Capillary Rise and Dimensionless Numbers

If the capillary is circular in cross-section with radius a, the meniscus will be approximately hemispherical with a constant radius of curvature, a/ cos 9, where 9 is the static contact angle. Departure from the hemisphericity is associated with the variation in liquid pressure over the surface due to the difference in gravitational force over the meniscus height, h. A measure of the hydrostatic gravitational force to the surface tension force is expressed by the nondimensional Bond number as 

For hydrostatic equilibrium, the balance between the pressure at the surface of the reservoir and the interface gives 

Note For mercury (9 = 140°), the capillary height will fall, not rise. 

For small capillaries, Hq is relatively large due to smaller value of capillary radius. For water, a = 13 mN/m. If a = 0.1 mm, we have //q = 0.15 m, which is a large value for accurate measurement of height. Thus, the capillary rise method is one of the most accurate means for the measurement of surface tension. Quite significant rise heights can be obtained in microchannels. Using surface tension data from Table 5.1, we find //q = 4.2 cm for water in a 100 pm radius PMMA polymer channel and Hq = 42 cm for radius, a =10 pm. 

Dividing a in both sides of equation (5.33), the above equation indicates pg a jcp- for a Hq. Thus, Bond number is very small, indicating that surface tension dominates over gravitation. For Bo = 1, the length scale of the capillary equals the capillary rise. 

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