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In Chapter 3 and Appendix G the energy levels and wavefunctions that describe the rotation of rigid molecules are described. Therefore, in this Chapter these results will be summarized briefly and emphasis will be placed on detailing how the corresponding rotational Schrodinger equations are obtained and the assumptions and limitations underlying them. 

Equations (7.40) and (7.41) suggest a second method, in addition to the copolymer composition equation, for the experimental determination of reactivity ratios. If the average sequence length can be determined for a feedstock of known composition, then rj and r2 can be evaluated. We shall return to this possibility in the next section. In anticipation of applying this idea, let us review the assumptions and limitation to which Eqs. (7.40) and (7.41) are subject ... 

Failing to acknowledge the importance of the analysis assumptions and limitations... 

What is meant by Rayleigh scattering What are the important assumptions and limitations of the Rayleigh theory ... 

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