Gary L. Bertrand
University of Missouri-Rolla
The early interpretation of capillarity considered the liquid to "climb" the walls of the tube to the point that the "grip" of the ring of the surface around the inner wall could just barely support the weight of the column of liquid beneath it. This surface tension was calculated as the weight of the column divided by the inner circumference of the tube. This was later refined to units of force/length. Thus, if a liquid of density (D) 0.800 g/cm3 rose to a height (H) of 5.00 cm in a capillary tube with an inner radius (r) of 0.0100 cm (in a gravitational field, g = 980 cm/sec2), the surface tension (g) was calculated:
Volume = Area x Height , Mass = Volume x Density , Force = Mass x Acceleration
V = P r2 H , M = P r2HD , F = P r2HDg
Circumference = 2P r
Surface Tension = g = Force/Circumference = P r2HDg / 2P r
g = rHDg / 2
g = (0.0100 x 5.00 x 0.800 x 980)/2 = 19.6 dyne/cm = 19.6 erg / cm2.
In SI units, this is expressed as: g = 19.6 mN / m = 19.6 mJ / m2 .
This idea of the surface tension supporting the weight of the liquid carried over to the detachment methods which provide alternative ways of determining the surface tension. In 1805, Young and LaPlace independently related surface tension to a pressure difference across a curved surface, providing the modern definition of surface tension. They used two radii of curvature (R1 and R2) in mutually perpendicular planes containing a line normal to the surface to describe the curvature at any point on a surface. These radii can be difficult to visualize, but are clear for two simple cases, a spherical surface and a cylindrical surface.
For a spherical surface, the two radii are identical R1 = R2:
For a cylindrical surface, one radius is infinite R2 = infinity, 1/R2 = 0 :
The equation of Young and LaPlace is:
DP = g (1/R1 + 1/R2) .
The application to capillary rise uses two relationships:
In a continuous fluid (a
liquid or air), the pressure is everywhere
the same at the same height in the gravitational field.
At different heights in the gravitational
field, the pressure within the
fluid differs (DP) by the density of the fluid (D) multiplied by the
difference in height (H) and the gravitational constant (g).
The height (H) at the bottom
of the meniscus is related to the pressure difference through the following
DP = P1 - P2
P2 = P3 - DliqgH
P1 = P4 - DairgH
P3 = P4
DP = (Dliq - Dair)gH
and the equation of Young and LaPlace becomes:
g (1/R1 + 1/R2) = (Dliq - Dair)gH .
At the bottom of the meniscus, the two radii are equal to each other R1 = R2 = R ,
and if the meniscus is a hemisphere, the radii are equal to the radius of the capillary R = r ,
allowing an approximate solution:
g = rHg (Dliq - Dair)/2 .
This is a good approximation for a capillary with a very small radius, but the error increases with the radius of the capillary because the meniscus tends to flatten at the bottom and the radius becomes greater than the hemisphere approximation. In 1915, Lord Rayleigh (see Physical Chemistry of Surfaces 5th Ed by Arthur W. Adamson, 1990, John Wiley and Sons, Ch 2) developed a better approximation:
g = (rg/2) (Dliq - Dair) (H + r/3 - 0.1288r2/H + 0.1312r3/H2 + ...) .
This provides a good approximation for small capillaries (such that H/r is large), provided that the height can be measured accurately. The location of the bottom of the meniscus can be measured very accurately, but the liquid level outside of the capillary is difficult to observe due to the thickness of the meniscus at the wall of the container. The twin capillary technique was developed to avoid this problem, using two capillaries of different radii (r1 , r2) and measuring the difference in the heights of the liquid (DH), giving a reasonable first approximation:
g = (gDH/2) (Dliq - Dair) / (1/r1 - 1/r2) .
This usually is accurate to within 0.1 dyne/cm for most liquids in a capillary with an inner diameter less than 1 mm.