Analysis and Review of Chemical Reactions and Transport Processes in Pulsed Electrical Discharge Plasma Formed Directly in Liquid Water

Abstract

Electrical discharges formed directly in liquid water include three general cases where (a) streamer-like plasma channels form in, but do not span, the electrode gap, (b) spark discharges produce transient plasma channels that span the electrode gap, and (c) arc discharges form plasma channels with relatively longer life times. Other factors including the input energy (from <1 J/pulse to >1 kJ/pulse) as well as solution properties and the rates of energy delivery affect the nature of the discharge channels. An understanding of the formation of chemical species, including the highly reactive hydroxyl radical and more stable molecular species such as hydrogen and hydrogen peroxide, in such plasma requires determination of temporal and spatial variations of temperature, pressure, plasma volume, and electrical characteristics including current, voltage (electric field), and plasma conductivity. In spark and arc discharges analysis of the physical processes has focused on hydrodynamic and thermal characterization, while only a limited amount of work has connected these physical processes to chemical reactions. On the other hand, the most successful model of the chemical reactions in streamer-like discharges relies on simple assumptions concerning the temperature and pressure in the plasma channels, while analysis of the physical processes is more limited. This paper reviews the literature on the mathematical modeling of electrical discharges in liquid water spanning the range from streamer-like to spark and arc discharges, and compares the properties and processes in these electrical discharges to those in electron beam radiolysis and ultrasound.

Introduction and Scope

The understanding of electrical discharge plasma formed either directly in liquid water or in a gas phase that contacts the liquid is important for many scientific and engineering topics, and interest in this field is growing, particularly for applications to biomedicine and to materials synthesis [1–12]. The process of converting electrical energy into useful chemical species is of specific interest for the biomedical and chemical processes where radicals and molecules formed from water initiate a wide variety of useful biochemical and chemical reactions. Previous reviews have covered in detail the chemical reaction pathways and some of the physical processes involved in electrical discharges formed directly in the liquid and in a wide range of cases where plasma generated in the gas phase contacts the liquid phase [7–9]. In contrast, the present contribution focuses on a detailed review of the mathematical modeling efforts that describe how the discharge leads to the formation of chemical products for the case where the plasma is formed within the liquid phase using pulsed electrical discharges. We focus on electrical discharges formed directly inside the liquid phase because these discharges are very different from those that are initiated in a gas phase that contacts the liquid surface, and a separate review is required for that topic.

Pulsed electrical discharges directly in the liquid phase can be characterized by the input energy of the pulse (very broadly grouped into >1 kJ/pulse, ~1 J/pulse, and <1 mJ/pulse), by the rate of delivery of the pulse (also very broadly classified as subsonic and supersonic [13]), and by the nature of the discharge (arc or spark vs. streamer-like [14–16]). Other important factors are the conductivity of the contacting solution (which can affect the rate of delivery of the discharge), the size of the electrode gap (i.e., sufficiently high input power in small gaps can lead to arcs and sparks), and the pulse characteristics (pulse energy, rise time, duration, and electrode polarity). Spark discharges produce conductive channels that bridge the gap between the electrodes; sparks are transitory while arcs generally maintain the plasma channel spanning the electrodes for longer periods of time [17]. In streamer-like discharges the plasma channel dissipates before it can completely span the electrode gap. Typically, kJ/pulse arc discharges, also referred to as electrohydraulic discharge, produce large thermal effects, significant UV emissions, and, characteristically, large shockwaves; these discharges have also been used for chemical pollution control and bio-decontamination [2]. Bubbles may be formed by the electrical heating from the discharge over a range of input power and discharge conditions, but bubbles are particularly important for the high energy kJ input discharges. Under some experimental conditions where the time scale for discharge formation is long (>10–100 μs) compared to the time scale to form bubbles by heating the liquid, the generated bubbles may affect electrical breakdown whereby the discharge propagates though the gas phase of the bubbles [18]. As the energy per pulse decreases and/or as the rate of discharge propagation increases, the magnitude of the thermal, UV, and hydraulic (shockwaves) effects are reduced and relatively more energy is directly converted to chemical processes, e.g., ionization and dissociation.

Most recently, extremely fast low energy mJ pulses have been suggested to initiate non-thermal plasma by electronic processes in the liquid where no bubbles or significant thermal effects occur [19]. Clearly, no single theory may be able to cover the large range of behavior from the mJ to kJ pulses over all solution and electrical conditions, even within this relatively specific range of types of pulsed discharges directly in water. In addition, while our understanding of the connections between the physical nature of the discharge and the chemical species production is perhaps more advanced for the J to kJ pulses (at least for sparks and arcs), there are still significant questions that need further study. It is the purpose of this review to address some of these challenges.

Pulsed electrical discharges in liquid water have similarities and differences with three other processes whereby high energy physical processes interact with the liquid state: (a) radiation chemistry of the liquid phase, (b) ultrasonic chemistry, and (c) pulsed laser breakdown. All of these processes have large characteristic spatial and temporal variation of energy input and are examples of so-called non-homogeneous processes [20]. In radiation chemistry, high energy particles interact with the liquid phase causing track-like structures with branching patterns, and these structures are key to determining the subsequent solution chemistry [21, 22]. While electron beams with energies from keV to MeV are the most common radiation technologies, there is significant interest in the interaction of protons, alpha particles and high energy photons (X-rays and γ-rays) with matter for radiation biology [23], and thus interactions of radiation with liquid water are particularly well studied [21, 24, 25]. High frequency (20 kHz–500 MHz) ultrasonic processes lead to cavitation and transient high temperature and pressure regions in the liquid [26]. Pulse laser breakdown, spanning a wide range of energy levels from μJ to mJ and J per pulse, is most importantly used in ocular surgery where thermal and mechanical effects predominate [27]. The chemistry of radicals formed in water by electrical discharges also has some common features with the advanced oxidation processes including UV, supercritical fluid, and other chemical based technologies [28].

In this paper we emphasize the roles of electrical discharge formation on initiating and sustaining chemical reactions and do not focus explicitly on breakdown theories. The emphasis is therefore on post-breakdown phenomena. However, since chemical reactions are generally strongly influenced by physical and chemical solution properties including temperature, pressure, the formation of UV emissions and the generation of plasma electrons, it is necessary to consider the physical and chemical processes generated by the electric discharge. There are some key issues and questions that must be addressed in developing accurate models of the chemical reactions in such a discharge. Many of these issues are not yet fully resolved and the purpose of this manuscript is to help frame the modeling efforts in order to stimulate further research on this topic. Some key questions that need further study are given below.

• (1) How are the chemical and physical phenomena coupled?

• (2) What is the role of plasma discharge channel structure on chemical reactions?

• (3) What are the appropriate temperatures, pressures, and volumes for the local chemical reactions?

• (4) How should the spatial transition region from the plasma domain to the bulk solution be modeled? What are the effects of this transition and the corresponding pressure and temperature gradients on chemical reactions and the distribution of chemical species?

• (5) What are the relative roles of thermal (pyrolysis or high temperature cracking) and non-thermal (direct electron collision) reactions?

• (6) What are the roles of the bulk phase chemical reactions and how do they compare and differ from the reactions in the plasma?

Physical Description of Discharges in Water

An example of a 1 J/pulse (60 W) [29, 30] streamer-like electrical discharge with a point-to-plane electrode configuration is shown in Fig. 1. The streamer-like plasma channels are formed in low conductivity (10⁻⁴–10⁻² S/m) water with a relatively large electrode gap and low input power. The corresponding current and voltage waveforms of approximately 1 μs duration are shown in Fig. 2. High speed images of similar discharges show that the velocities of propagation of the plasma channels vary from 3 × 10³ m/s to 4 × 10⁴ m/s for long pulses (>500 ns) and up to 5 × 10⁴ m/s for very short pulses (70–200 ns) with input energy from as low as 0.4 J/pulse to 176 J/pulse [16, 31–33]. For plasma channel velocities in the range of 10³–10⁵ m/s the times to reach an example plasma length or gap distance of 1 cm are 0.1–10 μs. The diameters of such plasma channels range from 10 to 100 μm [31, 33, 34]. Table 1 shows sample diffusion and thermal conduction times based upon such channel dimensions for gases and liquids. Gas phase diffusion and conduction time scales are 10 and 1 μs, respectively, indicating relatively fast transport, and therefore small concentration and temperature gradients within the discharge channels. (These are decidedly conservative estimates since the temperature in the plasma channel and therefore transport properties are generally much higher.) Mass and heat transport based upon liquid phase properties under ambient conditions is much slower where diffusion and thermal conduction time scales are 100 ms and 700 μs, respectively. The relatively lower rates of transport in the liquid phase suggest the formation of sharp gradients in temperature and concentration between the plasma region and the bulk solution and also that much of the transfer between the two occurs by mixing after the plasma has dissipated rather than during the plasma propagation; this is the reason for the commonly used adiabatic plasma channel assumption.

Fig. 1

Streamer-like electrical discharge in low conductivity water. Approximately 1 J/pulse and 60 W [119]. © 2009 WILEY–VCH Verlag GmbH & Co. KGaA, Weinheim. Reprinted, with permission, from (Shih and Locke [119])

Fig. 2

Current and voltage waveforms for streamer-like discharge in low conductivity water. Approximately 1 J/pulse and 60 W

Table 1 Typical time and length scales for various processes consider

While it was found that the solution conductivity (distilled to tap water; e.g., 1–5 μS/cm to >1 mS/cm) and input energy (1.6–0.4 J/pulse) did not affect the velocity of streamer-like discharges [16, 33], solution conductivity and input power strongly affect the size of the plasma channels [29] and can be very important in the electrical breakdown of water and the subsequent thermal and hydrodynamic phenomena and chemical reactions [35–37]. For example, the Maxwell relaxation time, ε/σ, or ratio of electrical permittivity to conductivity is 7 μs and 7 ns for water solutions of 1 μS/cm and 1 mS/cm, respectively [35, 36, 38]. This time scale characterizes the “charge relaxation due to ohmic conduction”, and thus for very short pulses relative to this relaxation time the fluid behaves as a dielectric [38]. For pulses shorter than this characteristic time the electric field induces polarization in the fluid before ion flow can induce conductivity current [35, 36, 38]. High solution conductivity, as implied in Fig. 3, leads to shorter, more intense plasma channels with more formation of bubbles and consequently larger roles of hydrodynamic and thermal processes. In addition, solution conductivity has a strong effect on UV emissions [39] and chemical species formation [37, 39, 40].

Fig. 3

Streamer-like discharges for various water conductivities. Approximately 1 J/pulse and 60 W [37]. © [2011] IEEE. Reprinted, with permission, from (Shih and Locke [119])

Hydrodynamic phenomena induced by the electric field and current include thermally induced bubble formation and bubble expansion. These processes are typically more important for high energy and arc or spark discharges (e.g., kJ/pulse) and may also be important in the lower (J/pulse) energy regimes, particularly for longer pulses, higher solution conductivity, and shorter discharge gaps where sparks occur. Table 2 shows various model and experimental results for velocities of plasma-bubble hydrodynamic expansions. These velocities range from 10² to 10³ m/s and are generally lower than the speed of sound in water (1,500 m/s). The corresponding time scales for these velocities are 10–100 μs for 1 cm gap distances. This time scale is larger than that for the lower power streamer-like, J/pulse discharges discussed above and shown in Fig. 1. The diffusion and conduction time scales for 1 mm gas and liquid regions (typical bubble sizes) shown in Table 1 are greater than 10 ms indicating much slower diffusive and conductive transport than the velocity of expansion.

Table 2 Summary of plasma arc and spark model and experimental properties

Of course chemical reaction time scales are strongly influenced by temperature, concentration of reactants, and rates of mixing or diffusion [41]. The reaction rate constant for a second order diffusion-limited reaction in a liquid phase under ambient conditions is 10¹⁰ M⁻¹s⁻¹, and if one reactant, for example a chemical probe, has a concentration of 1 mM, the time constant for the corresponding pseudo-first order reaction is 10⁻⁷ s. In the case of a gas phase, the diffusion-limited reaction rate constant is 7.8 × 10¹⁰ m³/mole-s, and thus for a probe at 0.1 % of the gas concentration at 100 °C and 1 atm the time constant for the pseudo-first order reaction is 10⁻¹¹ s. Many radical reactions, particularly those that involve the hydroxyl radical, OH^∙, react at or near the diffusion limitation [42]. Therefore, these short reaction time scales indicate that the very fast radical reactions can occur throughout the relatively longer (microsecond) times of the physical processes (streamer propagation and bubble expansion) mentioned above. Such fast radical reactions also occur locally in the discharge and at the interface between the discharge and the liquid solution. The formation of more stable products such as hydrogen (H₂) oxygen (O₂) and hydrogen peroxide (H₂O₂) occur by radical reactions in these regions and these more stable molecular products transport into the bulk solution by mixing.

Figure 4 [43] shows an example of ultrasound in water whereby small bubbles are induced in a track-like structure reminiscent of that shown in Fig. 1 for electrical discharges in water. Such filamentary track-like structures are commonly formed in ultrasound in water [43–46]. Some ultrasonic reactors utilize dissolved gases such as argon or air to facilitate the bubble formation [47]. As discussed further below, the physics of the thermal and flow processes in some electrical discharge processes are based upon the same foundation as those for ultrasound [26]. Nevertheless, ultrasound has significant differences with regards to measured chemical efficiencies for the formation of key indicator species, such as hydrogen peroxide and hydroxyl radicals [6, 48]. The energy yields, molecules/100 eV, for H₂O₂ production by a 1 J/pulse streamer-like discharge in water and by sonolysis are 7.51 × 10⁻² and 8.48 × 10⁻³, respectively, and the corresponding values for OH^∙ are 1.01 × 10⁻² and 4.92 × 10⁻³, respectively [48]. The hydrogen peroxide yields for a wide range of electrical discharges in water as well as in contact with water span three orders of magnitude (one order lower and two higher than the value given above), and data for direct streamer-like discharge in water cluster around the value given above but have not generally been reported for a wider range of conditions [6]. Quantitative measurements of hydroxyl radicals and hydrogen peroxide for arc and spark-like discharges are not available, and for hydrogen peroxide this is likely due to the high temperatures in the entire solution (not just the plasma) which cause significant degradation of this species.

Fig. 4

Photograph of ultrasonic pulses in water [43]. Parlitz et al. [43] by permission of the Royal Society

In contrast, the speed of light, 3 × 10⁸ m/s, characterizes the interaction of electrons from radiation sources with water. As the energy input increases from 1 keV to 10 MeV, the velocity of electrons interacting with water increases from 1.8 × 10⁷ m/s to near the speed of light, see Table 1. In contrast, the penetration depth can be quite small. For energy loss of 0.02 eV/Å, [22] the depth of penetration increases from 5 μm to 5 cm over the same energy input scale (see also Fig. 5 for summary of various simulations of penetration depth with electron energy [49]). The calculated time scales of 10⁻¹³–10⁻¹⁰ s are quite small and fall within the chemical stage (10⁻¹³–10³ s) of radiation chemistry [22]. Radiation processes typically lead to only very small temperature increases, and, thus, hydrodynamic effects are generally not important. The electron collisions with water molecules in the track structures lead to ionization and excitation along meandering pathways as the electrons penetrate into the liquid. Thereafter, the diffusion of the primary radiation chemical products from the tracks strongly influences the formation of subsequent species. Figure 6 shows an example of the tracks formed in liquid water by 10 keV electrons and the total electron dose for the same conditions [22]. The initial and 1 ps yields for OH^∙ in a 1 MeV electron are approximately 5.5 molecules/100 eV, however, these values decay with time as the OH^∙ react to form H₂O₂ and other species [50]. The energy yields for H₂O₂ production by electron beam radiolysis (5–25 eV tracks in 1 MeV) varied from 4 × 10⁻² at 10⁻¹² s to 8 × 10⁻¹ at 10⁻⁷ s [50]. Therefore, with regards to the formation of these two chemical species the streamer-like electrical discharge (with 1 J/pulse in deionized water with fast rise time) falls between radiation and sonolysis.

Fig. 5

Penetration depth for electrons into water as a function of the electron energy [49]. Reprinted with permission from Plante and Cucinotta [49]. IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

Fig. 6

Simulations of 10 keV electron tracks in water showing a primary electron track structure and b dose distribution [166]. Reprinted with permission from Pimblott et al. [166]. Copyright (1996) American Chemical Society

It has recently been proposed that for very short mJ per pulse, electrical discharges in water lead to non-thermal interactions [19, 51]. Figure 7 shows an example of such a discharge and the plasma is formed on the time scale of 10⁻¹⁰ s at voltages of 220 kV. The velocity of propagation of the plasma channel was found to be 5 × 10⁶ m/s with channel diameters of 50–100 μm. No evidence of bubble formation or hydrodynamic effects was observed for these very rapid rise (150 ps) mJ per pulse discharges. This discharge has very little heating and the chemical reactions may be more similar to radiation chemistry than sonochemistry.

Fig. 7

Discharge propagation for ultrafast mJ electrical discharge pulse in distilled water [51]. © (2011) IEEE. Reprinted, with permission, from Starikovskiy et al. [51]

Very low power (μJ to mJ per pulse) lasers generate some hydrodynamic effects, but little effects on chemical reactions have been reported. Lasers generate acoustic waves [52] and their interactions with matter lead to dielectric breakdown (10¹⁰ W/cm²), vaporization or material ablation, thermoelastic process, electrostriction, and radiation pressure [52]. Laser interactions with liquids have been studied [53] and pulsed lasers have been very successfully utilized in medicine. In such applications power densities and time scales of 10⁻³–10⁻¹⁵ W/cm² and seconds to 10⁻¹⁵ s, respectively, have been used for shock wave generation and tissue photo-destruction with very small thermal damage to adjacent tissue. For example, in a 1 mJ 6 ns laser pulse local temperatures can reach 10⁴ K and pressures 20–60 kbar in 100 μm plasma regions and 60 % of the total energy is converted to mechanical energy [27]. Some photochemical reactions have been reported for the case of photosensitive dyes, added to a tumor, to generate singlet oxygen, but other chemical effects have not been reported [54].

The chemistry and physics of electrical discharges in water also have similarities to a number of other processes including glow discharge electrolysis [55–58] and plasma electrolysis [10], supercritical and wet air oxidation chemistry (due to high temperatures and pressures) [59–64], and other advanced oxidation processes (due to common radical chemistry based on hydroxyl and other radicals) [28] as well as photochemical processes (due to significant UV formation by plasma). It is also interesting to consider another non-homogeneous process, that of microwave chemistry, where in contrast the length scale of the spatial non-homogeneities are much smaller that the micron-scale plasma channels discussed here [65].

Hydrodynamic Phenomena

The application of large electric fields to liquids can lead to significant coupled thermal, electrical, and hydrodynamic effects. The large electric fields can induce body forces that drive fluid motion by increasing the pressure or causing large fluctuations of pressure which in turn may generate bubbles or cause existing bubbles to expand. High electric current can also lead to high temperatures by Joule heating and the resulting high temperature fluid leads to bubble generation and expansion. Pressure, temperature, and the phase conditions (i.e., presence of gas bubbles and size of plasma region) can clearly affect the type and rates of chemical reactions induced by the plasma discharge in water. Some of the reported work that deals with hydrodynamic and thermal aspects focuses on the pre-breakdown events and discharge initiation and electric field propagation, others are focused on determination of post-breakdown physical properties of the fluid including spark and arc formation and plasma/bubble expansion. Analyses of hydrodynamic effects with electrical discharges in water have been utilized for determination of temperature and pressure in various discharges [66–72]. Analysis of the fundamental hydrodynamic behavior builds upon work from the general theories of detonation [73] and the specific aspects related to underwater explosions [74], and basic cavitation phenomena [75] with the addition of the effects of large electric fields. In this section we review literature related to these hydrodynamic and physical effects and in the following section we focus on the chemical reactions.

Equations of Motion and Continuity

Cauchy’s equation of motion, Eq. (1), and the continuity equation, Eq. (2), are the bases for analysis of the velocity, v, and mass density, ρ, in a fluid as functions of the pressure, P, body forces, f, and viscous stress, τ [76–78].

$$ \rho \frac{D \varvec{v}}{Dt} = \rho \varvec{f} - \nabla P + \varvec{\tau} $$

(1)

$$ \frac{\partial \rho }{\partial t} = \nabla \cdot \left( {\rho \varvec{v}} \right) $$

(2)

For compressible fluids these equations need to be supplemented by a suitable equation of state and constitutive relationships are needed for the body and viscous forces. In some plasma models the ideal gas law is used [69] while in some cases equations of state for compressible liquids have been used (a common form for water is the Tait equation of state [79]) while other empirical [80] and analytical [81] functions have been used. Typically, the viscous stress tensor for Newtonian fluids can be considered, although viscous effects are generally neglected in the plasma models reviewed here [77]. In the case of fluid dielectrics under the addition of an applied electric field, E, the additional force terms, Eq. (3), must be included in Cauchy’s equation to account for the electric field-induced forces on free charge carriers, ρ _f , spatial variations in the fluid dielectric constant, ε, and variation of the dielectric permittivity with density [38, 66, 82].

$$ \nabla \left[ {\frac{1}{2}\varepsilon_{\text{o}} \rho \left. {\frac{\partial \varepsilon }{\partial \rho }} \right|_{T} \varvec{E} \cdot \varvec{E}} \right] - \frac{1}{2}\varepsilon_{\text{o}} \varvec{E} \cdot \varvec{E}\nabla \varepsilon + \rho_{f} \varvec{E} $$

(3)

The electrical permittivity of a vacuum $\varepsilon_{\text{o}} = 8. 8 5\times 10^{ - 1 2}$ and the dielectric constant of water $\varepsilon = 80.$

If the dielectric constant varies with temperature and density, the gradient term in Eq. (3) can be given by [82]

$$ \nabla \varepsilon = \left. {\frac{\partial \varepsilon }{\partial T}} \right|_{\rho } \nabla T + \left. {\frac{\partial \varepsilon }{\partial \rho }} \right|_{T} \nabla \rho $$

(4)

Ushakov utilized the Onsager [83]-Kirkwood [84]-Frohlich [85] theory for the dielectric constant in polar liquids to show [66]

$$ \rho \frac{\partial \varepsilon }{\partial \rho } \approx K\varepsilon $$

(5)

The constant K is approximately equal to 1.5. Neglecting the spatial gradient term in the expression for dielectric constant, the viscous stress terms, and the free charge force in the equation of motion, Ushakov approximated Cauchy’s equation as [66]

$$ \rho \frac{D \varvec{v}}{Dt} = - \nabla \left[ {P - \varepsilon_{o} K\varepsilon E^{2} /2} \right] $$

(6)

The electrostatic field determined from the solution of the Laplace equation for a spherical tip electrode is ϕ _o /r ₁ where ϕ ₀ is the applied potential and r ₁ the radius of the electrode. For an applied potential of 20 kV with a 10⁻⁴ m radius of curvature needle, the electric field at the tip of the needle is 2 × 10⁸ V/m. This value is consistent with electrical breakdown voltages in water in the range of 10⁷–10⁹ V/m [35, 86–90]. For water the electrostriction force term can therefore be quite large, e.g. >1 GPa [66]. Utilizing the acoustic approximation of the equation of motion (i.e., neglecting the inertial and viscous terms in the equation of motion, Eq. (1), and considering local perturbations of pressure and density [78]) and the electrostriction force term, Korobeinikov [66, 91] shows that as the field increases from 1 × 10⁸ V/m to 20 × 10⁸ V/m the pressure increases from 3.5 MPa to 2.2 GPa. In addition, using Monte Carlo methods Joshi et al. [92] have found pressures of 1 GPa at fields of 5 × 10⁹ V/m. Such pressure changes have dramatic effects on the liquid and can induce cavitation as well as form shockwaves [26].

Ultimately the fluid velocity can be determined through solution of the equation of motion of the fluid which can account for the formation of shockwaves. However, if cavitation generates bubbles, it is necessary to determine the bubble size as well as the rate of bubble expansion (and/or contraction). The equations for the bubble wall motion are derived from a mechanical energy balance [26] including kinetic energy of expansion and pressure–volume work across the phase boundary that satisfy both the continuity equation and the equation of motion. One relatively general form of an equation to determine the radius of a spherical bubble, R, is given by the Rayleigh-Plesset Eq. (7) [26, 75]

$$ R\ddot{R} + \frac{3}{2}\dot{R}^{2} = \frac{1}{\rho }\left[ {P_{L} - P_{\infty } } \right] $$

(7)

The dot implies differentiation with respect to time, the subscript L indicates the liquid phase and infinity indicates pressure far from the bubble. A very simple approximation for bubble expansion in organic liquids was found using Eq. (7) neglecting the first term on the left hand side and including only the electrostatic contribution of the pressure [second term on the right hand side of Eq. (6)] for the case of an electric field arising from a spherical conductor [93]. The results showed good agreement with experiments for the bubble radius expansion in a limited range of cases where vapor pressure effects were small and for relatively low expansion velocity.

Various expressions for the pressure term on the right hand side as well as corrections to the left hand side of Eq. (7) include the effects of surface tension, hydrostatic pressure, vapor pressure in the bubble, viscosity, fluid compressibility, and electrostriction [26, 66, 94]. Korobeinikov derived an analytical approximation to the above expression with electrostriction, surface tension, and viscosity effects and the results were used mainly to analyze the bubble mechanism of electrical breakdown [94].

Alternative, and more general, forms of the bubble equation of motion, incorporate enthalpy, H, and the effects of high Mach number, i.e., the ratio of fluid velocity to speed of sound, c, following the Kirkwood-Bethe approximation given in Eq. (8) [26, 69, 74]

$$ \left( {1 - \frac{{\dot{R}}}{c}} \right)R\ddot{R} + \frac{3}{2}\dot{R}^{2} \left( {1 - \frac{{\dot{R}}}{3c}} \right) = \left( {1 + \frac{{\dot{R}}}{c}} \right)H + \frac{{\dot{H}}}{c}\left( {1 - \frac{{\dot{R}}}{c}} \right) $$

(8)

This equation was utilized in a model of spark discharge in water [95] where the enthalpy was determined by the Tait equation of state applied to the plasma (the plasma phase constituted the bubble phase). A modified version of Eq. (8) [96] developed for underwater explosions accounts for non-isolated bubbles [97].

Others [69, 98, 99] avoided solving the bubble equation of motion by using the Rankine-Hugoniot relation, which couples a mass balance at the front of a propagating shockwave with the equation of state for water (a modified Tait equation [69, 74]). The resulting equation established a relationship between pressure in the bubble and the velocity of the bubble wall.

$$ \left( {P + \beta } \right)^{3/7} = \left( {\frac{{3\sqrt {\rho_{o} } }}{{\sqrt {7\alpha^{1/14} } }}} \right)\dot{R} + \beta^{3/7} $$

(9)

For water, Kratel [69] used α = 3,001 atm, β = 3,000 atm, and ρ _o  = 1 g/cm³; this model was coupled to the energy and mass balances as discussed in following sections of this review. Equation (9) determines the rate of thermal expansion, through the temperature dependence of the pressure from the equation of state, as the major driving force for bubble expansion. For arc and spark electrical discharges the main energy input comes from Joule heating from the high current. For high current cases, thermally induced expansion may predominate while in high voltage short pulse discharge, i.e. high electric field cases, electrostriction may predominate. Full analysis including both electrical and thermal processes requires further study to determine the range of applicability of each.

Gidalevich and coworkers developed a hydrodynamic analysis of plasma expansion in liquids [70, 72]. They noted that direct solution of the equation of motion (coupled to the mass and energy balances) is quite difficult, and therefore they derived an alternative expression for the velocity of expansion utilizing the discontinuity across the shockwave in a manner similar to that used to derive Eq. (9) by Kratel [69]

$$ \dot{R} = c\left[ {\frac{{(P - P_{\infty } )\left( {\left( {1 + (P_{L} - P_{\infty } )/B} \right)^{1/7} - 1} \right)}}{{7B\left( {1 + (P_{L} - P_{\infty } )/B} \right)^{1/7} }}} \right]^{1/2} $$

(10)

B = 3,214 atm and c = 1.5 × 10³ m/s in water.

In another approach to consider sub and supersonic expansion, Gidalevich and coworkers [72] utilized directly the equations of motion and continuity to determine an equation for the radial expansion of a cylinder with a similar form to the Raleigh-Plesset equation, Eq. (7), that was derived for spherical expansion.

As the pressure and temperature increase inside the bubbles eventually at high enough temperature a plasma state is formed. The high temperatures and pressures affect chemical reactions as well as mass transfer of water and other reactants between the plasma-bubble and the liquid. These and other factors will be considered in following sections.

Electrohydrodynamics

More general developments in the field of electrohydrodynamics consider the full Maxell equations to determine time and spatial variations of the electric and magnetic fields [38] and detailed analysis of many problems in electric-field induced fluid motion have been reported. [100–103]. In many applications where the “electro-quasi-static” approximation [38] holds, the Maxwell equations reduce to

$$ \nabla \cdot \left( {\varepsilon \varepsilon_{o} \varvec{E}} \right) = \rho_{f} \quad \nabla \times \varvec{E} = 0\quad \frac{{\partial \rho_{f} }}{\partial t} + \nabla \cdot \varvec{J} = 0\quad \varvec{J} = \sigma \varvec{E} $$

(11)

where J is the current density and σ is the electrical conductivity. The second equation implies that the electric field can be represented as the gradient of a scalar potential. Analysis of the propagation of ionization waves requires the individual species mass balances for the positive and negative ions and electrons and the total current determined from the sum of the species motion due to the electric field, diffusion, and convection [38]

$$ \varvec{J} = \sum\limits_{i} {\varvec{J}_{i} = } \sum\limits_{i} {z_{i} F} \varvec{N}_{i} = \sum\limits_{i} {\left| {z_{i} } \right|Fu_{i} c_{i} } \varvec{E} + \sum\limits_{i} { - z_{i} FD_{i} \nabla c_{i} } + \sum\limits_{i} {z_{i} Fc_{i} } \varvec{v} $$

(12)

F is Faraday’s constant, v is the mass average velocity, and N _i , z _i , c _i , D _i , and u _i are the species total flux, charge, concentration, diffusion coefficient and electrical mobility, respectively. In most models of plasma discharge in liquids diffusion and convective transport at the average velocity are neglected. Solution of these electric field equations coupled with the species balances for electrons and positive and negative ions has been well developed for electrical discharge propagation in the gas phase [104–106] and for gases in contact with liquid water [107, 108]. This approach has been reported for streamer-like fast discharges in liquid water [87–89, 109] with specific implications on the breakdown in liquids [18, 110–112]. For local high electric fields on the electrodes of 10⁹ V/m with applied potential of 10⁵ V, simulations up to 300 ns found electric field propagation of 2 × 10⁵ m/s with high temperature plasma regions behind the front of the propagating electrical field of up to 10⁴ K and plasma channels with maximum conductivities of 0.2 S/m [87]. These velocities can be compared with experimentally determined plasma channel velocities in water that range from 10³ to 10⁴ m/s for long duration (>500 ns) pulses and 10⁴ to 10⁵ m/s for short (70–200 ns) high voltage pulses [16, 31, 32]. These supersonic velocities suggest minimal effects of hydrodynamic (bubble expansion) factors for these type of discharges, while for discharges with high conductivity solutions and much longer pulses, the speed of sound limits the rate of discharge propagation [13, 113–115].

Electroconvective and thermal instabilities were investigated using the momentum, continuity (for incompressible fluids), and thermal energy balances [109]. The effects of free charge were neglected, while the electrostriction term (first term in Eq. (3)) in the momentum balance, and the second two equations in Eq. (11) with a temperature, T, dependent conductivity given by Eq. (13) were used.

$$ \sigma (T) = \frac{A}{T}\exp ( - B/T) $$

(13)

A and B are constants. The characteristic time for the growth of thermal instabilities decreased with increasing applied voltage (to 40 kV) in agreement with experimental data [109].

For high power discharges and discharges with sufficient time for thermal bubble formation (particularly for arc and spark discharges), the hydrodynamic aspects of bubble formation and expansion are certainly important, but for the lower power very high voltage and very short pulse cases (streamer-like discharges), the electrical aspects may predominate. Further mathematical analysis is required to develop our understanding of the relative importance of hydrodynamic expansion and electric field propagation as the discharge and solution parameters are changed and to consider systems with characteristics between the two limits of arc and spark discharges and streamer-like discharges.

Comments on Phase Transition and Bubbles

Bubbles in liquids can be either microbubbles (<1 μm) that pre-exist in the liquid due to dissolved gases or macrobubbles (to mm or large) formed by heating or purposely injected into the liquid. Most analyses of the hydrodynamics of bubble expansion and contraction in electrical discharges presuppose existing small seed bubbles and in ultrasound gases may be intentionally introduced as small bubbles. While micron size bubbles are quite stable lasting several seconds in water under ambient conditions [116], very small impurities (10⁻⁶–10⁻⁹ m) can serve as nucleation sites for bubble formation, and in ultrapure, degased liquids, bubbles can form spontaneously. Models incorporating the nucleation of bubbles in water [115], organic [117] and other [118] liquids have been developed to analyze electrical breakdown in liquids. In one case for breakdown in dielectric liquids (n-hexane) under long pulse conditions, a theory has been proposed based upon homogeneous nucleation principles whereby Joule heating leads to “burst-like boiling” with subsequent electrical breakdown in a percolation network of gas bubbles [117]. In this case, with no impurities in the liquid, the liquid temperature can increase above the boiling point without the formation of bubbles until it reaches the thermodynamic stability limit, i.e., the spinodal temperature, whereby bubbles form spontaneously. The experimentally observed effects of applied pressure and pulse duration on the breakdown strength were supported by the model which included the role of impurities on the formation of nucleation sites and not superheating. Recent work also suggests that pre-existing gaseous or low density nucleation sites are important for breakdown in liquid water [18, 110, 111]. In the case of relatively large mm size bubbles in liquids, streamers propagate along the surface inside the bubbles provided the bubbles are in close contact with the electrode [107, 108, 119, 120].

It can be noted that analysis of bubble dynamics in liquids is very important for the design of industrial gas–liquid chemical reactors (e.g., bubble columns) and computational methods (3D front tracking, 2D volume of fluid, level set, and phase field) have been developed to handle the complex multiphase transport problems with steep gradients at the deforming phase boundaries [121]. It is out of the scope of the present review to go into detail on these computational approaches, however, it can be noted that (coupled level set with volume of fluid) simulations of bubbles of mm size formed in liquid water under the application of high velocity gas jets can accurately determine the interfacial motion, bubble formation, and bubble rise [122, 123]. The phase field approach has been used to simulate bubble formation and thermal conduction in solid nuclear fuels [124]. Future work remains to apply such techniques to electrical discharge formed bubbles in liquids.

Energy Balances

Overall Energy Balance

One formulation of the application of the first law of thermodynamics, the conservation of energy, to a control volume, V, balances the total internal and kinetic energies, with the flow term and the rate of heat transfer from the environment to the control volume, $\dot{Q}$, and the rate of work, $\dot{W}$, done by the surroundings on the control volume, V, by Eq. (14) [77].

$$ \frac{d}{dt}\int\limits_{V} {\rho \left( {\hat{U} + \frac{{v^{2} }}{2}} \right)} dV = - \int\limits_{S} {\rho \left( {\hat{U} + \frac{{v^{2} }}{2}} \right)} \varvec{v} \cdot \varvec{n}dS + \dot{Q} + \dot{W} $$

(14)

It should be noted that alternative derivations and more fundamental statements of the conservation of energy in electro-magnetic systems are available whereby electromagnetic energy terms are included [38, 125]. Here we include electromagnetic terms only as a source term due to Joule heating (in the heat flux, given below) because that is the only additional term included in all models reviewed here, even for models of very high electric fields. Analysis of the additional terms in the energy balance, for example terms analogous to the electrical force terms in Eq. (3), may be considered in future work.

The heat flux and work terms in Eq. (14) are given by integrals of the fluxes with the unit normal vector, n, over the surface area, S,

$$ \dot{Q} = - \int\limits_{S} {\varvec{q} \cdot \varvec{n}} dS $$

(15)

$$ \dot{W}_{S} = \int\limits_{S} {\varvec{v} \cdot \left({\varvec{n} \cdot \varvec{\sigma} } \right)} dS = -\int\limits_{S} {P\varvec{v} \cdot \varvec{n}dS} + \int\limits_{S}{\varvec{v} \cdot \left( {\varvec{n} \cdot \varvec{\tau} } \right)}dS $$

(16)

For multicomponent systems the total heat flux includes contributions from conduction, diffusion, radiation, and the Dufour effects [77], and we have added a source term for electrical heating, q ^S, (such sources can either be homogeneously distributed and directly incorporated into Eq. (14), or they can be determined from transport across the boundaries)

$$ \varvec{q} = - k\nabla T + \mathop \sum \limits_{i = 1}^{N} \varvec{J}_{i} \hat{H}_{i} + \varvec{q}^{rad} + \varvec{q}^{x} + \varvec{q}^{S} $$

(17)

Several models of spark discharges have neglected the conduction, diffusion, and Dufour terms in the total heat flux, the viscous terms in the work term, and kinetic energy in the overall energy balance [95, 97]. The energy balance over the plasma zone is then given by the time rate of change of total internal energy in terms of heat flow (source), pressure (P)-volume (V) work, and internal energy transfer by mass flow

$$ \dot{E} = \dot{Q} - P \dot{V} + \hat{U} \dot{N} $$

(18)

$$ \dot{E} = \frac{d}{dt}\int\limits_{V} {\rho \left( {\hat{U}} \right)} dV\quad \dot{V} = \int\limits_{S} {\varvec{v} \cdot \varvec{n}} dS\quad \dot{N} = - \int\limits_{S} {\rho \varvec{v} \cdot \varvec{n}} dS $$

(19)

These models [95, 97] included in the heat flow term the energy need to increase the temperature of the incoming vapor at constant pressure up to the temperature of the internal contents of the bubble

$$ \dot{Q} = \dot{Q}_{s} + \dot{Q}_{m} + \dot{Q}_{r} $$

(20)

$$ \dot{Q}_{m} = \dot{N}\left( {e_{evaporization} +\int\limits_{{T_{e} }}^{T} C_{P} \left( {T^{\prime } }\right)dT^{\prime } } \right) $$

(21)

The radiation term is given by

$$ \dot{Q}_{r} = - \int\limits_{S} {\varvec{q} \cdot \varvec{n}} dS = - 4\pi R^{2} \int\limits_{0}^{\infty } S_{\nu } d\nu $$

(22)

R is the bubble radius, S _ν is the spectral flux normal to the plasma bubble surface, e _vaporization is the specific energy of water evaporation, and C _P is the heat capacity at constant pressure.

Several approaches have been utilized to analyze the radiation term [69, 95, 97]. For example, Kratel [69] assumed that the plasma emits as a blackbody following the approximated Stefan-Boltzmann law

$$ \dot{Q}_{r} = - 4\pi R^{2} (1 - f)\sigma_{B} T^{4} $$

(23)

where T is the average temperature in the plasma channel,$\sigma_{B}$ is the Stefan-Boltzmann constant, and f is the blackbody radiation coefficient representing the fraction of energy absorbed in the thin liquid region surrounding the plasma (assumed to be 0.2). Cook et al. [97] consider a more detailed analysis with the Stefan-Boltzmann law whereby

$$ \begin{gathered} \dot{Q}_{r} = - 4\pi R^{2} \varepsilon_{r} \sigma_{B} \left( {T^{4} - T_{\infty }^{4} } \right) \hfill \\ \varepsilon_{r}^{ - 1} = \varepsilon_{water}^{ - 1} + \varepsilon_{sphere}^{ - 1} - 1\quad \varepsilon_{water} = 0.65 \hfill \\ \varepsilon_{sphere} = 1 - \tfrac{1}{2}\left( {\frac{{l_{mf} }}{R}} \right)^{2} \left[ {1 - \left( {1 + 2\frac{R}{{l_{mf} }}} \right)\exp \left( {\frac{ - 2R}{{l_{mf} }}} \right)} \right] \hfill \\ \end{gathered} $$

(24)

The emissivity of the spherical bubble, $\varepsilon_{sphere} ,$ depends upon the mean free path, $l_{mf} ,$ of photons in the bubble. Here it is assumed that the bubble is optically thick. In an optically thick plasma the mean free path is much smaller than the dimensions of the plasma, whereas for optically thin plasma the mean free path is much larger than the dimension of the plasma [126]. Integration of the spectral flux over all wavelengths as in Eq. (22) was performed in a third most general approach [95].

Kratel [69] included only the pressure–volume expansion work term, electrical heating and radiation loss terms in the energy balance.

$$ \dot{E} = - P\dot{V} - \left( {1 - f} \right)\sigma_{B} T^{4} S - \dot{Q}_{s} $$

(25)

For pulse energies from 144 to 900 J, Timoshkin et al. [127] neglected the radiation loss term in Eq. (25).

Thermal Energy Balance

Temperature can be determined either through a direct relationship with the internal energy or enthalpy or through derivation of the thermal energy balance by removal of the mechanical energy terms, obtained from the equation of motion, Eq. (1), from the total energy balance, Eq. (13) [76, 77]. Several of the key plasma models incorporate the relationship between the internal energy and the fluid state properties. For example, for an ideal gas with a constant ratio of specific heats, γ, the internal energy is given by Eq. (26) [68, 69, 127].

$$ U = \rho \hat{U}V = \frac{NkT}{\gamma - 1} = \frac{PV}{\gamma - 1} $$

(26)

where N = nV. Using the above equation for the internal energy, the overall energy balance, Eq. (25), for the plasma channel used by Kratel [69], for example, becomes

$$ \frac{dT}{dt} = \frac{{2I^{2} }}{{U_{T} \sigma_{cond} (t)2\pi^{2} R^{4} }} - \frac{{U_{n} }}{{U_{T} }}\frac{dn}{dt} - \frac{2(P + U)}{{U_{T} R}}\frac{dR}{dt} - \frac{2(1 - f)}{{U_{T} R}}\sigma_{B} T^{4} $$

(27)

U _T and U _N are defined by

$$ U_{T} = \left( {\frac{\partial U}{\partial T}} \right)_{n} ,U_{n} = \left( {\frac{\partial U}{\partial n}} \right)_{T} $$

The mass balance was determined from the continuity equation, Eq. (2), applied to the control volume. Including vaporization induced by radiation and volume expansion the mass balance used by Kratel [69] is

$$ \frac{dn}{dt} = \frac{2}{R}\left[ {\frac{{f\sigma_{B} T^{4} }}{{e_{vaporization } }} - n\frac{dR}{dt}} \right] $$

(28)

The plasma channel is assumed to continuously evaporate a thin layer of water surrounding the plasma which then gets incorporated back into the plasma. Other expressions for the mass flow [95, 97] have been used and will be discussed in a following section.

The internal energy of an ionized gas can be determined by extending the ideal gas law to account for ionization by Eq. (29) [126].

$$ \begin{gathered} \hat{U} = \frac{3}{2}N(1 + \alpha_{e} )kT + N\sum\limits_{m} {Q_{m}^{en} \alpha_{m} + N} \sum\limits_{m} {W_{m}^{ex} \alpha_{m} } \hfill \\ Q_{m}^{en} = I_{1} + I_{2} + \ldots + I_{m} \hfill \\ \end{gathered} $$

(29)

where α _e is the degree of ionization, α _m is the degree of dissociation of the mth species, N is the total number of atoms per mass of gas, W ^ex _m is the excitation energy of the mth species, and I are the ionization potentials. The degrees of dissociation and ionization are determined from the equilibrium Saha relationship—see following section. Lu et al. [95] used such an approach while Cook et al. [97] included the dissociation terms but neglected ionization in a form similar to Eq. (29). Lu et al. [95] used Eq. (29) with the energy balance, Eq. (18), to determine Eq. (30)

$$ \left( {\frac{\partial U}{\partial T}} \right)_{N,V} \dot{T} = \dot{Q} - \left[ {P + \left( {\frac{\partial U}{\partial V}} \right)_{T,N} } \right]\dot{V} $$

(30)

where $\dot{Q}$ is the total heat flux given by Eq. (20), V is the bubble volume calculated using the Kirkwood-Bethe model, Eq. (8), for the bubble growth, and P is the plasma pressure calculated using the Tait equation of state.

Another form of the thermal energy balance for a homogeneous fluid derived from the general energy balance for multicomponent mixtures is given by Eq. (31) [76, 77].

$$ \rho C_{P} \frac{DT}{Dt} = - \left( {\nabla \cdot \varvec{q}}\right) - \varvec{\tau} :\nabla \varvec{v} + \left({\frac{{\partial ln\hat{V}}}{\partial lnT}} \right)\frac{DP}{Dt} +\mathop \sum \limits_{i = 1}^{N} \hat{H}_{i} \left[ {\nabla \cdot\varvec{J}_{i} - R_{i} } \right] $$

(31)

This equation includes heating by viscous dissipation (second term on the right hand side), chemical reaction, and total flux by including all the mechanisms previously discussed. Additional terms, other than Joule heating, due to an electric field in polar (and non-polar) liquids have been included in more detailed thermal energy balance expressions [38]. However, even for large electric fields the contributions of these additional terms were generally negligible [38]. Further analysis may be useful to evaluate these approximations for electrical discharges in liquid water.

Stalder and coworkers [128, 129] utilized the Maxwell field equations, Eq. (11), with the thermal energy balance, Eq. (31), including only the conductive transport terms and the heat source due to current flow, to determine the time and spatial variation of the temperature and electric field in a highly saline solution with electrical discharges by Eq. (32).

$$ \rho \left( T \right)C_{p} \left( T \right)\frac{\partial T}{\partial t} - \nabla \cdot \left( {k\left( T \right)\nabla T} \right) = \varvec{J} \cdot \varvec{E} $$

(32)

Clearly for high temperature applications, the temperature dependence of the transport and material properties (including heat capacity, density, and thermal conductivity) need to be known and the phase transition from liquid to gas needs careful consideration.

Mededovic and Locke [130, 131] used a one dimensional form of Eq. (32) to calculate the temperature distribution within a plasma channel formed by a discharge of 1 J per pulse in water. Based upon an overall energy balance for a given plasma volume with known energy input, the model assumed that the temperature in the center of discharge channel is 5,000 K whereas the temperature at the plasma/bulk liquid boundary is 300 K. Due to the high thermal conductivity of the plasma region, very small gradients in temperature inside the plasma region were found.

Models of electrical discharges in water require physical property data from ambient to above 30,000 K and from atmospheric pressure to >1,000 atm. Various models and experimental measurements show that below 2,000 K, regardless of the pressure, the thermal conductivity of water collapses into a single curve. Matsunaga [132], for example, reports thermal conductivity data for the temperature range 300 to 2,000 K at 1 MPa. Wagner and Pruss [133, 134] report data for temperatures between 273 and 1,273 K and for a wide range of pressures (0.05–1,000 MPa). Vasic [135] gives the thermal conductivity of high temperature steam from 1,300 to 3,300 K for pressures of 0.01, 0.10, 1.00, 10.0, and 100 MPa. Data from Vasic indicates that at pressures higher than 1 MPa, changes in the thermal conductivity of water are very small. Thermal conductivity data for non-ideal water plasma ranging from 500 to 20,000 K and pressures 0.1, 0.2, 0.5, 1.0 and 2.0 MPa have been reported [136, 137]. Thermal conductivity, viscosity, and electrical conductivity were determined in a plasma based upon solution of Boltzmann’s equation for temperatures in the range 400–30,000 K and pressures from 0.5 to 10 atm and including the contributions of the various individual species formed from water dissociation at equilibrium [138]. Figure 8a shows an example of the contributions of the various species to the thermal conductivity of the mixture of the thermal water plasma. As shown in Fig. 8b, above 10,000 K the pressure had a strong effect on the total thermal conductivity.

Fig. 8

Simulations of the effects of temperature and pressure on thermal conductivity of an equilibrium water plasma at 1 atm. a contributions of each species to the total conductivity, b effects of pressure and temperature [138]. With kind permission from Springer Science + Business Media Aubreton et al. [138] Fig. 5 and Fig. 6. Copyright (2009) Spring Science + Business Media

Wagner [133] provides heat capacity data for pure water from 300 to 1,300 K for several pressures including 0.1, 0.25, 0.5, 1.0, and 5.0 MPa. The heat capacity data of steam for the pressures between 0.01 and 100 MPa and the temperature range 1,300 and 3,000 K are given in Vasic [135]. Above 3,000 K and up to 7,500 K, the heat capacity data can be found in [137]. Cook et al. [97] summarize in one empirical formula the effects of pressure from 1 to 1,000 atm and temperature up to 14,000 K for water vapor.

To account properly for the plasma composition, some thermodynamic properties such as the specific heat capacities can be directly determined from the internal energy by Eqs. (33) and (34).

$$ C_{V} (T) = (\partial U/\partial T)_{N,V} $$

(33)

$$ C_{P} (T) = \left( {\frac{\partial (H)}{\partial T}} \right)_{P,N} = \left( {\frac{\partial (U + PV)}{\partial T}} \right)_{P,N} $$

(34)

Utilization of the internal energy to determine heat capacity was used in several plasma spark models [95, 97], and ultimately the validity of this approach depends upon the accuracy of the internal energy function. Coufal determined the thermodynamic properties of a water thermal plasma accounting for the 33 species including most of the products of dissociation and ionization for temperatures from ambient up to 50 kK and 1, 5, and 20 atm through minimization of Gibbs free energy [139].

The effects of pressure and temperature on the dielectric constant of pure water have been described building upon the Kirkwood theory [84] for pressures of 5,000 atm and temperatures to 3,000 K [140, 141]. Further analysis of the dielectric constant of water can be found in the literature [83–85, 142–145].

Circuit Balance, Conservation of Charge, and Plasma Conductivity

Analysis of pulsed capacitor discharges requires Kirchhoff’s equation for an RLC circuit with inductance, L, capacitance, C, and resistance, R _el . In a more general case, the total resistance includes the circuit resistance, R _c , and liquid resistance R _l , added in series to the plasma resistance, R _p and the resistance in the liquid solution R _l . In spark or arc discharges the dominating influence is likely the plasma resistance while in streamer-like discharges the resistance in the liquid can be important and can vary from that of highly purified water to salt water. In terms of voltage, Kirchhoff’s law is written as

$$ \frac{{d^{2} V}}{{dt^{2} }} + \frac{{R_{el} }}{L}\frac{dV}{dt} + \frac{V}{LC} = 0\quad R_{el} = R_{c} + R_{p} + R_{l} $$

(35)

In some cases it is more convenient to write Eq. (35) in terms of current and charge [69]. Various expressions have been used for the plasma resistance (or conductivity) and these should account for the formation and presence of ions in the plasma. For example, for a fully ionized gas where electrons do not interact and positive ions are stationary, the plasma resistivity is given by [146].

$$ \rho_{e} = \frac{{\alpha \pi^{3/2} m_{e}^{1/2} e^{2} \ln \Uplambda }}{{2(2kT)^{3/2} }}\quad \Uplambda = \frac{3}{{2e^{3} }}\left( {\frac{{k^{3} T^{3} }}{{\pi n_{e} }}} \right)^{1/2} $$

(36)

where m _e and e are the mass and the charge of the electron, respectively, T is the temperature, α is a dimensionless coefficient equal to 1.8, and n _e is the electron density. Ioffe [68] used Eq. (36) and assumed that the total resistance is governed by the resistance of the discharge plasma spark channel by

$$ R_{el} = \frac{{\rho_{e} (t)l}}{{S_{c} (t)}} $$

(37)

where l is the plasma channel length and S _c is the cross-sectional area of the plasma. Robinson [80] proposed a semi-empirical relationship based upon Eq. (36) but with the addition of an exponential term and experimental coefficient $\xi$.

$$ \frac{1}{{\rho_{e} }} = \sigma = \xi T^{3/2} e^{ - 5000/T} $$

(38)

For arc and spark discharges, respectively, Kratel [68] utilized Eq. (38) with the temperature raised to the ½ power while Lan et al. [99] used the 3/2 power. Another approach relating the resistance of the plasma to the internal energy has been used [127, 147, 148].

$$ R_{p} (t) = \frac{{A_{sp} l^{2} }}{U(t)} $$

(39)

where A _sp is a constant which characterizes the content of the spark breakdown channel, l is the channel length, and the internal energy is given in Eq. (26) [88]. Equation (13) gives another empirical form for conductivity.

In streamer-like discharges the conductivity has been related to the electric field and model estimates found the maximum plasma conductivity to vary from approximately 10⁻⁴–10⁻¹ S/m as the field increased from 2 × 10⁸ to 10⁹ V/m [88]. Figure 9 shows schematic diagrams of arc and streamer-like discharges. In the case of streamer-like discharges the plasma does not span the entire electrode gap and resistance in both the plasma region and the liquid solution may be important. For a plasma channel propagating through a liquid the resistance of the highly pure water (for conductivity of 10⁻⁶ S/m) would certainly play an important role in Eq. (35). While modeling efforts to determine the plasma resistance in such cases have been developed [88], models including both the resistances in the plasma and the solution, which may both vary with time, have not been reported. The solution conductivity may vary due to chemical reactions. Analysis of these conductivity effects is very important to further develop models of streamer-like discharges. In such cases, thermal energy balances in the bulk liquid solution should also be included in order to determine the relative amounts of thermal energy dissipation by Joule heating in the plasma and the solution. On the other hand, in spark and arc discharges the plasma conductivities can vary from 10⁴ to 10⁵ S/m [127, 149, 150], but the plasma spans the electrode gap and the resistance in the liquid phase is not important. In these cases, the plasma channel, i.e. electrode gap, is fixed, but the radial expansion is considered as discussed in the previous sections concerning hydrodynamics.

Fig. 9

Schematic of plasma arc and streamer-like discharges in water

Discussion of Various Hydrodynamic Model Results: Temperature and Pressure Estimates

Most of the modeling efforts dealing with hydrodynamic effects apply to spark and arc discharges over a wide range of input energy (1 J/pulse to kJ/pulse) in water where the discharge plasma channels bridge the electrode gaps. In such discharges, the temperatures and pressures tend to be very high and shockwaves and UV emissions are generated. Since these models presuppose the formation of the arc channels that bridge the electrode gap, they do not simulate streamer-like discharges.

For arc discharges Kratel [69] divided the energy deposited in the plasma into the mechanical energy of the shock waves, the radiation energy, the discharge energy, and the energy of the bubble through

$$ \begin{gathered} E_{shockwave} = \int {pdV} \quad E_{radiation} = (1 - f)\int {\sigma_{B} T(t)^{4} dt} \quad E_{discharge} = \int {R(t)I(t)^{2} dt} \hfill \\ E_{discharge} - E_{shockwave} - E_{radiation} = \frac{4}{3}\pi p_{0} a_{\max }^{3} \hfill \\ \end{gathered} $$

(40)

Variation of the capacitance from 15 to 135 μF with fixed voltage of 5 kV gave total energy (1/2CV²) input of 190 J to 1.7 kJ. Solving the energy balance (Eq. (27)), mass balance (Eq. (28)), and circuit balance (in terms of current) with Eq. (9) for the velocity of the plasma bubble radius, the maximum temperature and pressure were shown to increase from 29,000 K to 31,000 K and 5.0–5.6 kbar while the percentage of total input energy delivered to the discharge varied from 80 to 45 %. At the same time, the relative amounts of energy converted to shock waves to discharge and radiation to discharge varied from 44 to 40 % and 12.8–11 %, respectively, as the input energy was increased. For a 62 J input Lu et al. [95] separated the acoustic energy of the shock waves from the PV work and found that after 10 ms 45 % of the energy was converted to the work term, 15 % to acoustic energy in the shockwaves, and 15 % to total thermal radiation. Applying the same model to a 1 J pulse input with 0.63 J delivered energy they found 38 % of the input energy went into shockwaves. Figure 10 shows example results for this discharge whereby temperatures exceed 50,000 K and pressure reaches 5,500 atm, and both gradually decay over the time of 0.5 μs computed in the model. The model of Lu et al. [95] is generally similar to that of Kratel, however, Lu et al. used Eq. (8) for the bubble expansion, a more detailed radiation model, and a different internal energy function, Eq. (29). Since the models were not computed at the same energy input conditions, it is difficult to make direct comparison, and further work should be conducted to compare these two models under similar conditions to identify the major factors that control the process in situations where the input powers are the same, and both modeling approaches need further validation and comparison with experimental data.

Fig. 10

Model results for spark discharge in water. 1.25 nF and 4 × 10⁴ V [95]. Reprinted with permission from Lu et al. [95]. Copyright (2002), American Institute of Physics

Table 2 summarizes model and some experimental results for various electrical discharge plasmas formed directly in the liquid phase. Results span input energies from 1 J/pulse to approximately 3 kJ/pulse and the input energy is varied by both changes in input voltage from 5 to over 82 kV and capacitance from 1.25 nF to 150 μF. Peak temperatures range from 9,000 K to 80,000 K. Peak pressure can also be quite high, 25,000 atm at the center, to relatively low values 5 atm measured ½ to 1 meter from the discharge. Plasma expansion velocities also vary considerably from a low of 10 m/s to 1,000 m/s. From these results there do not appear to be any simple correlations between, for example, higher temperatures or pressures and higher input energy. Ioffe’s [68] and Kratel’s [69] models suggest that higher temperature occurs at higher input voltage for the same input energy, while Lan’s [99] and Lu’s [95] results show higher temperature at higher input voltage even if the input energy is lower. It is important to note that the spark and arc plasma discharges have a number of applications in chemical and biological decontamination because of the high temperatures and pressures and significant UV emissions and shock waves. Such high energy and power systems may be particularly useful for solutions with highly concentrated contaminants, but for dilute solutions the efficiency may be low.

In all cases of the models above, there is limited experimental data comparison with the models and much further work is needed to validate the models with experimental data over a wide range of operating conditions. For streamer-like discharges in water, the major results of electrohydrodynamic modeling provide information on the velocity of the channel propagation and the electric fields, although some models suggest temperatures in the plasma channel to be 10⁴ K [88]. On the other hand, experimental approaches have been used to estimate electron densities (10²⁴–10²⁵ m⁻³) [32, 35, 151], electron temperatures (ca. 1 eV) [32, 151], OH^∙ rotational temperatures (2,000–5,000 K) [32, 35], and local pressures (1,000 atm by 20 ns) [32]. Detailed experimental and modeling work is needed to more fully determine the time and spatial variations of the plasma temperature and other properties in such plasma [152].

A partially related electrical discharge with water termed “electrodynamic explosion” or “electric arc explosion” is generated in highly saline solutions either at the gas–liquid interface or directly in the liquid leading to formation of high velocity water droplets and expansion of pressure waves into the gas and generation of supercritical states [153–157]. Pressures to 50 k atm can be generated, however, they cannot be explained by purely thermal processes [155]. The streamer-like, spark, and arc discharges discussed in the present review are different from these explosions in that the plasma is confined by the surrounding liquid and reactor vessel and the surrounding liquid is not mechanically and electrically fragmented. All of these processes can lead to supercritical fluid conditions, but in sparks, arcs, and streamer-like discharges the supercritical regions are somewhat localized while the explosive discharges can lead to supercritical regions over larger volumes.

Chemical Species Formation

The hydrodynamic, thermal, and electrical processes discussed in the previous sections are necessary to establish the temporal, and in some cases spatial variations of temperatures, pressures, electric fields, and volumes of the plasma (or plasma-bubble regions where the main chemical reactions between plasma generated species occur). High temperatures as well as direct electron collisions and interactions with UV light formed by the plasma can lead to ionization, and a variety of other reactions including excitation, neutral dissociation, electron attachment and dissociative attachment can occur in these processes. The relative importance of the various pathways depends upon the temperature of the plasma and the energy distribution of the electrons.

Ionization Equilibrium and Kinetics

Ionization of molecules usually begins at temperatures around several thousand to ten thousand degrees. Application of the law of mass action to an ionization process

$$ A_{m} \Leftrightarrow A_{m + 1} + e $$

allows determination of the number of ions in the ionization state, m + 1, with respect to the less ionized state m. The Saha equation [158] gives the relative number of atoms in two states as a function of the electron density n _e and temperature T by Eq. (41) [126]

$$ \frac{{n_{m + 1} n_{e} }}{{n_{m} }} = 2\frac{{u_{m + 1} }}{{u_{m} }}\left( {\frac{{2\pi m_{e} kT}}{{h^{2} }}} \right)^{3/2} e^{{ - \frac{{I_{m + 1} }}{kT}}} = K_{m + 1} (T) $$

(41)

where n _m and n _m+1 are the number densities of atoms in the ionization state m and the ionization state m +1 of a given element, u _m and u _m+1 are the partition functions of the two states, m _e is the electron mass, and I _m+1 is the ionization potential from state m to m + 1. Although the Saha equation is generally used for quasi-equilibrium (thermal) plasmas it has also been applied to non-thermal plasmas. However, in the latter case it significantly overestimates the degree of ionization [159]. The Saha equation has been used in underwater discharge to estimate the densities of H^∙, H⁺, O^∙, O⁺, O⁺⁺, and e^- [95, 98, 99], but connection between the local concentrations in the plasma to the final products and the bulk phase chemistry have only been made in the ultrasonic chemistry literature [47]. (Of note, in many ultrasonic applications the degree of ionization is found to be quite low, partly due to the effect of high pressure to suppress ionization [47, 160].)

The kinetics of ionization by electrical discharge in water were suggested to follow [87, 89]

$$ rate = v\exp ( - (A - 2\sqrt {e^{3} E/\varepsilon } )/kT) $$

(42)

where A is the activation energy, ν is the ionization frequency. This expression is based upon the Schottky effect of the lowering of the work function of a charged particle by an electric field [161–163]. Rates of formation of OH^∙ and H₂O₂ as functions of the average electric field for 1 J/pulse discharges in water generally followed Eq. (42), however analysis over a wider range of conditions and utilizing local electric fields requires further work [30].

Radiation Pathways

When ionizing radiation such as high energy electrons passes through an aqueous solution, it electronically excites and ionizes the water molecules along track-like pathways. Low-energy secondary electrons released by the ionization may further ionize and excite or dissociate water in the vicinity of the track, however, these electrons rapidly slow down (thermalize) below 7.4 eV and become trapped as solvated electrons. For both the primary charged particles and the secondary electrons, this slowing down process is accomplished by transfer of energy to the medium in a sequence of discrete events.

The initial event is the transfer of ~7–100 MeV, an amount of energy sufficient to cause ionization and excitation of water molecules by R1 and R2.

$$ {\text{H}}_{{\text{2}}} {\text{O}}\mathop \to \limits^{{rad}}{\text{H}}_{2} {\text{O}}^{{ \cdot + }} + e^{ - } \quad {\text{E}} \sim 13 {\text{eV}} $$

(R1)

$$ {\text{H}}_{{\text{2}}} {\text{O}}\mathop \to \limits^{{rad}}{\text{H}}_{{\text{2}}} {\text{O}}^{*} \quad {\text{E}} \sim 7.4 {\text{eV}} $$

(R2)

The time scale for the formation of H₂O^∙+, H₂O^*, and subexcitation electron species is on the order of 10⁻¹⁵ s [164–167]. In this pre-chemical stage, the radical ion of the water molecule, H₂O^∙+ (~ 10⁻¹⁴ s) collides with another water molecule to produce a hydroxyl radical and a hydronium ion

$$ {\text{H}}_{2} {\text{O}}^{ \cdot + } + {\text{H}}_{2} {\text{O}} \to {\text{H}}_{3} {\text{O}}^{ + } + {\text{OH}}^{ \cdot } $$

(R3)

The exited water molecule H₂O^* collides and splits another water molecule into hydroxyl and hydrogen radical

$$ {\text{H}}_{2} {\text{O}}^{*} + {\text{H}}_{2} {\text{O}} \to {\text{H}}^{ \cdot } + {\text{OH}}^{ \cdot } + {\text{H}}_{2} {\text{O}}\quad {\text{E}} \sim 5 {\text{eV}} $$

(R4)

In addition, H₂O^* can dissociate into hydrogen and oxygen species

$$ {\text{H}}_{2} {\text{O}}^{*} + {\text{H}}_{ 2} {\text{O}} \to {\text{H}}_{2} + {\text{O}} + {\text{H}}_{ 2} {\text{O}} $$

(R5)

The subexcitation electrons having energies <7.4 eV are captured by the surrounding water molecules and become an aqueous or solvated electron in less than 4 × 10⁻¹³ s by R6.

$$ e^{ - } + {\text{H}}_{ 2} {\text{O}} \to e_{aq}^{ - } $$

(R6)

Overall, at 10⁻¹¹s, the products of the initial radiation events (H₃O⁺, H^∙, OH^∙, O^∙, H₂ and e_aq) are clustered together within small regions called spurs. Although still in the same vicinity, all the species begin to diffuse randomly about their initial positions. During the diffusion process, they either react with each other to form molecular and secondary radical byproducts or they escape into the bulk liquid. At t > 10⁻¹¹ s (the chemical stage [168–170]), H₂ does not react further whereas O^∙ reacts with H₂O to give H₂O₂. The other species react to give stable molecular byproducts.

At about 10⁻⁷ s, the major products of radiolysis of water are OH^∙, e_aq, H^∙, H₃O⁺, H₂, and H₂O₂. The relationship between the chemical yields and the track structure during electron radiolysis of water and aqueous systems has been investigated by numerous researchers using stochastic modeling (Monte Carlo methods) [167–177]. These models show very good agreement with experimental data and reveal that the spatial distribution of reactants is determined by the structure of the radiation track.

Unlike in the electrical discharges discussed previously, thermal and hydrodynamic effects are generally not important in the chemical reactions of electron beams interacting with water. It has been proposed that regions of high temperatures, up to several thousand degrees, called “thermal spikes”, form from ionizing radiation interacting with water [178, 179] and in some cases from interactions of high LET radiation with solids [180] and biomedical materials. For example, increased LET (Po²¹⁰ α-particles and Co γ-rays) affected the decomposition of nitrate crystals [181]. In addition, the existence of thermal spikes and the understanding of how radiation damages DNA is extremely important in radiation cancer therapy. While studying the radiation damage caused by the irradiation of a biological tissue with energetic ions, the temperature in the vicinity of ion tracks was found to increases up to 1,900 K very early after an ion’s passage (10⁻¹⁵ and 10⁻⁹ s) [182]. The presence of such high temperatures suggests the possibility of thermo-mechanical pathways as one of the dominant pathways for the disruption of irradiated DNA. Nevertheless, while thermal spikes may be important in some high LET radiation (protons, etc.) in biomedical interactions of radiation and in solids, most analyses of electron beam interactions with water include only reaction and diffusion effects.

Analysis of the chemical kinetics of radiation chemistry requires information on ionization and excitation in liquid water [183] as well as electron impact cross section (Fig. 2 in [166]) including elastic scattering, ionization, and excitation. Codes and experimental data for cross section data are reviewed [174].

Other Reaction Pathways

While electron impact ionization is critical in electrical discharge, ultrasound, and the irradiation of water, there are also nonionizing pathways including simple excitation, neutral dissociation, electron attachment and dissociative attachment. For example, in the photolysis of water in the gas phase [184, 185] water molecules are dissociated via two pathways

$$ {\text{H}}_{ 2} {\text{O}} + h\nu \to {\text{H}}^{ \cdot } + {\text{OH}}^{ \cdot } $$

(R7)

$$ {\text{H}}_{ 2} {\text{O}} + h\nu \to {\text{H}}_{ 2} + {\text{O}} $$

(R8)

For photons having energies between 6.70–8.54 eV, the branching ratio between reactions (R7) and (R8) is 0.99:0.01 [185]. In liquid water the quantum yield for the formation of the geminate pair (H^∙ + OH^∙) is 0.45 at room temperature and 184.9 nm [186]. On the other hand, no experimental evidence for the occurrence of reaction R8 was found even though it has been reported that about 25 % of the primary process can be attributed to the direct splitting of water into molecular hydrogen and atomic oxygen [184].

Sokolov and Stein [187] also postulated that the very small quantum yield of about 0.05 at 184.9 nm (8.54 eV) is attributed to the reactions

$$ ({\text{H}}_{ 2} {\text{O}})_{aq} + h\nu \to ({\text{H}}_{ 2} {\text{O}})_{aq}^{**} \to e_{aq}^{ - } + {\text{OH}}^{ \cdot } + {\text{H}}_{ 3} {\text{O}}_{aq}^{ + } $$

(R9)

A plot of quantum efficiency vs. photon energy [183] reveals that the photo-dissociation quantum yield exhibits a maximum at photon energies of ~8 eV. Because of the high probability of non-radiative transition to the ground state, this yield is very low for energies below 6.70 eV and above 10.0 eV because of the high ionization probability. The quantum efficiency of the photoionization increases with an increase in the photon energy and at the photon energy of ~11.7 eV it reaches a value of 1.

For primary electrons having energies between 10³–10⁶ eV, Mozumder [183] calculated that the G value (molecules/eV) for neutral dissociation in liquid water induced by electrons has a maximum of <0.01. The computed dissociation yield is approximately an order of magnitude lower than the experimentally measured G value [188]. To account for the difference, Mozumder proposed dissociative electron attachment as an additional channel for the dissociation of water into neutral fragments. The process results in the formation of H⁻ and OH^∙ by

$$ e^{ - } + {\text{H}}_{ 2} {\text{O}} \to {\text{H}}^{ - } + {\text{OH}}^{ \cdot } $$

(R10)

after which hot H^- looses the electron in a collision with another water molecule by

$$ {\text{H}}^{ - } + {\text{H}}_{ 2} {\text{O}} \to {\text{H}}^{ \cdot } + {\text{OH}}^{ \cdot } $$

(R11)

The net result of reactions (R10) and (R11) is the dissociation of water molecules into H and OH^∙.

An estimation of the total neutral dissociation yield based on the dissociative electron attachment is in agreement with the experimentally measured value of 0.8. As a result, the direct neutral dissociation process (G < 0.01) is negligible compared to the dissociative electron attachment. The G value for the ionization of liquid water by primary electrons of energies between 10³–10⁶ eV calculated by Mozumder [183] is approximately 1 molecule/100 eV.

Species Mass Balances: Diffusion and Reaction

The molar species continuity equation is given by Eq. (43) [76]

$$ \frac{{\partial c_{i} }}{\partial t} + \nabla \cdot \varvec{N}_{i} = R_{i} $$

(43)

where c _i is the molar species concentration, N _i is the molar flux, and R _i is the net rate of production of species by all chemical reactions. The constitutive relationship for the molar flux can include various transport mechanisms including diffusion (Fickian), convection (hydrodynamics), electroconvection, thermal diffusion, and pressure diffusion (e.g., Eq. (12)) [76].

Early models of radiation chemistry utilized solution of the species balance with diffusive transport by Eq. (44) [22]

$$ \frac{{\partial c_{i} }}{\partial t} = D_{i} \nabla^{2} c_{i} + R_{i} $$

(44)

In the prescribed diffusion approach [21] the transport part was approximated by a Gaussian function and various radical reactions considered. The diffusion limitations on chemical reactions and the effects of force and electrostatic interactions have been determined [168, 169]. More modern approaches use large numbers of chemical reactions (see for example [175]) with the appropriate species balances to simulate the chemical phase (10⁻¹³–10⁻⁶ s) after the radiation track structure is determined (ending at 10⁻¹³ s). Reactions in radiation chemistry are typically evaluated at the constant temperature of the bulk solution.

In contrast, ultrasonic chemistry utilizes extensive coupling of the species balances to the hydrodynamic effects through the generation of high temperatures and high pressures [189, 190]. In some cases equilibrium is assumed for the chemical reactions [191] while in others the full kinetic expressions are utilized [192]. In ultrasound transport from the bubble (plasma phase) to the bulk fluid is considered and this also requires utilization of interface transport expressions as well as the accommodation coefficient [193].

Interfacial transport between a vapor bubble and the liquid phase based upon the kinetic theory of gases assuming a Maxwellian velocity distribution gives the mass flux, $\dot{m}$, by Eq. (45) [47, 97, 194–196]

$$ \dot{m} = \alpha_{acc} \sqrt {\frac{{M_{wt} }}{{2\pi R_{g} }}} \left[ {\frac{{\Upgamma P_{v} }}{{T_{v}^{1/2} }} - \frac{{P_{L} }}{{T_{L}^{1/2} }}} \right] $$

(45)

where $\alpha_{acc}$ is the accommodation coefficient, R _g gas constant, M _wt molecular weight, and $\Upgamma$is a factor that accounts for the motion in the bulk of the gas phase. The accommodation coefficient was first introduced by Maxwell to account for the possibility that some molecules colliding with an interface are reflected while some transfer into the surface [193]. Molecular and radical transport into aerosol particles have been extensively studied in atmospheric chemistry, and accommodation coefficients at water–gas interfaces have been studied for a wide range of compounds including the hydroxyl and hydroperoxyl radicals [197, 198]. Equation (45) was used in models of spark discharges in water [97] as well as in models of ultrasonic chemistry [47].

Chemical Reaction Models of Electrical Discharge in Water

Streamer-like Discharge Reaction Models

The first reactor model to analyze the production rates of molecular species and the degradation of phenol using pulsed electrical streamer-like discharge in water (Fig. 1, 1 J/pulse, 60 W) was developed by Joshi et al. [30] building upon kinetics from radiation chemistry and using batch kinetics for a single phase well mixed isothermal solution without diffusion or other transport effects under ambient conditions. The mass balance of the well mixed reactor in terms of the average concentration of a specific species is given by Eq. (46) where N _species is the number of species and N _Ri is the number of reactions for that species.

$$ \frac{{d\left\langle {c_{i} } \right\rangle }}{dt} = \sum\limits_{j}^{{N_{Ri} }} {\left\langle {r_{ij} } \right\rangle } \quad i = 1 \ldots N_{species} \quad j = 1 \ldots N_{Ri} $$

(46)

The primary chemical reactions in the plasma were assumed similar to those in radiolytic processes. The model included three overall water dissociation reactions (R12–R14) that correspond to the production of hydroxyl radical, molecular hydrogen and hydrogen peroxide and aqueous electrons.

$$ {\text{H}}_{ 2} {\text{O}}\mathop{\longrightarrow}\limits{{k_{OH} }}{\text{H}}^{ \cdot } + {\text{OH}}^{ \cdot } $$

(R12)

$$ {\text{H}}_{ 2} {\text{O}}\mathop{\longrightarrow}\limits{{k_{{H_{2} O_{2} }} }}1/2{\text{H}}_{ 2} {\text{O}}_{ 2} + 1/2{\text{H}}_{ 2} $$

(R13)

$$ {\text{H}}_{ 2} {\text{O}}\mathop{\longrightarrow}\limits{{k_{{e_{aq}^{ - } }} }}{\text{H}}_{ 2} {\text{O}}^{ + } + e_{aq}^{ - } $$

(R14)

These reactions were assumed and experimentally found to be zero-order and dependent only upon the energy input (the applied potential was varied to change the input energy). Propagation and termination reactions of ions and radicals formed in the primary reactions were taken from UV photolysis, electron beam, and γ-radiation kinetic models. In addition to the reactions of water, the model also included chemical reactions of phenol and its primary degradation byproducts (e.g., hydroquinone) with hydroxyl radicals. The reaction rate constants for the formation of hydroxyl radicals (k _OH) and hydrogen peroxide ($k_{{{\text{H}}_{ 2} {\text{O}}_{ 2} }}$) as functions of discharge power were determined experimentally with carbonate ion radical scavenging of radicals and a pseudo-steady-state approach. Comparison of the model results with the experimental data revealed that the model reasonably predicted the degradation of hydroquinone.

Grymonpre et al. [199] extended this isothermal batch kinetic model to analyze phenol degradation in the presence of iron salts by a similar pulsed corona electrical discharge treatment as used by Joshi et al. [30]. The Fenton’s regent (dissolved iron (II) ions) reactions were added to the basic model; these species to form OH^∙ from hydrogen peroxide. The rate constant for the reaction (R12) was fit to experimental data from experiments on hydrogen peroxide formation in the absence of iron salts and phenol. The resulting model described the effects of initial iron and phenol concentrations on the degradation of phenol and the formation of byproducts. A full sensitivity analysis [200] on reaction rates was performed with these models to determine the key reactions in the phenol oxidation. In an additional study, Grymonpre et al. [201] added the effects of mass transfer, adsorption, and catalytic reactions on activated carbon particles and also included the effect of ozone transferred from the gas phase. The model results were supported by the experimental evidence where it was shown that the addition of suspended activated carbon particles to the reactor solution enhances the removal rate of phenol. Reaction rate constants at the catalyst surface were determined by fitting the model to the experimental data. The model was found to closely simulate the experimentally measured phenol concentration and byproduct formation.

The models discussed above compared well with experimental data because the major reactions for the organic degradation under the given conditions were direct OH^∙ attack and/or reactions with OH^∙ formed from hydrogen peroxide through the Fenton reactions. UV and high temperature were not major factors for the species studied under these conditions of 1 J/pulse streamer-like discharge. Figure 11 shows comparison of model and experiments for example organic compound degradation where the model used OH^∙ rates of formation based upon independent experiments and literature values for the organic compound reaction with OH^∙. Clearly for these species the hydroxyl radical pathway can describe the degradation [202, 203], however, the role of reductive reactions [204] and electrode effects [205–208] should be considered in future modeling efforts.

Fig. 11

Sample organic compound destruction by stream-like discharge in water showing model results based upon independently determined hydroxyl radical rates of formation [202, 203]. a and b Reprinted with permission from Sahni and Locke [202, 203] Copyright (2006) with permission from Elsevier. c Reprinted with permission from Sahni [202]

To further assess the understanding of the reaction pathways, particularly that given by R13, Kirkpatrick and Locke [209] measured the formation rates of H₂, O₂, and H₂O₂ during a similar electrical discharge in water. They found that for various solution conductivities, applied voltages, and discharge powers, the molar rate of H₂ formation is twice that of H₂O₂ and that the rate of O₂ formation is approximately 20–25 % that of H₂O₂. The overall stoichiometry of molecular species formed by the pulsed electrical discharge in water follows the stoichiometry represented by

$$ 6{\text{H}}_{ 2} {\text{O}} \to 4{\text{H}}_{ 2} + 2{\text{H}}_{ 2} {\text{O}}_{ 2} + {\text{O}}_{ 2} $$

(R15)

It was also found that the production rates of H₂, O₂ and H₂O₂ increase with the applied power but the stoichiometry did not change. The yield of H₂ relative to H₂O₂ was thus not the same as that initially postulated by Joshi et al. [30].

Following initial studies with emissions spectroscopy that demonstrated the existence of OH^∙ radicals in liquid phase electrical discharge [29, 35, 210], chemical methods with various radical scavengers were utilized to determine the presence of OH^∙ radicals in electrical discharges in water [210]. In further work to quantify the formation of OH^∙ radical production in a pulsed electrical discharge in water, Sahni and Locke used two different chemical probes (dimethyl sulfoxide and disodium salt of terephthalic acid) (Fig. 12) [48]. They investigated a wide range of operational parameters on the production rate of OH^∙ and found zero order kinetics for both probes. The rate of OH^∙ formation was found to be strongly dependent on the probe concentration; as the concentration of the probe increases, the measured rate of production of OH^∙ also increases. However, at high enough probe concentrations, the production rate of OH^∙ reaches a limit due to the diffusion of the probe molecules into the plasma channel and complete scavenging of OH^∙. Such radical probe methods have been commonly used in the radiation chemistry in water and Fig. 13 shows the yield of H₂O₂ as a function of radical scavenger bromide ion (Br⁻) and the time dependence of the yield based on a diffusion-kinetic model [211, 212].

Fig. 12

Effects of probe concentration on hydrogen peroxide and hydroxyl radical production in pulsed streamer-like electrical discharge in water [48]. Reprinted with permission from Sahni and Locke [48]. Copyright (2006) American Chemical Society

Fig. 13

Yields of hydrogen peroxide as functions of a probe concentration and b time for electron beam radiolysis in water [212]. Reprinted with permission from LaVerne and Pimblott [212]. Copyright (1991) American Chemical Society

To quantify reductive species (HO ^∙₂ and O ^∙−₂ ), Sahni and Locke [204] again used two different chemical probes, tetranitromethane (TNM) and tetrazolium chloride (NBT). TNM reacts with a high reaction rate with both H^∙ and O ^∙−₂ whereas NBT reacts only with O ^∙−₂ . The results showed that different probes yield significantly different production rates of radical species as was observed with the OH^∙ scavengers. It was postulated that radical scavengers with higher volatility were more likely to penetrate into the plasma zone and affect quenching. The results also showed that the rate of reductive species formation in the pulsed streamer discharge increases as the input power to the system increases. Experiments indicated that O ^•-₂ may be the primary reductive species that transfers from the plasma into the bulk of the aqueous solution.

Quantification of the oxidative and reductive radicals with the various probes led to the hypothesis that the plasma could be divided into two domains whereby in the inner plasma core the radicals are generated by electron impact and thermal processes and that this core is surrounded by a recombination zone where molecular species, such as H₂O₂ and H₂, are formed by radical recombination (see Fig. 14) [48, 204, 213]. Mededovic and Locke [130, 131] developed a model to predict the experimentally measured concentrations of molecular and radical species in plasma based upon this two domain conception. The model took into account the pulsed nature of the electrical discharges and assumed that chemical reactions in the plasma are entirely thermal. Due to the repetitive nature of the plasma pulses, the model was developed only for the duration of one 1 μs pulse. The model coupled energy and mass balances with the assumption that the temperature in the center of the plasma channel is 5,000 K and at the plasma-liquid boundary it is 300 K. Table 3 shows the reactions used in the core discharge region and Table 4 shows the reactions used in the recombination zone and it can be noted that charged species were not included. It was found that the main reactions responsible for the formation of hydrogen, oxygen and hydrogen peroxide are the following

$$ {\text{H}}_{ 2} {\text{O}} + {\text{M}} \to {\text{H}}^{ \cdot } + {\text{OH}}^{ \cdot } + {\text{M}} $$

(R16)

$$ {\text{H}}^{ \cdot } + {\text{H}}^{ \cdot } + {\text{M}} \to {\text{H}}_{2} + {\text{M}} $$

(R17)

$$ {\text{OH}}^{ \cdot } + {\text{OH}}^{ \cdot } \to {\text{H}}_{2} {\text{O}}_{2} $$

(R18)

$$ {\text{OH}}^{ \cdot } + {\text{OH}}^{ \cdot } \to {\text{H}}_{ 2} {\text{O}} + {\text{O}} $$

(R19)

$$ {\text{O}} + {\text{OH}}^{ \cdot } \to {\text{O}}_{ 2} + {\text{H}}^{ \cdot } $$

(R20)

M is a collision third body. Numerical results accurately described the experimentally measured concentrations of molecular and radical (OH^∙) [48] byproducts in the plasma. However, this model had one adjustable parameter, namely the pressure inside the plasma channel. The pressure is directly related to the number of water molecules in the plasma channel. While the value used corresponded approximately to that measured in comparable discharges [32], further model improvements are needed to predict the pressure and to account for spatial and temporal variations of the temperature, pressure, and reacting species, to include the effects of UV on the chemical reactions, and to span a broader range of input power and solution properties, e.g. solution conductivity. While the role of electron collision reactions should also be included in future models [152], this model, and the corresponding experiments, nevertheless suggests that the exact mechanism of water dissociation may not be critical as long as the discharge has sufficient energy and/or temperature to lead to very high rates of water dissociation. In such a case the recombination of radicals to form stable molecular species may control their rate of formation [159, 214].

Fig. 14

Schematic of two-zone reaction model for streamer-like discharge in water

Table 3 Chemical reactions and rate functions occurring in the core of the two-zone model of electrical discharge streamer-like channels in water [130, 131]

Table 4 Recombination zone reactions occurring in the outer zone of the two-zone model of electrical discharge streamer-like channels in water [130, 131]

Reactions Induced by UV Emissions in Arc Discharges

The effect of UV on photochemical reactions in general is well studied [215] and photochemical reactor analysis, particularly for advanced oxidation, has advanced to the design stage [216–219]. In the case of electrical discharges in water, UV emissions have been measured [210] and quantified [39] for various power input and solution conditions for streamer-like and arc discharges and the UV light has been shown to have significant effects on chemical reactions [220, 221]. A model of photolysis reactions in electrical arc discharges in water, based upon batch kinetics, utilizes the rate equation for species R

$$ - r_{R,photo} = \Upphi (\lambda )I_{o} (\lambda ,t)(1 - \exp ( - 2.303\varepsilon_{ex} /c_{R} ) $$

(47)

where $\Upphi (\lambda )$ is the photolysis quantum yield, I _o is the UV source intensity, $\varepsilon_{ex}$ is the extinction coefficient, and λ is the radiation path length [69, 220]. To analyze UV emissions in a batch pulsed electrohydraulic arc discharge reactor under constant discharge conditions, the reaction rate in Eq. (47) was approximated by a zero order rate constant, k _o , determined from fitting to the experimental data. In further analysis to consider the effects of multiple pulses and complete mixing with the bulk solution between pulses, the batch kinetic degradation of a target compound was found to be

$$ \begin{gathered} \frac{{dc_{R} }}{{dN_{P} }} = - k_{1} c_{R} - k_{o} \hfill \\ k_{1} = - \ln \left( {\frac{{V_{T} - V_{R} }}{{V_{T} }}} \right) \hfill \\ \end{gathered} $$

(48)

where V _T is the total solution volume, N _P is the number of pulses, and V _R is the plasma channel volume.

Coupled Hydrodynamics and Chemical Reaction Models of Electrical Discharge Plasma

Lan et al. [99] included chemical reactions in an arc discharge model that had the same hydrodynamic model used by Kratel [69]. The electrons, injected from the cathode surface, were assumed to cause the ionization of water molecules to form H⁺, O⁺, H^∙, O^∙ and O²⁺. The densities of e^-, H⁺, O⁺, H^∙, O^∙ and O²⁺ were obtained from the Saha equation, Eq. (41), for thermal equilibrium. Numerical simulations were performed as a function of the pulse time (2 × 10⁻⁵ s) for the capacitor charge voltage of 3.2 kV and pulse current of 6.86 kA which corresponded to the discharge energy of 5.8 kJ. The model predicts the maximum temperature inside the plasma channel is 2.2 × 10⁴ K at 1 × 10⁻⁶ s. At that same time the pressure reaches a value of ~9×10⁸ Pa. The density of water molecules reaches the maximum of 3.8 × 10²⁷ m⁻³ at 2 × 10⁻⁶ s. The model also predicts that the electron is the species with the highest density followed by H⁺, O⁺, O²⁺, H^∙, and O^∙. Comparison with experimental data of such a model is needed and in order to do so, the model should be coupled to mass balances for the bulk solution.

Mukasa et al. [222] described the chemistry inside a plasma generated at the tip of an electrode using high-frequency irradiation at 13.56 MHz. Based on experimental observations, it was assumed that the plasma takes the form of a sphere and expands only in the radial direction. The temperature at the bubble surface was fixed at 373 K and the bubble pressure was assumed uniform and equal to 101 kPa. Chemical reactions for the water decomposition inside the plasma were taken from sonochemistry and assumed to be caused only by high temperatures. The diffusion coefficients of species inside the plasma were calculated using the Fuller-Schettler-Giddings equation and in the bulk liquid using the Wilke–Chang equation [223]. At the plasma-liquid interface, molecular species such as H₂, O₂ and H₂O₂ were assumed to dissolve into the bulk liquid whereas OH^∙, H^∙ and O^∙ do not transport into the bulk phase. In the development of the thermal energy balance, in addition to the heat supplied by the power deposition into the plasma, the model takes into account conduction between the plasma and the surrounding liquid. However, convection and radiation are neglected.

Depending on the input power, the maximum temperature in the interior of the bubble was determined to be between 3,500 K (110 W) and 4,000 K (310 W). Approximately 90 % of the energy deposited into the plasma was found to be lost to heating the surrounding liquid. The bubble radius increases from an initial condition of 3 mm to approximately 8 mm. Concentration profiles of molecular and radical species indicate that in the center of the bubble where the temperature is the highest H^∙ and O^∙ are dominant. With increasing distance from the bubble center (r > 2 mm) OH^∙ dominate. Additionally, at r = 2 mm H₂ and O₂ reach their maximum concentrations. As the molecules diffuse towards the bubble–liquid boundary (r = 8 mm), the concentrations of all the radicals and O₂ decrease. However, the concentration of hydrogen decreases only slightly. This result was explained by the stability of H₂ and the high value of its diffusion coefficient. The ratio of H₂ to O₂ predicted by the model was 1:0.3 which is much higher than the experimentally measured result of 1:0.04. The concentrations of other species including H₂O₂ are negligibly small; the calculated rate of H₂O₂ production (1 × 10⁻¹⁰ mol/s) is four orders of magnitude lower than the measured one (1.3 × 10⁻⁶ mol/s).

Comments on Ultrasonic Chemistry and Hydrodynamics Models

Mathematical models of ultrasonic reactors that couple hydrodynamics and chemical reactions have been reported [47, 189, 224–226]. An imposed pressure or sound field drives the equation of motion of the bubble (e.g. Eq. (7) or (8)) which is coupled to the thermal energy balance and the species balances. Because of the possibly very high temperatures, radiation is included in some models. Typical models include reactions of neutrals, radicals, and molecular species and not charged ions. Figure 15 shows an example simulation demonstrating the pulsed nature of the bubble expansion and contraction similar to that shown in Fig. 10 for the spark model. Figure 15 also shows the temporal variation of temperature and radicals and molecular species near the time of bubble collapse [225]. Transport of radicals from the bubble to the liquid phase has been considered through Eq. (44), however, the availability of radicals for reaction in the liquid phase may also arise after the collapse of the bubble and release of the radicals into solution [47]. It was suggested that both mechanisms play a role. In the analysis of pollutant treatment by sonochemical processes, the transport of the pollutant into the bubble was considered of importance and an equilibrium model for the chemical reactions was found adequate to describe the reactions within the bubble [191].

Fig. 15

Hydrodynamic model simulations of ultrasonic pulse in water showing a time dependence of bubble radius, b temperature near region of bubble collapse c formation of chemical species [225]. Reprinted with permission from Yasui et al. [225]. Copyright [2005], American Institute of Physics

Summary and Recommendations

Both radiation and electrical discharges are characterized by the presence of electrons whereas ultrasound involves only the presence of highly transient localized regions of very high temperature and pressure. The energy of primary electrons during radiation varies from keV to MeV and these electrons are not involved in the chemical transformations of solutes present in the bulk liquid far from the initial radiation track. They are only responsible for the creation of secondary electrons which, depending on the energy, either ionize or excite water molecules. In electrical discharges, the electron energy increases with the applied voltage and is dependent on the nature of the pulse and solution conditions. The average energy of electrons formed during streamer-like electrical discharges in water is estimated to be 0.5–2 eV which are much lower compared to gas-phase discharges where electron energy can be as high as 10 eV [227]. From a chemical standpoint, the upper limit of 2 eV is sufficient to cause only vibration and rotational excitation. However, the overall energy of electrons in plasma is represented by the Electron Energy Distribution Function (EEDF) which implies that a very small percentage of electrons have energies higher than 2 eV. Since streamer-like plasma in water is assumed to be weakly ionized, the energy of some electrons in plasma must be higher than 10 eV. It may also be possible to explain the ionization in liquid water using the successive excitation concept introduced by Magee and Burton for electrical discharges (applied to methane [228]). They suggested that such discharges are characterized by a series of successive electronic excitations by low energy electrons to higher and higher levels which lead to the creation of energy levels high enough for the dissociation and ionization of gaseous molecules [229].

The roles of thermal reactions in radiation and electrical discharge chemistry are quite different and they may vary significantly between spark or arc discharges and streamer-like discharges. While during streamer electrical discharge the temperature of the plasma can rise several thousand Kelvin and in arc and spark discharges temperatures can reach 50,000 K, the maximum temperature rise observed during irradiation of water by high-energy electrons is only a few degrees. On the other hand, sonochemical reactions are driven by the implosion of cavitation bubbles whose temperature and pressure can reach values as high as 5,000 K and 500 atm, respectively. These conditions assure that the water molecules inside the plasma or bubble are dissociated into H^∙ and OH^∙. Streamer-like discharges, at least those above J/pulse, therefore have temperatures closer to ultrasound than radiation chemistry.

While quantifying OH^∙ produced by the electrical discharge in water, Sahni and Locke [48] postulated that the types of chemical reactions occurring inside the streamer-like plasma channel are very similar to the ones occurring during ultrasonic processing. In the center of the plasma channel high temperature causes the dissociation of water molecules into radicals (i.e. thermal degradation). A thin layer (~200 nm) adjacent to the center of the plasma channel (interfacial region) comprises supercritical water and divides the plasma interior from the bulk liquid. The recombination of the plasma radicals into stable byproducts occurs in this layer. Finally, less than 10 % of the radicals formed in the plasma center diffuse into the bulk liquid to react with solutes.

In addition to the chemical effects initiated by the plasma channel formed at the tip of the high-voltage electrode, electrical discharges produce shock waves which cause cavitation in the bulk liquid similar to the one observed in ultrasound. Nevertheless, despite these apparent similarities, the chemical yields (as already discussed) for streamer-like discharges are significantly different from those of ultrasound and fall between those of electron beam radiation and ultrasound. It is possible that control of the discharge properties, including rate and amount of energy input as well as sparking, may allow variation of the chemical yields of electrical discharges to span the limits of electron beams and ultrasound. Further experimental and modeling studies are necessary to address this matter.

Table 5 summarizes the major chemical and physical processes that occur in streamer-like, spark, and arc discharges directly in water. In spark and arc discharges analysis of the physical processes has focused on hydrodynamic and thermal characterization, while only a limited amount of work has connected these physical processes to chemical reactions. There is also a limited amount of data on the chemical processes, including formation or active reactive species, in such discharges. On the other hand, the most successful model of the chemical reactions in streamer-like discharges relies on simple assumptions concerning the temperature and pressure in the plasma channels, while analysis of the physical processes is more limited.

Table 5 Summary of key physical and chemical processes in streamer-like, spark, and arc discharges