Key examples — free energy & phase equilibria

Where the chapter’s machinery shows up across the bookshelf.

Example 1: homogeneous nucleation of a vapour bubble

A pure liquid under tension $\Delta p = p_v - p_\infty > 0$ is thermodynamically unstable — vapour is the lower-free-energy phase. But to nucleate a vapour bubble requires creating a liquid-vapour interface, costing $4\pi R^2 \sigma$, which dominates at small $R$. The barrier $\Delta G^* = 16\pi\sigma^3/(3\Delta p^2)$ for water at room temperature with realistic tensions ($\sim 10\,\text{MPa}$) is hundreds of $k_BT$. The Boltzmann-weighted rate is so small that homogeneous nucleation is essentially forbidden — the cavitation puzzle of Cavitation Ch 1.3. The resolution is heterogeneous nucleation at pre-existing crevices, where the geometry reduces the effective surface area and barrier.

Example 2: heterogeneous nucleation at a crevice

If a small gas-filled crevice on a surface exists, the meniscus inside it has a much smaller surface-area cost than a free-floating bubble. The barrier scales as $\sin^3\theta_c / (\Delta p)^2$ where $\theta_c$ is the contact angle — and for hydrophobic crevices ($\theta_c \to 180°$) the barrier vanishes. This explains the gap between the in-principle tensile strength of water ($\sim 100\,\text{MPa}$, from the intermolecular-forces chapter spinodal) and the measured tensile strength ($\sim 1\,\text{MPa}$, governed by the worst crevice in your sample). See Cavitation Ch 2.2.

Example 3: vapour pressure inside an oscillating bubble

The pressure of vapour inside a Rayleigh–Plesset bubble depends on temperature, and the temperature depends on the polytropic compression of the gas inside. The Clausius–Clapeyron equation closes the loop: $p_v(T)$ rises exponentially in temperature, so a small temperature change during compression produces a large vapour-pressure shift, which in turn affects the next bubble-wall acceleration. This positive-feedback between $T$ and $p_v$ is what drives the violent collapse phase of sonoluminescence. See Cavitation Ch 3.2.

Example 4: MET-channel mechanotransduction

The mechanically gated MET channels in hair-cell stereocilia switch between open and closed states under tip-link tension. With the open-vs-closed energy difference linear in tip-link displacement $x$ ($\Delta G = \Delta G_0 - K_\text{TL}\,d\,x$), the open probability is exactly the Fermi function from the chapter:

Measured stereocilia data fit this sigmoid quantitatively. The width parameter $k_B T / (K_\text{TL}\,d) \approx 100\,\text{nm}$ matches the actual displacement range over which hair-cell responses go from near-zero to saturation. See Hearing Ch 4.6.

Example 5: spinodal as the in-principle tensile limit of water

If you compress a liquid below its equilibrium density (apply tension), the pressure becomes increasingly negative — until $(\partial p/\partial \rho)_T = 0$, the spinodal, where mechanical stability is lost and homogeneous nucleation becomes barrierless. For water using a Lennard-Jones-like potential (intermolecular-forces chapter), this spinodal pressure is $\sim -100\,\text{MPa}$ — roughly 1000× atmospheric. This is the bound the free-energy curvature argument places on what a perfect bubble-free sample of water could withstand. Real samples fail orders of magnitude earlier.

Example 6: polytropic gas inside the bubble — Clausius–Clapeyron at work

The vapour pressure $p_v(T)$ enters the bubble interior equation of state. Small temperature swings during oscillation drive proportionally larger swings in $p_v$, and the resulting effective gas equation of state is neither isothermal nor adiabatic but intermediate-polytropic. The Cavitation book’s bubble-contents chapter uses this combined polytropic + Clausius–Clapeyron model.

Cross-book backlinks

• Cavitation Ch 2.1 — homogeneous nucleation: the barrier from this chapter, kinetic prefactor.
• Cavitation Ch 2.2 — heterogeneous nucleation & crevices: the same barrier with geometric corrections.
• Cavitation Ch 1.2 — tensile strength in principle: spinodal as free-energy curvature limit.
• Cavitation Ch 3.2 — bubble contents: Clausius–Clapeyron + polytropic gas.
• Hearing Ch 4.6 — hair-cell transduction: MET-channel Fermi function.