3.2 Bubble contents: gas, vapour, polytropic exponent

The Rayleigh–Plesset equation derived in the previous lesson is driven on the right-hand side by $p_B(t)$ , the pressure inside the bubble. This lesson works out what $p_B$ actually is. It has two physically distinct contributions — vapour from the liquid and any permanent gas trapped inside — that combine in different ways depending on the timescales of the bubble’s motion.

The clean separation is

$p_B(t) \;=\; p_v(T_B) \;+\; p_G(t),$

where $p_v$ is the saturation vapour pressure at the bubble’s interior temperature $T_B$ and $p_G$ is the partial pressure of any permanent (non-condensable) gas — typically air dissolved in the surrounding liquid, or hydrocarbons in industrial samples.

The vapour partial pressure

Liquid water at any temperature $T$ has an equilibrium vapour pressure $p_v(T)$ — the pressure at which the rate of evaporation of water molecules from a flat liquid surface into the gas phase exactly balances the rate of condensation back from the gas. The Clausius–Clapeyron relation gives

$\frac{d p_v}{dT} = \frac{L \rho_v}{T},$

with $L$ the latent heat of vaporisation per unit mass and $\rho_v$ the saturated vapour density. For water at room temperature, $L \approx 2.45 \times 10^6$ J/kg and $p_v(20\ °\text{C}) \approx 2.34$ kPa = 0.023 atm. The vapour pressure rises rapidly with temperature — by $\sim 7$ % per Kelvin near room temperature — and reaches 1 atm at the boiling point of 100 °C.

If the bubble interior maintains thermal equilibrium with the surrounding liquid (a good approximation for slowly varying $R$ ), then $T_B = T_\infty$ and $p_v(T_B)$ is a constant throughout the bubble’s motion. The vapour contribution to $p_B$ is then a fixed offset that simply augments the gas-pressure or replaces it in the absence of any permanent gas.

The thermal-equilibrium assumption fails dramatically during rapid bubble dynamics — especially during the final stages of inertial collapse, when bubble wall speeds approach the speed of sound and the compressed gas inside reaches temperatures of thousands of Kelvin. That regime — bubble collapse, with cavitation noise and sonoluminescence as the most striking manifestation of the breakdown of thermal equilibrium — is the subject of later chapters not yet drafted. For the Rayleigh–Plesset analysis of typical-amplitude bubble oscillation, treating $p_v$ as constant is adequate.

The permanent gas pressure

The permanent gas inside the bubble — typically air dissolved from the surrounding liquid, accumulated during whatever process originally formed the bubble — has its own pressure $p_G$ . Unlike vapour, which freely evaporates into and condenses out of the bubble (so its partial pressure tracks the temperature), permanent gas can leave or enter the bubble only by diffusion across the bubble-liquid interface. Diffusion is slow: the characteristic time for dissolved-gas equilibration across a 10 μm bubble in water is many milliseconds. For bubble motions faster than that — most of the dynamics we care about — the mass of permanent gas inside the bubble can be treated as constant.

If the gas mass is constant, the ideal gas law gives

$p_G \, V = m_G \frac{k_B T_B}{m_\text{mol}} \implies p_G(t) = \frac{3 m_G k_B T_B(t)}{4 \pi m_\text{mol} R^3(t)},$

where $m_G$ is the total mass of permanent gas in the bubble and $m_\text{mol}$ is the mass of a single gas molecule. For an isothermal compression/expansion the temperature stays constant and

$p_G(t) = p_{G,0} \left(\frac{R_0}{R(t)}\right)^3,$

with $p_{G,0}$ the permanent-gas pressure at some reference radius $R_0$ . For an adiabatic compression/expansion (no heat exchange with the surrounding liquid), the gas obeys $p V^\gamma = \text{const}$ with $\gamma = c_p / c_v$ (1.4 for diatomic gases like air), giving

$p_G(t) = p_{G,0} \left(\frac{R_0}{R(t)}\right)^{3 \gamma}.$

The polytropic exponent

Real bubble dynamics sit between these limits. The thermal conduction inside the bubble has a finite characteristic time, and depending on whether bubble oscillations are slower or faster than that time the gas behaves more isothermally or more adiabatically. The standard approximation is to use a polytropic relation

$p_G(t) \, V^\kappa(t) = \text{const} \implies p_G(t) = p_{G,0} \left(\frac{R_0}{R(t)}\right)^{3 \kappa},$

with $\kappa$ between 1 (isothermal) and $\gamma$ (adiabatic). The polytropic exponent $\kappa$ depends on the bubble’s oscillation frequency relative to its thermal time constant:

$\tau_\text{th} \approx \frac{R^2}{\alpha_\text{gas}},$

with $\alpha_\text{gas}$ the thermal diffusivity of the gas (air at room temperature: $\alpha_\text{gas} \approx 2.2 \times 10^{-5}$ m²/s). For a 10 μm air bubble in water, $\tau_\text{th} \approx 5\ \mu$ s — comparable to the period of a 30 kHz acoustic drive.

The Prosperetti analysis (1977) gives a frequency-dependent effective polytropic constant $\kappa(\omega)$ that smoothly interpolates between isothermal at low frequencies ( $\omega \tau_\text{th} \ll 1$ , so $\kappa \to 1$ ) and adiabatic at high frequencies ( $\omega \tau_\text{th} \gg 1$ , so $\kappa \to \gamma$ ). For most practical analyses the simpler constant- $\kappa$ approximation is used, with $\kappa$ chosen from a look-up table based on the bubble size and the dominant drive frequency.

When the polytropic approximation fails

The polytropic relation $p_G V^\kappa =$ const breaks down in two regimes:

Mass transfer across the interface. Dissolved gas exchange between the bubble and the surrounding liquid alters the total gas mass inside the bubble on timescales much longer than the dynamics. This is the rectified diffusion effect: a bubble driven at acoustic frequencies accumulates gas during the expansion phase (when the bubble surface area is large and the gas concentration gradient pulls inward) and loses gas during compression (when the surface area is small and the gradient pushes outward). The asymmetry of the surface area between expansion and compression means net gas accumulates. The bubble grows over many cycles. A full treatment belongs in a later chapter on driven oscillating bubbles.
Strong compression with finite thermal conduction. During inertial collapse the gas inside the bubble reaches extreme temperatures and pressures within microseconds, and the assumption that the gas behaves as a single-temperature volume — implicit in the polytropic relation — breaks down. The gas develops internal temperature gradients, finite-rate chemistry begins (water-vapour dissociation, ionisation), and the simple ideal-gas treatment fails. Detailed bubble-collapse modelling (Chapters 5–6) must include the finite-thermal-conductivity gas dynamics.

For most of the bubble-dynamics regimes we care about — oscillation at moderate amplitudes, growth and collapse at modest pressure variations — the polytropic approximation with constant $\kappa$ is adequate. The interactive in the next lesson lets you adjust $\kappa$ and see how its value affects the bubble’s dynamics.

Putting it together: $p_B(t)$ in the Rayleigh–Plesset equation

For typical analyses we write

$\boxed{p_B(t) = p_v + p_{G,0} \left(\frac{R_0}{R(t)}\right)^{3 \kappa}.}$

The two parameters $p_v$ (constant, set by liquid temperature) and $p_{G,0}$ (constant, set by initial conditions and gas mass) plus the polytropic exponent $\kappa$ (set by oscillation frequency and bubble size) fully specify the internal-pressure history of the bubble in terms of the radius $R(t)$ alone. The Rayleigh–Plesset equation becomes a closed second-order ODE for $R(t)$ :

$\rho \left[R \ddot R + \frac{3}{2} \dot R^2 \right] = p_v + p_{G,0} \left(\frac{R_0}{R}\right)^{3 \kappa} - p_\infty(t) - \frac{2 \sigma}{R} - \frac{4 \mu \dot R}{R}.$