2.3 Nucleation site populations and dissolved gas

A real liquid sample contains a distribution of nucleation sites, not a single one. The cavitation threshold of the sample is set by the weakest site, but the cavitation behaviour — the rate at which bubbles appear once tension is sustained, the volume density of bubbles in the resulting bubble cloud, the spectral content of the noise — depends on the entire distribution.

This lesson introduces the standard parameterisation of the distribution and discusses the role of dissolved gas in stabilising it.

The size distribution $N(R)$

The number density function $N(R)$ gives the population of free nuclei in the liquid: $N(R) \, dR$ is the number per cubic metre of free nuclei with equilibrium radius in $[R, R + dR]$ . Both spherical free microbubbles and effective-radius equivalents for crevice gas pockets (the radius of the crevice mouth) contribute to $N(R)$ . The distribution is usually broadband — it covers several decades in $R$ , from $\sim 0.1$ μm up to tens of μm, with a power-law tail.

For most laboratory and oceanographic water samples, the distribution is well-fit by

$N(R) \;\sim\; R^{-4},$

a power law that has been measured in many independent experiments. The exponent of 4 is empirical, not derived from any first-principles theory — it appears to reflect a balance between the rate at which small nuclei are created (by turbulent breakup, dissolution of dissolved gas, etc.) and the rate at which they are destroyed (by buoyant rise, dissolution back into solution, coalescence). The overall normalisation varies enormously: ocean water at the surface has $N(10\ \mu\text{m}) \approx 10^4$ m $^{-3}$ , while degassed laboratory water can have $N(10\ \mu\text{m}) < 10^{-2}$ m $^{-3}$ — six orders of magnitude of variation in the same nominal size class.

Critical tension as a function of nucleus radius

For each nucleus, the critical tension at which it becomes unstable and releases a free bubble depends on its size and on whether it is a free microbubble or a crevice gas pocket. For free microbubbles, the stable equilibrium against dissolution requires that the bubble’s gas content satisfy

$p_g + p_v - p_\infty = \frac{2 \sigma}{R_0},$

with $R_0$ the equilibrium radius. The critical tension at which Young–Laplace stability is lost and the bubble runs away is approximately

$p_{\infty, \text{crit}} \approx p_v - \frac{4}{3 \sqrt{3}} \cdot \frac{2 \sigma}{R_0} \cdot \left(\frac{2 \sigma}{3 p_g R_0}\right)^{1/2}.$

The key qualitative feature: larger nuclei have smaller critical tensions. A 10-μm nucleus might trigger at $-0.05$ atm; a 1-μm nucleus might require $-0.5$ atm; a 0.1-μm nucleus might require $-5$ atm. The most-easily-triggered nuclei are the largest ones, even if they are vastly outnumbered by smaller ones (because $N(R) \sim R^{-4}$ ).

Cavitation susceptibility

The combination of the population $N(R)$ and the size-dependent critical tension defines a sample’s cavitation susceptibility — the function $f(p_\infty)$ giving the expected number of cavitation events per unit volume per unit time as a function of ambient pressure. Approximately:

$f(p_\infty) = \int_{R_\text{crit}(p_\infty)}^\infty N(R) \, dR,$

where $R_\text{crit}(p_\infty)$ is the smallest radius that becomes unstable at the given tension. The integration runs from this minimum size up to infinity.

Pulling water from $+1$ atm to $-0.05$ atm activates only the largest nuclei (very few per unit volume) and produces sparse cavitation. Pulling further to $-1$ atm activates a much larger swath of the population and produces dense cavitation. Pulling to $-10$ atm activates essentially all preexisting nuclei in the sample and produces saturation — every site that can release a bubble has done so.

This is the source of the engineering tradeoff: aggressive degassing and surface preparation strip out the large-radius end of $N(R)$ and shift the susceptibility curve to higher tensions, but every order of magnitude of improvement costs disproportionate effort.

The role of dissolved gas

A nucleus’s stability against dissolution into the bulk liquid depends on the concentration of dissolved gas in that bulk. Henry’s law gives the equilibrium dissolved-gas concentration as proportional to the partial pressure of the gas in any phase in contact:

$c_g = k_H \cdot p_{g,\text{gas-phase}},$

with $k_H$ Henry’s constant (for air in water at 20 °C, the saturation concentration is about 22 mg of dissolved gas per kg of water at 1 atm of air).

A free microbubble of equilibrium radius $R_0$ contains gas at pressure $p_g = p_\infty + 2\sigma/R_0$ — higher than the ambient pressure by the Laplace term. By Henry’s law, the local equilibrium concentration in the liquid surrounding the bubble is correspondingly higher. If the bulk liquid is below this concentration (undersaturated), the bubble loses gas by diffusion and shrinks; if the bulk is above (supersaturated), the bubble gains gas and grows.

For a sample in long-term equilibrium with a gas atmosphere at pressure $p_\infty^*$ , the equilibrium concentration is $c_\infty = k_H \cdot p_\infty^*$ . A free microbubble of radius $R_0$ in this sample equilibrates only if its internal gas pressure equals $p_\infty^*$ , i.e., $p_\infty + 2\sigma/R_0 = p_\infty^*$ , i.e., the bubble is at the Laplace radius for the prevailing supersaturation. Smaller bubbles dissolve away; larger bubbles grow indefinitely. The equilibrium is therefore unstable in the bulk liquid, and free microbubbles cannot persist on long timescales — they either dissolve or rise out.

This is why the bulk of the long-lived $N(R)$ population in any settled liquid sample consists of crevice gas pockets and not free microbubbles — the crevices’ geometry prevents the gas pockets from being either dissolved away (the contact-angle condition keeps the pocket from flooding) or carried out (the pocket is mechanically trapped). Free microbubbles do appear during agitation, turbulent mixing, and rapid pressure variations, but they are typically transient on minute-to-hour timescales.

Engineering implications

The dependence on $N(R)$ has three engineering consequences:

Degassing reduces cavitation susceptibility, but only down to the level of the crevice population. No amount of bulk degassing removes the crevice-trapped gas pockets that dominate long-lived $N(R)$ . To go below the crevice limit requires surface treatment of the container.
Pressurisation prior to a cavitation event dissolves any free microbubbles into the bulk solution (Henry’s law: the equilibrium dissolved gas concentration scales with ambient pressure). A few minutes of pre-pressurisation followed by rapid tension can dramatically reduce the active $N(R)$ , because the free microbubbles have all been dissolved away and the crevice pockets are sometimes flooded too. This is the basis of the Reynolds pressurisation test in cavitation tunnels.
The same sample can have different cavitation thresholds on different days — depending on how it has been recently agitated, how long it has settled, and what the local gas atmosphere has been. The variability of the threshold is the variability of $N(R)$ .