Surfaces cost energy: surface tension, work of cohesion and why drops are round

Cut a solid up small enough and almost all of it is surface

Intuition

Think of a loaf of bread. Slice it into crumbs and every crumb has a crust, so you now have far more crust than before. The bread is the same amount. Surfaces work the same way: split something into tiny pieces and the surface area explodes while the volume stays put.

Lecture 32 opens with why surfaces matter. A water strider walks on water. A desert beetle collects drops. A lotus leaf stays clean. Coatings, Goretex, Teflon, inkjet printing and lab-on-a-chip devices all depend on how liquids behave at a surface. Nanoscience studies materials with one or more dimensions in the nanometre range, where surfaces dominate and the main forces are intermolecular.

The slide’s example makes the point with numbers. See the worked example below.

Try it: worked example

A cube of side d = 1 cm is divided into 10^18 cubes whose sides are 10^6 times smaller (d = 10 nm). Compare the total surface area before and after.

Show answer and steps

Answer6 cm^2 (6 x 10^-4 m^2) before, 600 m^2 after. The area rises by a factor of one million.

1. A cube has six faces, each of area $d^2$. So $A = 6 \times d^2 = 6\ \text{cm}^2 = 6\cdot10^{-4}\ \text{m}^2$.
2. Make each side $10^6$ times smaller: $1\ \text{cm} / 10^6 = 10\ \text{nm}$.
3. The number of small cubes is $(10^6)^3 = 10^{18}$, so the volume is unchanged.
4. One small cube has area $6 \times (10\ \text{nm})^2 = 600\ \text{nm}^2$.
5. Total area: $10^{18} \times 600\ \text{nm}^2 = 600\ \text{m}^2$.
6. Compare: $600\ \text{m}^2 / 6\cdot10^{-4}\ \text{m}^2 = 10^6$. Most of the material is now at a surface.

A surface costs energy because surface molecules have lost neighbours

Picture a crowd. Someone in the middle is squeezed by people on every side. Someone on the edge is only squeezed from the inside. A molecule in a liquid is the same. This pull between like molecules is called cohesion. A molecule in the bulk is pulled equally in all directions. A molecule at the liquid surface is pulled only inwards and sideways. To make more surface you have to drag molecules up from the inside, and that takes energy.

Surface tension, $\gamma$, is the excess energy of a system because it has a surface, per unit area. It depends on how strong the intermolecular forces are. Stronger forces give a bigger $\gamma$.

Liquid	$\gamma$ (mN/m) at 20 °C	Boiling point (°C)
ethyl ether	17	34.5
acetone	24	56
ethanol	23	78
cyclohexane	25	81
water	73	100
mercury	485	357

The table shows the link: liquids that hold together more strongly also have the higher boiling points and the higher surface tensions.

Now the exact definition. Surface tension is the free energy needed to create a unit of new surface area, at constant temperature and volume.

$\gamma = \left(\frac{\partial G}{\partial A}\right)_{T,V}$

$G$ is the Gibbs free energy and $A$ the area. Making a new area $\Delta A$ takes work

$\text{work} = \gamma \cdot \Delta A$

For liquids $\gamma$ is called the surface tension (N m⁻¹). For solids it is called the surface energy (J m⁻²). The two units are the same thing, and the name is only convention. You can use $\gamma$ in equilibrium calculations where $G$ is minimised.

Surface tension is also a sideways pull you can measure as a force

Intuition

Stretch a soap film on a wire frame and it tugs back, like a thin stretched rubber sheet. The tug along each centimetre of edge is the surface tension.

The lecture’s picture is a liquid film in a U-shaped frame, closed by a movable rod of length $l$. Pulling the rod a distance $dx$ makes new surface. The work done is $\gamma \cdot l \cdot dx$, and this equals force times distance, $F \cdot dx$. So

$\text{work} = \gamma \cdot l \cdot dx = F \cdot dx \quad\Rightarrow\quad \gamma = F/l$

$\gamma$ is the force per unit length, normal to the rod, in the plane of the surface and pointing toward the liquid. That is why the units are N m⁻¹. Putting the two views together gives the unit check:

$\frac{J}{m^2} = \frac{N\,m}{m^2} = \frac{N}{m}$

Exam detail

Be ready to give $\gamma$ both ways: energy per area (J m⁻²) from $\gamma = (\partial G/\partial A)_{T,V}$, and force per length (N m⁻¹) from $\gamma = F/l$. Lecture 32’s learning outcomes name both.

Splitting a block gives the work of cohesion, and interfaces have their own tension

The work of cohesion, $W$, is the work per unit area needed to pull a material into two parts. This creates two new surfaces, and the parts end up beyond the range of their forces of interaction. For a block made of A and B joined along an A–B interface:

$W_{AB} = \gamma_{AV} + \gamma_{BV} - \gamma_{AB}$

Read it in plain words. You make an A surface and a B surface against vapour (V), which costs $\gamma_{AV} + \gamma_{BV}$. You lose the A–B interface, which gives back $\gamma_{AB}$. For a block of one material A the two new surfaces are identical and no interface is lost:

$W_{AA} = 2\gamma_{AV}$

When two liquids that do not mix touch, the free energy to expand their shared area is the interfacial tension, $\gamma_{ab}$. Interfacial tensions are usually much lower than surface tensions. The slide’s example is water against chloroform, $\gamma_{ab} = 28\ \text{mN m}^{-1}$. Surface tension of a solid or liquid against vapour is written $\gamma_{SV}$ or $\gamma_{LV}$.

Pitfall

Do not mix up $W_{AB}$ and $W_{AA}$. The factor 2 belongs to the one-material case, because both new surfaces are the same. The two-material case subtracts $\gamma_{AB}$.

Drops are round because a sphere has the least surface for its volume

A free liquid wants the smallest possible free energy $\gamma A$. For a fixed volume, a sphere has the smallest surface area. So a droplet is spherical unless gravity squashes it. The slide’s counter-example is an oblate (flattened) mercury drop. It has a larger area, because atoms have been moved from the inside of the drop to the surface, and that costs $\text{work} = \gamma \cdot \Delta A$.

Lecture 32 lists four visible effects of surface tension. Partial spreading and a contact angle at a liquid edge. Spherical droplets. Break-up of a liquid cylinder into droplets, where water’s high $\gamma$ makes a long neck at a tap. Capillary effects: capillary bridges stick fibres together by Laplace pressure, and wetting liquids rise higher in thinner capillaries. The next topic builds on the first and last of these.

Recall

Why would a drop of mercury sitting flat (oblate) have higher free energy than a spherical drop of the same volume?

The flat drop has more surface area. Each extra bit of area costs $\gamma$ per unit area, so the free energy $\gamma A$ is larger. The sphere has the least area, so the lowest energy.