On the question of the fine fragmentation of liquid-metal droplets during steam explosions

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Introduction
Despite more than 50 years of research on the causes of the fine fragmentation of liquid-metal droplets during steam explosions, the mechanism of this phenomenon remains inadequately understood.This is evidenced by the diversity of physically distinct hypotheses addressing this problem.Figure 1 provides a schematic representation of the principles underlying some of these hypotheses.
One of these hypotheses, known as the cavitationacoustic hypothesis [1], is actively used to interpret the dispersion process of metallic lead droplets under the influence of pulsed laser radiation [2][3][4].Such a process is applied in modern microchip manufacturing technology using extreme ultraviolet lithography [5].The cavitation-acoustic hypothesis is also employed to describe the fragmentation of water droplets under the impact of a sharp mechanical shock [6].In the latter case, the amplitude of the external pressure pulses, with a duration of several microseconds, is approximately ~ 15 MPa.
In the context of the mentioned studies, the following results present experimental and computational assessments regarding the capability of hot liquid-metal droplets to undergo fragmentation through the cavitation-acoustic mechanism when subjected to pressure pulses generated by the collapse of vapor bubbles.Such vapor bubbles are formed when the coolant liquid comes into contact with a significantly overheated surface.

Experimental investigation of the pressure pulses
Experiments investigating pressure pulsations generated by the collapse of vapor bubbles during the crisis of nucleate boiling of distilled water were conducted using a specially designed setup (Figure 2).In the experiments, hot steel rods with hemispherical end caps were used as the working sections.The rods, heated by electric current, were immersed in water to a depth equal to the radius of the hemisphere.The temperatures of the samples and coolant, measured by chromel-alumel thermocouples, varied in the ranges of (190-600) °C and (15-95) °C, respectively.Signal measurement and processing were performed using the LabVIEW software environment.As secondary equipment, an instrument complex based on National Instruments hardware was employed.The pressure pulsations were measured by piezoelectric sensors (PCB model HSM 113A28) in conjunction with a signal conditioning (amplification) module of the PCB 482A22 type.The amplified signal was transmitted to the National Instruments PXI-5122 oscilloscope via coaxial cables.The typical sampling rate of the signal was 7´10 5 samples per second.As evident from these photographs, the fragments exhibit a sponge-like (porous) structure, which can be attributed to the process of internal cavitation and the release of dissolved gases.

Results of numerical evaluations
Based on the aforementioned experimental data, let us perform an approximate calculation of pressure field changes in different media: coolant (water), tin and lead droplets.Additionally, we will estimate the amplitude of the initiating pulse on the molten surface at the point of water boiling.To do this, we will utilize the scalar Helmholtz wave equation for acoustic pressure, pt, in the temporal domain, which can be expressed as follows: Here, t represents time, ρ and c denote the density and speed of sound in the investigated medium, and δ represents the sound absorption coefficient, determined by the formula: where μ and   represent the viscosity and second viscosity of the medium, cp is the isobaric heat capacity, γ and к are the adiabatic index and thermal conductivity coefficient of the medium.To determine the second viscosity   , the results from reference [7] are utilized.
It is assumed that the entire system melt-liquid is enclosed within a container of large dimensions, and the waves reflected from its walls do not affect the calculation results: where n is the unit vector normal.From the symmetry of the problem formulation, it follows that: The initial temporal conditions can be expressed as follows: Using experimental data obtained in water with a piezoelectric sensor, an approximate numerical calculation was performed to estimate the pressure amplitude at the collapse point of a vapor bubble on a hot surface.For this purpose, single pressure pulses were applied at the collapse location, and the pressure field throughout the water cavity was calculated based on the amplitude value.The calculations continued until a match between the calculated and experimental values was observed at the desired distance (6 mm) from the heated sample.Based on the calculation results presented in Figure 5, it can be inferred that the pressure on the hot surface may reach approximately 20 MPa.The impulse of this amplitude was approximated by a Gaussian curve and was used as the excess pressure, simulating the effect of bubble collapse: The duration of this impulse was set as t0 = 3 μs, and the amplitude value of the pressure was p0 = 185 atm.
The density-pressure relationship for tin at a temperature T = 683 K was determined using the Mie-Grüneisen equation of state.
To simplify the calculation procedure, the following assumptions were made: 1) The surface of the droplet does not deform under the influence of the wave front -it is assumed that the liquid metal is surrounded by a solid oxide shell (as observed during the experiment); 2) The metal droplet initially has a perfect spherical shape; 3) The material of the droplet is considered pure, while in reality, it may contain impurities and trapped air, water vapor, or other gas impurities.
At the initial stage, the calculations will be performed for liquid tin and lead.
Below are the instantaneous fields of acoustic pressure in the non-stationary problem for liquid lead at its melting temperature (600 K) with one and two sources of disturbance, as well as an impact on the entire surface (Fig. 6).As seen from the provided illustrations (Figures 7 -9), regions of negative pressure are indeed generated within the volume of the droplet, and the depth of these regions is largely determined by the parameters of the external impulse and the area of impact of this impulse load.The maximum (in magnitude) pressure is observed when hypothetically striking the entire surface of the sphere.With an initial impulse of around 20 MPa, the value of this pressure can reach several thousand atmospheres, which is comparable to the cavitation inception pressure.

Conclusion
The obtained results are consistent with the assumption of the possibility of fine fragmentation of liquid metal droplets through the cavitation-acoustic mechanism.It should also be noted that the numerical calculations were conducted without considering gases dissolved in liquid metals.This effect contributes to the formation of cavitation and is likely to be utilized to control the process of fine fragmentation of the melt.

Fig. 2 .Fig. 3 .Fig. 4 .
Fig. 2. Experimental setup diagram: 1 -National Instruments oscilloscope (NI); 2 -NI analog-to-digital converter (ADC); 3 -computer for recording data from high-speed camera; 4 -crate PXI N1; 5 -signal processing unit with pressure sensor, including charge amplifier; 6 -thermocouple amplifier; 7,8 -piezoelectric pressure sensor; 9 -working section with heater and embedded thermocouples; 10 -coordinate device; 11thermocouples; 12 -video camera; 13 -water container.The characteristic waveforms of the measured pressure pulsations at various temperatures of the hot surface are presented in Figure3a and 3b.Maximum pulsation amplitudes at a distance of 6 mm from the heated surface were observed in individual pulses and reached values of approximately 1 MPa (Figure3a).The duration of such pulses was approximately 10 microseconds, and the temporal interval between individual intense pulses was around 1 millisecond.

Fig. 5 .
Fig. 5. Pressure variation as a function of distance to the vapor bubble collapse location.