2.1. Set-up for the generation of spherical bubbles, plasma photography and oscillation tracking

The experimental arrangements for the investigation of the behaviour of spherical laser-induced cavitation bubbles in water are depicted in figure 1. We used different laser systems and detection schemes, depending on the investigation task. Small, highly spherical bubbles were generated and monitored by fs laser pulses using the set-up of figure 1(a). Their initial size was identified with the size of the laser-induced plasma and determined by photographing the plasma luminescence, whereas their oscillations were tracked by recording the scattering signal of a cw probe beam. Previous work showed excellent agreement between photographically determined R(t) curves and the predictions of spherical bubble models, including the ratio R_max/T_osc (Kroeninger et al. Reference Kroeninger, Koehler, Kurz and Lauterborn2010; Obreschkow et al. Reference Obreschkow, Tinguely, Dorsaz, Kobel, de Bosset and Farhat2013). Therefore, it is sufficient to measure T_osc to characterize spherical bubble oscillations.

Figure 1. Experimental arrangements for the investigation of the behaviour of spherical laser-induced bubbles in water. (a) Set-up with confocal adjustment of three microscope objectives enabling us to generate highly spherical bubbles, record their oscillations with a cw probe laser beam and take high-resolution images of plasma luminescence. (b) Illustration of the directly transmitted and multiply reflected parts of the probe laser beam that interfere behind large bubbles. Light scattering and interference are detected by the AC-coupled photoreceiver in (a) and recorded using a digital oscilloscope. (c) Set-up for time-resolved photography of bubble formation and shock wave emission at times up to t = 120 ns. An optically delayed frequency-doubled portion of the pump laser pulse is used for illumination. Hydrophone signals of breakdown and collapse shock waves are recorded to monitor the bubble oscillation time for each shot.

Energetic ns laser pulses were employed for time-resolved pump-probe photography of bubble wall formation and initial shock wave emission using the set-up of figure 1(c).

In figure 1(a), the laser source is a Ti:sapphire fs laser (Spectra Physics Spitfire) pumping a travelling-wave optical parametric amplifier of superfluorescence (TOPAS; Light Conversion, TOPAS 4/800) as described by Linz et al. (Reference Linz, Freidank, Liang and Vogel2016). At a wavelength of λ = 775 nm and at 1 kHz repetition rate, this laser system delivers pulses of 265 fs duration and up to 20 μJ pulse energy.

The core of the set-up for plasma photography and probe beam measurements of the subsequent bubble oscillations is a water-filled cuvette with three confocally adjusted water immersion microscope objectives (Leica HCX APO L U-V–I) built into the cuvette wall. The pump laser beam is focused into deionized and filtered (0.2 μm) water by either a $\times$40, NA = 0.8 objective with 3.3 mm working distance, or a $\times$63, NA = 0.9 objective with a working distance of 2.2 mm. The rear entrance pupil of the objectives was overfilled to create a uniform irradiance distribution corresponding to an Airy pattern in the focal plane. A cw probe laser beam (CrystaLaser, 658 nm, 40 mW) is aligned collinear and confocal with the fs-pump beam. The transmitted probe laser light is collected by a $\times$10, NA = 0.3 objective built into the opposite cell wall and imaged onto a photoreceiver (Femto HCA-S-200M-SI) connected to a digital oscilloscope (Tektronix DPO 70604). The photoreceiver is protected from the fs laser irradiation by a blocking filter.

Plasma luminescence was photographed through a third microscope objective ($\times$20, NA = 0.5, 3.5 mm working distance) that was oriented perpendicular to the optical axis of the pump and probe beams, and recorded by a digital SLR camera (Canon EOS 5D). The intermediate image formed by the $\times$20 objective and tube lens was further magnified 8 times using a Nikkor objective (63 mm/1 : 2,8). This way, we achieved a total magnification factor of 162 and a diffraction-limited spatial resolution of 0.5 μm. A confocal arrangement of all three water immersion objectives could only be achieved when the $\times$40 objective was used to focus the fs-pulses. For tighter focusing with the $\times$63 objective, the $\times$20 imaging objective had to be removed and we could only perform probe beam scattering measurements.

The absorbed fraction $E_{abs}$ of the laser energy $E_L$ was obtained from measurements of the plasma transmittance T_tra using the relation ${E_{abs}} = {E_L}(1 - {T_{tra}})$. For transmission measurements, the photoreceiver was exchanged by a calibrated energy meter, and the $\times$10 objective was replaced by a $\times$63 water immersion objective (NA = 0.9) that collected all transmitted light. Calibration accounted for light losses by reflections at optical surfaces and by absorption in the microscope objective and in water.

2.2. Single-shot recording of bubble oscillations via probe beam scattering

Figure 1(b) depicts the path of the probe laser beam through a bubble produced by a confocal pump laser pulse. The confocal adjustment guarantees that even very tiny bubbles at the optical breakdown threshold can be detected with high sensitivity (Vogel et al. Reference Vogel, Linz, Freidank and Paltauf2008; Linz et al. Reference Linz, Freidank, Liang and Vogel2016). For larger spherical bubbles, the probe beam passes perpendicularly through the bubble wall, and most of the probe laser light is transmitted through the focal region. Only a small portion is scattered or reflected at the bubble walls and interferes with the directly transmitted beam. For bubbles larger than the Rayleigh range, up to 96 % of the incident light is transmitted and 0.04 % interferes with the transmitted beam after being reflected at the rear and then the anterior bubble wall. The bias is removed by AC coupling of the photoreceiver, which had a signal bandwidth reaching from 25 kHz to 200 MHz. The coherent mixing of multiply reflected light with the transmitted beam that is shown in figure 1(b) causes a small interference modulation of the probe beam signal at the detector. Additionally, changes in the angular distribution of Mie forward scattering lead to fluctuations of the light amount transmitted through the collection aperture. These fluctuations are most pronounced for bubbles larger than the beam waist diameter but smaller than the Rayleigh range. During the oscillation of large bubbles, this leads to transient strong signal modulations shortly after the start and towards the end of each oscillation, and these modulations enable us to determine the oscillation period, T_osc (Vogel et al. Reference Vogel, Linz, Freidank and Paltauf2008). The collection NA should be chosen such that these modulations are maximized while the direct light transmission is attenuated as much as possible. In our experiments, a value of NA ≈ 0.1 provided the best results.

Due to buoyancy, small bubbles move a little upward during their oscillations although they retain a spherical shape. Nevertheless, late bubble oscillations are still detectable if the buoyancy is small enough such that the bubble stays within the probe beam focus. According to (2.1), a bubble with R_max = 100 μm moves approximately 0.63 μm during the first oscillation but much less during later oscillations, when the radius is significantly smaller. The diffraction-limited focus diameter d = λ/NA in our set-up is 0.97 μm. Thus, the bubble moves less than the focus diameter, and late oscillations can be detected. Because the light passage through the bubble becomes slightly asymmetric after the large initial oscillation during which buoyancy is strongest, the forward Mie scattering lobe will later be obliquely oriented. Therefore, both width and orientation of the central lobe change during bubble oscillations, which enhances the intensity fluctuations of the light transmitted through the collecting aperture of NA ≈ 0.1. For 30 μm < R_max < 50 μm, more than 100 oscillations could be traced in ‘lucky shots’, and radius variations well below 1 nm during late oscillations were detected.

2.3. Time-resolved photography of bubble formation and shock wave emission

Energetic 10 mJ and 20 mJ nanosecond laser pulses focused at NA = 0.25 were used to create large high-density plasmas, as shown in figure 1(c). Such plasmas enable us to visualize the bubble wall formation in the early phase of plasma expansion as well as shock-wave-induced phase transitions, which may enlarge the vaporized liquid volume and shift the bubble wall location. Bubbles were generated by a Nd:YAG laser (Continuum YG 671-10), which delivers pulses of 1064 nm wavelength with 6 ns duration at pulse energies of up to 250 mJ. The pulse energy was measured using a pyroelectric energy meter (Laser Precision Rj 7100).

A collimated and optically delayed frequency-doubled portion of the pump laser beam was used for illumination at short delay times up to 120 ns. For each shot, the bubble oscillation time was monitored by recording hydrophone signals of breakdown and collapse shock waves using a polyvinylidene fluoride (PVDF) hydrophone (Ceram) with 12 ns rise time. With the collimated illumination beam, we could not use the $\times$20 microscope objective for imaging as done in figure 1(a) because its back focal plane lies inside the objective and it can easily be damaged when the collimated laser beam is focused on an interior lens. Instead, we used an external $\times$7 macro objective (Leitz Photar) for photography that provided a spatial resolution of 2 μm.

3.1. Equations governing the cavitation bubble dynamics

We used the Gilmore model of cavitation bubble dynamics (Gilmore Reference Gilmore1952; Lauterborn & Kurz Reference Lauterborn and Kurz2010) to calculate the temporal development of the bubble radius and the pressure inside the bubble, as well as the pressure distribution in the surrounding liquid. The model considers the compressibility of the liquid surrounding the bubble, viscosity and surface tension. Sound radiation into the liquid from the oscillating bubble is incorporated based on the Kirkwood–Bethe hypothesis (Cole Reference Cole1948). The Gilmore model assumes a constant gas content of the bubble, neglecting evaporation, condensation, gas diffusion through the bubble wall and heat conduction. For strong oscillations, i.e. strong compression of the contents inside the bubble, the model is augmented by a van der Waals hard core law to account for a non-compressible volume of the inert gas inside the collapsing bubble (Löfstedt, Barber & Putterman Reference Löfstedt, Barber and Putterman1993; Lauterborn & Kurz Reference Lauterborn and Kurz2010).

The bubble dynamics is described by the equation

(3.1)\begin{equation}\left( {1 - \frac{U}{C}} \right)R\dot{U} + \frac{3}{2}\left( {1 - \frac{U}{{3C}}} \right){U^2} = \left( {1 + \frac{U}{C}} \right)H + \frac{U}{C}\left( {1 - \frac{U}{C}} \right)R\frac{{\textrm{d}H}}{{\textrm{d}R}}.\end{equation}

Here, R is the bubble radius, $U = \textrm{d}R/\textrm{d}t$ is the bubble wall velocity, an overdot means differentiation with respect to time, C is the speed of sound in the liquid at the bubble wall and H is the enthalpy difference between the liquid at pressure p(R) at the bubble wall and at hydrostatic pressure

(3.2)\begin{equation}H = \int_{p{|_{r \to \infty }}}^{p{|_{r = R}}} {\frac{{\textrm{d}p{\kern 1pt} (\rho )}}{\rho }} ,\end{equation}

whereby ρ and p are the density and pressure within the liquid, and r is the distance from the bubble centre. The driving force for the bubble motion is expressed through the difference between the pressure within the liquid at the bubble wall and at a large distance from the wall (static pressure). Assuming an ideal gas inside the bubble, the pressure P at the bubble wall is given by

(3.3)\begin{equation}P = p\left|{_{r = R} = \left( {{p_\infty } + \frac{{2\sigma }}{{{R_n}}}} \right)} \right.{\left( {\frac{{R{\kern 1pt}_n^3 - R{\kern 1pt}_{v\,\textrm{d}W}^3}}{{{R^3} - R{\kern 1pt}_{v\,\textrm{d}W}^3}}} \right)^\kappa } - \frac{{2\sigma }}{R} - \frac{{4\mu }}{R}U,\end{equation}

where σ denotes the surface tension, μ the dynamic shear viscosity and κ the ratio of the specific heat at constant pressure and volume. The symbol R_n denotes the equilibrium radius of the bubble at which the bubble pressure balances the hydrostatic pressure. The term $R_{v\,\textrm{d}W}^3 = {(b{R_n})^3}$ describes the size of the van der Waals hard core, with van der Waals radius $R_{v\,\textrm{d}W}$ and van der Waals coefficient b. The pressure is assumed to be uniform throughout the volume of the bubble. The pressure far away from the bubble is $p{|_{r \to \infty }} = {p_\infty }$. The equation of state (EOS) of water is approximated by the Tait equation, with B = 314 MPa, and n = 7 (Ridah Reference Ridah1988)

(3.4)\begin{equation}\frac{{p + B}}{{{p_\infty } + B}} = {\left( {\frac{\rho }{{{\rho_\infty }}}} \right)^n},\end{equation}

which leads to the following relationships for the sound velocity C and enthalpy H at the bubble wall:

(3.5)\begin{gather}C = \sqrt {c_\infty ^2 + (n - 1)H} ,\end{gather}

(3.6)\begin{gather}H = \frac{{n({p_\infty } + B)}}{{(n - 1){\rho _\infty }}}\left[ {{{\left( {\frac{{P + B}}{{{p_\infty } + B}}} \right)}^{(n - 1)/n}} - 1} \right],\end{gather}

with ${c_\infty }$ and ${\rho _\infty }$ denoting the sound velocity and mass density in the liquid at normal conditions. The term dH/dR in (3.1) can be derived from (3.3) and (3.6) by calculating (dH/dP) × (dP/dR). It reads

(3.7)\begin{equation}\frac{{\textrm{d}H}}{{\textrm{d}R}} = \frac{1}{{{\rho _0}}}{\left( {\frac{{{p_\infty } + B}}{{P + B}}} \right)^{1/n}} \times \left( { - 3\kappa {R^2}\left( {{p_\infty } + \frac{{2\sigma }}{{{R_n}}}} \right)\frac{{{{(R_n^3 - R_{v\,\textrm{d}W}^3)}^\kappa }}}{{{{({R^3} - R_{v\,\textrm{d}W}^3)}^{\kappa + 1}}}} + \frac{{2\sigma }}{{{R^2}}} + \frac{{4\mu U}}{{{R^2}}}} \right).\end{equation}

3.2. Description of laser-induced bubble initiation

Following Vogel et al. (Reference Vogel, Busch and Parlitz1996a), we neglect details of the breakdown process and refer only to the plasma size at the end of the laser pulse, and to the maximum radius reached by the cavitation bubble as a consequence of plasma expansion. Calculations start with a bubble nucleus with radius R ₀, whereby the volume of this nucleus is identified with the photographically determined plasma size in the liquid. At t = 0, the nucleus contains liquid water which is then heated by the laser pulse, expands and forms a bubble when temperature and pressure have dropped below the critical point. For the sake of convenience, one usually denotes the outer border of this nucleus also already as the ‘bubble wall’. The energy input during the laser pulse is simulated by raising the value of the equilibrium radius R_n from its small initial value R_n = R ₀ at the beginning of the pulse to a much larger final value R_nbd. The underlying assumption is that the absorbed laser energy is proportional to the amount of liquid vaporized by the laser pulse, which in turn is proportional to the equilibrium volume of the laser-induced bubble given by $4/3{\rm \pi} R_{nbd}^3$. This assumption holds when the energy deposited into the plasma is significantly larger than the sum of the energy needed for heating the vaporized liquid volume to the boiling temperature plus the latent heat of vaporization.

The equilibrium radius R_n is a measure of the total gas content of the cavitation bubble and does not distinguish between vapour and non-condensable gas. An increase of the R_n value beyond R ₀ implies that the pressure inside the bubble rises and that the bubble starts to expand. Iteratively, we determine the R_nbd value for which the calculation yields the same oscillation time T_{osc 1} or maximum radius R_{max 1} as determined experimentally.

While energy deposition by ultrashort laser pulses can be regarded as quasi-instantaneous, the finite duration of the laser pulse must be considered for ns breakdown. The temporal evolution of the laser power P_L during the pulse is modelled by a sin² function with duration τ_L (full-width at half-maximum) and total duration 2τ_L

(3.8)\begin{equation}{P_L} = {P_{L0}}{\sin ^2}\left( {\frac{\rm \pi}{{2{\tau_L}}}t} \right),\quad 0 \le t \le 2{\tau _L}.\end{equation}

Assuming that the cumulative volume increase of the equilibrium bubble at each time t during the laser pulse is proportional to the laser pulse energy E_L absorbed up to this time, Vogel et al. (Reference Vogel, Busch and Parlitz1996a) derived an equation for the temporal development of the equilibrium radius R_n during the laser pulse

(3.9)\begin{equation}{R_n}(t) = {\left\{ {R_0^3 + \frac{{R_{nbd}^3 - R_0^3}}{{2{\tau_L}}}\left[ {t - \frac{{{\tau_L}}}{\rm \pi}\sin \left( {\frac{\rm \pi}{{{\tau_L}}}t} \right)} \right]} \right\}^{1/3}}.\end{equation}

For laser-induced bubble formation in an incompressible liquid, the pressure discontinuity at the plasma border would be felt throughout the entire volume of the liquid once it is allowed to take effect in the simulation. The cavity wall then starts to be accelerated outward from rest, i.e. $U = 0$ at $t = 0$. By contrast, for a compressible liquid, the shock front represents an ‘event horizon’ up to which the breakdown effects are ‘felt’ by the liquid. As the plasma border is sharp, a shock front will form immediately, and the initial bubble wall velocity U ₀ equals the initial particle velocity u_p behind the shock front, whereby the shock pressure is identical with the initial plasma and bubble pressure, p_s = P.

Gilmore considered the case where the internal bubble pressure, P_i, is suddenly changed to a new constant value, which produces a finite velocity jump in an infinitesimal time. Considering only large terms in the equation of motion for the bubble wall and using the Tait equation that links H and C to P, he derived the first-order approximation

(3.10)\begin{equation}{U_0} = \int_0^H {\frac{{\textrm{d}h}}{c}} \approx \frac{H}{C} \approx \frac{{P - {p_\infty }}}{{{\rho _\infty }{c_\infty }}}.\end{equation}

The approximate expression is accurate when $|H|\ll {C^2}$, which for water corresponds to $|P_{i} - {p_\infty }|\ll 2000\ \textrm{MPa}$ (Gilmore Reference Gilmore1952). This is not sufficient for modelling laser-induced breakdown, where much larger plasma pressures may be involved. Therefore, we will present a derivation of U ₀ based on the Hugoniot curve data from Rice & Walsh (Reference Rice and Walsh1957). It yields (3.10) as first-order approximation and enables us to formulate a second-order approximation, which is accurate up to much higher pressure values.

Rice and Walsh fitted their Hugoniot curve data by the analytical expression

(3.11)\begin{equation}{u_p} = {c_1}({10^{({u_s} - {c_\infty })/{c_2}}} - 1),\end{equation}

where u_s is the shock wave velocity and the constants are c ₁ = 5190 m s⁻¹, c ₂ = 25 306 m s⁻¹ and c ₀ is the sound velocity, c_∞ = 1483 m s⁻¹. By rearranging (3.11), u_s can be expressed as a function of u_p

(3.12)\begin{equation}{u_s} = {c_\infty } + {c_2}{\log _{10}}({u_p}/{c_1} + 1).\end{equation}

Using (3.11) and the conservation of momentum at a shock front, ${p_s} - {p_\infty } = {u_s}{u_p}{\rho _\infty }$ (Duvall & Fowles Reference Duvall and Fowles1963), one can link p_s to u_s

(3.13)\begin{equation}{p_s} = {c_1}{\rho _\infty }{u_s}({10^{({u_s} - {c_\infty })/{c_2}}} - 1) + {p_\infty },\end{equation}

where ${\rho _\infty } = 998\ \textrm{kg}\;{\textrm{m}^{ - 3}}$ is the mass density of water and ${p_\infty } = {10^5}\ \textrm{Pa}$ is the hydrostatic pressure. Inserting (3.12) into (3.13), one finally obtains a relation between p_s and u_p

(3.14)\begin{equation}{p_s} = {\rho _\infty }{u_p}[{c_\infty } + {c_2}{\log _{10}}({u_p}/{c_1} + 1)] + {p_\infty }.\end{equation}

If ${u_p} \ll {c_1}$, the second term in the bracket can be dropped, which leads to

(3.15)\begin{equation}{p_s} = {\rho _\infty }{u_p}{c_\infty } + {p_\infty }.\end{equation}

Immediately after breakdown, u_p = U ₀ and p_s = P. After resolving (3.15) for U ₀, we get

(3.16)\begin{equation}{u_p} = {U_0} = (P - {p_\infty })/{\rho _\infty }{c_\infty },\end{equation}

which equals Gilmore's first-order approximation in (3.10).

Before deriving a higher-order approximation of (3.14), let us first see how we can integrate the rapid start of the bubble wall velocity during the laser pulse into the equation of motion (3.1). For this purpose, we rewrite the equation such that it describes the evolution of $\dot{U}$ and add a term ${\dot{u}_p}$ that expresses the evolution of the particle velocity at the bubble wall driven by the energy deposition during the laser pulse

(3.17)\begin{equation}\dot{U} ={-} \frac{3}{2}\frac{{{U^2}}}{R}\frac{{C - U/3}}{{C - U}} + \frac{H}{R}\frac{{C + U}}{{C - U}} + \frac{U}{C}\frac{{\textrm{d}H}}{{\textrm{d}R}} + {\dot{u}_p}.\end{equation}

The term ${\dot{u}_p}$ is derived from (3.16) as

(3.18)\begin{equation}{\dot{u}_p} = \left\{ {\begin{array}{*{20}{@{}l}} {\dfrac{{\dot{P}{\kern 1pt} }}{{{\rho_\infty }{c_\infty }}}}&{\textrm{for}\;0 \le t \le 2{\tau_L},}\\ 0&{\textrm{otherwise}\textrm{.}} \end{array}} \right.\end{equation}

The time interval $0 \le t \le 2{\tau _L}$ corresponds to the duration of the laser pulse as defined by (3.8). We shall now look at the pressure evolution. For very short pulse durations, energy deposition is inertially confined and we can neglect the bubble wall movement during the pulse and use the approximation R = R ₀. Since the fluid does not yet move, we can neglect also viscosity. Assuming κ = 4/3, we obtain from (3.3) for the time evolution of the bubble pressure during the laser pulse

(3.19)\begin{equation}P = \frac{{{p_\infty }}}{{R_0^4}}R_n^4(t) + \frac{{2\sigma }}{{R_0^4}}R_n^3(t),\end{equation}

with R_n(t) given by (3.9). The time derivative of (3.19) reads

(3.20)\begin{equation}\dot{P} = \frac{{4{p_\infty }{R_n}(t) + 6\sigma }}{{R_0^4}}R_n^2(t){\dot{R}_n}(t),\end{equation}

and the time derivative of (3.9) is

(3.21)\begin{equation}{\dot{R}_n}(t) = \frac{1}{3}R_n^{ - 2}(t)\frac{{R_{nbd}^3 - R_0^3}}{{2{\tau _L}}}\left[ {1 - \cos \left( {\frac{\rm \pi}{{{\tau_L}}}t} \right)} \right].\end{equation}

By inserting (3.21) into (3.20) one gets

(3.22)\begin{equation}\dot{P} = \frac{{2{p_\infty }{R_n}(t) + 3\sigma }}{{3{\tau _L}R_0^4}}(R_{nbd}^3 - R_0^3)\left[ {1 - \cos \left( {\frac{\rm \pi}{{{\tau_L}}}t} \right)} \right],\end{equation}

and by inserting (3.22) into (3.18) one finally obtains

(3.23)\begin{equation}{\dot{u}_p} = \frac{{2{p_\infty }{R_n}(t) + 3\sigma }}{{3R_0^4{\tau _L}{\rho _\infty }{c_\infty }}}(R_{nbd}^3 - R_0^3)\left[ {1 - \cos \left( {\frac{\rm \pi}{{{\tau_L}}}t} \right)} \right],\end{equation}

which enables us to numerically integrate (3.17).

A simulation of the bubble wall movement based on the first-order approximation is shown as dash-dotted curve in figure 2, together with simulation results not considering the ‘jump start’ of the bubble wall and the results of the second-order approximation that will be presented below. The first-order approximation yields a start velocity ${U_0} = 886\ \textrm{m}\;{\textrm{s}^{ - 1}}$, which is much higher than the particle velocity behind a shock front having a pressure equal to the bubble pressure at the end of the laser pulse. For the starting conditions of figure 2, this pressure is P_max = 1.31 GPa, and the corresponding particle velocity is only approximately $500\ \textrm{m}\;{\textrm{s}^{ - 1}}$ (Rice & Walsh Reference Rice and Walsh1957).

Figure 2. Time evolution of bubble wall velocity during the early expansion phase for three modelling approaches: (i) particle velocity behind the shock wave front is not considered (no jump start); (ii) the first-order approximation of ${\dot{u}_p}$ in (3.23) is used to consider the evolution of particle velocity during the laser pulse; (iii) the second-order approximation in (3.30) is used. For all simulations, the input parameters are R ₀ = 1.33 μm and R_nbd = 13.88 μm, which correspond to the signal in figure 9 that will later be analysed in detail.

Besides overestimating U ₀, the first-order approximation predicts a continuous drop of the bubble wall velocity after the jump start, which contradicts the physical picture of the sequence of events. Although the shock front immediately detaches from the plasma, the bubble wall continues for a while to be accelerated by the internal bubble pressure, which leads to a peak of the U(t) curve a short while after the jump start. Later, the bubble wall velocity decreases although the bubble pressure is still higher than the hydrostatic pressure because the kinetic energy imparted to the liquid is distributed among an ever-larger liquid mass.

In order to improve the accuracy of the model predictions, we go back to the relationship between P and u_p in (3.14) and formulate a second-order approximation considering the second term in the bracket through its Taylor expansion

(3.24)\begin{equation}{\log _{10}}\left( {\frac{{{u_p}}}{{{c_1}}} + 1} \right) = \frac{{{u_p}}}{{\log (10){c_1}}} - \frac{{u_p^2}}{{\log (100)c_1^2}} + \frac{{u_p^3}}{{\log (1000)c_1^3}} - \cdots .\end{equation}

If u_p is well below ${c_1} = 5190\ \textrm{m}\;{\textrm{s}^{ - 1}}$, higher-order terms of the Taylor expansion can be dropped. Keeping the first term and inserting (3.24) into (3.14), we obtain

(3.25)\begin{equation}P = {\rho _\infty }{c_\infty }{u_p} + \frac{{{\rho _\infty }{c_2}}}{{\log (10){c_1}}}u_p^2 + {p_\infty }.\end{equation}

For $P \gg {p_\infty }$, we can ignore p_∞, which enables us to formulate a quadratic equation of type $(A{x^2} + Bx - P = 0)$

(3.26)\begin{equation}\underbrace{{\frac{{{\rho _\infty }{c_2}}}{{\log (10){c_1}}}}}_{A}u_p^2 + \underbrace{{{\rho _\infty }{c_\infty }}}_{B}{u_p} - P = 0.\end{equation}

For A, B, P > 0, such equations have a positive real and a negative imaginary root

(3.27)\begin{equation}{u_p} = \frac{{ - B \pm \sqrt {{B^2} + 4AP} }}{{2A}}.\end{equation}

Inserting A and B in (3.26) into the positive root of (3.27), we obtain

(3.28)\begin{equation}{u_p} = \frac{{\sqrt {\rho _\infty ^2c_\infty ^2 + 4\frac{{{\rho _\infty }{c_2}}}{{\log (10){c_1}}}P} - {\rho _\infty }{c_\infty }}}{{\frac{{2{\rho _\infty }{c_2}}}{{\log (10){c_1}}}}}.\end{equation}

The time derivative of this equation is

(3.29)\begin{equation}{\dot{u}_p} = \frac{{\dot{P}}}{{\sqrt {\rho _\infty ^2c_\infty ^2 + \frac{{4{\rho _\infty }{c_2}}}{{\log (10){c_1}}}P} }}.\end{equation}

Equation (3.29) equals the first-order approximation result in (3.18) when the second term in the denominator is neglected. Inserting (3.22) into (3.29), we finally obtain

(3.30)\begin{equation}{\dot{u}_p} = \frac{1}{{\sqrt {\rho _\infty ^2c_\infty ^2 + \frac{{4{\rho _\infty }{c_2}}}{{\log (10){c_1}}}P} }}\frac{{2{p_\infty }{R_n}(t) + 3\sigma }}{{3R_0^4{\tau _L}}}(R_{nbd}^3 - R_0^3)\left[ {1 - \cos \left( {\frac{\rm \pi}{{{\tau_L}}}t} \right)} \right],\end{equation}

with P given by (3.19).

Numerical integration of (3.17) with ${\dot{u}_p}$ from (3.30) yields the solid curve in figure 2, with start velocity ${U_0} = 513\ \textrm{m}\;{\textrm{s}^{ - 1}}$ in good agreement with Hugoniot data, and a time evolution U(t) that corresponds well to the expected physical scenario described above. In the following, the second-order approximation of the jump start of the bubble wall velocity will be used in all numerical simulations, if not otherwise mentioned.

3.3. Acoustic and shock wave emission

The solution of (3.17) with ${\dot{u}_p}$ from (3.30) and R_n(t) from (3.9) was used to calculate the pressure distribution in the liquid surrounding the cavitation bubble (Gilmore Reference Gilmore1952; Knapp, Daily & Hammitt Reference Knapp, Daily and Hammitt1970). The calculation is based on the Kirkwood–Bethe hypothesis, which expresses that the quantity $y = r(h + {u^2}/2)$ propagates outward along a ‘characteristic’, traced by a point moving with velocity $c + u$. Here, c is the local velocity of sound in the liquid, u is the local liquid velocity and h is the enthalpy difference between liquid at pressures p and ambient pressure p_∞ (Cole Reference Cole1948). The Kirkwood–Bethe hypothesis leads to the differential equations

(3.31)\begin{gather}\dot{u} ={-} \frac{1}{{c - u}}\left[ {(c + u)\frac{y}{{{r^2}}} - \frac{{2{c^2}u}}{r}} \right],\quad \dot{r} = u + c,\end{gather}

(3.32)\begin{gather}\textrm{with}\;c = {c_\infty }{\left( {\frac{{p + B}}{{{p_\infty } + B}}} \right)^{(n - 1)/2n}}.\end{gather}

The pressure p at $r = r(t)$ is given by

(3.33)\begin{equation}p = ({p_\infty } + B){\left[ {\left( {\frac{y}{r} - \frac{{{u^2}}}{2}} \right) \cdot \frac{{(n - 1){\rho_\infty }}}{{n({p_\infty } + B)}} + 1} \right]^{n/(n - 1)}} - B.\end{equation}

Numerical solution of (3.17) and (3.31) with the bubble radius R, the bubble wall velocity U and the quantity $y = R(H + U{{\kern 1pt} ^2}/2)$ at the bubble wall as initial conditions yields the velocity and pressure distribution in the liquid along one characteristic. Solution of the equation for many initial conditions, i.e. along many characteristics, allows computation of u and p for a network of points (r, t). To determine u(r) and p(r) at a certain time, one has to collect a set of points with t = constant from this network.

When the bubble pressure is high, the pressure profiles in the liquid become steeper with time until a shock front is formed. Afterward, the calculations yield ambiguous pressure values, because they do not consider the energy dissipation at the shock front. The ambiguities have no physical meaning but simply indicate the presence of a discontinuity. The position of the shock front and the peak pressure at the front can be determined using the conservation laws for mass, impulse and energy flux through the discontinuity. As illustrated in supplementary figure S1 available at https://doi.org/10.1017/jfm.2022.202, it is defined by a vertical line in the u(r) plots cutting off the same area from the ambiguous part of the curve as that added below the curve (Rudenko & Soluyan Reference Rudenko and Soluyan1977; Landau & Lifschitz Reference Landau and Lifschitz1987). The location of the front was determined in the u(r) plots and transferred to the p(r) plots. The progressive reduction of peak pressure values going along with this procedure represents dissipation effects at the shock front, which are associated with an abrupt temperature rise (Brinkley & Kirkwood Reference Brinkley and Kirkwood1947; Cole Reference Cole1948; Rice & Walsh Reference Rice and Walsh1957; Duvall & Fowles Reference Duvall and Fowles1963; Müller Reference Müller2007).

We employed a commercial Matlab software package for the numerical integration of (3.17) and (3.31). The constants used for water at a temperature of 20 °C are: density of water ${\rho _\infty } = \textrm{ }998\ \textrm{kg}\;{\textrm{m}^{ - 3}}$, surface tension $\sigma = 0.073\ \textrm{N}\ {\textrm{m}^{ - 1}}$, adiabatic exponent for water vapour κ = 4/3, coefficient of the dynamic shear viscosity $\mu = 0.001\ \textrm{N s}\ {\textrm{m}^{ - 2}}$, velocity of sound ${c_\infty } = 1483\ \textrm{m}\;{\textrm{s}^{ - 1}}$, static ambient pressure ${p_\infty } = 100\ \textrm{kPa}$, vapour pressure ${p_v} = 2.33\ \textrm{kPa}$ and van der Waals coefficient b = 1/9. A van der Waals hard core is used in the calculations of bubble collapse but it is not needed for modelling the bubble expansion. Therefore, the van der Waals radius reads ${R_{v\,\textrm{d}W}} = 1/9{R_{nc}}$, where R_nc is the equilibrium radius of the bubble relevant for the collapse phase. It is considerably smaller than R_nbd immediately after optical breakdown because most of the water vapour produced during bubble generation condenses during the oscillation (Ebeling Reference Ebeling1978). The R_nc value is chosen such that the calculation yields the same oscillation time of the rebounding bubble, T_{osc 2}, as determined experimentally.

3.4. Choice of the adiabatic exponent

We use the room temperature value of the adiabatic exponent for water vapour, κ = 4/3, although the adiabatic exponent of water drops with increasing temperature and the temperature during breakdown and bubble collapse reaches much higher values than room temperature. This simplification is justified by the fact that part of the water molecules will dissociate for T > 3000 K (Mattsson & Desjarlais Reference Mattsson and Desjarlais2006, Reference Mattsson and Desjarlais2007), resulting in diatomic molecules such as H₂ and O₂, which have a larger adiabatic exponent of 1.4 at room temperature (Fujikawa & Akamatsu Reference Fujikawa and Akamatsu1980). Since both changes will, at least partly, compensate each other, the choice κ = 4/3 appears reasonable even for a large temperature range.

Bubble oscillation is isothermal (κ = 1) most of the time and adiabatic only during early expansion and late collapse/early rebound (Prosperetti & Hao Reference Prosperetti and Hao1999; Brenner, Hilgenfeldt & Lohse Reference Brenner, Hilgenfeldt and Lohse2002). Therefore, some researchers use a continuously changing time-dependent value of κ (Brenner et al. Reference Brenner, Hilgenfeldt and Lohse2002), and others switch from the isothermal to the adiabatic value, when the bubble radius passes the equilibrium radius (Barber et al. Reference Barber, Hiller, Lofstedt, Putterman and Weninger1997; Yuan et al. Reference Yuan, Ho, Chu and Leung2001). Nevertheless, following Lauterborn & Kurz (Reference Lauterborn and Kurz2010), we use a constant value because a variation of κ will largely complicate the tracking of energy partitioning during the bubble oscillations while it has little influence on R(t) and P(t).

3.5. Indirect consideration of condensation in the transition from nonlinear to linear bubble oscillations

During laser-induced plasma formation, liquid water within the plasma volume is vaporized and partially dissociated into gaseous products (Roberts et al. Reference Roberts, Cook, Rogers, Gleeson and Griffy1996; Mattsson & Desjarlais Reference Mattsson and Desjarlais2006; Elles et al. Reference Elles, Shkrob, Crowell and Bradforth2007; Müller et al. Reference Müller, Bachmann, Kroninger, Kurz and Helluy2009). Atomic hydrogen and oxygen will largely recombine to form water but some molecular hydrogen and oxygen remain as long-lived gaseous products (Nikogosyan, Oraevsky & Rupasov Reference Nikogosyan, Oraevsky and Rupasov1983; Barmina, Simakin & Shafeev Reference Barmina, Simakin and Shafeev2016, Reference Barmina, Simakin and Shafeev2017). Unlike for single bubble sonoluminescence (SBSL), where a sequence of many acoustically driven oscillations allows for rectified diffusion of dissolved air into the bubble (Brenner et al. Reference Brenner, Hilgenfeldt and Lohse2002), diffusion of dissolved gas into a laser-induced cavitation bubble is negligibly small (Akhatov et al. Reference Akhatov, Lindau, Topolnikov, Mettin, Vakhitova and Lauterborn2001). The degree of water dissociation depends on temperature and, thus, on the initial plasma energy density (Mattsson & Desjarlais Reference Mattsson and Desjarlais2006; Sato et al. Reference Sato, Tinguely, Oizumi and Farhat2013). Therefore, the vigour of the bubble collapse, which depends on the amount of non-condensable gas contained in the bubble, is correlated to the properties of the laser plasma.

The vapour produced during optical breakdown largely condenses during the nonlinear bubble oscillations, only the non-condensable gas remains, and finally the bubble exhibits small-amplitude linear oscillations around the equilibrium radius of the residual gas bubble, R_res. The use of different R_n values for the calculation of laser-induced bubble expansion and for its dynamics during the first collapse and later collapse events parametrizes vapour condensation during the first few oscillation cycles (Ebeling Reference Ebeling1978). The R_nbd, R_{nc 1} and R_{nc 2} values are chosen by fitting the predicted bubble dynamics to measured values of T_{osc 1}, T_{osc 2} and T_{osc 3}, respectively. After the second collapse, the R_n value is kept constant because we assume that condensation is now approximately complete and that the residual bubble is mainly filled with non-condensable gas.

A reduction of R_n between breakdown and collapse seems to contradict the assumption of adiabatic expansion and collapse implied in (3.3). In fact, expansion and collapse can be approximated as adiabatic processes only during the initial expansion and the final collapse phase. In the expanded stage, heat and mass transfers at the bubble wall resemble an isothermal scenario. It is usually assumed that at R = R_max the vapour inside the bubble is at equilibrium with the liquid outside the bubble such that the bubble pressure corresponds to the equilibrium vapour pressure at room temperature (Fujikawa & Akamatsu Reference Fujikawa and Akamatsu1980; Prosperetti & Hao Reference Prosperetti and Hao1999; Akhatov et al. Reference Akhatov, Lindau, Topolnikov, Mettin, Vakhitova and Lauterborn2001). However, the change between adiabatic and isothermal conditions hardly affects the bubble motion during the expanded stage because after the initial expansion phase, the ongoing bubble expansion is driven by inertia. Thus, condensation and heat exchange take place without any major influence on R(t) during inertially controlled oscillations but they are crucial for the final collapse phase, the collapse pressure and the rebound amplitude. Therefore, the polytropic equation (3.3) together with a reduction of R_n at the stages of maximum bubble expansion provides a realistic description of the bubble oscillation with implicit consideration of the net amount of condensation taking place during the first oscillations. The actual decrease of R_n is gradual and not stepwise as assumed in our simulations, where R_n is reduced at R_{max 1} and R_{max 2}, but the stepwise reduction does not influence the predicted dynamics.

Fujikawa & Akamatsu (Reference Fujikawa and Akamatsu1980) and Yasui (Reference Yasui1995) demonstrated that the bubble collapse is more vigorous when mass transfer by condensation and heat conduction are considered in the simulations because the reduction of the bubble's gas content by condensation reduces the buffering effect of the gas. Heat conduction reduces the temperature at collapse, which facilitates condensation and leads to higher collapse velocity and peak pressure. In our approach, the influence of both condensation and heat conduction is indirectly accounted for by fitting R_nc to match measured oscillation times.

The mass reduction of the bubble content during the collapse phase must be considered for obtaining realistic values of the collapse temperature. This is done by assuming that the collapse proceeds as adiabatic process from R = R_max starting at room temperature with a virtual bubble pressure ${p_{R\,max,virt}}$ corresponding to the amount of gas represented by R_nc. This virtual starting pressure is lower than the real pressure given by the equilibrium vapour pressure at R = R_max. For determining ${p_{R\,max,virt}}$, we must consider that R_nc refers to a bubble with internal pressure p = p_∞ at room temperature. Since ${p_{R\,max,virt}}$ also refers to room temperature, we must relate the pressure in bubbles of different size at equal temperature containing different amounts of gas, which is described by Boyle's law. That leads to

(3.34)\begin{equation}{p_{R\,max,virt}} = {p_\infty }{\left( {\frac{{{R_{nc}}}}{{{R_{max}}}}} \right)^3}.\end{equation}

For an adiabatic collapse, pressure and temperature are linked by

(3.35)\begin{equation}p_1^{(1 - \kappa )}T_1^\kappa = p_2^{(1 - \kappa )}T_2^\kappa ,\end{equation}

which provides

(3.36)\begin{equation}{T_{coll}} = 293\ \textrm{K}{\left( {\frac{{{p_{R\,max,virt}}}}{{{p_{coll}}}}} \right)^{(1 - \kappa )/\kappa }},\end{equation}

for the collapse temperature. With κ = 4/3 and by inserting (3.34) into (3.36) we get

(3.37)\begin{equation}{T_{coll}} = 293\ \textrm{K}{\left( {\frac{{{p_{coll}}}}{{{p_\infty }}}} \right)^{1/4}}{\left( {\frac{{{R_{max}}}}{{{R_{nc}}}}} \right)^{3/4}}.\end{equation}

This approach provides an upper estimate of the collapse temperature, as heat conduction is neglected.

At a later stage, when it exhibits small-amplitude linear oscillations, the bubble is filled mostly with non-condensable gas. It originates largely from water dissociation in the laser plasma; Akhatov et al. (Reference Akhatov, Lindau, Topolnikov, Mettin, Vakhitova and Lauterborn2001) showed that rectified diffusion of dissolved air into the laser-induced cavitation bubble is negligibly small. Besides the non-condensable gas, a small fraction of vapour will also be present in the residual bubble. Its amount is given by the equilibrium pressure corresponding to the temperature of the liquid at the bubble wall. The linear resonance frequency of the residual bubble reads as (Lauterborn & Kurz Reference Lauterborn and Kurz2010)

(3.38)\begin{equation}{\nu _0} = \frac{1}{{2{\rm \pi}{R_{nres}}\sqrt {{\rho _\infty }} }}\sqrt {3\kappa \left( {{p_\infty } + \frac{{2\sigma }}{{{R_{nres}}}} - {p_v}} \right) - \frac{{2\sigma }}{{{R_{nres}}}} - \frac{{4{\mu ^2}}}{{{\rho _\infty }R_{nres}^2}}} .\end{equation}

Measurement of the bubble oscillation time at late stages yields ${\nu _0}$, and by inserting this value into (3.38), ${R_{nres}}$ can be determined with high precision. Comparison of the radius ${R_{nres}}$ of the residual gas bubble and R_nc then enables us to discriminate between the gas and vapour content at the first bubble collapse.

3.6. Energy balance for laser-induced bubble formation and oscillations

Already decades ago theoretical studies have shown that, during the expansion of large bubbles driven by underwater explosions or induced by optical breakdown, the largest part of the initial energy is radiated away as a shock wave and degraded into heat by dissipative processes as the wave propagates outward (Cole Reference Cole1948; Ebeling Reference Ebeling1978). A smaller fraction of the initial energy remains as bubble energy, and upon collapse and rebound of the bubble the largest part of the remaining energy is again radiated away acoustically.

Research in the 1980s and 1990s (Vogel & Lauterborn Reference Vogel and Lauterborn1988; Vogel et al. Reference Vogel, Busch and Parlitz1996a; Vogel et al. Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999b) focused on an experimental investigation of energy partitioning for millimetre-sized bubbles by measuring the bubble's potential energy

(3.39)\begin{equation}{E_{pot}} = (4/3){\rm \pi} R_{max}^3({p_\infty } - {p_v}),\end{equation}

and shock wave energy

(3.40)\begin{equation}{E_{SW}} = \frac{{4{\rm \pi} R_m^2}}{{{\rho _\infty }{c_\infty }}}\int {p_s^2\,\textrm{d}t} ,\end{equation}

with R_m denoting the distance of the measurement location from the emission centre. Measurement of the temporal shock wave profile needed for the determination of E_SW is challenging (Vogel & Lauterborn Reference Vogel and Lauterborn1988; Tinguely et al. Reference Tinguely, Obreschkow, Kobel, Dorsaz, de Bosset and Farhat2012; Lauterborn & Vogel Reference Lauterborn and Vogel2013), especially close to the source. However, both near- and far-field data are needed to assess the total emitted shock wave energy and the rate of energy dissipation upon wave propagation (Cole Reference Cole1948; Vogel et al. Reference Vogel, Busch and Parlitz1996a; Vogel et al. Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999b). One way out was to determine near-field pressure profiles by numerical calculations using the Gilmore model and far-field profiles by hydrophone measurements (Vogel et al. Reference Vogel, Busch and Parlitz1996a). Another alternative was to determine the energy dissipation from the decay of shock wave pressure with propagation distance that was obtained by measuring u_s(r) (Vogel et al. Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999b). These investigations revealed that up to a distance of 10 times the plasma radius, 80 %–90 % of the initial shock wave energy is dissipated.

Tinguely et al. (Reference Tinguely, Obreschkow, Kobel, Dorsaz, de Bosset and Farhat2012) established an energy balance of bubble collapse and rebound by identifying the emitted shock wave energy with the difference of bubble energies before collapse and after rebound (i.e. at R_{max 1} and R_{max 2}). Considering the change of internal energy ΔU_int arising from the work done by the liquid on the gas in the bubble between the two stages, the shock wave energy is ${E_{SW}} = E_{pot}^{max1} - E_{pot}^{max2} - \Delta {U_{int}}$. Experimentally, static pressure and gas pressure were varied in the ranges ${p_{stat}} \in [1,100]\ \textrm{kPa}$ and ${p_{R\,max1}} \in [1,100]\ \textrm{Pa}$, respectively. It turned out that ΔU_int was negligible (<1 %) for the range of parameters investigated, which justified the approximation ${E_{SW}} \approx E_{pot}^{max1} - E_{pot}^{max2}$. The above approach is adequate for large bubbles but too simple for ${R_{max}} \to 0$, where viscous damping and surface tension must be considered. Moreover, it neglects the energy flow by water vaporization and condensation, and provides no information on the energy partitioning between shock wave emission and bubble formation after breakdown, Finally, it would be interesting to track the energy flow through the collapse phase itself, distinguishing between the energy stored in the compressed bubble content and in the liquid surrounding the bubble. In the following, we present a complete treatment of the energy flow and partitioning for laser-induced bubbles based on the Gilmore model.

3.6.1. Overview over energy partitioning

Figure 3 presents a flow diagram for the partitioning of the absorbed laser energy, E_abs. During optical breakdown, the energy absorbed in the plasma volume ${V_P} = (4/3){\rm \pi} R_0^3$ partitions into vaporization energy

(3.41)\begin{equation}{E_v} = {\rho _\infty }(4/3){\rm \pi} R_0^3[{C_p}({T_2} - {T_1}) + {L_V}],\end{equation}

and an internal energy gain ΔU_int of the heated, pressurized gas volume. Here, T ₁ and T ₂ denote the room temperature (20°) and boiling temperature of water (100 °C), respectively, C_p = 4187 J (K kg)⁻¹ is the isobaric heat capacity of water at 20 °C and L_V = 2256 kJ kg⁻¹ is the latent heat of vaporization at 100 °C. An equation for ΔU_int will be given further below.

Figure 3. Energy partitioning pathways for laser-induced cavitation bubbles separated into four phases: bubble expansion, first collapse, first rebound and afterbounces. The respective phases are indicated by the superscripts $exp$, $coll$, $reb$ and $res$ in the symbols denoting the energy fractions. Solid arrows indicate the conversion of energy fractions that enter the next bubble oscillation phase, which include the vaporization energy ${E_v}$, the internal energy ${U_{int}}$ of the bubble content and the potential energy ${E_{pot}}$ of the expanded or collapsed bubble. The potential energy is, at each stage, given by the work done against (or by) hydrostatic pressure, ${W_{stat}}$, plus the work done against (or by) surface tension, ${W_{surf}}$. The dashed arrows represent energy dissipation in each phase via viscous damping, ${W_{visc}}$, vapour condensation, ${E_{cond}}$ and shock wave emission, ${E_{SW}}$. The rebound shock wave emission is driven partly by the internal energy of the collapsed bubble, and partly by the energy stored in the liquid compressed during the collapse phase, $E_{compr}^{coll1}$. Correspondingly, the rebound shock wave energy is composed of two fractions: ${E_{SWB}}$ arising from the bubble rebound, and ${E_{SWL}}$ arising from the re-expansion of the compressed liquid. A complete energy balance can be established only at particular times, when the kinetic energy is zero, i.e. at R_{max 1}, R_{min 1} and R_{max 2}. The energy of the shock wave emitted after optical breakdown is obtained by subtracting the balance established for R = R_{max 1} from ${E_{abs}}$, and the energy of the shock wave emitted during the rebound is evaluated by comparing the balance for R = R_{max 2} with the total energy of the compressed bubble and liquid upon collapse.

The expanding bubble content does work, W_gas, on the surrounding liquid, and the internal energy decreases accordingly. The index ‘gas’ refers here to both water vapour and the non-condensable gas produced by plasma-mediated water dissociation. The total energy involved in the bubble oscillation is

(3.42)\begin{equation}{E_{abs}} = {E_v} + \Delta {U_{int}}(t) + {W_{gas}}(t).\end{equation}

For isochoric energy deposition with ultrashort laser pulses, ${E_{abs}} = {E_v} + \Delta {U_{int}}$, and the work on the liquid starts only after the end of the laser pulse, when the energy of the free electrons in the laser plasma has been thermalized. In the general case, however, conversion of ΔU_int into W_gas starts already during the laser pulse. During bubble expansion, the gas does work on the liquid, whereas during collapse the inrushing liquid does work on the bubble content.

During bubble expansion, the gas compresses the surrounding liquid and overcomes the liquid viscosity, the hydrostatic pressure p_∞ and the pressure p_surf arising from the surface tension at the bubble wall. In doing this, it creates kinetic energy E_kin of the accelerated liquid, potential energy E_pot of the expanding bubble, drives the emission of a shock wave with energy E_SW and does the work W_visc by overcoming viscous damping. Altogether, the work done on the liquid involves the components

(3.43)\begin{equation}{W_{gas}} = E_{SW}^{bd} + {E_{kin}} + {W_{visc}} + {E_{pot}},\quad \textrm{with}\;{E_{pot}} = {W_{stat}} + {W_{surf}},\end{equation}

where W_stat and W_surf denote the work done against hydrostatic pressure and surface tension.

At R = R_max, the kinetic energy is zero and the potential energy reaches its maximum value $E_{pot}^{max1}$. At this stage, the energy of the breakdown shock wave, $E{\kern 1pt} _{SW}^{bd}$, can be obtained by evaluating all other terms in (3.42) and (3.43) and subtracting them from E_abs.

The vaporization energy E_v is needed for the phase transition itself and cannot be converted into mechanical energy of shock wave emission and bubble oscillation. It is released during the bubble oscillations by condensation of vapour at the bubble wall and heat conduction into the surrounding liquid. Besides from latent heat, the energy dissipated by condensation originates also from internal energy stored in the vapour. The total amount lost during expansion is $E_{cond}^{exp}$. The release of latent heat can be assessed by comparing the amount of vapour from the liquid in the plasma volume with the amount contained in the expanded bubble at R_{max 1}.

During collapse, the potential energy of the expanded bubble partitions into energy needed to overcome liquid viscosity, a part $E_{compr}^{coll1}$ consumed for compression of the liquid surrounding the bubble and another part increasing the internal energy during the compression of the bubble content. That increase of internal energy is, however, counteracted by losses from vapour condensation. Together with the latent heat released during condensation, it constitutes the energy fraction $E_{cond}^{coll1}$. The release of latent heat is assessed by comparing the amount of vapour in the bubble at R_{max 1} with the amount contained in a bubble with equilibrium radius R_{nc 1}. We can distinguish between water vapour and non-condensable gas content of the bubble at first collapse by assuming that the residual bubble contains mostly non-condensable gas and that all vapour is condensed at second collapse. The gas content of the residual bubble is obtained by evaluating the late bubble oscillations with the help of (3.38), and the vapour content at first collapse is given by the volume difference R_{nc 1} and R_{nc 2} = R_nres.

The energy partitioning during rebound resembles the events after optical breakdown, with one crucial difference: shock wave emission is now driven not only by the re-expanding bubble content but also by the re-expansion of the compressed liquid surrounding the bubble. Therefore, a much larger energy fraction is radiated away acoustically during rebound than after optical breakdown. The parts of the rebound shock wave energy, $E{\kern 1pt} _{SW}^{reb}$, which originate from the re-expansion of the bubble and liquid are denoted E_SWB and E_SWL, respectively.

During later bubble oscillations (‘afterbounces’), shock wave emission turns into linear acoustic emission. We treat the energy of the acoustic waves emitted after second collapse and during later oscillations as one entity, denoted as E_acoust. Finally, a small bubble remains that contains non-condensable gas and vapour at equilibrium conditions. Since the potential energy of this bubble is zero, its energy is solely given by the residual internal energy $U_{int}^{res}$. As already mentioned, the residual bubble with radius R_nres contains mostly non-condensable gas and little vapour. The vapour content is determined by the equilibrium vapour pressure at the temperature of the liquid at the bubble wall. We will see in § 4.2.3 that this temperature is significantly larger than room temperature due to heat dissipation from the bubble content and at the front of the outgoing shock waves. However, for simplicity, we assume room temperature of the residual bubble's content when we establish the energy balance.

In the following, we provide a step-by-step account of energy partitioning.

3.6.2. Changes of internal energy and latent heat released into the liquid

For a spherical oscillating bubble containing an ideal gas under adiabatic conditions, the change of internal energy from state 1 (p_{gas 1}, V ₁) to state 2 (p_{gas 2}, V ₂) is

(3.44)\begin{equation}\Delta {U_{int}} = \frac{{4{\rm \pi}}}{{3(\kappa - 1)}}({p_{gas2}}R_2^3 - {p_{gas1}}R_1^3).\end{equation}

The energy absorbed during plasma formation is deposited into a small volume with equivalent spherical radius R ₀ that is regarded as initial size of the laser-induced cavitation bubble. We assume a pressure

(3.45)\begin{equation}{p_{gas|R = {R_0}}} = {p_\infty } + 2\sigma /{R_0},\end{equation}

for the bubble nucleus at time t ₀ = 0 before the laser pulse, and express the time-dependent gas pressure during the pulse through the time evolution of equilibrium radius R_n as

(3.46)\begin{equation}{p_{gas}}(t) = \left( {{p_\infty } + \frac{{2\sigma }}{{{R_n}(t)}}} \right){\left( {\frac{{R_n^3(t)}}{{{R^3}(t)}}} \right)^\kappa }.\end{equation}

The internal energy of the bubble at time t during bubble expansion is then given by

(3.47)\begin{equation}\Delta U_{int}^{exp}(t) = \frac{{4{\rm \pi}}}{{3(\kappa - 1)}}\left\{ {\left( {{p_\infty } + \frac{{2\sigma }}{{{R_n}(t)}}} \right)\left( {\frac{{R_n^{3\kappa }(t)}}{{{R^{3\kappa }}(t)}}} \right){R^3}(t) - \left( {{p_\infty } + \frac{{2\sigma }}{{{R_0}}}} \right)R_0^3} \right\}.\end{equation}

Since the second term in (3.47) is very small, we will neglect it in the following.

For the bubble collapse and subsequent bubble oscillations, the van der Waals hard core must be considered. In the numerical integration of (3.17) and in the calculation of ${U_{int}}(t)$, it is introduced at the time corresponding to R = R_{max 1}. After deleting the second term in (3.47) and considering the van der Waals hard core, it reads

(3.48)\begin{equation}\Delta U_{int}^{coll1}(t) = \frac{{4{\rm \pi}}}{{3(\kappa - 1)}}\left\{ {\left( {{p_\infty } + \frac{{2\sigma }}{{{R_n}(t)}}} \right){{\left( {\frac{{R_n^3(t) - R_{v\,\textrm{d}W}^3}}{{{R^3}(t) - R_{v\,\textrm{d}W}^3}}} \right)}^\kappa } \times ({R^3}(t) - R_{v\,\textrm{d}W}^3)} \right\}.\end{equation}

In order to quantify the loss of internal energy of the bubble by condensation of vapour at the bubble wall during bubble expansion and collapse we first define the internal energy at the time of maximum bubble expansion for which equilibrium conditions at ambient temperature, i.e. isothermal conditions are assumed

(3.49)\begin{equation}U_{int}^{max1}{|_{p = {p_v}}} = \frac{{4{\kern 1pt} {\rm \pi}}}{{3(\kappa - 1)}}{p_v}R_{max1}^3 = 4{\rm \pi}{p_v}R_{max1}^3.\end{equation}

The internal energy loss during bubble expansion is given by the difference between the energy of an adiabatically expanding bubble at R_max (which is obtained by evaluating (3.47) at the time of maximum bubble expansion for R_n = R_nbd) and the energy corresponding to isothermal conditions at R_max from (3.49)

(3.50)\begin{equation}\Delta U_{int,cond}^{exp} = {U_{int}}{|_{{R_n} = {R_{nbd}}}} - U_{{\mathop{\rm int}} }^{max1}{|_{p = {p_v}}}.\end{equation}

In a similar fashion, the internal energy lost during bubble collapse is calculated by subtracting the energy of an adiabatically collapsing bubble with gas content corresponding to the equilibrium bubble radius at collapse (which is obtained by evaluating (3.48) for R_n = R_{nc 1}) from the energy corresponding to isothermal conditions at R_{max 1}

(3.51)\begin{equation}\Delta U_{int,cond}^{coll1} = U_{int}^{max1}{|_{p = {p_v}}} - {U_{int}}{|_{{R_n} = {R_{nc1}}}}.\end{equation}

The changes of internal energy during rebound and second collapse are

(3.52)\begin{equation}\Delta U_{int,cond}^{reb} = {U_{int}}{|_{{R_n} = {R_{nc1}}}} - U_{int}^{max2}{|_{p = {p_v}}},\end{equation}

and

(3.53)\begin{equation}\Delta U_{int,cond}^{coll2} = \Delta U_{int,cond}^{res} = U_{int}^{max2}{|_{p = {p_v}}} - {U_{int}}{|_{{R_n} = {R_{nc2}}}},\end{equation}

respectively. We assume that condensation is complete after second collapse such that a constant amount of residual internal energy remains.

The internal energy lost by condensation is dissipated as heat into the surrounding liquid. In addition, we need to look at the latent heat released into the liquid. For this purpose, we express the amount of vapour contained in the bubble at different instants in time (directly after the laser pulse, at R_{max 1}, R_{min 1} and R_{max 2}) through the radius of a vapour bubble at room temperature with pressure 0.1 MPa. The vapour bubble radius after breakdown is calculated considering conservation of mass during vaporization of the liquid in the plasma volume. With

(3.54)\begin{equation}{\rho _\infty }{\textstyle{4 \over 3}}{\rm \pi} R_0^3 = {\rho _v}{\textstyle{4 \over 3}}{\rm \pi}{(R_v^{bd})^3},\quad \textrm{we}\ \textrm{obtain}\;R_v^{bd} = {R_0}{({\rho _\infty }/{\rho _v})^{1/3}}.\end{equation}

The mass density of vapour at p_v = 0.1 MPa and T = 20 °C is ${\rho _v} = 0.761\ \textrm{kg}\;{\textrm{m}^{ - 3}}$. The amount of vapour in the expanded bubble can be assessed by assuming that the vapour pressure at R_{max 1} and R_{max 2} equals the equilibrium vapour pressure at room temperature, p_v = 2.33 kPa (Lauterborn & Kurz Reference Lauterborn and Kurz2010). The corresponding bubble radii for vapour at ambient pressure are then

(3.55)\begin{equation}R_v^{max\,i} = {R_{max\,i}}{({p_v}/{p_\infty })^{1/3}},\quad \textrm{with}\;i = 1\ \textrm{and}\;2.\end{equation}

The loss of latent heat by condensation during bubble expansion is given by

(3.56)\begin{equation}\Delta E_v^{exp} = {E_v} - E_v^{max1},\quad \textrm{with}\;E_v^{max1} = {\left( {\frac{{R_v^{max1}}}{{R_v^{bd}}}} \right)^3}{E_v},\end{equation}

and E_V from (3.41). Note, that this does not include the loss of internal energy by condensation, which will be presented in the next section. The energy transfer during the first collapse is

(3.57a,b)\begin{equation}\Delta E_v^{coll1} \!=\! E_v^{max1} - E_v^{coll1},\quad \textrm{with}\;E_v^{coll1} = {\left( {\frac{{R_v^{coll1}}}{{R_v^{bd}}}} \right)^3}{E_v}\quad \textrm{and}\quad R_v^{coll1} \!=\! {(R_{nc1}^3 - R_{nc2}^3)^{1/3}}.\end{equation}

The calculation of $R_v^{coll1}$ is based on the assumption that at second collapse the condensation process is completed and the bubble contains only non-condensable gas. The latent heats released during rebound and second collapse are given by

(3.58)\begin{equation}\Delta E_v^{reb} = E_v^{coll1} - E_v^{max2},\quad \textrm{with}\;E_v^{max2} = {\left( {\frac{{R_v^{max2}}}{{R_v^{bd}}}} \right)^3}{E_v},\end{equation}

and

(3.59)\begin{equation}\Delta E_v^{coll2} = E_v^{max2},\end{equation}

respectively. The latent heat remaining at R_{max 2} is dissipated during the transition into linear bubble oscillations. For each stage, the total condensation loss E_cond is the sum of internal energy loss and release of latent heat

(3.60)\begin{equation}{E_{cond}} = \Delta {U_{int,cond}} + \Delta {E_v}.\end{equation}

3.6.3. Work done by the gas, and shock wave energies

The work done by the gas during the bubble oscillations is given by

(3.61)\begin{equation}{W_{gas}}(t) = \int {{p_{gas}}(t)\,\textrm{d}V} = \int {4{\rm \pi}\,{R^2}{p_{gas}}(t)\,\textrm{d}R} .\end{equation}

For numerical integration, this relation must be rewritten as time integral. By transforming the differential variable from dR to dt’ using dR = (dR/dt′) × dt′ = U(t′)dt′, we obtain

(3.62)\begin{equation}{W_{gas}}(t) = \int_0^t {4{\rm \pi} R{{(t^{\prime})}^2}U(t^{\prime}){p_{gas}}(t^{\prime})\,\textrm{d}t^{\prime}} ,\end{equation}

where p_gas is given by (3.46).

The work W_gas done on the liquid during bubble expansion (or by the liquid on the collapsing bubble) is composed of W_stat, W_surf, and W_visc, as expressed by (3.43). The work done at a given time to overcome the hydrostatic pressure is

(3.63)\begin{equation}{W_{stat}}(t) = \int {{p_\infty }\,\textrm{d}V} = \int {4{\rm \pi}{R^2}{p_\infty }\,\textrm{d}R} = \int_0^t {4{\rm \pi}R{{(t^{\prime})}^2}U(t^{\prime}){p_\infty }\,\textrm{d}t^{\prime}} .\end{equation}

The work done to overcome the surface tension is

(3.64)\begin{equation}{W_{surf}}(t) = \int {{p_{surf}}\,\textrm{d}V} = \int {4{\rm \pi}{R^2}{p_{surf}}\,\textrm{d}R} = 8{\rm \pi}\sigma \int_0^t {R(t^{\prime})U(t^{\prime})\,\textrm{d}t^{\prime}} ,\end{equation}

and the potential bubble energy at any given time is therefore

(3.65)\begin{equation}{E_{pot}}(t) = {W_{stat}}(t) + {W_{surf}}(t).\end{equation}

Finally, the work required to overcome viscosity is

(3.66)\begin{equation}{W_{visc}}(t) = \int {{p_{visc}}\,\textrm{d}V} = \int {4{\kern 1pt} {\rm \pi}{R^2}{p_{visc}}\,\textrm{d}R} = 16{\rm \pi}\mu \int_0^t {R(t^{\prime})U{{(t^{\prime})}^2}\,\textrm{d}t^{\prime}} .\end{equation}

The energy balance for the conversion of the absorbed energy E_abs during the expansion process is obtained by numerical integration of the above equations up to the time t_{max 1}

(3.67)\begin{equation}{E_{abs}} = \underline {E_{SW}^{bd}} + \underline {E_{cond}^{exp\,}} + \underline {{W_{visc}}} + E_{pot}^{max1} + U_{int}^{max1}{|_{p = {p_v}}} + E_v^{max1}.\end{equation}

The underscored terms are energy parts that are dissipated during expansion, the other terms remaining at t = t_{max 1} are parts of the total bubble energy $E_B^{max1}$, which is the sum of the bubble's internal and potential energy. The parts of $E_{pot}^{max1}$related to hydrostatic pressure and surface tension are given by (3.59) and (3.60). Note that the energy change by condensation in (3.63) refers only to the part related to the change of internal energy; the loss of latent heat by condensation is already covered by (3.54). The breakdown shock wave energy, $E{\kern 1pt} _{SW}^{bd}$, cannot be calculated directly but it can be determined from (3.63) since all other terms are known.

The energy balance for the collapse process starts with the energy remaining at R_{min 1} and tracks their dissipation and conversion during collapse. It is obtained by numerical integration of (3.58)–(3.62) up to t = t_{min 1}

(3.68)\begin{align}E_B^{max1} = E_{pot}^{max1} + U_{int}^{max1}{|_{p = {p_v}}} + E_v^{max1} = \underline {E_{cond}^{coll1}} + \underline {{W_{visc}}} + E_{compr}^{coll1} + U_{int}^{min1} + E_v^{coll1}.\end{align}

The underscored terms represent energy parts that are dissipated during collapse, the other terms remaining at t = t_{min 1} describe parts of the total amount of energy contained in the compressed liquid and gas, $E_{compr}^{total}$. The energy stored in the compressed liquid, $E_{compr}^{coll1}$, cannot be calculated directly but it can be obtained from (3.64) because the other terms are known.

The energy balance for the rebound starts with the energy contained in the compressed bubble and liquid and tracks its dissipation and conversion during re-expansion. The balance at R_{max 2} is determined by numerical integration up to t = t_{max 2}

(3.69)\begin{align}E_{compr}^{total} = E_{compr}^{coll1} + U_{int}^{min1} + E_v^{coll1} = \underline {E_{SW}^{reb}} \!+\! \underline {E_{cond}^{reb}} + \underline {{W_{visc}}} \!+\! E_{pot}^{max2} \!+\! U_{int}^{max2}{|_{p = {p_v}}} + E_v^{max2}.\end{align}

The total rebound shock wave energy, $E{\kern 1pt} _{SW}^{reb}$, is determinable from (3.69), since all other terms are known. We can even distinguish between the parts of the shock wave energy originating from the re-expansion of the compressed bubble with energy $U_{int}^{min1}$ and the compressed liquid with energy $E_{compr}^{coll1}$, which are denoted ${E_{SWB}}$ and ${E_{SWL}}$, respectively,

(3.70a,b)\begin{equation}{E_{SWL}} = E_{compr}^{coll1}\quad \textrm{and}\quad {E_{SWB}} = E_{SW}^{reb} - {E_{SWL}}.\end{equation}

In order to obtain the energy balance for the afterbounces, the numerical integration is conducted up to a time at which the bubble oscillations have ceased and only a residual gas bubble remains. For this time period we have

(3.71)\begin{equation}E_B^{max2} = E_{pot}^{max2} + U_{int}^{max2}{|_{p = {p_v}}} + E_v^{max2} = \underline {{E_{acoust}}} + \underline {E_{cond}^{coll2}} + \underline {{W_{visc}}} + U_{int}^{res}{|_{R = {R_{nc2}}}}.\end{equation}

The energy E_acoust of the acoustic radiation after second collapse and during later oscillations is calculated from the other known values in (3.67). Assuming that condensation is completed at second collapse, we have $R_{nres}^{} = R_{nc2}^{}$. Note that we need to know R_{max 3} to determine R_{nc 2}. For nanobubbles, experimental values of R_{max 3} may not be available. In that case, we identify the equilibrium radius during afterbounces and the residual bubble radius with R_{nc 1}.

The above approach for establishing an energy balance of laser-induced bubble oscillations is valid as long as heat conduction out of the energy deposition volume during the laser pulse can be neglected and when the laser pulse duration τ_L is much shorter than the bubble oscillation time. The characteristic thermal diffusion time for a spherical absorber is ${\tau _D} = {d^2}/8{\kern 1pt} \kappa $, where d is the focal diameter and κ is the thermal diffusivity. For pulse durations ${\tau _L} \ge {\tau _D}$, the dynamics changes from approximately adiabatic towards isothermal conditions, and the bubble expands significantly already during laser pulse. As a consequence, less work is done by the expanding gas and both acoustic radiation and the overshoot over the equilibrium radius are largely reduced. Models of such dynamics must explicitly consider heat and mass transfer. The approach presented here is valid only as long as energy deposition is thermally confined and ${\tau _L} \ll {T_{osc}}$.