9.6.5. Capillary Rise

One consequence of the pressure difference across a curved liquid-air interface is the well-known effect that water rises up in a narrow tube in spite of gravity. The word capilla means hair in Latin; if the tube were hair thin, the rise would be very large.

Capillary rise occurs due to the pressure difference created at the curved surface of a liquid, like water, in contact with the sides of a narrow tube. This is primarily influenced by surface tension, a property of liquids that results in an excess force at the liquid's surface. The narrower the tube, the greater the rise of the liquid against gravity. This phenomenon is especially prominent in very narrow tubes (capillaries), where the height of the liquid column can be significantly higher than in wider tubes.

An example of capillary rise is seen in how water travels through the tiny tubes of plants, allowing them to distribute water from their roots to leaves. Just like how a paper towel can absorb water by drawing it up through its fibers, plants use capillary action to move water upwards even against the force of gravity.

To see this, consider a vertical capillary tube of circular cross section (radius a) inserted into an open vessel of water. The contact angle between water and glass is acute. Thus the surface of water in the capillary is concave. This means that there is a pressure difference between the two sides of the top surface. This is given by (Pi – Po) =(2S/r) = 2S/(a sec θ) = (2S/a) cos θ.

When a capillary tube is immersed in water, the water adheres to the glass, forming a meniscus that is concave. This concave shape results in a pressure difference between the liquid inside the tube and the atmospheric pressure outside. The formula (Pi - Po) = (2S/r) relates the pressure difference to the radius of the tube (r) and the surface tension (S) of the liquid. The smaller the radius of the tube, the greater the impact of surface tension, resulting in higher capillary rise.

Imagine a thin straw in a drink. When you place the straw in the liquid, you may notice the liquid level inside the straw is higher than the liquid level outside it. This is the same principle; the thinner the straw (similar to a capillary), the higher the liquid rises inside due to surface tension pulling it upwards.

Thus the pressure of the water inside the tube, just at the meniscus (air-water interface) is less than the atmospheric pressure. Consider the two points A and B. They must be at the same pressure, namely P0 + h ρ g = Pi = PA. Using Eq. (9.27) and (9.28), we have h ρ g = (Pi – P0) = (2S cos θ)/a.

By using the relation of pressures at points A and B, we can establish that the height to which water rises in the tube (h) can be calculated using the equation hρg = (Pi - P0). When rearranged, this gives an expression for the height of rise as h = (2S cos θ)/(ρg a). This indicates that the height depends not only on the liquid's properties (density and surface tension) but also on the angle of contact θ and the radius of the tube.

Consider a cotton wick in an oil lamp. The wick draws oil upward against gravity through capillary action, allowing the oil to reach the flame. The height to which the oil rises in the wick depends on the wick's thickness and the oil's surface tension. The narrower the wick, the higher the oil rises due to the principles of capillary action.

Typically, the capillary rise is of the order of a few cm for fine capillaries. For example, if a = 0.05 cm, using the value of surface tension for water, we find that h = 2S/(ρ g a).

Capillary action can lead to significant rises in liquid levels even within small tubes. By substituting actual values into the derived formula, one can calculate the height of liquid rise in practical scenarios. For instance, when the radius is about 0.05 cm, the height of the rise can be calculated to demonstrate how small adjustments in the radius of the tube can lead to measurable differences in the height of the liquid column.

This principle is especially relevant in water conservation where small tubes are used in irrigation to allow water to flow slowly into the soil, effectively utilizing capillary rise to ensure that plants get enough water without wastage. This shows how understanding capillary action can lead to sustainable practices in agriculture.

Notice that if the liquid meniscus is convex, as for mercury, i.e., if cos θ is negative then from Eq. (9.28) for example, it is clear that the liquid will be lower in the capillary.

When dealing with mercury, which exhibits a convex meniscus, the behavior of liquids in capillaries differs from that of water. In this case, the liquid does not rise in the tube because the forces between the mercury molecules are stronger than the adhesive forces between mercury and the glass, leading to a negative value in the equation used for capillary rise. Thus, in such instances, the mercury level will be lower than the surrounding liquid.

Think about using a mercury thermometer. Unlike water which climbs up a thin tube, mercury contracts back down. This occurs because the cohesive forces in mercury are stronger than the adhesive forces with the glass, causing the mercury to form a convex shape in the tube rather than rising like water. Understanding this helps us use mercury effectively in thermometers and other scientific instruments.

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