# Physics Tutorial: Adhesive and Cohesive Forces. Surface Tension and Capillarity

In this Physics tutorial, you will learn:

• The meaning of adhesive and cohesive forces
• The conditions for the existence of adhesive and cohesive forces
• What shape does a certain surface take due to adhesive and cohesive forces?
• What is surface tension?
• Where does surface tension differ from surface tension force?
• What is capillarity?
• How to calculate the maximum height a liquid can rise in narrow tubes?

## Introduction

Why water drops have a spherical shape?

Why mercury looks like a liquid when it is in glass containers while when it falls on the ground looks like grains taking spherical shapes?

How does water rises up to the highest leaves in trees?

How does blood circulates in all our body even if we are standing up?

How do the soap bubbles form?

Why water seems as sticking on the sides of a glass container?

Why water surface in narrow tubes has a round shape instead of plane shape it has in wider containers?

I am sure all these questions have arisen in your mind at least once. Nature is interesting; not always things are as they look. Therefore, here we are to explore its hidden aspects and explain the mysteries of micro-world.

This tutorial discusses concepts like adhesive and cohesive forces, surface tension and capillarity, which are related to the behaviour of liquids and other properties of matter we have explained earlier in the previous sections. Although such concepts are somehow bypassed by the actual school curriculum, they are very important in explaining the behaviour of liquids. Therefore, along with the explanation of these new concepts, we will also review some matter properties you already know. But as a proverb says: "Repetition is the mother of skills", isn't it?

As explained in Physics tutorial "Density. Density of Fluids", matter is made up by microscopic particles called atoms. Sometimes, two or more atoms pair together in molecules. In solids, particles are strongly bonded to each other, while in gases their bonding is very weak. Liquids are an intermediate category between solids and gases.

By definition, cohesion is the property of like particles to stick to each other. Obviously, these like particles belong to the same substance. To make possible such a mutual attraction, particles interact with each other through certain attraction forces known as cohesive forces. For example, water droplets take a spherical shape in the lower part as cohesive forces exerted by inner molecules to those in the surface, cause a shrink on water volume and as a result, all water molecules tend to gather around the centre of the droplet. On the other hand, when substances are different, they stick to each other because of adhesive forces. Thus, we can define adhesion as the ability of different substances or surfaces to cling to each other. For example, water wets the inner part of a metal container because water molecules cling to the molecules of the metal container by means of adhesive forces. Another example: water bubbles gather to the inner faces of the container because of adhesive forces of water. The existence of adhesive and cohesive forces depends in great extent by the densities of surfaces in contact. Thus, in solids (which usually have high densities), cohesive forces prevail over adhesive ones as molecules of solids are closer (and as a result, stronger bonded) to each other than to the molecules of the other substance they are in contact. On the other hand, gases show more adhesive than cohesive properties. This is why gases stick easily to other surfaces.

Liquids, as an intermediate category, manifest different behaviour depending on the surfaces they are in contact with. Thus, water (as the most widespread liquid) manifests cohesive behaviour when it is in contact with less dense materials such as plastics (most plastics have a smaller density than water, i.e. less than 1000 kg/m3). For example, when we pour some water inside a narrow tube made of plastics, water molecules stick to themselves more than to the plastics surface. As a result, the water surface curves down as shown in the figure. On the other hand, when liquids are in contact with a denser substance, they manifest more adhesive than cohesive properties. As an example, we can mention water when poured in a glass tube (glass has a density of 2500 kg/m3). In such cases, the meniscus curves up, as water tends to stick on container walls. When adhesive and cohesive forces are equal, the liquid surface is entirely horizontal.

## Surface Tension

As stated above, molecules in a liquid pull each other by means of cohesive forces. There is an equilibrium between these forces inside the liquid because any molecule inside the liquid is pulled in all direction by other similar molecules, and therefore the resultant of all cohesive forces acting on that molecule is zero.

Only on the molecules that are on the upper layer of the liquid the resultant of cohesive forces is not zero, as they do not have other molecules above to pull them. Molecules of the upper layer are pulled by other molecules below and as a result, a non-zero resultant force in the inward direction is produced. This force is known as the force of surface tension, F, and it causes liquids to shrink and take a spherical shape (as in water droplets we discussed earlier).

Do not confuse surface tension and surface tension force. Surface tension, γ is an intrinsic property of the liquid itself (like density, freezing temperature, colour, texture, etc.) It depends neither on environmental conditions nor on the material structure but only on the surface tension force F and the length L of the part of liquid surface in contact with the surrounding substance. The equation of surface tension is:

γ = F/L

Obviously, the unit of surface tension γ is [N/m]. From the above formula, we obtain for the surface tension force,

F = γ × L

### Example 1

The surface tension of water in 20°C is 72 8 mN/m (milli Newtons per metre). What is the surface tension force exerted by water if it is poured inside a narrow tube of diameter 8 mm? ### Solution 1

From theory, we know that surface tension force is calculated by the formula

F = γ × L

where γ is the surface tension (here γwater = 72.8 mN/m = 0.0728 N/m) and L is the length of the part of water surface in contact with the tube.

Since the tube is cylindrical-shaped, the above length L represents the circumference of cylinder's base, i.e.

L = C = 2 × π × r
= 2 × π × d/2
= π × d

where d is the diameter of the tube (d = 8 mm = 0.008 m).

Hence, we obtain for the surface tension force F:

F = γwater × π × dtube
= 0.0728 N/m × 3.14 × 0.008 m
= 0.0018 N

As you see, the force of surface tension is very small. However, when the surface increases this force helps heavy objects such as ships to float more easily on water due to the compactness that water molecules produce during their shrinkage.

## Capillarity

As stated in the Physics tutorial "Gas Pressure", capillarity is the phenomenon of a liquid's raise through a narrow tube due to the change in air pressure.

Mercury barometers are examples of the capillarity effect application in practice. When there is a difference in pressure above the surface of mercury in the narrow tube when we remove the air from it by creating a vacuum, mercury rises at a higher position compared to the surrounding surface in contact with air. Other examples of capillarity include the raise of water from the roots to the highest parts of a tree, the circulation of blood throughout the human body even when we are standing up, the use of straws to drink juices by removing the air in the upper part of the straw, etc.

When liquids are in narrow tubes, their level increases until the weight of the raised liquid balances the force of surface tension. This helps us calculate the height of a liquid inside a narrow tube in respect to the surrounding part of liquid whose upper surface is in contact with the atmosphere, such as in the case of the difference in heights of a mercury barometer as the one shown above.

### Example 2

What is the height of a tree if we assume its capillary tubes starting from the roots and going up to the highest leaves as uniform (they have the same thickness everywhere)? All capillary tubes are used to carry water (ρwater = 1000 kg/m3 = 103 kg/m3) and they are assumed as cylindrical shaped of base diameter equal to 0.005 mm. Take the surface tension of water equal to 72.8 mN/m and g = 9.81 m/s2. ### Solution 2

We take capillaries as the cylindrical tube shown below. As explained in theory, the balance is reached when the weight of the raised water is equal to the surface tension force. Thus,

Wwater = Fsurface tension
mwater × g = γwater × 2 × π × rb
ρwater × Vwater × g = γwater × 2 × π × rb
ρwater × Vwater × g = 2 × γwater × π × rb
ρwater × π × rb2 × hwater × g = 2 × γwater × π × rb
ρwater × rb × hwater × g = 2 × γwater
hwater = 2 × γwater/ρwater × rb × g

Given that rb = d / 2 = 0.005 mm / 2 = 0.0025 mm = 0.0000025 m = 2.5 × 10-6 m and γ = 72.8 mN/m = 0.0728 N/m = 7.28 × 10-2 N/m, we obtain

hwater = 2 × 7.28 × 10-2/103 × 2.5 × 10-6 × 9.81
= 5.94 m

Thus, the tree is 5.94 m high as this is the maximum height the water can reach when raising through the capillary tubes.

## Summary

Cohesion is the property of like particles to stick to each other. These like particles belong to the same substance. To make possible such a mutual attraction, particles interact with each other through certain attraction forces known as cohesive forces. For example, water molecules are pulled with each other through cohesive forces making water droplets take a spherical shape (occupying thus the smallest possible volume).

On the other hand, when substances are different, they stick to each other because of adhesive forces. Thus, we can define adhesion as the ability of different substances or surfaces to cling to each other. For example, water wets the inner part of a metal container because water molecules cling to the molecules of the metal container by means of adhesive forces.

In solids (which usually have high densities), cohesive forces prevail over adhesive ones as molecules of solids are closer (and as a result, stronger bonded) to each other than to the molecules of the other substance they are in contact. On the other hand, gases show more adhesive than cohesive properties. This is why gases stick easily to other surfaces.v

Liquids, as an intermediate category, show a dualistic behaviour depending on the density of the substance they are in contact with. They manifest cohesive behaviour when it is in contact with less dense materials and adhesive behaviour when they are in contact with less dense materials.

Molecules of the upper layer of a liquid are pulled by other molecules below, and as a result, a non-zero resultant force in the inward direction is produced. This force is known as the force of surface tension, F, and it causes liquids to shrink and take a spherical shape.

Do not confuse surface tension and surface tension force. Surface tension, γ is an intrinsic property of the liquid itself (like density, freezing temperature, colour, texture, etc.) It depends neither on environmental conditions nor on the material structure but only on the surface tension force F and the length L of the part of liquid surface in contact with the surrounding substance. The equation of surface tension is:

γ = F/L

Obviously, the unit of surface tension γ is [N/m]. From the above formula, we obtain for the surface tension force,

F = γ × L

The force of surface tension is very small. However, when the surface increases this force helps heavy objects such as ships to float more easily on water due to the compactness that water molecules produce during their shrinkage.

Capillarity is the phenomenon of a liquid's raise through a narrow tube due to the change in air pressure.

Examples of capillarity include: the raise of mercury in barometer columns, the raise of water from the roots to the highest parts of a tree, the circulation of blood throughout the human body even when we are standing up, the use of straws to drink juices by removing the air in the upper part of the straw, etc.

When liquids are in narrow tubes, their level increases until the weight of the raised liquid balances the force of surface tension. This helps us calculate the height of a liquid inside a narrow tube.

## Adhesive and Cohesive Forces. Surface Tension and Capillarity Revision Questions

1. The surface tension of methanol in 20° C is 22.50 mN/m. What is the radius of the tube is a methanol sample poured in this tube, produces a 1 mN force of surface tension?

1. 7 cm
2. 70 cm
3. 7 mm
4. 0.7 mm

2. What would be the value of the tube radius in a mercury barometer in normal atmospheric pressure to measure correctly the air pressure, if the air from the upper part of the tube was not removed? Take the density of mercury equal to 13 600 kg/m3 and gravitational field strength as 9.81 m/s2. Also, take the surface tension of mercury equal to 0.486 N/m. 1. 9.586 × 10-6 m
2. 9.59 mm
3. 4.793 × 10-6 m
4. 4.79 mm

3. A 0.15 g water mixed with soap produces a soap film when inserted into a rectangular frame whose one end can shift freely. As a result, the frame surface can be enlarged or reduced as needed. The frame is kept in a horizontal position. Given that the moveable side (and its opposite side as well) is 4 mm long, what is the maximum length of the other side xmax so that the soap film does still exist? Take the surface tension of soap film equal to 0.03 N/m, g = 10 m/s2.

1. 21 mm
2. 5 mm
3. 1 mm
4. 2 mm