SURFACE TENSION 

When a small quantity of water is poured on a clean glass plate, it spreads in all directions in the form of a thin film. But when a small quantity of mercury is poured on the glass plate, it takes the form of a spherical drop. Similarly, if a small quantity of water is poured on a greasy glass plate, it also takes the form of small globules like mercury. This shows that the behavior of liquids is controlled not only by gravitational force (weight) but some other force also acts upon it which depends upon the nature of the surfaces in contact. If the weight of the liquid is negligible then its shape is perfectly spherical. For example, rain drops and soap bubbles are perfectly spherical. We know that for a given volume, the surface area of sphere is least. Hence may say that the free surface of a liquid has a tendency to contract to a minimum possible area.

Definition of Surface Tension

Let an imaginary line AB be drawn in any direction in a liquid surface. The surface on either side of this line exerts a pulling force on the surface on the other side. The force lies in the plane of the surface and is at right angles to the line AB. The magnitude of this force per unit length of AB is taken as a measure of the surface tension of the liquid. Thus if F be the total force acting on either side of the line AB of length l, then the surface tension is given by
                                   T = F/l
If l = 1 then T = F. Hence the surface tension of a liquid is defined as the force per unit length in the plane of the liquid surface, acting at right angles on either side of an imaginary line drawn in that surface. Its unit is Newton/meter and the dimensions
are [MT^-2].
The value of the surface tension of a liquid depends on the temperature of the liquid, as well as on the medium on the other side of the surface. It decreases with rise  in temperature and becomes zero at the critical temperature.

Example 1

There is a horizontal film of soap solution. On it a thread is placed in the form of a loop. The film is pierced inside the loop and the thread becomes a circular loop of radius R. If the surface tension of the loop be T, then what will be the tension in the thread?

Solution

Consider an element of length dl which is making an angle dq at the center.
Tdl=2Fsindθ2
T.dl=2Fdθ2   (sinθ≅θ)when θ is very small.
Tdl=FdlR
F=RT

Inter atomic Cohesive and Adhesive Forces

According to the molecular theory, matter is made of minute particles called ‘molecules’ which can remain in free state and attract each other. The force of attraction between the molecules of the same substance is called cohesive force, and that between the molecules of different substances is called adhesive force. These forces are different from the gravitational forces and do not obey the inverse-square law. If the distance between the molecules is greater than 10^-9 meter, the force of attraction between them is negligible, but within this limit the force increases very rapidly as the distance between the molecules decreases. The maximum distance (≈ 10^-9 meter) upto which two molecules attract each other is called molecular range. It is usually denoted by c.

The effects of cohesive and adhesive forces are observed in daily life. It is due to the cohesive force that two drops of a liquid when brought in mutual contact coalesce into one. It is difficult to separate two sticky plates of glass wetted with water because quite a -large force has to be applied against the cohesive force between the molecules of water. The definite shape of solid substances is also due to the cohesive force present between its molecules. In general, we cannot adhere two pieces of solid simply by pressing them together. The reason is that ordinary pressure cannot bring the molecules of the two pieces so close (≈ 10^-9 meter) that cohesive forces may becomes effective between them. But if their surfaces in contact are melted by heating the molecules in the liquid state fill up the space between the solid surfaces. Then, on cooling, the surface adhere together. This is the process to join metals by welding. However, by special machines two pieces of metal can be pressed to an extent that their molecules come within the molecular range and stick together. This is called ‘cold welding’.

Adhesive forces come into play when two different substances are brought in contact. When we pour water on a glass plate, the plate becomes wet because the molecules of water stick to the molecules of glass under adhesive forces. In order to dry the wet plate it should be wiped by a substance whose adhesive for water molecules is greater than of glass, for example rough dry cloth. Silken and nylon cloths cannot be used to dry wet glass plate because their adhesive for water is less.

Water wets the glass surface, but mercury does not:

The adhesive force between water molecules and glass molecules is greater than the cohesive force between the molecules of water. Hence when water is poured on glass, the water molecules cling with the glass molecules and the glass surface is wetted. On the other hand, the adhesive force between mercury molecules and glass molecules is less than the cohesive force between mercury molecules. Hence mercury molecules do not cling with glass molecules, that is, mercury does not wet the glass. If, however, the glass surface is greasy, then water also does not wet the glass because the adhesive force between water and grease is less than the cohesive force between water molecules themselves.

The adhesive force between oil and water is less than the cohesive force of water, but greater than the cohesive force of oil. Therefore, a water drop poured on the surface of oil contracts to take the form of a globule, while a  drop of oil poured on the surface of water spreads to a large area in the form of a thin film.

The adhesive force between ink and paper is greater than the cohesive force of ink. That is why ink sticks on paper. Writing on blackboard by chalk is also possible due to adhesive force.

Explanation of Surface-tension

Laplace explained the phenomenon of surface-tension on the basis of inter-molecular forces. We know that if the distance between two molecules is less than the molecular range c(≈ 10^-9 meter) then they attract each other, but if the distance is more than this then they attract each other, but if the distance is more than this then the attraction becomes negligible.
Therefore, if we draw a sphere of radius c with a molecule as center, then only those molecules which are enclosed within this sphere can attract, or be attracted by, the molecule at the center of the sphere. This is called ‘sphere of molecular activity’.

In order to understand the tension acting in the free surface of a liquid; let us consider four liquid molecules like A, B, C and D along with their spheres of molecular activity. The molecule D is well inside the liquid and so it is attracted equally in all directions. Hence the resultant force acting on it is zero. The sphere of molecule C is just below the liquid surface and the resultant force on it is also zero. The molecule B which is a little below the liquid surface has its sphere of molecular activity partly outside the liquid. Thus the number of liquid molecules in upper half (attracting in downward). Hence the molecule B experience a resultant downward force. The molecule A is in the surface of the liquid, so that its sphere of molecular activity is half outside the liquid and half inside. As such it experiences a maximum downward force. Thus all the molecules situated between the surface and a plane XY, distance c below the surface, experience a resultant downward cohesive force.

When the surface area of liquid is increased, molecules from the interior of the liquid rise to the surface. As these molecules reach near the surface, work is done against the (downward) cohesive force. This work is stored in the molecule in the form of potential energy. Thus the potential energy of the molecules lying in the surface is greater than that of the molecules in the interior of the liquid. But a system is in stable equilibrium when its potential energy is minimum. Hence, in order to have minimum potential energy, the liquid surface tends to have minimum number of molecules in it. In other words, the surface tends to contact to a minimum possible area. This tendency is exhibited as surface tension. Thus surface tension is explained.

Surface Energy

When the surface area of a liquid is increased, the molecules from the interior rise to the surface. This requires work against the force of attraction of the molecules just below the surface. This work is stored in the form of potential energy in the newly formed surface. Besides this, there is cooling due to the increase in surface area. Therefore, heat flows into the surface from the surroundings to keep its temperature constant and is added to its energy. Thus the molecules in the surface have some additional energy due to their position. This additional energy per unit area of the surface is called surface energy.

Relation between Surface Tension and Work done in increasing the Surface Area:

Let a  liquid film be formed between a bent wire ABC and a straight wire PQ which can slide on the bent wire without friction. As the film surface tends to contract, the wire PQ moves upward. To keep PQ in equilibrium, a uniform force F (which includes the weight of  the wire) has to be applied in the downward direction.
It is found that the force F is directly proportional to the length l of the film in contact with the wire PQ. Since there are two free surfaces of the film, we have
F∝2l   or   F=T×2l
where T is a constant called ‘surface tension’ of the liquid. Now, suppose the wire is moved downward through a small distance Dx. This results in an increase in the surface area of the film. The work done by the force F(= force ×distance) is given by
W=F×Δx=T×2l×Δx
But 2l×Δx is the total increase in area of both the surfaces of the film. Let is be ΔA.Then
W=T×ΔA
or T=WΔA
If ΔA then T = W. Then the work done in increasing the surface area by unity will be equal to the surface tension T. Hence, the surface tension of a liquid is equal to the work required to increase the surface area of the liquid film by unity at constant temperature. Therefore, surface tension may also be expressed in joule/meter².

Example 2

There are 1000 droplets of mercury of 1 mm diameter on a glass plate. Subsequently they merge into one big drop. How will the surface energy of the surface layer change? The process is isothermal. Surface tension of
mercury = 0.475 N m^-1.

Solution

Surface energy of 1000 droplets=1000(2π×0.052×10−6)×0.475
=1.49×10−5J
Volume of 1000 droplets = volume of the drop Volume of 1000 droplets = volume of the drop Volume of 1000 droplets = volume of the drop Volume of 1000 droplets = volume of the drop Volume of 1000 droplets = volume of the drop
1000×4π3(0.05)2=4π3r3  or r=10×0.5mm
Surface energy of drop==4π×0.52×10−6×0.475=1.49×10−6=1.34×10−5J
Loss in surface energy  =1.49×10−5−1.49×10−6=1.34×10−5J

Shape of Liquid Meniscus in a Glass Tube

When a liquid is brought in contact with a solid surface, the surface of the liquid becomes curved near the place of contact. The nature of the curvature (concave or convex) depends upon the relative magnitudes of the cohesive force between the liquid molecules and the adhesive force between the molecules of the liquid and those of the solid.

In the figure, water is shown to be in contact with the wall of a glass tube. Let us consider a molecule A on the water surface near the glass. This molecules is acted upon by two forces of attraction.(i) The resultant adhesive force P, which acts on A due to the attraction of glass molecules near A. Its direction is perpendicular to the surface of the glass.
(ii) The resultant adhesive force Q, which acts on A due to the attraction of neighboring water molecules. It acts towards the interior of water.

The adhesive force between water molecules and glass molecules is greater than the cohesive force between the molecules of water. Hence, the force P is greater than force Q. Their resultant R will be directed outward from water (fig.a).

In fig.(b), mercury is shown to be in contact with the wall of glass tube. The cohesive force between the molecules of mercury is far greater than the adhesive force between the mercury molecules and the glass molecules. Hence, in this case, the force Q will be much greater than the force P and their resultant R will be directed towards the interior of mercury.

The resultant force R acts on all the molecules on the surface of water or mercury. For the molecules more and more away from the wall, the adhesive force P goes on decreasing while the cohesive force Q becomes more and more vertical. Consequently, the resultant R also becomes more and more vertical. In the middle of the surface, P becomes zero and Q becomes vertical. Hence the resultant R becomes exactly vertical.

If the surface of the liquid is in equilibrium, the resultant force acting on any molecule in the surface must be perpendicular to the surface. Hence the liquid surface sets itself perpendicular to the resultant force everywhere. This is why the water surface assumes a concave shape while the mercury surface assumes a convex shape in a glass tube. In either case the resultant force in the middle is vertical and the surface there is horizontal.

Angle of Contact

When the free surface of a liquid comes in contact of a solid, it becomes curved near the place of contact. The angle inside the liquid between the tangent to the solid surface and the tangent to the liquid surface at the point of contact is called the angle of contact for that pair of solid and liquid.The angle of contact for those liquids which wt the solid is acute. It is zero for pure water and clean glass; for ordinary water and glass it is about 8^o. The liquids which do not wet the solid have obtuse angle of contact. For mercury and glass the angle of contact is 135^o. In figure (a) and (b) are shown the angles of contact q for water-glass and mercury-glass.The angle of contact for water and silver is 90^o. Hence in a silver vessel the surface of water at the edges also remains horizontal.

Pressure difference between the two sides force a Curved liquid surface

A molecule lying in the surface of a liquid is attracted by other molecules in the surface in all directions. If the surface is plane then the molecule is attracted equally in all directions. Hence the resultant, force on the molecule due to surface tension is zero. If the surface is convex, then a resultant component of all the forces of attraction acting on every molecule acts normal to the surface is directed inward. Similarly, if the surface is concave, then every molecule experiences a resultant force due to surface tension acting normally outward.

Obviously, for the equilibrium of a curved surface, there must be a difference of pressure between its two sides so that the excess pressure force may balance the resultant force due to surface tension. Hence the pressure on the concave side must be greater than the pressure on the convex side. This difference of pressure is equal to 2T/R, where T is the surface tension and R is radius of curvature of the surface.

Excess pressure inside a drop

Let us consider a spherical drop of liquid of radius R. If the drop is small, the effect of gravity may be neglected and shape may be assumed to be spherical.
If the pressure just outside the surface is P₁ and just inside the surface is P₂.
P₂ – P₁ = 2T/R
The pressure inside the surface is greater than the pressure outside the surface.
Note:
The pressure on the concave side is greater than the pressure on the convex side.
If there is an air bubble inside the liquid as shown in the figure, is single surface is formed. There is air on the concave side and liquid on the convex side. The pressure in the concave side is greater than the pressure in the convex side, by an amount 2T/R.
p2−p1=2TR

Excess Pressure Inside a Soap Bubble

Let the pressure of the air outside the bubble be P₁, within the soap solution be P¢ and that in the air inside the bubble be P₂.
p′−p1=2TR
Similarly, looking at the inner surface,
p2−p′=2TR
Adding these two equations,
p2−p1=4TR

Example 3

A soap bubble of radius r is placed on another soap bubble of radius R. What is the radius of the film separating the two bubbles?

Solution

Excess of pressure inside the first bubble = 4T/r
Excess of pressure in the second = 4T/R
So, excess of pressure on the two sides of the separating film
=4T(1r−1R)
If R′ is the radius of the film then excess of pressure inside the film is 4TR′.
4TR′=4T(1r−1R)   ⇒1R′=1r−1R