3.3 Static equilibrium and the Blake threshold

Before solving the full Rayleigh–Plesset equation, it is worth understanding its static structure: the radii at which the right-hand side vanishes ($\dot R = \ddot R = 0$) and the bubble can sit at rest. These equilibria — their location, their stability, and the conditions under which they cease to exist — anchor the qualitative understanding of every dynamic problem in the rest of the book.

The most consequential feature is the Blake threshold: a critical ambient pressure below which no static equilibrium exists at all, and any bubble must grow without bound. The Blake threshold is the cleanest mathematical statement of what makes inertial cavitation begin.

The equilibrium condition

Setting $\dot R = \ddot R = 0$ in the Rayleigh–Plesset equation eliminates the inertial and viscous terms, leaving

$p_B - p_\infty - \frac{2 \sigma}{R} = 0.$

Using the gas-content form $p_B = p_v + p_{G,0} (R_0 / R)^{3\kappa}$ from the previous lesson — and for static equilibrium taking $\kappa = 1$ (isothermal, since the bubble is at rest in thermal contact with the liquid):

$p_\infty \;=\; p_v + \frac{K}{R^3} - \frac{2 \sigma}{R},$

where $K \equiv p_{G,0} R_0^3$ is a gas-content parameter that summarises the total mass of permanent gas in the bubble. $K$ is conserved as long as no diffusion across the interface happens — which it doesn’t on the timescales we care about. The equilibrium condition is therefore a single algebraic relation between $p_\infty$ and $R$, parameterised by the temperature-set quantities $p_v$ and $\sigma$ and the gas-content quantity $K$.

Stability of the equilibria

The equilibrium curve $p_\infty(R)$ has a definite shape. At small $R$ the gas-pressure term $K / R^3$ dominates and the curve falls steeply with increasing $R$; at large $R$ the surface-tension term $-2\sigma/R$ dominates and the curve rises with increasing $R$. Somewhere between the two regimes the curve has a maximum — the Blake radius.

The second derivative $\frac{d^2 p_\infty}{dR^2}\big|_{R_B}$ is negative, so the Blake radius is a maximum of the equilibrium curve. For a given ambient pressure $p_\infty < p_{\infty,\text{crit}}$ — anywhere below the maximum — two equilibria exist: one at $R < R_B$ and one at $R > R_B$. At $p_\infty = p_{\infty,\text{crit}}$ they merge; above the threshold no equilibrium exists at all.

Stability of the small-$R$ branch

To check the stability of the small-$R$ equilibrium, perturb $R$ slightly to $R + \delta R$ (with $\delta R > 0$) and ask whether the net pressure force on the wall is restoring or destabilising. From the Rayleigh–Plesset equation, the linearised dynamics about an equilibrium gives

$\rho R_0 \, \delta \ddot R \;=\; \left[\frac{dp_B}{dR} - \frac{2 \sigma}{R^2}\right]_{R_0} \cdot \delta R = -\left[\frac{dp_\infty}{dR}\right]_{R_0} \delta R.$

On the small-$R$ branch, $\frac{d p_\infty}{dR} > 0$ (the equilibrium curve is rising), so the right-hand side is negative — perturbations decay. The equilibrium is stable.

On the large-$R$ branch, $\frac{d p_\infty}{dR} < 0$, the right-hand side is positive — perturbations grow exponentially. The equilibrium is unstable.

This is a striking conclusion: any bubble in a static-equilibrium configuration is either at the small-$R$ stable equilibrium or it grows without bound and there is no equilibrium that returns it to a finite radius. There is no stable large-$R$ equilibrium for a gas bubble in liquid under any conditions.

The Blake threshold

The Blake threshold is the maximum tension a bubble can sustain in stable equilibrium:

$p_{\infty,\text{crit}} = p_v - \frac{4}{3} \sqrt{\frac{2 \sigma^3}{3 K}}.$

Three features deserve emphasis:

1. The threshold is finite and pressure-dependent. Any preexisting bubble with gas content $K$ has a definite critical tension below which it cannot survive in equilibrium. Larger bubbles (larger $K$, larger $R_B$) have higher critical tensions — they are more easily destabilised by tension. Smaller bubbles have lower critical tensions and are more stable.

2. The critical tension is set entirely by the gas content and surface tension. For a 10 μm bubble at 1 atm equilibrium pressure, $K = (1 \text{ atm}) (10 \mu\text{m})^3 \approx 10^{-13}$ J, $R_B \approx 12$ μm, and the Blake threshold is $p_v - 0.024$ atm ≈ $-0.02$ atm. Just a slight tension destabilises a moderate-sized free bubble.

3. The critical radius at threshold ($R_B$) is not the equilibrium radius $R_0$. A bubble that was in stable equilibrium at $R_0$ before tension was applied moves along the equilibrium curve toward $R_B$ as $p_\infty$ is reduced. At $p_\infty = p_{\infty,\text{crit}}$, the stable and unstable equilibria coalesce at $R = R_B$. Below threshold, no static equilibrium exists and the bubble starts to grow inertially — the onset of inertial cavitation.

Connection to the heterogeneous-nucleation picture

The Blake threshold for a single bubble of gas content $K$ is the cleaner expression of what we previously called the heterogeneous nucleation threshold in Lesson 2.3. A sample with a distribution $N(R_0)$ of preexisting bubbles will exhibit a cavitation event whenever $p_\infty$ falls below the Blake threshold of any of its constituent bubbles. The first events occur at the highest-$R_0$ (largest gas content) bubbles, whose Blake thresholds are closest to zero. Pulling tension further activates progressively smaller bubbles whose Blake thresholds are correspondingly lower.

The Blake threshold is also the cleanest characterisation of what is called the transient or inertial cavitation threshold in acoustic cavitation. A bubble driven by an oscillating sound field oscillates stably as long as the negative half-cycles of the drive stay above the Blake threshold; below threshold the bubble cannot follow the drive quasi-statically and inertial collapse begins. The full nonlinear analysis of this regime — including frequency dependence and dissipative damping that shifts the practical threshold above the static Blake value — belongs to a later chapter on driven oscillating bubbles, not yet drafted.

What we have built

The static analysis has identified the qualitative structure of the bubble’s phase space:

• One stable equilibrium at each $p_\infty > p_{\infty,\text{crit}}$, found on the small-$R$ branch of the equilibrium curve.
• A coexisting unstable equilibrium on the large-$R$ branch.
• Coalescence at the Blake radius $R_B = \sqrt{3K/2\sigma}$ when $p_\infty = p_{\infty,\text{crit}}$.
• No equilibrium at all for $p_\infty < p_{\infty,\text{crit}}$, beyond which the bubble enters inertial growth.

The next lesson moves from static to dynamic: how does the bubble actually move as $p_\infty(t)$ varies through these regimes? Some analytical regimes (Rayleigh inertial collapse, the linearised small-amplitude oscillation) admit closed-form solutions; the general case requires numerical integration. We do both.