Method 1. Experimental

The processes of basalt melting and plasmoid ejection therefrom are initiated in this study by directing localized microwave power into the basalt specimen. This localization effect is implemented by the microwave-drill mechanism20, which creates the hotspot from which plasma is ejected in a form of a fire-column or a fireball, into the cavity.

The experimental setup shown in Fig. 2A,B includes a microwave cavity fed by a 2.45-GHz, 1-kW magnetron. The cavity is made of a 10 × 5 cm2 cross-section waveguide, shorted at the end by a mirror made of a periodic array of metallic vanes, under cutoff. Similar arrays are also installed as the sidewalls of the cavity. This cage structure prevents microwaves leaks, and yet enables a broad view into the waveguide with no disturbance to the various in-situ diagnostic means (such as the thermal camera and optical spectrometer which require direct lines of sight).

In this experiment we use basalt stones (from the Golan heights) either in their original natural shapes or cut to (3-cm)3 cubic bricks. This load is placed inside the cavity in an optimized position whereas the (optional) movable electrode directs the microwave energy locally into the substrate. Various level of focusing are optionally available in this setup, including (i) a single-electrode focusing as in Fig. 2A, (ii) no-electrode focusing, and (iii) a multi-electrode array focusing. Each electrode in this array expedites the intentional excitation of the hotspot in its vicinity, when brought into contact with the basalt-stone surface, and hence enables the stimulation of the LMH and fire-column (and consequently of a fireball) in a more controlled fashion. Typically, electrodes were applied to cubical bricks in these experiments, whereas no electrodes were used with natural stones.

The magnetron’s switched-mode power supply (MagDrive-1000, Dipolar Ltd., Sweden) provides an adjustable input microwave power up to 1 kW. The microwave power is delivered to the cavity via an isolator and an impedance auto-tuner (Homer, S-Team Ltd., Bratislava, Slovak republic), as depicted in Fig. 2B. The incident and reflected waves are recorded by the impedance-analyser mode of the auto-tuner, which also enables adaptive impedance matching, and optimal transmission of the microwave power into the cavity. The adaptive tuning is applied to intensify the hotspot heating in order to eject the plasma (whereas the synchronized de-embedded reflection coefficient measurements are presented and stored for offline analysis). In order to get more accurate measurements once the plasma is ejected, we turn off the adaptive tuning in order to measure the actual reflection coefficient instead of the de-embedded signal. This is possible due to the self-tuning effect of the plasmoid, as an alternative to the external impedance matching.

The scattered microwave is analysed in order to find the plasmoid impedance, and hence to estimate its dielectric and plasma properties. First, the reflections of the 2.45-GHz input power are measured during the various stages presented in Fig. 1A–D. The reflection measurements (as in Fig. 5C) enable us to find the impedance and dielectric properties of the fire-column by simulating an equivalent load, with the same shape, position and size.

The diagnostic means also include video recording, thermal imaging, optical emission spectroscopy (OES), and plasma I-V measurements, all conducted via the vane openings along the waveguide. A LabVIEW code controls the system, synchronizes the time between all the diagnostic means and accumulates the experimental data.

The video camera recording (at 200 fps) is used to monitor, record times and synchronize between effects such as inner hotspot formation, cracks, breakage, lava eruption and plasma ejection to other diagnostic means used in the experiment. Once ejected, the plasma column’s size, shape and position can be detected by the video recording. Also, evolution of fireballs, and their shape and movement are monitored by the video camera.

The FLIR-SC300 thermal camera monitors the spatial temperature evolution of the sample, the hotspot formation, and its temperature and size. The thermal imaging data is later compared to simulation results in order to validate the physical model.

The optical spectra emitted from the lava and plasma eruptions (Fig. 6A) are captured by an optical spectrometer (Avaspec-3648) with 0.3-nm resolution in the range 200–1000 nm. The results are analyzed for element identification and blackbody radiation, as well as radical emission curve fitting and Boltzmann plot (in order to find the rotational and excitation temperatures, respectively).

The excitation temperature is estimated, as in refs18,23,27, by using a Boltzmann plot of the line intensity, assuming a partial local thermal-equilibrium (pLTE) state of the plasma. The intensity I ki of the transition from the upper k to the lower i energy level is given in the Boltzmann equation,

$$\mathrm{ln}\,({I}_{ki}{\lambda }_{ki}/{g}_{k}{A}_{ki})=-\,(1/{k}_{B}{T}_{exc}){E}_{k}+{\rm{const}}.,$$ (M1)

where λ ki and A ki are the transition wavelength and probability, respectively, k B is the Boltzmann constant, and E k is the upper energy state with a g k degeneracy (as in ref.23, the relatively large scatter of the measurement results from the fit line in Fig. 6C is due to the 0.3-nm spectral resolution and the consequent overlap of the detected atomic lines).

A Langmuir-like probe is inserted into the plasma in order to measure its I-V characteristics. As shown in Fig. 7A(a), the 1.6-mm diameter, 50-mm long probe is fed by a 50-Hz, ~100-V alternating voltage. Valid results are only observed when the melt exists as illustrated in Fig. 7A(a), and closes the electric circuits to enable the current flow (otherwise much smaller currents are erroneously measured). A presumable reason is the isolating powder oxide accumulated on the metal surfaces, which adds a significant resistant to the plasma current loop. The hysteresis observed in the I-V curve may also indicate a contamination of the electrode, which leads to a capacitive effect.

The powder products are ex-situ examined, as in refs18,23,27, by a FEI Quanta 200FEG environmental SEM. The chemical element composition is analyzed using energy dispersive spectroscopy (EDS) with a Si(Li) liquid-nitrogen cooled Oxford INCA X-ray detector.

Method 2: LMH model and simulation

The LMH process (prior to the plasma ejection) is modelled by a set of coupled EM-thermal equations4,5 for a time-varying (temperature-dependent) inhomogeneous lossy medium in a bounded region (a basalt stone in a cavity, in this case). The effective dielectric permittivity of the medium in the microwave regime is represented as a complex variable by

$${\varepsilon }_{eff}(T)={\varepsilon }_{0}[{\varepsilon }_{r}^{{\rm{^{\prime} }}}(T)-j({\varepsilon }_{r}^{{\rm{^{\prime} }}{\rm{^{\prime} }}}(T)+{\sigma (T)/\omega \varepsilon }_{0})],$$ (M2)

where ε 0 is the permittivity of vacuum, ε′ r and ε″ r are the real and imaginary components, respectively, of the relative temperature-dependent permittivity of the medium (where T(x, y, z; t) is the temperature, spatially and temporally evolved during the process), and σ(T) is the temperature-dependent electric conductivity of the medium.

The EM-wave equation is given in the frequency domain by

$$

abla \times (

abla \times \tilde{{\bf{E}}})-{\varepsilon }_{eff}(T){k}_{0}^{2}\tilde{{\bf{E}}}=0,$$ (M3)

where \(\tilde{{\bf{E}}}\) is the electric-field vector of the EM wave, ω is the angular frequency of the EM-wave, and \({k}_{0}=\sqrt{{\varepsilon }_{0}{\mu }_{0}}\omega \) is its wavenumber.

The heat equation is given in the time domain by

$$\rho {c}_{p}\frac{\partial T}{\partial t}-

abla \cdot ({k}_{th}

abla T)=Q,$$ (M4)

where ρ is the (local) medium density, c p and k th are its heat capacity and thermal conductivity, respectively, and Q(x, y, z; t) is the absorbed power density in the medium, as evolved during the LMH process. The latter is given by

$$Q=\frac{1}{2}\omega {\varepsilon }_{0}{\varepsilon }_{r}^{{\rm{^{\prime} }}{\rm{^{\prime} }}}{|\mathop{{\bf{E}}}\limits^{ \sim }|}^{2}+\frac{1}{2}\sigma {|\mathop{{\bf{E}}}\limits^{ \sim }|}^{2},$$ (M5)

for the dielectric and ohmic losses, respectively, and it couples the EM-wave and heat equations (Eqs M2 and M3, respectively, also coupled by the temperature dependence of the medium’s EM parameters). The single-frequency operation and the relatively slow-time variation of the temperature (in a > 1 ms scale) with respect to the EM-wave variation (in a ~1 ns time scale) enable us to apply a two-time scale approximation, hence to solve the EM-wave equation in the frequency domain and the heat equation in the time domain. The two solvers are iteratively applied, each with its relevant boundary conditions. The temperature-dependent parameters of basalt, employed in this analysis, are given with the relevant references in Table 2.

Table 2 The temperature-dependent basalt’s properties employed in the numerical simulations, P N (p 0 , p 1 , p 2 , … p N ) denotes polynoms of the N-th order, \({\sum }_{n=0}^{N}{p}_{n}{T}^{n}\) . Full size table

Referring to the material properties essential for LMH, our previous microwave-drill analyses (e.g.4) show that the intentional-LMH effect is feasible in materials such as silicon, germanium, glass, and various ceramics (e.g. mullite, cordierite, zirconia, alumina of 86% purity, and clay) but not for instance in sapphire or pure alumina (due to their small dielectric losses). Also, the expansion coefficient and mass density may affect the material’s brittleness and cracking (which seem to appear in our basalt experiments after the inner melting). In glass, the optical transparency might be useful to visually observe the molten core inside, but it is not necessary for the hotspot evolution (the LMH effect may occur as well in opaque dielectric materials).

Method 3. Dusty plasma interaction with microwaves

Following ref.26, the effective dielectric permittivity of the dusty plasma, ε r , consists of the plasma complex permittivity, \({\varepsilon }_{p}^{^{\prime} }\) − \(j{\varepsilon }_{p}^{^{\prime\prime} }\), and of the dust conductivity σ ed 26, as follows

$${\varepsilon }_{r}={\varepsilon }_{p}^{{\rm{^{\prime} }}}-j{\varepsilon }_{p}^{{\rm{^{\prime} }}{\rm{^{\prime} }}}-j{\sigma }_{ed}/{\varepsilon }_{0}\omega ,$$ (M6a)

where ε 0 is the vacuum permittivity and ω is the angular frequency. The dielectric permittivity of the plasma is given by its approximated real and imaginary components,

$${\varepsilon }_{p}^{{\rm{^{\prime} }}}=1-\frac{{\omega }_{p}^{2}}{{\omega }^{2}+{\upsilon }^{2}},$$ (M6b)

$${\varepsilon }_{p}^{{\rm{^{\prime} }}{\rm{^{\prime} }}}=\frac{{\omega }_{p}^{2}\upsilon }{\omega ({\omega }^{2}+{\upsilon }^{2})},$$ (M6c)

respectively, where \({\omega }_{p}=\sqrt{{e}^{2}{n}_{e}/{m}_{e}{\varepsilon }_{0}}\) and υ are the plasma and the collision frequencies, respectively, and e, n e and m e are the electron charge, density and mass, respectively. The dusty plasma conductivity is given in similar conditions by ref.26, as

$${\sigma }_{ed}\cong {\eta }_{ed}\frac{\omega }{\hat{k}}[\frac{{\omega }^{2}-\upsilon {\upsilon }_{ch}}{({\omega }^{2}+{\upsilon }_{ch}^{2})({\omega }^{2}+{\upsilon }^{2})}+j\omega \frac{\upsilon +{\upsilon }_{ch}}{({\omega }^{2}+{\upsilon }_{ch}^{2})({\omega }^{2}+{\upsilon }^{2})}],$$ (M7)

where η ed is the charging factor, presented in terms of the dust collision length factor \({l}_{d}={({n}_{d}\pi {r}_{d}^{2})}^{-1}\), as \({\eta }_{ed}={\omega }_{p}^{2}{\varepsilon }_{0}/{l}_{d}\), with the dust grain density and the average particle radius denoted by n d and r d , respectively. In this analysis, \(\hat{k}\) represents the effective spatial angular frequency. The electron collision frequency is approximated by, υ = V Te σ n N n where \({V}_{Te}=\sqrt{{k}_{B}{T}_{e}/{m}_{e}}\) is the electron thermal velocity, σ n and N n are the neutrals cross-section and density, respectively, T e is the electron temperature, and υ ch is the dust charging frequency.

As in refs18,23, the simulation here includes a loaded microwave cavity as shown in Fig. 2A. The plasma column inside is represented by a dielectric cylinder of h = 50mm height and d PC = 15mm diameter. In the heuristic transmission-line model presented in ref.23, the plasma column is modeled by a lumped element in a transmission line (having admittances of Y PC = G PC + jB PC and Y c , respectively). The conductance and susceptance of the plasma column are approximated by G PC = ωε 0 \({\varepsilon }_{r}^{^{\prime\prime} }\) A/h and B PC = ωε 0 \({\varepsilon }_{r}^{^{\prime} }\) A/h, respectively, where A is the effective cross-section area of the plasma-column. In this analysis (as in ref.18), the complex ε r space is numerically scanned in order to find the conditions that provide reflection values as measured in the experiments. The real part of ε r is initially chosen as 0.2 while the imaginary part is searched between 3 to 100. The simulation and the experimental observations, presented in the Smith chart in Fig. 5C(b), result in an estimate of \({\varepsilon }_{r}^{^{\prime\prime} }\) ~ 25 for the effective dissipation factor (including conductivity). Using the excitation temperature found by Boltzmann plot (~0.3–0.6 eV) and the dust particle size obtained by SAXS analyses of similar dusty plasmas18,23,27,28 with the assumptions therein (e.g., υυ ch ≪ ω2 and υ ch ≪ ω2), the dielectric constant and the corresponding electron density are estimated in this case by

$${\varepsilon }_{r}\cong 1+\frac{{\omega }_{p}^{2}}{{\omega }^{2}+{

u }^{2}}[-1+\frac{

u }{\widehat{k}{l}_{d}}-j(\frac{

u }{\omega }+\frac{1}{\widehat{k}{l}_{d}})],$$ (M8)

and

$${n}_{e}\sim \frac{{m}_{e}{\varepsilon }_{0}{\varepsilon }_{r}^{{\rm{^{\prime} }}{\rm{^{\prime} }}}}{{e}^{2}}\frac{{\omega }^{2}+{\upsilon }^{2}}{\upsilon /\omega +1/\hat{k}{l}_{d}},$$ (M9)

respectively, where \(\hat{k}\) is approximated by ~π/d PC for the finite transverse profile of the plasma column. Equation (M9) yields n e ~ 1017–1018m−3, similarly to other weakly-ionized plasmas and flames at atmospheric pressure in air.