Measurement of Capillary Driving Forces

FIGURE 6.10. In the capillary rise method of surface tension measurement, surface tension effects canse the wetting hquid to rise in the small capillary to a height that just balances the hydrodynamic force dne to gravity (a). For non-wetting liquids such as mercury, a depressing effect is observed (Z ). 

If the contact angle of the liquid on a capillary surface is 6, the radius of the tube is r, and it is assumed that r is sufficiently small that the liquid surface in the tube is spherical, then the radius of curvature of the liquid-air interface, R, = r/cos 6. According to Equation (6.1), the capillary pressure at point B will be (-2cr cos 9lr) and the net driving force for capillary rise will be Pcap(A) cap(B), sincc at A, Pcap = 0, the total driving force comes from the curvature of the interface in the tube. The liquid movement will continue until the hydrostatic head of liquid in the tube, Apg/i, is equal to Pcap(B), where Ap is the difference in density between the liquid and the vapor, g is the acceleration due to gravity, and h is the height of the meniscus in the tube above the liquid level at A. Then 

Equation 6.9 is the classic equation for determining the surface tension of a liquid by the capillary rise method. When 9 = 0°, the equation simplifies to 

FIGURE 6.11. In a zero-gravity situation, or in horizontal capillary tubes where gravity can be ignored, capillary flow will occur spontaneously to take the system to equilibrium. In (a) the two radii or curvature are unequal causing a pressure differential between A and B. Capillary flow will occur until the pressure has equalized (f ) or until other forces inhibit further movement. 

In such a thermodynamic analysis, it is necessary to use Equations (6.4) and (6.5), and the fact that in the system the sum of the areas of solid-vapor (Asv) and solid-hquid (Asl) interfaces remains constant. Using the equations, it can be shown that the change in free energy SG caused by a change in position of the three-phase boundary by a distance 8s can be represented as 

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