Wetting: spreading, contact angle, Young's equation and measuring surface tension

Whether a drop spreads depends on the energy of the dry solid against a wet one

Intuition

Pour water on clean glass and it flattens into a thin film. Pour it on a waxed car and it beads up into balls. Each surface either welcomes the liquid or refuses it. The question is which is cheaper in energy: leaving the solid bare, or covering it with liquid.

The lecture turns that question into a number, the spreading parameter $S$. It is the difference in surface energy per unit area between the dry solid and a solid completely covered by a liquid film.

$S = \gamma_{SV} - (\gamma_{SL} + \gamma)$

$\gamma_{SV}$ is the solid against vapour. $\gamma_{SL}$ is the solid against liquid. $\gamma$ is the liquid against vapour. If $S < 0$, covering the solid costs energy, so you get partial wetting: a drop with a contact angle $\theta_E$. If $S > 0$, covering the solid lowers the energy, so you get total wetting: the liquid spreads into a film.

High-energy surfaces are wetted and low-energy surfaces are not

High-energy surfaces have $\gamma_{SV}$ of about 500 to 5000 mN/m. Ionic, covalent and metallic solids are examples, and glass is fairly high-energy. Most liquids wet them, because that lowers the total energy. Water on clean glass spreads into a thin film with $\theta = 0°$.

Low-energy surfaces have $\gamma_{SV}$ of about 10 to 60 mN/m. Molecular crystals, plastics and Teflon (PTFE) are examples. Few liquids wet them, because spreading would raise the energy. Liquids do not spread on a surface with a lower surface energy than their own. Water on PTFE forms a drop with $\theta \gg 0$.

Young’s equation balances three pulls at the edge of the drop

Intuition

At the edge of a drop three surfaces meet: solid and vapour, solid and liquid, liquid and vapour. Each one pulls along its own surface, like three people in a tug of war on a rope knot. The drop settles at the angle where the sideways pulls cancel.

At the three-phase line, the pull to the left must equal the pull to the right. The solid-vapour term pulls one way along the solid. The solid-liquid term pulls the other way. The liquid surface pulls at the angle $\theta_E$, so only the part $\gamma\cos\theta_E$ acts along the solid. Balancing them gives Young’s equation:

$\gamma_{SV} = \gamma_{SL} + \gamma\cos\theta_E$

The lecture then substitutes this into $S$. The $\gamma_{SL}$ terms cancel, which gives the Young-Dupre equation:

$S = \gamma_{SL} + \gamma\cos\theta_E - (\gamma_{SL} + \gamma) = \gamma(\cos\theta_E - 1)$

So $S$ depends only on the liquid’s surface tension and the contact angle. Because $\cos\theta_E \le 1$, $S$ is zero or negative for any drop that has an angle. At $\theta_E = 0°$, $S = 0$, the boundary between partial and total wetting.

Exam detail

Know the order: spreading parameter $S$, then Young’s equation (a force balance), then Young-Dupre (put Young into $S$). Be able to do the substitution yourself and say which terms cancel.

Real surfaces stick, and rough ones can repel water almost completely

Real surfaces have defects that pin the three-phase line, so you cannot measure $\theta_E$ directly. When the line advances (liquid added) you see the advancing angle $\theta_A$. When it recedes (liquid removed) you see the smaller receding angle $\theta_R$.

$\Delta\theta = \theta_A - \theta_R$

This difference is contact angle hysteresis. The lecture notes that often $\Delta\theta \sim \theta_E$, which can be frustrating.

The sacred lotus leaf shows the opposite extreme. It has double-scale roughness, $\theta_E \ge 150°$, low $\Delta\theta$ and very low contact area, about 1%. Drops roll off and carry dirt with them, so the leaf is self-cleaning. Artificial superhydrophobic surfaces need a contact angle of at least $150°$, very low hysteresis and a very low roll-off angle. The double-scale roughness traps air under the drop.

Plates and capillaries turn wetting into a measurement of surface tension

Two methods in the lecture measure $\gamma$ of a liquid.

The Wilhelmy method measures the force on a partly submerged object, such as a plate or fibre, hung from a balance. The total force is weight plus buoyancy plus the surface tension force. You know the weight. You can measure the perimeter of the object. You can make the buoyancy zero by putting the bottom edge at the liquid surface ($d = 0$). That leaves the surface tension force, which gives one of $\gamma$ and $\theta$ when you know the other.

$\text{surface tension force} = p\,\gamma\cos\theta$

$p$ is the perimeter of the contact line, $\gamma$ the surface tension and $\theta$ the equilibrium contact angle.

In capillary rise, a wetting liquid climbs a narrow tube until the pull of the surface balances the weight of the column. The height $H$ in a capillary of radius $R$ is

$H = \frac{2\gamma\cos\theta_E}{\rho g R}$

$\rho$ is the liquid density and $g \approx 9.8\ \text{m/s}^2$. A thinner tube gives a higher rise, as in the lecture photo of capillaries standing in water. The derivation is in the extra reading and not on the slide.

Pitfall

Do not read $\theta$ in the Wilhelmy force as always 0. The slide says the surface tension force gives one of $\gamma$ and $\theta$. For a fully wetting plate you assume $\theta = 0$ to get $\gamma$.

Recall

Why does Young's equation have cos theta_E and not theta_E?

Only the part of the liquid’s pull that runs along the solid counts in the force balance. That part is $\gamma\cos\theta_E$, the component of $\gamma$ in the direction of the solid surface.