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# Localized microwave-heating (LMH) of basalt – Lava, dusty-plasma, and ball-lightning ejection by a “miniature volcano”

## Abstract

This paper presents various phenomena obtained by localized microwave-heating (LMH) of basalt, including effects of inner core melting, lava eruption and flow (from the molten core outside), plasma ejection from basalt (in forms of fire-column and ball-lightning), and effusion of dust (deposited as powder by the plasma). The experiments are conducted by irradiating a basalt stone (~30-cm3 volume, either naturally shaped or cut to a cubic brick) in a microwave cavity, fed by an adaptively-matched magnetron (~1 kW at 2.45 GHz). Effects of LMH and thermal-runaway instability in basalt are observed and compared to theory. Various in- and ex-situ diagnostics are used in order to characterize the dusty-plasma observed and its nanoparticle products. The resemblance of the experimental phenomena obtained in small laboratory scale to gigantic volcanic phenomena in nature is noticed, and its potential relevance to further volcanic studies is discussed. In particular, we show that LMH could be instrumental for laboratory demonstrations and simulations of miniature-volcano effects, such as lava flows, formation of volcanic glass (obsidian), eruption of dusty-plasma and volcanic ash, and ejection of ball lightning. These findings might be significant as well for various applications, such as drilling and mining, microwave-induced breakdown spectroscopy (MIBS), mineral extraction, and powder production directly from basalts.

## Introduction

Microwave heating is unique, compared to other heating techniques, because it may initially generate the thermal energy inside the irradiated object (due to the penetration of the microwave radiation inside, and its absorption and conversion to heat therein)1. Furthermore, localized microwave-heating (LMH) and the consequent formation of hotspots by thermal-runaway instability may cause local melting in various materials (e.g. silicates2), and even plasma ejection outside.

The thermal-runaway instability3, undesired in most microwave applications, may intensify the heat localization effect4,5. It occurs in materials with temperature-dependent properties, which cause the hotter region (typically the core) to become more susceptible to further microwave heating than the outer, cooler parts. This positive-feedback instability localizes the microwave heating and concentrates the thermal energy, and hence eventually leads to creating of a molten hotspot.

Microwave-heating effects in basalts, such as melting, cracking and crushing6,7,8,9,10,11,12, have been studied in various contexts mainly related to mining and construction applications. The thermal-runaway effect in basalt5,9 leads to a hotspot formation and melt inside the stone6,9,10, as illustrated in Fig. 1A. Further irradiation of the basalt stone by a regulated microwave power can stably maintain the molten core inside, in a thermal equilibrium. However, a slight increase in the power may elevate the temperature and increase the pressure inside the stone. Cracks may appear then on the rock surface; the inner melt may erupt outwards and flow as a lava stream over the surface (Fig. 1B), like a “miniature volcano”. Here we have also found that a further irradiation of the basalt heats the hotspot up to a point in which plasma is ejected from its surface, as shown in Fig. 1C13.

Plasmas are usually generated from basalts by various other means, not involving microwaves. The experimental studies reported in the literature include, for instance, plasma generated from basalt by CO2 and Nd:YAG lasers (for laser-induced breakdown spectroscopy, LIBS)14; inductively-coupled plasmas of basalt in various forms (rock, fiber and glass) analyzed by atomic emission spectroscopy15; and a plasma spray of basalt in powder form implemented in a basalt coating technique16.

As observed in the present work13, starting from an LMH hotspot in basalt, the plasma is first ejected and evolves as a fire-column, which is stably maintained by the microwave irradiation. Then, it may also emit a fireball, buoyant in the air atmosphere within the cavity, as shown in Fig. 1D. This basalt-originated fireball is either adjacent to the initiating fire-column or detached from it, tending to hover towards the microwave source (as in refs2,17,18,19).

This paper presents observations of volcano-like phenomena in basalt-LMH experiments, also in comparison to theory; characterizes the dusty-plasma ejected and its nano-powder products; and discusses the LMH potential as a means for experimental laboratory-scale simulations of volcanic effects, and for other material-processing applications.

## Experimental Setup

The LMH experiments presented in this paper have been conducted in the experimental setup illustrated in Fig. 2A,B [Method 1]. The microwave cavity (Fig. 2A) is made of a 10 × 5 cm2 cross-section waveguide, shorted at the end by a microwave reflector. This cage-like structure of the cavity, with wide openings (~22 mm) between the vanes, enables direct access and broad view from various directions into the interaction region (with immediate lines of sight for the various diagnostic means). The cavity is fed by a 2.45-GHz magnetron (regulated up to 1-kW input power).

By applying LMH, the inner core of the basalt is first melted6,8,9,10. Further irradiation leads to the eruption of the molten basalt, from the core to the surface (similarly to lava eruption and flow in volcanoes). The molten basalt cools down and solidifies to a black glass (obsidian). A further irradiation causes the basalt to emit a plume of dusty plasma, which evolves to a confined plasma column, and further to a plasmoid similar to a natural ball-lightning. These fire-column and fireball-like plasmoids are characterized here by various in- and ex-situ means.

The optional use of a movable electrode (as in the microwave-drill device20 and in previous plasma studies2,21) may expedite the eruption process by intensifying the local electric field. However, similar effects may occur in basalt with no electrode as demonstrated here as well (though in a less controllable fashion). The electrode enables reaching the melting temperature more locally and rapidly within a hotspot underneath the surface, which provides a path for lava eruption from the molten core. Here we also examined the use of an electrode array in order to induce multiple hotspots.

## Results

### Core melting and lava eruption

Effects of core melting, followed by lava eruption and flow (and further solidification to obsidian) have been observed in LMH of natural basalt stones in these experiments, as shown in Fig. 3A (with no electrode). As an alternative to the naturally-shaped stones, cubic bricks (of 3-cm side length) cut from natural basalts have also been experimented, since their well-defined geometry enables better comparison with numerical simulations. The LMH experiments with basalt cubes (using electrodes) resulted in similar effects of core melting, and lava eruption and flows, as shown in Fig. 3B. The thermal image in Fig. 3B(e) indicates that the molten lava temperature exceeded ~1,200 K. The lava erupts from the core outwards, and then flows downwards by gravity with a front-flow velocity of ~1 cm/s, while its color varies from luminous white to orange and red [Supplementary Video S1a and S1b]. It eventually cools down and vitrifies to black volcanic glass, namely obsidian. Various other volcano-like phenomena are observed (as also shown in Fig. 3A,B), such as formations of bubbles and domes, fissures and craters, vents, blasts, bursts of lava, and lava tunneling. The latter is observed for instance when two hotspots are simultaneously created by electrodes in two opposite faces of the basalt brick. An inner lava tunneling effect is evolved then, in the core between the two hotspots, as observed via the porous outer surface of the basalt brick [Supplementary Video S2].

The theoretical analysis of the basalt’s LMH interaction ([Method 2] incorporates the thermal-runaway instability and the interactive temperature-dependence of the basalt’s properties9. The numerical simulations presented in Fig. 4A–C show the inner hotspot formation and core melting prior to the lava eruption, in agreement with the experimental results (with no electrode). The additional effect of the sharp geometry on the LMH evolution is demonstrated in Fig. 4C(a–c), by simulations of various shapes such as a sharp stone (a) and single- and multiple-electrode schemes (b, c).

### Dusty-plasma ejection in forms of fire-columns and fireballs

In this study we have found that a further microwave irradiation of the molten basalt causes a plasma ejection from it. In this process, the microwave localization effect continues to evolve and to heat up the melt, and hence plasma is ejected from it as shown in Fig. 5A(a–d) and Supplementary Video S3. Starting from a molten hotspot (a), the plasma is first emitted (b), and then forms a stable fire-column (c). In some cases, another plasmoid is also evolved in a form of a fireball, either alongside the fire-column (d) or detached from it as shown in Fig. 5B(d) and Supplementary Video S4. The detached fireball is typically less stable than the fireball adjacent to a fire-column, and it tends to move towards the magnetron. The fireball disappears within a few seconds, similarly to natural ball-lightning22. Other modes of plasma ejection from basalts observed in this study are a fire-column eruption from a dome made of a thin layer of obsidian (blown up by the lava eruption), and a simultaneous ejection of two plasma columns from two different hotspots, as shown in Fig. 5B(b,c), respectively (both effects are presented in Supplementary Video S5).

### LMH and plasma characterization

Various diagnostic techniques were applied in order to characterize the LMH of basalt, as well as the plasma ejected from it, and its products. These diagnostic means include microwave-scattering analyses, optical spectroscopy, and scanning electron microscopy (SEM). The analyses of the microwaves scattered from the plasmoid results in an estimate of its microwave impedance, which provides a measure for its dielectric and plasma properties. The reflections of the 2.45-GHz microwave input are detected during the various stages of the LMH process. The complex reflection coefficient Γ (Fig. 5C(a)), also presented in a Smith-chart form in Fig. 5C(b), shows that the microwave reflection tends to decrease (from |Γ| ~ 1 to ~0.4) after the plasma ejection, and to further decrease (to |Γ| ~ 0) when a fireball emerges (Supplementary Video S6). At the stable fire-column stage, the reflection coefficient is bouncing around ~0.4, while being attracted by the Smith-chart origin. The corresponding power reflection is smaller than 20% at this stage, hence the plasmoid interacts with its microwave source as an adaptively matched load. At the fireball stage, this self-tuning mechanism tends to maximize the microwave power absorbed by the fireball, by optimizing its intensity, position and size. The estimate of the plasma impedance and dielectric properties is based on a comparison of the reflection measurements with simulation results of a dummy load, with equivalent shape, position and size18,19,23.

The optical spectrum of the light emitted by the basalt plasma-column (Fig. 6A,B) reveals its atomic lines, which are identified using24 as of Fe, P, C, Si, Al, Mg, Ti, and Ca. All these elements are known as ingredients of basalt rocks. The Boltzmann plot technique, implemented on the Fe and Ti lines, results in excitation temperature estimates of Texc ~ 0.3 and ~0.6 eV, respectively (Fig. 6C). The spectral emission of OH radicals is identified by fitting to LifBase data25. The rotational temperature is also roughly estimated in the same range of Trot ~ 0.3–0.6 eV (Fig. 6D).

A Langmuir probe is inserted into the plasma in order to measure its I-V characteristics (Fig. 7A(a)). Valid results are only observed when the melt flows to the grounded chamber floor, and closes the electric circuits to enable a current loop (otherwise, smaller currents are measured due to the high impedance of the powder layer deposited by the plasma on the ceiling) [Supplementary Video S7]. The hysteresis observed in the I-V curve (Fig. 7A(b)) is attributed to the capacitive effect created by the powder deposited on the electrode. A rough estimate of the electron temperature by the curves’ slopes results in Te ~ 0.3 eV and ~0.6 eV, for the voltage rise and fall, respectively. This range coincides with the results estimated by optical spectroscopy.

Microwave scattering measurements are also performed at other frequency ranges, by irradiating the plasmoid by another probing signal at low power. The latter is coupled via additional input and output ports (marked 1 and 2, respectively, in the inset of Fig. 7B(a)). Scattering results for the microwave transmission coefficients (measured and simulated) at a lower frequency band (0.8–1.5 GHz) are shown in Fig. 7B(a), with and without the plasma. The results show that near the anti-node frequency of the coupling between Ports 1 and 2 (~1.25 GHz, corresponding to L ~ λ/2, where L is the coupler arm length), the transmission coefficient S21 is increased by >20 dB in the presence of plasma. Indeed, below the plasma frequency, the fire-column behaves like a conductor, which shorts the coupler arm length by half (~L/2 ~ λ/4) and hence significantly increases the microwave transmission between Ports 1 and 2. A numerical simulation of this coupler, with an equivalent column-shaped load inside, enables us to estimate the plasma dielectric permittivity. As shown in Fig. 7B(b,c), the transmission increase at the anti-node frequency follows the abrupt temperature rise, in which the plasma switches from the decoupled to the coupled state of Ports 1 and 2.

### Dust effusion and ex-situ analyses

A significant amount of dust is produced by the basalt’s plasma, deposited as a white powder on the collector shown in Fig. 2A. The surface morphologies of the particles created by the plasma are characterized by a scanning electron microscope (SEM) [Method 1]. The accumulated particles are observed in various shapes, mostly aggregates in few micro-meter sizes. Figure 8A(a–f) show aggregates of such particles on the collector surface as observed by SEM. Analyses of energy dispersive spectroscopy (EDS) reveal that these aggregates are composed of the elements listed in Table 1, also typical to the original basalt content. In addition, the basalt’s plasma generates larger spherical particles, in sizes of tens of microns and larger, as seen on the collector (Fig. 8A,B(a)). Holes of similar sizes are also noticed on the collector surface (this finding may either hint particle bombardment or a plasma localization effect). Island-like regions are observed upon the white powder deposition on the collector, as seen in Fig. 8A(c). EDS shows that they mostly consist of aluminum, as the collector itself, hence they may indicate an etching effect.

EDS analyses were also conducted on the basalt stone in its original form and its molten state (Fig. 8B(b)), in order to compare the compositions of the vitrified product (obsidian) and the natural stone. Table 1 shows the relative presence of the constitutive elements in the various phases of this process, namely the natural stone, the melt, and the deposited powder. The weight ratios differ between the various phases. However, it seems that the deposited powder consists of elements provided solely by the basalt (e.g. Si and Fe). These ingredients of the powder were most likely extracted from the basalt, and then evolved and transported by the plasma to the collector. Consequently, these elements were also detected within the plasma by optical spectrometry.

X-ray diffraction (XRD) pattern of massive samples of the molten basalt show low intensity of diffraction peaks, suggesting that the great part of the sample consists of glass. From the position of the peaks, it is possible to determine that the two major crystalline phases are an oxide of aluminum, magnesium and iron, and a silicate of magnesium and iron. The diffraction pattern of the white deposition (Fig. 8B(a)) enables to determine the presence of crystalline phases in the powder, including three silicate phases (Enstatite (Mg,Fe)2Si2O6, Augite (Ca,Mg,Fe)2Si2O6 and Mullite 3Al2O3·2SiO2) and two oxides (Periclase MgO and Wustite Fe0.9O). Peak observed are related to a phase with phosphor, possibly in amorphous phase (as confirmed by the modulated background).

A simplified model of microwave interaction with dusty plasma26 [Method 3] yields an estimate for the electron density in the basalt-originated plasma as ne ~ 1017–1018 m−3, assuming an electron temperature of Te ~ 0.3–0.6 eV (estimated by the various measurements above), and dust number density of nd ~ 1016 m−3, as measured by small-angle X-ray scattering (SAXS) of similar elemental dusty plasmas18,23,27,28. Therefore, this plasma is characterized as a weakly-ionized, low-density dusty plasma, which contains components of the original basalt stone (e.g. silicon and iron oxides).

## Discussion

Modeling capabilities of volcanic phenomena are essential for both field- and laboratory-based studies. The contemporary research of volcanic processes incorporates a variety of theoretical and experimental models in an interdisciplinary approach, including thermodynamics, solid mechanics, fluid dynamics, and material sciences29. Theoretical models, e.g. of pyroclastic flows30, elastoplastic ground deformation31 and volcanic ash transport forecasting32, improve the understanding and the predictability of these processes. Laboratory models of volcanoes and lava phenomena employ analogues materials in order to experimentally study for instance lava domes33, submarine lava dynamics34 and mantle flows35.

In this interdisciplinary approach, the LMH-basalt study presented here may open new possibilities for experimental modeling in volcanology and geosciences. The controllable basalt core-melting and the lava-flow effects observed here in a miniature laboratory scale, and their resemblance to giant volcanic phenomena in nature, can be utilized for experimental small-scale simulations and demonstrations of various volcanic phenomena.

The natural ball-lightning phenomena, occasionally associated with volcanic eruptions, can also be experimentally modeled by the LMH-generated fire-column and fireball plasmas. A large number of fireball sightings has been recorded in the last centuries22, many occurring near volcanos or at times of heavy air pollution caused by volcanic eruptions. Spectrograph measurements of natural ball-lightning reveals the presence of Si, Fe, Ca and Al36 as in soil, in line with the theory suggesting that ball lightning is generated by lightning striking the soil37. These elements appear as well in the spectroscopy measurements of the basalt-generated plasma (along with other basalt components), in resemblance to the natural ball-lightning studies22,36,37. This finding adds another aspect to the visual similarity of microwave-generated fireballs to natural ball-lightning phenomena2,38).

The phenomena of lightning within eruptions of volcanic ash plumes in nature has also been recorded and studied39. This high-energy lightning chemically modifies and melts the volcanic ash40, generating spherical shapes similar to those observed in the powder accumulated in this study. For instance, the composition of the volcanic ash sampled in Mount St. Helens eruption on May 18, 198041 includes SiO2 (65%), Al2O3 (18%), FetO3 (5%), and smaller amounts of MgO, CaO, Na2O, K2O, TiO2 and P2O. For comparison, the powder accumulated from the basalt-generated plasma consists of similar elements, as listed in Table 1. The relatively high concentration of phosphorous (and of Na and K as well) detected in the powder can be attributed to its relatively low melting temperature. This environmentally significant effect also features volcanic ash42.

Major challenges have remained for further research, including in-depth investigations and modeling of (a) the LMH interaction with basalt and phase transitions, (b) plasma ejection mechanisms and effects of thermionic emission, (c) microwave interactions with dusty plasma and particle agglomeration therein, (e) plasma-surface interactions, and plasma-chemical reactions occurring in this process.

In a more technology-oriented approach, this study of LMH in basalts could be relevant to various potential applications, including for instance (a) microwave-assisted drilling in basalts20, (b) cracking and crushing of basalts for mining and construction operations43, (c) basalt vitrification, glass production and coating, (d) joining of basalt bricks, (e) direct extraction of minerals in a powder forms by means of dusty-plasma production from basalts, and (f) analyses and identification of rock contents in the field by portable microwave-induced breakdown spectroscopy (MIBS) systems44 (as a variation of the laser-based LIBS technique45).

## Methods

### 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 Iki 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 λki and Aki are the transition wavelength and probability, respectively, kB is the Boltzmann constant, and Ek is the upper energy state with a gk 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 ε0 is the permittivity of vacuum, εr and εr 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

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

$$\rho {c}_{p}\frac{\partial T}{\partial t}-\nabla \cdot ({k}_{th}\nabla T)=Q,$$
(M4)

where ρ is the (local) medium density, cp and kth 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.

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, εr, consists of the plasma complex permittivity, $${\varepsilon }_{p}^{^{\prime} }$$ − $$j{\varepsilon }_{p}^{^{\prime\prime} }$$, and of the dust conductivity σed26, as follows

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

where ε0 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, ne and me 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 ηed 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 nd and rd, respectively. In this analysis, $$\hat{k}$$ represents the effective spatial angular frequency. The electron collision frequency is approximated by, υ = VTeσnNn where $${V}_{Te}=\sqrt{{k}_{B}{T}_{e}/{m}_{e}}$$ is the electron thermal velocity, σn and Nn are the neutrals cross-section and density, respectively, Te is the electron temperature, and υch 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 dPC = 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 YPC = GPC + jBPC and Yc, respectively). The conductance and susceptance of the plasma column are approximated by GPC = ωε0$${\varepsilon }_{r}^{^{\prime\prime} }$$ A/h and BPC = ωε0$${\varepsilon }_{r}^{^{\prime} }$$ A/h, respectively, where A is the effective cross-section area of the plasma-column. In this analysis (as in ref.18), the complex εr space is numerically scanned in order to find the conditions that provide reflection values as measured in the experiments. The real part of εr 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., υυchω2 and υchω2), the dielectric constant and the corresponding electron density are estimated in this case by

$${\varepsilon }_{r}\cong 1+\frac{{\omega }_{p}^{2}}{{\omega }^{2}+{\nu }^{2}}[-1+\frac{\nu }{\widehat{k}{l}_{d}}-j(\frac{\nu }{\omega }+\frac{1}{\widehat{k}{l}_{d}})],$$
(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/dPC for the finite transverse profile of the plasma column. Equation (M9) yields ne ~ 1017–1018m−3, similarly to other weakly-ionized plasmas and flames at atmospheric pressure in air.

When the dusty-plasma region coexists with the molten hotspot in the cavity, most of the microwave power is absorbed by the plasma. This domination is visually observed, since the molten rock tends to cool down once the plasma is ejected, and it is also evidenced by the thermal-camera measurements (e.g. in Fig. 7B(c)). This effect is also verified by simulation and by an analytical estimate (taking into account the effective values of the dielectric permittivity and electrical conductivity of both media, σeff = σ + ωε″ in a transmission-line model with lumped loads23). According to this estimate, the microwave power absorbed by the plasma column is more intense by at least one order of magnitude than the power absorbed then by the rock.

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## Acknowledgements

This research is supported by The Israel Science Foundation (Grant No. 1896/16). The authors also wish to acknowledge Drs. Zahava Barkay and Davide Levy for the SEM and XRD analyses, respectively, and Profs. Shmuel Marco, Ehud Heyman, and Rachman Chaim for helpful discussions.

## Author information

Authors

### Contributions

E.J. conceived the fundamental concept, experimentally discovered the LMH effect of plasma emission from basalt, and supervised the research project. Y.S. participated in the experimental design, developed the data acquisition system, and performed the simulations and analyses. Both authors carried out the experimental work and wrote the paper together.

### Corresponding author

Correspondence to Eli Jerby.

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Jerby, E., Shoshani, Y. Localized microwave-heating (LMH) of basalt – Lava, dusty-plasma, and ball-lightning ejection by a “miniature volcano”. Sci Rep 9, 12954 (2019). https://doi.org/10.1038/s41598-019-49049-5

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