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Contact Angle & Capillarity - Liquid in a Vertical Tube

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Contact Angle & Capillarity - Liquid in a Vertical Tube

Determining the contact angle, theta, of liquid in a vertical tube.

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Contact Angle

Surface tension occurs during a gas-liquid interface, but if that interface comes in contact with a solid surface - such as the walls of a container - the interface usually curves up or down near that surface. Such a concave or convex surface shape is known as a meniscus

The contact angle, theta, is determined as shown in the picture to the right.

The contact angle can be used to determine a relationship between the liquid-solid surface tension and the liquid-gas surface tension, as follows:

gammals = - gammalg cos theta

where

  • gammals is the liquid-solid surface tension
  • gammalg is the liquid-gas surface tension
  • theta is the contact angle
One thing to consider in this equation is that in cases where the meniscus is convex (i.e. the contact angle is greater than 90 degrees), the cosine component of this equation will be negative which means that the liquid-solid surface tension will be positive.

If, on the other hand, the meniscus is concave (i.e. dips down, so the contact angle is less than 90 degrees), then the cos theta term is positive, in which case the relationship would result in a negative liquid-solid surface tension!

What this means, essentially, is that the liquid is adhering to the walls of the container and is working to maximize the area in contact with solid surface, so as to minimize the overall potential energy.

Capillarity

Another effect related to water in vertical tubes is the property of capillarity, in which the surface of liquid becomes elevated or depressed within the tube in relation to the surrounding liquid. This, too, is related to the contact angle observed.

If you have a liquid in a container, and place a narrow tube (or capillary) of radius r into the container, the vertical displacement y that will take place within the capillary is given by the following equation:

y = (2 gammalg cos theta) / (dgr)

where

  • y is the vertical displacement (up if positive, down if negative)
  • gammalg is the liquid-gas surface tension
  • theta is the contact angle
  • d is the density of the liquid
  • g is the acceleration of gravity
  • r is the radius of the capillary
NOTE: Once again, if theta is greater than 90 degrees (a convex meniscus), resulting in a negative liquid-solid surface tension, the liquid level will go down compared to the surrounding level, as opposed to rising in relation to it.
Capillarity manifests in many ways in the everyday world. Paper towels absorb through capillarity. When burning a candle, the melted wax rises up the wick due to capillarity. In biology, though blood is pumped throughout the body, it is this process which distributes blood in the smallest blood vessels which are called, appropriately, capillaries.
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