Electron multiplication in nanovoids at the initial stage of nanosecond discharge in liquid water
VVersion xx as of February 2, 2021Primary authors: Petr B´ılek, J´an Tungli, Milan ˇSimek, Zdenˇek BonaventuraTo be submitted to PSST
Electron multiplication in nanovoids at the initial stage of nanosecond discharge inliquid water
Petr B´ılek,
1, 2
J´an Tungli, Milan ˇSimek, and Zdenˇek Bonaventura Department of Physical Electronics, Faculty of Science, Masaryk University, Brno, Czech Republic Department of Pulse Plasma Systems, Institute of Plasma Physics CAS, Prague, Czech Republic (Dated: February 2, 2021)The process of electron multiplication through the bouncing-like accelerated motion of electronsinside nanovoids formed owing to external electric fields in bulk liquid water is investigated usingMonte Carlo simulations in Geant4-DNA. Our results show that the initial charge developed at themetal/liquid interface can be multiplied and expanded along the direction of the external electricfield on a picosecond timescale, owing to collision-free interiors of the nanoruptures. Characteristicfeatures of two different electron multiplication mechanisms are revealed and characterized. We findthat electrons can be accelerated inside cylindrical nanoruptures while bouncing off the void/waterinterface. Simulations predict geometric conditions leading to charge multiplication along the void,rather than electron capture or thermalization in bulk liquid. Our results are consistent withthe recent verification of the causal relation between electrostriction-induced perturbations in bulkliquid, and the subsequent formation of luminous filaments evidencing the presence of energeticelectrons.
PACS numbers:
Introduction .—The processes of electron generationand multiplication govern the basic physics of electricaldischarges, including discharges in polar liquids such aswater. In discharges produced by highly non-uniform andtime-dependent electric fields (usually produced throughhigh-voltage pulses of nanosecond duration applied tothe pin-plane electrode geometry), ultrafast processeson metal/liquid water interfaces and in H-bonded net-work structure of bulk liquid water provide competingmechanisms that determine the dynamics of dischargeinitiation. Investigating principal mechanisms responsi-ble for electron multiplication and acceleration in suchnon-homogeneous and highly collisional environments isextremely challenging and of fundamental importance.The main difficulties in studying such phenomena in-clude the very complex behavior of water exposed tostrong and rapidly changing electric fields and extremelyshort characteristic temporal (sub-nanosecond) and spa-tial (sub-micrometer) scale. Another poorly understoodphenomenon is the origin and evolution of the chargedspecies leading to discharge initiation. Considerable un-certainty remains about processes leading to the multi-plication of principal charged species in external electricfield, including their spatiotemporal expansion and theformation of branched structures in bulk liquid water onmillimeter scale.It should be noted that owing to H O autoionization[1, 2], even chemically pure liquid water contains a cer-tain number of H O + (hydronium) and OH − (hydroxide)ions, which are responsible for minimal electrical conduc-tivity of ≈ × − µ S/cm. Autoionization occurs be- cause of the fluctuations of local electric fields acting be-tween neighboring molecules that can drive proton trans-fer, leading to the formation of hydronium and hydroxideions. The nascent ions either recombine within tens offemtoseconds or get separated by the Grotthuss mecha-nism [3, 4]. Shneider and Pekker [5] estimate equilibriumconcentration of OH − ions in pure water at room tem-perature and density nanopores formed due to the elec-trostriction at sub-ns timescales to be ≈
60 and ≤ per µ m , respectively. In real experiments, deionized waterwith initial conductivity between 0.5–1 µ S/cm is typi-cally used. This implies characteristic OH − /nanoporeratio of ≈ × − and a reasonable probability that OH − occurs on the nanopore/liquid interface (considering thedynamic nature of the H O autoionization process). TheOH − ions occuring on the nanopore/liquid interface thenprovide a source of free electrons. The electron densitycan be estimated [5] as n e ∼ w ( I n , E )[OH − ]∆ t , where w ( I n , E ) is the rate of electron tunnelling from OH − as afunction of I n the electron affinity energy in negative ionand E is the electric field. ∆ t denotes the characteristictime (typically < a r X i v : . [ phy s i c s . p l a s m - ph ] J a n the formation of the double layer structure on the in-terface (metal surface–Helmholtz layer–diffused layer–bulk liquid) [8]. For positive potentials, H O tends tore-orient with its hydrogen atoms away from the sur-face, and the ionization mechanism based on tunnelingand proton transfer (H O → H O + +e − ; H O + +H O → H O + +OH) becomes effective for electric fields aboveseveral GV/m [8].The processes that alone, or through their interplaymay explain the multiplication of charged species areimpact ionization in the liquid bulk, field-assisted elec-tron emission (i.e., Zener tunneling) [9–11], and the ac-celeration of electrons through low-density regions andruptures resulting from electrostrictive forces [5, 12, 13].Pure impact ionization seems to be dubious, as electronsin the liquid are unable to gain sufficient energy to ionizewater molecules. This is because the energy delivered bythe applied electric field dissipates efficiently because ofthe frequent scattering of electrons on water molecules[14–17]. Moreover, the action of Zener tunneling as theenabling process seems improbable since it requires verylarge electric fields and a large pressure in the liquid vol-ume [18, 19]. Nevertheless, Zener tunneling is supposedto play a role [11] at the metal/liquid interface owingto the electric field enhancement (tens of GV/m) due tolocal metallic surface asperities.Currently, the prevailing opinion is that the multi-plication mechanism is associated with the appearanceof nanoruptures or nanovoids in the bulk liquid, whichoccurs as a result of the ponderomotive electrostrictiveforces induced by highly non-uniform time-dependentelectric fields [20–23]. These voids then provide suffi-cient collision-free space for acceleration of initial andsecondary electrons to energies exceeding the ionizationpotential of water molecules. The initial seed electronsare supposed to be provided by the detachment fromOH − ions present at the moment of void formation atthe void/water interface. Some theories even assume thathydroxide density is increased at the cathode-side end ofvoids, which can further increase the injection of initialelectrons into the voids [24].Recent theoretical studies of nanosecond discharge inpolar liquids based on electrostriction usually assumethat cavities are initiated from fluctuations determinedby the very fast switching dynamics of H-bonded H Omolecules. Conditions under which these fluctuationscan grow have been investigated assuming that sphericalsymmetry accounts for cavity expansion dynamics. Theability of electrons to ionize water molecules is then inves-tigated based on the electron energy, after the diameterof the cavity is traversed. It is important to stress thatthe spherical geometry of the cavities in external elec-tric fields is a rather oversimplified approximation. Thestrong electric field is not only responsible for the expan-sion of the fluctuations but also causes rapid stretchingin the direction of the electric field. As a result, voids in the form of long fibrous hollow structures are created inthe bulk of water [21, 25].
Model and methods .—Up to now, theoretical studiesmostly considered the spherical cavity geometry and theone step acceleration-ionization mechanism. In this work,we propose a new scenario for electron multiplicationthat can occur inside significantly stretched voids. Weshow that electrons can propagate inside the cavity whilebouncing off the water surface. This bouncing-like mo-tion allows for the repeated acceleration of electrons and,thus, enhances the production of secondary electrons inthe cavity. We study this phenomenon through MonteCarlo simulation, and we formulate the minimal condi-tions that need to be fulfilled for electrons to multiply.To the best of our knowledge, such a study has not beentackled before, and the presented results shed light onone of the missing steps in electrostriction-based theoriesfor fast discharge initiation in liquid water.In the present model we consider cavities as long cylin-drical voids of radius R with a homogeneous electric fieldof strength E oriented along the axis. Note that theelectron motion in the cylinder is analogous to projectilemotion in a vertical tube subjected to the action of thegravitational force. It can be shown that the product ofthe electric field strength and the cavity radius, E · R , isa scaling parameter for discussion of simulation resultsin different cylinder radii and electric field strengths.We suppose that electrons in the void accelerate freely,i.e., without collisions, and interact only with watermolecules when they penetrate the water surrounding thevoid. This assumption is justified because the character-istic spatial scale of these voids is in the order of 100 nm,i.e., much shorter than the mean free path of electrons,which is approximately 6 µ m in the corresponding equi-librium water vapor pressure.The electron interaction with water near the surfaceof the void implies two basic outcomes: (a) the electronpenetrates the water bulk and terminates there, or (b)the electron is bounced back to the void and is accel-erated by the electric field again. These situations areschematically shown in Figure 1. FIG. 1: Electron interactions at the void/water interface: (a)Termination: The primary electron thermalizes in the liquidand terminates there (b) Bouncing: the primary electron isbounced back to the void as a result of collisions with watermolecules. New electrons can be emitted to the void duringboth processes (red dashed arrows).
In both cases, a certain number of secondary electronsmay be emitted from the surface of water to the void.In the case of an electron penetrating the void/water in-terface, the electron’s energy may dissipate through in-elastic collisions. Ionizing collisions in the bulk liquidwill produce H O + ions and secondary electrons. TheH O + ions are converted to H O + +OH, while electronsare hydrated in picosecond time scales [26]. Excitationreactions driven by the low-energy electrons ( (cid:54)
10 eV)will produce H O ∗ (where ∗ stands for excited electronicstates, e.g., ˜a B , ˜A B , ˜b A , ˜B A , . . . ) [27, 28]. Alltransient species formed during electron-driven reactionswill be thermalized in the bulk liquid, forming a varietyof species (e.g., H , H O and OH − ) [26].In the following, we investigate conditions under whichthe bouncing of electrons leads to an increase in the num-ber of electrons propagating along the void. For the de-scription of electron interactions with water, we use thestate-of-the-art simulation framework Geant4-DNA [29–32]. This framework offers a variety of models to sim-ulate the physical interactions of electrons in liquid wa-ter. We use the Geant4-DNA physics ‘option 4’ construc-tor, which includes the Emfietzoglou-Kyriakou dielectricmodel for inelastic scattering and the Uehara screenedRutherford model for elastic scattering of electrons [33].The Sanche model is included for vibrational excitation[34], and the Melton model for attachment [35]. We as-sume that sub-excitation electrons, i.e., electrons withenergy below ε se =7.4 eV, cannot contribute to furtherionization in water, as the acceleration of electrons dueto the electric field is ineffective because of their high col-lisionality in water. Thus, no tracking of sub-excitationelectrons immersed in water is performed in the simula-tion. Note that this energy cutoff is a parameter of thesimulation. Results .—We compare two distinct scenarios for elec-tron propagation along the void in terms of the efficiencyof electron multiplication. The first scenario, shown inFigure 2(a), considers bouncing of electrons with theirpossible multiplication along and at the end of the void.The second scenario, shown in Figure 2(b), considers a di-rect flight through the void with ionization possible onlyat the end of the void. Concerning the model param-eters, we choose E · R to vary between 15–25 V, whichincludes a break-even point between non-multiplying andmultiplying cases. For long cylinders (e.g., R = 30 nm),it corresponds to the homogeneous electric field variedbetween 0.5 GV/m and 0.83 GV/m.Let us first focus on the scenario with bouncing-like prop-agation of electrons. In the Geant4-DNA simulation, anensemble of primary electrons is launched from the sur-face of water to the void with an isotropic velocity distri-bution at initial energy 7 . E · R ∈ { , , , , , } V. Notethat the stationary growth or the decay in the numberof electrons can be characterized by the exponential law N ( z/R ) = N exp ( Rαz/R ), where N is the number ofelectrons at z = 0, and 1 / ( αR ) is a characteristic e-foldmultiplication distance. The transient phase of electronpropagation with respect to the initial condition is notshown here and the length of the void is assumed to besufficiently long ( L = 150 R ) to ensure that the interac-tion of electrons with the end of the void has no impacton observed results. The parameter αR shows a lineardependence on E · R for E · R >
17 V, as seen in Fig-ure 3(b). The break-even point value of E · R = 19 . αR = 0. For E · R > . ε and theangle of incidence ψ ∈ [0 , π/ E · R required for the electron avalancheto be developed inside the cavity. Therefore, the valueof 19 . E · R that guarantees the occurrence of an avalanche.As discussed above, the electrons that impact thevoid/water interface will either terminate in water orbounce back to the void. The probability of the elec-tron bouncing back to the void as a function of the inci-dent electron energy and angle of incidence ψ ∈ [0 , π/ FIG. 2: Two scenarios for electron propagation in long ruptures: (a) bouncing and multiplication of electrons along the surfaceof the void, (b) direct flight with ionization at the end of the rupture. Red lines and dots represent the profile of the cylindricalvoid, blue lines show electron trajectories, and black dots describe locations of electron collision events. Geant4-DNA simulationwas performed for R = 30 nm, E · R = 25 V and length L of the void 25 R for a single initial electron with energy 7 . E · R ∈ { , , , , , } V follows the exponentiallaw N ( z/R ) = N exp ( αRz/R ), where N is the number ofelectrons at z = 0, and 1 /αR is a characteristic e-fold mul-tiplication distance. (b) The coefficient αR as a function of E · R . For E · R >
17 V, αR is a linear function of E · R .The value of E · R required for the conservation of electronspropagating along the tube is 19.4 V, i.e., when αR = 0. the probability of electron bouncing is complementary tothe probability that the electron will terminate in bulkwater.Another important quantity that describes interac-tions of the primary electron with water is the numberof secondaries that are created in the bulk of water andthermalize inside. This quantity is shown in Figure 6,and it increases with increasing energy of the incidentelectron. Note that it decreases with the incident anglebecause of the increasing bouncing probability, as shownin Figure 5.In the following, we compare the electron bouncing-like propagation scenario with the direct flight scenarioin terms of the total number of ionizing events generatedin bulk water. We denote the total number of ionizationsrecorded in case of bouncing-like propagation as N bp . FIG. 4: Number of secondary electrons emitted from waterby primary electron impact as a function of incident electronenergy for five incident angles ψ ∈ [0 , π/
2) calculated usingGeant4-DNA.
Note that this number is a sum of all ionizations that werecaused by impacting electrons on the void/water inter-face, both along the cavity surface, and at the end of thecavity. For the direct flight scenario, the same numberof initial electrons at energy ε se is launched with velocityoriented directly along the axis of the tube. These elec-trons are accelerated by the applied electric field, andpenetrate into the bulk of water at the end of the tube.All ionization events are recorded, and we denote thisnumber as N df . Figure 7 shows the ionization gain ratio N bp / N df for E · R ∈ { , , , , , , } V, asa function of the cavity length
L/R . For E · R higherthan the break-even point value, the bouncing-like prop-agation of electrons along the cavity is more efficient interms of the total number of ionizations produced in bulkwater. Note that this efficiency is higher for higher valuesof E · R .Upon analyzing the various simulation results, wefound that the bouncing-like propagation scenario can FIG. 5: Probability of the primary electron bouncing backto the void P B ( ε, ψ ) as a function of the electron energy forfive incident angles ψ ∈ [0 , π/
2) calculated using the Geant4-DNA.FIG. 6: Number of secondary electrons generated by the im-pact of primary electrons inside the liquid, as a function of theelectron energy for five incident angles ψ ∈ [0 , π/
2) calculatedusing Geant4-DNA. be characterized by a certain effective propagation veloc-ity that is in the order of 10 m/s. Considering particularconditions illustrated in Figure 2(a) (for R = 30 nm and L/R = 25), this velocity reaches about 3 × m/s. Conclusions .—We analyzed two scenarios for elec-tron propagation/multiplication in nanovoids createdand stretched by the action of strong pulsed electric fieldsin liquid water. We show that an electron avalanche inthe void can be created. This avalanche is fed by the sec-ondary electron production on the void/water interfaceand is enhanced by the possibility of electrons bouncingoff the cavity surface. The avalanche is able to grow ifthe product of the cavity radius and the electric field in the cavity is higher than 19.4 V. Comparing the totalnumber of ionizing events that occur in the bulk waterbecause of this electron avalanche to the total number ofionizations when electrons directly fly through the cavityshows that the bouncing-like propagation ensures higherionization yield when the condition for electron avalanchegrowth is satisfied. Note that with respect to the value ofthe cutoff energy for electron tracing, all conclusions arevalid. However, lowering the energy cutoff would leadto the enhancement of secondary emission of electronsin the cavity. As a result, the electron ionization gainwould be even higher for the bouncing-like propagationscenario.
FIG. 7: Ionization gain ratio: N bp / N df , where N bp is the totalnumber of ionizations for bouncing-like propagation, and N df is the total number of ionizations for the direct flight, for E · R ∈ { , , , , , , } V as a function of the cavitylength
L/R . We observe that the effective velocity of bouncing-likepropagation is in the order of 10 m/s, and this shouldrepresent the upper limit for the velocity of expansionof luminous fronts driven by avalanche electrons. It isimportant to note that this 10 m/s limit is in agreementwith recent experimental results evidencing an approxi-mately linear initial expansion of the luminous fronts ondistances in the order of hundreds of micrometers, witha propagation velocity ∼ × m/s [36].Moreover, the results of this work are consistentwith recent experimental observations on the causalrelation between two coupled dark (non-luminous) andluminous discharge phases [37]. Experimental resultsconfirmed that the luminous phase (evidencing thepresence of high-energy electrons) implies the prioroccurrence of the dark non-luminous phase (evidencingbulk liquid perturbed by the electrostriction). Observeddelays between the two phases in the order of hundredsof picoseconds support the present concept based onthe onset and growth of electron avalanches insidenanoruptures aligned along the applied external electricfield. Further insights might be gained by analyzingthe characteristic parameters of the avalanches (electrondensity and energy distribution function) developing inlonger voids (tens of micrometers). Acknowledgments [1] P. L. Geissler, C. Dellago, D. Chandler, J. Hutter, andM. Parrinello, Science , 2121 (2001).[2] A. Volkov, V. Artemov, A. Volkov, and N. Sysoev, Jour-nal of Molecular Liquids , 564 (2017).[3] S. Cukierman, Biochimica et Biophysica Acta (BBA)-Bioenergetics , 876 (2006).[4] S. Imoto and D. Marx, Physical Review Letters ,086001 (2020).[5] M. N. Shneider and M. Pekker,
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