Surface Energy and Surface Tension

Surface Tension ($\gamma$) — Force per unit length. $N/m$. Tendency to minimize surface area.
Surface Energy ($U_s$) — Work done per unit area. $J/m^2$. Numerically $\gamma = U_s$.
Molecular Origin — Net inward pull on surface molecules due to stronger cohesive forces than adhesive forces with air.
Temperature Effect — $\gamma$ decreases with increasing temperature.
Impurities — Detergents decrease $\gamma$. Highly soluble salts slightly increase $\gamma$.
Angle of Contact ($\theta$) — Angle between liquid tangent and solid inside liquid.

- Wetting ($\theta < 90^\circ$): Adhesive > Cohesive (e.g., water on glass). - Non-wetting ($\theta > 90^\circ$): Cohesive > Adhesive (e.g., mercury on glass).

Capillary Rise/Fall (Jurin's Law) — $h = \frac{2\gamma \cos\theta}{\rho g r}$. $h \propto 1/r$.
Excess Pressure ($\Delta P$)

- Liquid Drop / Air Bubble in Liquid (1 surface): $\Delta P = \frac{2\gamma}{R}$ - Soap Bubble in Air (2 surfaces): $\Delta P = \frac{4\gamma}{R}$

Work Done in Area Change — $W = \gamma \Delta A$. For a film, $\Delta A = 2 \times \text{change in area of one side}$.
Splitting Drop — $W = 4\pi\gamma R^2 (n^{1/3} - 1)$ (for $n$ small drops from one large drop).

To remember the excess pressure formulas for drops and bubbles:

Drop has Double (2) gamma: $\Delta P = \frac{2\gamma}{R}$ Bubble has Both (2 surfaces, so 2x double) gamma: $\Delta P = \frac{4\gamma}{R}$

(Think of 'D' for Drop, 'B' for Bubble. 'D' is like 'two' in 'double', 'B' is like 'both' surfaces, so double the 'double'.)