# Surface tension

#### Prerequisites

One result of internal cohesion of a liquid is the tendency of the liquid to stick together at an interface with a gas (or a different liquid).  For specificity, we will describe this as a liquid-gas interface such as water-air, but it could be a liquid-liquid interface such as oil and water.

At the interface, the liquid molecules are attracted to the other liquid molecules below, but have only weak interactions with the lower-density gas. As a result, the net interaction with the rest of the liquid for a molecule on the surface is inward, pulling the surface molecules in closer until the shorter-range repulsions (atoms can't overlap) are encountered. One of the liquid molecules in the surface and its cohesive attractions are shown at the right where the molecule of interest has been colored red. Molecules in the body of the liquid are attracted by the cohesive attractions of the other liquid molecules in all directions. They tend to cancel since there are forces in all directions.

As a result of the unbalanced forces, such an interface is quite sharp and results in the liquid surface being pulled together tautly over the whole surface —  a surface tension. Essentially the molecules of the liquid would rather be completely surrounded by other molecules of that liquid, rather than gas, and so they minimize the area that is in contact with the gas. In the absence of gravity, this results in liquid blobs taking on a spherical shape since this shape has the minimum surface area (or in oil drops in water taking on a spherical shape).

Surface tension is the two-dimensional analog of  tension (1D) or pressure (3D): a "force acting in all directions" (therefore not a vector but a scalar) that can potentially pull if only one side is being considered. For a tension (1D), this means we are considering either the end of a string or how one part is pulling on the other.  For pressure (3D), this means that we are considering how a fluid exerts pressure on a wall or on another part of the fluid. For surface tension (2D), this means we are considering how a surface attaches to something else (the formation of a meniscus in a tube) or how it pulls on another part of a fluid.

If we now look down on the surface of a liquid from within the gas, we see an array of molecules. Suppose we now draw an imaginary line across the surface. The molecules at that boundary are drawn back to their side by the molecules on that side, but also drawn across by the molecules on the other side. An example molecule is shown in red. There clearly is an applied force drawing it towards the other side.

Since the force is molecule by molecule, what we expect is that the total force on a line in the surface will be proportional to the length.

This is analogous to pressure -- except pressure is repulsive and is in 3D, so it acts on 2D surfaces. Surface tension is attractive and is in 2D, so it acts on 1D lines. We therefore write the force of the fluid on the left side of the line (LFluid) pulling across the line on the fluid on the right side of the line (RFluid) as

$$F^{ST}_{LFluid \rightarrow RFluid} = \gamma L$$

where $\gamma$ (Greek "gamma") is the coefficient of surface tension (in N/m) and $L$ is the length of the line. The force is along the surface and perpendicular to the line and pulls the two edges of the surface together.

Surface tension has lots of interesting implications, such as capillary action and the Laplace Bubble law. (The latter has relevance to the danger of an enlarged heart.)

Karen Carleton and Joe Redish 10/25/11

Article 419