3D particle ‐ in ‐ cell simulation of positive streamer initiation in highly pressurized gaseous, liquid and supercritical CO 2 with field ionization

A 3D particle ‐ in ‐ cell/Monte Carlo collision model is used to investigate the streamer discharge inception in CO 2 at elevated pressures including gaseous, liquid and supercritical phases. Generation of free electrons is a prerequisite for initiation and development of positive streamers. Field ionization from impurity molecules of low density and low ionization energy is assumed as the source of primary ionization in the model. The field dependent generation rate of electrons is calculated by Zener's model. Tree ‐ like streamer with filamentary shape presents at the needle tip and propagates towards the plane electrode in all three phases of CO 2 , with differing in length and propagation speed. Furthermore, the influence of different parameters such as applied voltage, electrode tip size, and ionizable density and ionization potential of impurity molecules on the evolution of streamer discharge in supercritical CO 2 are investigated. Simulation results obtained from the 3D model agree well with experimental observa-tions in the literature, which show that field ionization might be one of possible ionization sources in pressurized CO 2 in the presence of high electric field.

is the most widely used gaseous dielectric in high voltage equipment owing to its high dielectric strength and excellent arc quenching properties [1,2]. However, SF 6 is a strong greenhouse gas and designated as regulated gas in third session of the conference of the Parties in the United Nations Framework Convention on Climate Change because of its high global warming potential, which is 24,000 times greater than that of CO 2 [3][4][5]. Therefore, development of new environmentally friendly insulating gas or mixtures is urgently required to replace SF 6 [6][7][8].
One approach that has been adopted is to use supercritical fluids (SCFs) as SF 6 substitute in high voltage switching and insulation technology because of their special thermodynamic and transport characteristics superior to those in gaseous and liquid phase [9]. The discharge in SCFs has attracted attention because of various applications in chemical industry [10].
Significantly, supercritical CO 2 is non-toxic, inflammable, and easily available and its critical point (P c ¼ 73.8 bar, T c ¼ 304.15 K) is easily accessible at ambient temperature compared to other SCFs [11].
The thermodynamic state of CO 2 can be identified and varied by pressure P and temperature T near its critical point. Pressure-density curves are drawn at three different temperatures near critical point (73.8 bar, 304.15 K) in Figure 1. CO 2 exists in supercritical phase at or above the critical point. If the temperature is below the critical temperature but pressure is above the critical pressure (80 bar), CO 2 exists in liquid phase. Below critical pressure, CO 2 exists in gaseous phase. Note that the value of different temperatures and pressures associated with gaseous, liquid and supercritical phase of CO 2 used in this work are also shown in Figure 1.
The initiation, propagation and branching of streamer discharge are fundamentals of breakdown phenomena of any dielectric medium [12]. The mechanism of discharge initiation in highly dense media is highly complicated and related to electric field distribution, chemical composition, applied voltage, thermodynamic state, and physical properties of dielectric medium. Note that the theory of breakdown initiation in gases is relatively well understood [13], but far from conclusive in dense media.
In last decades, many experimental research works have been performed to investigate the streamer discharge properties in pressurized CO 2 . Ito et al. studied the breakdown of CO 2 in parallel-plane gap of 1-2 µm up to 90 bar and found that plasma can be generated in CO 2 near its critical point with relatively low voltage ranging from 180 to 850V [14]. Seeger et al. investigated breakdown fields in CO 2 , CO 2 /O 2 , synthetic air, and CF 4 for conditions associated with high voltage system under impulse voltage up to 100 bar and found that breakdown field strength in above gases tend to saturate above a certain pressure [15]. The pre-breakdown and breakdown voltage of pressurized CO 2 including supercritical phase was studied for the purpose of development of plasma reactor with supercritical (SC)CO 2 in Kiyan et al [16]. Lock et al. studied the initiation of direct current (DC) pulsed discharges in SC-CO 2 in point-plane electrode [17]. Ihara et al. investigated the initiation mechanism of positive streamer discharge in pressurized CO 2 including supercritical phase with the consideration of direct ionization as a primary ionization mechanism [18]. They suggested different direct ionization processes, responsible for the initiation of positive streamer discharge in SC-CO 2 need further investigation, such as hole-emission at the metal-fluid interface, Auger release of electrons, detachment of electrons from negative ions, and field ionization.
Numerical simulations provide us microscopic details of the discharge that is difficult to measure through experiments and help to understand the discharge initiation mechanisms. Levko et al. built fluid and particle models to investigate the branching of negative streamer discharge in CO 2 at atmospheric pressure without considering any discharge initiation mechanism [19]. Tian et al. developed a theoretical model of breakdown in SC-CO 2 with molecular cluster formation [20]. They calculated electron energy distribution function using ionization and attachment coefficients through Boltzmann analysis. Jadidian et al. used fluid model to investigate the influence of impulse voltage polarity, peak and rise time on streamer discharge evolution in transformer oil under a needle-to-plane electrode [21]. However, to our best knowledge, no clear explanation of discharge initiation in CO 2 at elevated pressures including SC phase was elucidated in the literature.
In this study, field ionization is assumed as the primary source of free electrons to investigate the initiation and branching of positive streamer discharge around sharp anode in pressurized gaseous, liquid, and supercritical CO 2 , by a predeveloped 3D particle model. Moreover, the influence of electrode voltage, tip radii, and Zener's parameters on streamer discharge evolution in SC-CO 2 is presented. The study is organized as follows: Section 2 gives an overview of the 3D model and the details of computational domain used in the model. In Section 3, field ionization process with Zener's model is introduced. Simulation results obtained from a 3D particle in cell/Monte Carlo collision (PIC/MCC) model are presented and discussed in Section 4. Section 5 presents the summary and conclusion.

| OVERVIEW OF 3D PIC/MCC MODEL
Previously, a 3D PIC/MCC model was developed to study the discharge inception in N 2 /O 2 mixtures at atmospheric pressure [22]. Here, the 3D PIC/MCC model is further extended for discharge inception in highly dense mediums with field ionization process.   motion. Two different kinds of particle movers: leapfrog and Velocity scheme are usually used in plasma particle simulation. Two schemes are quite similar however the difference is at which time velocity and position of particles are calculated. In leapfrog scheme, the velocity and position of the particles are calculated with an offset in time while in Velocity Verlet scheme, the velocity and position of particles are calculated at the same time. In this mode, 'Velocity Verlet' scheme is adopted in this model to integrate the equation of motion, with advantages illustrated in Sun et al [23].
In a particle model, large number of particles has to be tracked individually; therefore, super-particles are commonly used. In the model, an adaptive particle management scheme was introduced to adjust the weight of the particles [24]. In addition, for the purpose of better resolutions, adaptive mesh refinement scheme was used for avoiding the stochastic errors during simulations [23]. Collisions between electrons and neutral molecules were included only because of low ionization degree plasma considered in this work. MCC technique with null collision method was used to model the collision processes. The collisional cross-section data of CO 2 was taken from Siglo database and is available on www.lxcat.net [25], which contains attachment, elastic, vibrational excitation, electronic excitation, and ionization reactions, as shown in Figure 2.
The schematic diagram of computational domain used in the present study is shown in Figure 3. A space domain of (1 mm) 3 was used, and a needle electrode was placed at the centre of the domain with a tip radius of r tip ¼ 10 µm. A positive dc voltage was supplied to the needle electrode and all other sides of the domain boundary were kept at zero potential. In this model, it was assumed that the particle flux reaching the plane electrode in front of needle electrode absorbed totally and no emission of secondary electrons from the plane electrode was considered. In the present simulation model, the bottom area of the domain was not considered for mesh refinement to avoid the discharges for the sake of saving computation time as previously considered, for example in Teunissen and Ebert [22].
The detail of the 3D PIC/MCC model is given elsewhere and the source code is available online [26]. In next section, the concept and implementation of field ionization as a primary ionization mechanism in pressurized CO 2 including supercritical phase is discussed.

| FIELD IONIZATION WITH ZENER'S MODEL
As mentioned above, discharge initiation theory in highly dense mediums is still not clear. In liquids or SCFs, the possible weak points where breakdown may initiate could be due to photoionization, density fluctuations, and the presence of conducting particles, field ionization and so forth. Reference [19] presented that the photoionization probability in CO 2 is very low due to two-photon process. Detachment of electron from negative ions is also one of the sources of free electrons in electronegative gases, but in case of dc voltage the source and number of negative ions prior to discharge is somehow questionable [27]. It is evident that direct ionization might be one of the primary ionization mechanism in liquid and SC-CO 2 [28]. Besides, field ionization could be implemented as primary ionization mechanism in highly dense mediums at high pressures where bubble formation is prohibited [29]. In field ionization process, electrons are elevated from valance band to conduction band in neutral molecules that results into formation of free electron and positive ion in the presence of high electric field.
In this study, field ionization which is a direct ionization process is assumed as the primary ionization mechanism in pressurized CO 2 near the critical point. The source of free electrons is impurity molecules with a low density and a low ionization threshold. The field dependent generation of electrons was calculated using Zener's model. Note that similar approach was used to investigate streamer inception in liquid dielectrics and highly pressurized N 2 [21,30,31]. Remark that Zener's model was first proposed for breakdown in solid dielectrics [32] and given by: where G 1 ðE → Þ represents the field dependent electron generation term, q is the electronic charge, n o is small amount of ionizable impurity density, |E → | is applied field, h is Plank's constant, m * is effective mass of electron, a is separation distance between the molecules, and Δ is the ionization potential.
The main challenge in implementing the Zener's model is the appropriate selection of its parameters because of the incomplete macroscopic knowledge available for highly pressurized CO 2 . The values of these parameters are well known for various solid dielectrics. For liquid hydrocarbons, most of the parameters are not quantified yet, and some of the parameters such as effective mass of electron (m*) are only known. As shown in the Equation (1), the generation rate increases with the increase in ionizable density, and electric field strength but decreases with the increase in ionization potential, and separation distance between the impurity molecules. Different values of parameters were selected and will be discussed in result section. SCFs are aggregated of clusters of different size with low ionization potentials. These clusters with low ionization potential could be the source of free electron generation. In SC-CO 2 , it was estimated that each cluster on average has 12 molecules with low ionization potential and the cluster size was measured about 5.6-8 � 10 À 10 m [12]. The values of Zener's parameters used in literature and in this study are given in Table 1.

| Streamer initiation in gaseous, liquid, and supercritical CO 2 with field ionization
As mentioned above, free electron generation is a prerequisite for initiation and propagation of streamers. The growth of the discharge depends on the density of free electrons ahead of it and if the density of free electrons is large enough, individual avalanches don not extinguish and the discharge grows smoothly, as illustrated in Sun et al [34,35]. For positive polarity streamers, different mechanism could be responsible for generation of source free electrons. In this model, field ionization with Zener's model is investigated as a primary ionization mechanism in highly pressurized gaseous, liquid, and supercritical CO 2 . The electron generation profile depends on Zener's parameters and electric field strength. The temperatures and pressures associated with the gaseous, liquid and SC phase of CO 2 used in this model are shown in Table 2.
The comparison of streamer discharge inception and branching for an anode potential of 12 kV are shown in Figure  4. Discharge was triggered from a seed containing 100 electron-ion pairs with Gaussian distribution; the seed was placed close to the tip of needle anode. The propagation of the discharge was simulated up to 0.5 ns. The radius of curvature of electrode tip was first selected 10 µm in order to produce sufficient electric field around the anode, prerequisite for field ionization process.
Tree-like streamer appears at the tip of the needle electrode, which can divide into three different stages. In the first stage, during the time 0-0.05 ns, an inception cloud appeared at the needle tip. The inception cloud also called diffusion discharges sometimes, which have been observed in air through experiments [36]. 2D plane cross-section of formation of inception cloud until 0.05 ns with local parameters such as length is presented in Figure 5. At the given times in each cross-section, the apparent length and diameter of the inception cloud is the maximum in gaseous phase comparing with liquid and SC phases. It takes relatively longer time for inception cloud destabilization into separate streamer channels in liquid and SC phase. The shape of the inception cloud changes from spherical to elliptical in liquid and supercritical phase.
Note that we define those thick separate streamer channels as the second stage of the streamer discharge. The number and apparent length of the thick streamer channels, that is primary streamers evolving from the inception cloud is larger in gaseous phase than that in liquid and SC phases shown in Figure 4. The second stage of the streamer discharge continues until 0.2 ns in our simulations. In the third stage, the thick streamer channels are further divided into multiple subsequent branches. The diameter of streamer channel is thick however the head of streamer thin and flattened due to which streamer main channels breaks out into subsequent branches. The morphology of the streamer discharge is quite similar in gaseous, liquid, and SC-CO 2 . The average velocity of propagating streamer is estimated as 200 km/s in our simulations, which is too fast to be measured in experiments, therefore, comparison of streamer discharge velocity is quite challenging. Even so, the morphology of streamer discharge in our simulation results agrees well with the that observed in experiments, for example, in Ihara et al [37]. The average electron density in gaseous, liquid, and SC-CO 2 is calculated as 3.8 � 10 19 m À 3 , 4.5 � 10 19 m À 3 , and 4.4 � 10 19 m À 3 accordingly. The above physical mechanism and different stages of streamer discharge initiation with field ionization process is explained in Figure 6.
In our simulations, when a positive dc voltage is supplied to the needle electrode, a high electric field is generated around the needle electrode. Impurity molecules with a density of n 0 ¼ 1 � 10 20 m À 3 and a low ionization threshold of Δ ¼ 4.6 eV are ionized in the presence of such a high electric field. Then, free electrons are generated from the impurity molecules near the needle anode. Due to high mobility, these electrons quickly move towards anode and are absorbed, leaving positive ions behind them. The accumulation of positive ions around the electrode tip decreases the field near the electrode and enhances it away from the electrode. The enhanced electric field further ionizes the impurity molecules and new free electrons are generated. Again, new electrons generated on the new position far away from the electrode travel towards to the anode and leave positive ions behind. Due to this process, the position of the enhanced field region changes the zone and the streamer propagates further into the gap following the field ionization wave. The shifting of high electric field zones away from the needle anode and different stages are evident as presented in Figure 6. In addition, Figure 7 shows the 2D cross-section of electric field distribution in streamer discharge in gaseous, liquid, and SC-CO 2 at different times. It is distinguished in that electric field insides the streamer channel is smaller than the electric field at the streamer head. Electric field at the surface of the inception cloud approximates to U/R and the maximum radius of the spherical shape inception cloud is R 0 ¼ U/E c , where E c is critical field strength [22]. When the inception cloud expands and reach the maximum size it destabilized into branches due to Laplacian instability [38]. Here R 0 plays an important role in destabilization of inception cloud that depends on the generation of free electrons ahead of the discharge associated with Zener's parameters such as ionizable density of impurity molecules. The maximum electric field in gaseous, liquid and SC-CO 2 are calculated as 3.9, 4.5, and 4.1 MV/cm, respectively.

| Effects of electrode voltage and electrode tip radius
Discharges at high pressures in the presence of a strong electric field, that is (E > 10 6 V/cm) are affected by electrode characteristics such as electrode voltage amplitude and electrode tip curvature, as changing these parameters results in different field distribution at electrode tip. On applying different voltages to needle electrode, electric field distribution around the electrode tip varies, which eventually has the effect on the generation of free electrons from the impurity molecules in the model. We here tested four voltage amplitudes in the simulations. As shown in Figure 8, the tip radius in all the case is 10 µm. For a 9 kV electrode voltage, the first stage of streamer discharge, that is inception cloud formation occurred at 0.3 ns, which further propagated into the gap and only rare branches occurred until 0.5 ns. With increasing the voltage to 10 kV, the size of the inception cloud increased. Three stages of streamer discharge as described above were clearly identified when voltage increased to 11 and 12 kV. The length scale and number of branches of streamer discharge increased with increasing electrode voltage.
In Figure 9, the influence of needle-electrode tip on streamer discharge initiation was investigated. The breakdown voltage decreases when a sharp tip of needle electrode is used because of maximum electric field enhancement around it. In this study, the needle electrode with three different tip radii is adopted under a fixed electrode voltage of 12 kV. The size of inception cloud decreased with increasing the tip radius. Formation of inception cloud and streamer branching occurred earlier if the electrode has a sharper tip of 5 µm radius. It is evident through the simulation results that number of branches and apparent length scale of streamer discharge decreased with increasing tip radius.

| Ionizable density of impurity molecules (n 0 )
As mentioned early, SC fluids are aggregated of clusters with low ionization potential and different sizes. These clusters could act as free volume and might be the source of free electrons in SC-CO 2 [29]. In this study, Zener's model is used to implement field ionization process. However, Zener's model includes different parameters such as ionizable density of impurity molecules, their separation distance and ionization potential, which have not been measured precisely. We assume that the impurity molecules are analogous to the clusters in SC-CO 2 , therefore the effect of their density and ionization potential on discharge evolution can be investigated. Figure 10 shows the influence of ionisable density (n 0 ) of impurity molecules n o ¼ 1 � 10 16 m À 3 , 1 � 10 17 m À 3 , 1 � 10 18 m À 3 , 1 � 10 19 m À 3 , 1 � 10 20 m À 3 , 1 � 10 21 m À 3 under a 12 kV applied voltage were investigated. According to Equation (1), increasing of the ionizable density n o results in electron generation rate G 1 increases and more free electrons can be generated. The apparent length and width of streamer channels increased with increasing amount of impurity molecules. For n o ≤ 1 � 10 19 m À 3 , streamer discharge propagated more axially into the gap and the deflection between the streamer channels reduced considerably. The size of the F I G U R E 6 (a) Physical illustration of discharge initiation, propagation, and (b) different stages of streamer discharge with field ionization using Zener's model ABBAS ET AL.
-21 inception cloud became smaller with a small n 0 . The inception cloud only destabilized into few thick streamer channels and with rare branches later on. The ionization zone decreases and becomes more confined to needle electrode tip with the decrease in ionizable density.
The limiting values of ionizable density below which electron generation was insufficient for triggering a streamer discharge was tested, for pressures ranging from 60 to 90 bar under three different applied voltages, as shown in Figure 11. Each value of impurity molecules on different pressures was tested five times approximately. For low pressures, that is in gaseous phase, the mean free path of electron is larger and discharge initiates with lower number of impurity molecules. The number of impurity molecules showed similar trend and tend to saturate in gaseous, near supercritical region and in the region far above than critical point.
With increasing pressure, the medium phase changes from gaseous to SC or liquid phase, therefore, an increasing number of impurity molecules is needed for discharge initiation. Remark that the limiting value of n o saturates at a certain pressure, for all the tested voltages.

| Ionization potential (Δ) of impurity molecules
The influence of ionization potential (Δ) in Zener's model on discharge evolution is shown in Figure 12. Three different values of ionization potential 4.6, 5.4, and 6 eV were selected. It was shown the with increasing the ionization potential, the size of the inception cloud decreased and changed from spherical to elliptical shape. With a high ionization potential, streamer discharge developed more axially rather than radially into the gap, the apparent length and number of branches of streamer discharge decreased. F I G U R E 7 2D cross-sections of electric field distribution in (a) gaseous, (b) liquid, and (c) SC-CO 2 at 12 kV F I G U R E 8 The effect of electrode voltage on streamer discharge evolution around needle anode with r tip ¼ 10 µm in supercritical CO 2 F I G U R E 9 The Influence of electrode tip on streamer discharge evolution in SC-CO 2 (P ¼ 80 bar, T ¼ 305 K, U ¼ 12 kV). SC, supercritical Field ionization with Zener's model is adopted to investigate the positive needle-to-plane streamer discharge in pressurized CO 2 including gaseous, liquid and supercritical phase using a 3D PIC/MCC model. We obtained tree-like streamer discharges, which are independent of medium phase. The streamer development consists of three stages as follows: (1) inception cloud, (2) thick streamer channels and (3) multiple branches. Simulated streamer morphologies are similar as observed in previous experiments. Our simulations show that field ionization might be an important source of free electrons at dense mediums close to critical point of CO 2 .
The electrode voltage and tip radius can significantly affect streamer discharge evolution in SC-CO 2 , and properties of the streamer discharge can be controlled by varying several parameters in Zener's model by controlling the generation of free electrons. F I G U R E 1 0 The influence of ionizable density of impurity molecules in Zener's model on streamer discharge evolution in SC-CO 2 (P ¼ 80 bar, T ¼ 305 K, U ¼ 12 kV). SC, supercritical F I G U R E 1 1 The limiting value of n o below which minimum or no electron generation occurs in given time scale for different pressures under different applied voltages F I G U R E 1 2 Influence of ionization potential of impurity molecules on streamer discharge evolution in SC-CO 2 (P ¼ 80 bar, T ¼ 305 K, U ¼ 12 kV). SC, supercritical